Abstract group 1536.408494914: $C_2\times D_4^2:D_6$

• Label: 1536.408494914
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{25} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{19} \)
• Perm deg.: not computed
• Trans deg.: $96$
• Rank: $7$

Group information

Description:	$C_2\times D_4^2:D_6$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	Group of order \(100663296\)\(\medspace = 2^{25} \cdot 3 \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.

Group statistics

Order	1	2	3	4	6	12
Elements	1	511	2	512	254	256	1536
Conjugacy classes	1	103	1	72	51	36	264
Divisions	1	103	1	72	51	36	264
Autjugacy classes	1	14	1	8	8	4	36

Dimension	1	2	4	8
Irr. complex chars.	128	96	32	8	264
Irr. rational chars.	128	96	32	8	264

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$96$
Rank:	$7$
Inequivalent generating 7-tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{4}=d^{2}=e^{4}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $96$ $\langle(1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(D_4^2:C_2)$
Semidirect product:	$(Q_8\times D_6)$ $\,\rtimes\,$ $C_2^4$	$(Q_8:D_6)$ $\,\rtimes\,$ $C_2^4$ (2)	$(D_6:Q_8)$ $\,\rtimes\,$ $C_2^4$	$(D_6:Q_8)$ $\,\rtimes\,$ $C_2^4$	all 402
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_4\times D_6)$ . $C_2^4$ (6)	$(D_6.D_4)$ . $C_2^4$ (2)	$D_6$ . $(D_4:C_2^4)$	$D_6$ . $(D_4\times C_2^4)$ (3)	all 72
Aut. group:	$\Aut(C_{12}.C_4^2)$	$\Aut(C_{12}.C_4^2)$	$\Aut(C_{12}.C_4^2)$	$\Aut(C_{12}.C_4^2)$	all 13

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{7} $
Schur multiplier:	$C_{2}^{20}$
Commutator length:	$1$

Subgroups

There are 574442 subgroups in 149300 conjugacy classes, 34953 normal (59 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $D_6\times C_2^4$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_2^7$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $D_6\times C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times D_4^2:C_6$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2\times D_4^2:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^5$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1536: $C_2\times D_4^2:D_6$

Order 768: $D_4^2:D_6$ x 64, $C_2\times C_3:(C_2^4.C_2^3)$ x 8, $C_2\times C_3:(C_2^4.C_2^3)$ x 8, $C_2\times (C_2\times D_4):C_2^2\times S_3$ x 8, $D_6\times D_4^2$ x 8, $C_2\times C_3:(C_2^4.C_2^3)$ x 4, $C_3:(C_2^3.C_2^5)$ x 4, $C_2\times C_2^4:C_2^2\times S_3$ x 4, $C_2^6:D_6$ x 4, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times (C_2^2\times D_4):C_2\times S_3$ x 2, $C_4^2:C_2^2\times D_6$ x 2, $C_2\times D_4^2:C_6$, $D_{12}:C_2^5$, $C_{12}:C_2^6$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_2^3:(C_2\times C_4)\times S_3$

Order 512: $C_4^2:C_2^5$

Order 384: $D_4:(S_3\times D_4)$ x 256, $S_3\times D_4:D_4$ x 256, $D_4^2:S_3$ x 256, $S_3\times D_4^2$ x 256, $C_2^5:D_6$ x 196, $C_2^3:(S_3\times D_4)$ x 128, $S_3\times C_2^3:D_4$ x 128, $C_2^5:D_6$ x 128, $C_{12}:C_2^5$ x 104, $D_{12}:C_2^4$ x 64, $(C_2\times D_{12}):D_4$ x 64, $C_4^2:C_2^2\times S_3$ x 64, $(C_2\times D_{12}):D_4$ x 64, $C_4^2:C_2^2\times S_3$ x 64, $D_4^2:C_6$ x 32, $(C_6\times D_4):D_4$ x 32, $C_2^5:D_6$ x 32, $C_6:D_4^2$ x 32, $C_6:D_4^2$ x 32, $C_6:D_4^2$ x 32, $(C_2\times D_4):D_{12}$ x 32, $C_4^2:C_2^2\times S_3$ x 32, $C_4:D_4\times D_6$ x 28, $(C_2\times D_{12}):C_2^3$ x 16, $(C_2\times D_{12}):C_2^3$ x 16, $(C_4\times D_6):C_2^3$ x 16, $(C_2\times D_{12}):C_2^3$ x 16, $C_2^5.D_6$ x 16, $C_2^5:D_6$ x 16, $C_6:D_4^2$ x 16, $C_4\times D_4\times D_6$ x 16, $C_2^5.D_6$ x 16, $C_2^5:D_6$ x 12, $C_2^2.D_4\times D_6$ x 12, $C_2^4:D_{12}$ x 12, $C_2^5.D_6$ x 12, $D_{12}:C_2^4$ x 9, $C_6\times D_4:D_4$ x 8, $C_6\times D_4^2$ x 8, $(C_6\times D_4):D_4$ x 8, $C_2^5:D_6$ x 8, $C_2^5.D_6$ x 8, $C_6:D_4^2$ x 8, $C_2\times D_{12}:D_4$ x 8, $C_2\times D_{12}:D_4$ x 8, $C_2\times D_{12}:D_4$ x 8, $(C_4\times D_6):D_4$ x 8, $C_2^5.D_6$ x 8, $C_2\times D_{12}:D_4$ x 8, $C_2\times D_4:D_{12}$ x 8, $C_2\times C_4^2:D_6$ x 8, $C_2\times C_4^2:D_6$ x 8, $D_{12}:C_2^4$ x 6, $D_{12}:C_2^4$ x 6, $C_6\times C_2^3:D_4$ x 4, $C_2^6:C_6$ x 4, $C_2^5:D_6$ x 4, $C_2\times C_4^2:D_6$ x 4, $C_4:D_4\times D_6$ x 4, $C_2\times C_4^2:D_6$ x 4, $C_2\times C_4^2:D_6$ x 4, $(C_4\times D_6):C_2^3$ x 4, $C_2^2:Q_8\times D_6$ x 4, $(C_4\times D_6):D_4$ x 4, $(C_4\times D_6):D_4$ x 4, $C_2^5.D_6$ x 4, $C_2^4:D_{12}$ x 4, $C_2^5.D_6$ x 4, $D_6\times C_2^5$ x 2, $D_6:C_2^5$ x 2, $C_4^2:C_2^2\times C_6$ x 2, $C_4^2:C_2^2\times C_6$ x 2, $(C_2^2\times C_{12}):D_4$ x 2, $(C_2^2\times C_{12}):D_4$ x 2, $C_2^5.D_6$ x 2, $C_2^5.D_6$ x 2, $C_2\times C_4^2:D_6$ x 2, $C_2\times C_4^2:D_6$ x 2, $C_2\times C_4^2:D_6$ x 2, $C_2\times C_4^2:D_6$ x 2, $C_{12}.C_2^5$, $C_{12}:C_2^5$, $C_2^4\times D_{12}$, $C_{12}:C_2^5$, $C_4^2:C_2^2\times C_6$, $C_2^5.D_6$

Order 256: $C_4^2:C_2^4$ x 96, $C_4^2:C_2^4$ x 8, $C_2^2\times D_4^2$ x 8, $C_2^5:D_4$ x 4, $C_2^5:D_4$ x 4, $C_4^2:C_2^4$ x 2, $C_4^2:C_2^4$ x 2, $D_4:C_2^5$, $D_4\times C_2^5$, $C_4^2:C_2^4$

Order 192: $C_{12}:C_2^4$ x 1910, $C_2^4:D_6$ x 1088, $D_{12}:C_2^3$ x 512, $C_2^4:D_6$ x 512, $D_{12}:D_4$ x 512, $C_2^4:D_6$ x 512, $S_3\times C_4:D_4$ x 448, $C_2^4:D_6$ x 300, $C_2^3:D_{12}$ x 300, $D_{12}:C_2^3$ x 288, $C_2^4.D_6$ x 284, $(D_4\times D_6):C_2$ x 256, $(C_2\times D_6):D_4$ x 256, $(C_2\times D_4):D_6$ x 256, $(C_2\times D_4):D_6$ x 256, $C_2^4.D_6$ x 256, $C_2^4:D_6$ x 256, $D_4\times D_{12}$ x 256, $C_4^2:D_6$ x 256, $C_2^4.D_6$ x 256, $D_{12}:C_2^3$ x 192, $D_{12}:C_2^3$ x 192, $S_3\times C_2^2.D_4$ x 192, $C_3\times D_4:D_4$ x 128, $C_3\times D_4^2$ x 128, $(C_2\times D_4):D_6$ x 128, $C_2^4.D_6$ x 128, $C_2^4:D_6$ x 128, $C_3:D_4^2$ x 128, $D_{12}:D_4$ x 128, $D_{12}:D_4$ x 128, $D_{12}:D_4$ x 128, $(C_4\times S_3):D_4$ x 128, $C_2^4.D_6$ x 128, $D_4:D_{12}$ x 128, $D_{12}:D_4$ x 128, $C_4^2:D_6$ x 128, $C_4^2:D_6$ x 128, $D_6\times C_2^4$ x 127, $D_6:C_2^4$ x 120, $C_2^5:C_6$ x 100, $C_2^4:D_6$ x 100, $C_3\times C_2^3:D_4$ x 64, $C_4^2:D_6$ x 64, $C_4^2:D_6$ x 64, $C_4^2:D_6$ x 64, $C_4^2:D_6$ x 64, $(C_4\times D_6):C_2^2$ x 64, $(C_4\times S_3):D_4$ x 64, $S_3\times C_2^2:Q_8$ x 64, $(C_4\times S_3):D_4$ x 64, $C_2^4.D_6$ x 64, $C_2^3:D_{12}$ x 64, $C_2^4.D_6$ x 64, $C_{12}:C_2^4$ x 60, $C_2^3\times D_{12}$ x 60, $C_{12}:C_2^4$ x 60, $C_2^4.D_6$ x 56, $C_2^4.D_6$ x 56, $C_{12}.C_2^4$ x 32, $C_3\times C_4^2:C_2^2$ x 32, $C_3\times C_4^2:C_2^2$ x 32, $(C_2\times C_{12}):D_4$ x 32, $(C_2\times C_{12}):D_4$ x 32, $C_2^4.D_6$ x 32, $C_2^4.D_6$ x 32, $C_2^4.D_6$ x 32, $C_4^2:D_6$ x 32, $C_4^2:D_6$ x 32, $C_4^2:D_6$ x 32, $C_4^2:D_6$ x 32, $C_2^4.D_6$ x 32, $C_2\times C_{12}:D_4$ x 28, $C_2^4.D_6$ x 28, $C_2^3:D_{12}$ x 28, $C_2\times C_4:D_{12}$ x 28, $C_2^4.D_6$ x 24, $C_2^3.D_{12}$ x 24, $C_2\times D_6.D_4$ x 24, $C_2^4.D_6$ x 24, $C_{12}.C_2^4$ x 18, $D_{12}:C_2^3$ x 18, $C_3\times C_4^2:C_2^2$ x 16, $C_2\times D_4\times C_{12}$ x 16, $C_2^4.D_6$ x 16, $C_2^4.D_6$ x 16, $C_2\times D_{12}:C_4$ x 16, $C_2^4.D_6$ x 16, $C_4^2:D_6$ x 16, $(C_2^3\times C_{12}):C_2$ x 12, $C_2^4:C_{12}$ x 12, $C_2^4.D_6$ x 12, $C_2^4.D_6$ x 12, $C_2^4.D_6$ x 12, $C_4:C_4\times D_6$ x 12, $C_2^3.D_{12}$ x 12, $C_2\times D_6:Q_8$ x 8, $C_2^4.D_6$ x 8, $C_{12}.C_2^4$ x 6, $D_{12}:C_2^3$ x 6, $C_2\times C_{12}:D_4$ x 4, $C_2\times C_4^2:C_6$ x 4, $C_2^2:Q_8\times C_6$ x 4, $C_2\times C_{12}.D_4$ x 4, $C_2\times D_6:Q_8$ x 4, $C_2^4.D_6$ x 4, $C_2^4.D_6$ x 4, $C_2\times D_6:Q_8$ x 4, $(C_4\times D_6):C_4$ x 4, $C_2^4.D_6$ x 4, $C_4^2:D_6$ x 4, $C_4^2:D_6$ x 4, $C_4^2:D_6$ x 4, $D_6\times C_4^2$ x 4, $C_2^5\times C_6$ x 2, $C_{12}.C_2^4$ x 2, $C_2\times C_4^2:C_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4\times C_{12}$, $C_6.C_2^5$

Order 128: $D_4^2:C_2$ x 1024, $C_4^2:C_2^3$ x 384, $C_2\times D_4^2$ x 384, $C_2^4:D_4$ x 228, $C_2^4:D_4$ x 192, $D_4\times C_2^4$ x 108, $C_4^2:C_2^3$ x 96, $C_4^2:C_2^3$ x 96, $D_4:C_2^4$ x 80, $C_4^2:C_2^3$ x 48, $C_2^4:D_4$ x 28, $C_4^2:C_2^3$ x 16, $C_2^4.D_4$ x 12, $C_2^5:C_4$ x 12, $D_4:C_2^4$ x 6, $C_4^2:C_2^3$ x 4, $C_4^2:C_2^3$ x 4, $C_2^4:Q_8$ x 4, $C_2^7$ x 2, $C_4^2:C_2^3$ x 2, $C_2^5\times C_4$

Order 96: $D_4\times D_6$ x 7264, $C_2^3\times D_6$ x 1890, $C_2^3:D_6$ x 1212, $D_4:D_6$ x 1152, $C_2^2:D_{12}$ x 864, $D_6:D_4$ x 864, $D_4:D_6$ x 768, $D_4:D_6$ x 768, $D_6.D_4$ x 736, $C_{12}:C_2^3$ x 612, $C_{12}:C_2^3$ x 606, $C_2^2\times D_{12}$ x 606, $D_6:D_4$ x 448, $C_2^3.D_6$ x 448, $D_4:D_6$ x 288, $D_{12}:C_2^2$ x 288, $C_2^4:C_6$ x 288, $C_2^4:S_3$ x 288, $C_2^2.D_{12}$ x 280, $C_2^3.D_6$ x 256, $C_{12}:D_4$ x 256, $C_{12}:D_4$ x 224, $C_{12}:D_4$ x 224, $C_{12}:D_4$ x 224, $C_4:D_{12}$ x 224, $D_6.D_4$ x 192, $C_{12}:D_4$ x 192, $D_6.D_4$ x 192, $C_2^3:C_{12}$ x 140, $C_2^3.D_6$ x 140, $C_2^3.D_6$ x 128, $C_4\times D_{12}$ x 128, $C_6.C_2^4$ x 128, $D_4\times C_{12}$ x 128, $C_2^3.D_6$ x 128, $D_{12}:C_4$ x 128, $D_6.D_4$ x 96, $C_2^2.D_{12}$ x 96, $C_6.C_2^4$ x 96, $Q_8:D_6$ x 96, $(C_2^2\times C_{12}):C_2$ x 96, $C_2^3.D_6$ x 96, $C_2^3.D_6$ x 96, $C_2^4\times C_6$ x 65, $C_2^3.D_6$ x 64, $D_6:Q_8$ x 64, $C_2^3\times C_{12}$ x 38, $C_6.C_2^4$ x 38, $(C_4\times S_3):C_4$ x 32, $C_2^3.D_6$ x 32, $C_4^2:S_3$ x 32, $C_4:D_{12}$ x 32, $C_4^2:S_3$ x 32, $S_3\times C_4^2$ x 32, $Q_8\times D_6$ x 32, $C_{12}:D_4$ x 32, $C_4^2:C_6$ x 32, $C_{12}.D_4$ x 32, $C_{12}.D_4$ x 32, $D_6:Q_8$ x 32, $C_{12}.D_4$ x 32, $C_{12}.D_4$ x 32, $D_6:Q_8$ x 32, $C_2^3.D_6$ x 24, $C_4^2:C_6$ x 16, $C_2^3.D_6$ x 16, $C_2\times C_4:C_{12}$ x 12, $C_2^2.D_{12}$ x 12, $C_6:C_4^2$ x 12, $C_6.C_2^4$ x 6, $C_2\times C_4\times C_{12}$ x 4, $C_6.C_2^4$ x 2

Order 64: $D_4\times C_2^3$ x 2150, $D_4:D_4$ x 2048, $D_4^2$ x 2048, $C_2^3:D_4$ x 1888, $D_4:C_2^3$ x 1024, $C_2^3:D_4$ x 1024, $C_2^3:D_4$ x 672, $C_4^2:C_2^2$ x 512, $C_4^2:C_2^2$ x 512, $C_4^2:C_2^2$ x 384, $C_2^4:C_4$ x 332, $C_2^3.D_4$ x 288, $C_4^2:C_2^2$ x 256, $D_4:C_2^3$ x 240, $C_2^6$ x 129, $C_4^2:C_2^2$ x 96, $C_4^2:C_2^2$ x 96, $C_2^3:Q_8$ x 96, $C_2^4\times C_4$ x 62, $C_4^2:C_2^2$ x 48, $C_2^3.D_4$ x 12, $C_2^2\times C_4^2$ x 4, $Q_8\times C_2^3$ x 2

Order 48: $C_2^2\times D_6$ x 7156, $S_3\times D_4$ x 4800, $C_6:D_4$ x 2432, $C_4\times D_6$ x 1264, $C_6\times D_4$ x 1216, $C_2\times D_{12}$ x 1216, $D_4:S_3$ x 576, $D_{12}:C_2$ x 576, $C_2^3\times C_6$ x 503, $D_6:C_4$ x 352, $C_2^2\times C_{12}$ x 212, $C_6.C_2^3$ x 212, $D_4:C_6$ x 192, $D_{12}:C_2$ x 192, $C_2^2:C_{12}$ x 176, $C_6.D_4$ x 176, $C_6.D_4$ x 96, $S_3\times Q_8$ x 64, $C_6:Q_8$ x 48, $C_4:C_{12}$ x 48, $C_{12}:C_4$ x 48, $C_{12}:C_4$ x 48, $C_6\times Q_8$ x 16, $C_4\times C_{12}$ x 16

Order 32: $C_2^2\times D_4$ x 9688, $C_2^2\wr C_2$ x 2304, $D_4:C_2^2$ x 2048, $C_2^5$ x 1955, $C_4:D_4$ x 1792, $D_4:C_2^2$ x 1536, $C_2^3:C_4$ x 1296, $C_4\times D_4$ x 1024, $C_2^2.D_4$ x 768, $C_2^3\times C_4$ x 688, $C_4:D_4$ x 256, $C_4^2:C_2$ x 256, $C_2^2:Q_8$ x 256, $C_2^2.D_4$ x 144, $C_4^2:C_2$ x 128, $C_2\times C_4^2$ x 48, $C_2^2\times Q_8$ x 40

Order 24: $C_2\times D_6$ x 6508, $C_2^2\times C_6$ x 975, $C_3:D_4$ x 800, $C_4\times S_3$ x 448, $D_{12}$ x 400, $C_3\times D_4$ x 400, $C_2\times C_{12}$ x 220, $C_6:C_4$ x 220, $C_3:Q_8$ x 48, $C_3\times Q_8$ x 16

Order 16: $C_2\times D_4$ x 9664, $C_2^4$ x 7659, $C_2^2\times C_4$ x 1688, $D_4:C_2$ x 1536, $C_2^2:C_4$ x 704, $C_4:C_4$ x 192, $C_2\times Q_8$ x 128, $C_4^2$ x 64

Order 12: $D_6$ x 1340, $C_2\times C_6$ x 455, $C_{12}$ x 36, $C_3:C_4$ x 36

Order 8: $C_2^3$ x 7483, $D_4$ x 1600, $C_2\times C_4$ x 888, $Q_8$ x 64

Order 6: $S_3$ x 52, $C_6$ x 51

Order 4: $C_2^2$ x 1795, $C_4$ x 72

Order 3: $C_3$

Order 2: $C_2$ x 103

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $C_2\times D_4^2:D_6$

Order 768: $C_2\times D_4^2:C_6$, $D_{12}:C_2^5$, $C_{12}:C_2^6$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_3:(C_2^3.C_2^5)$, $C_2\times C_3:(C_2^4.C_2^3)$, $D_4^2:D_6$, $C_2\times (C_2\times D_4):C_2^2\times S_3$, $D_6\times D_4^2$, $C_2\times (C_2^2\times D_4):C_2\times S_3$, $C_2\times C_2^4:C_2^2\times S_3$, $C_4^2:C_2^2\times D_6$, $C_2^6:D_6$, $C_2\times C_2^3:(C_2\times C_4)\times S_3$

Order 512: $C_4^2:C_2^5$

Order 384: $C_{12}:C_2^5$ x 13, $C_2^5:D_6$ x 8, $D_{12}:C_2^4$ x 3, $C_6:D_4^2$ x 3, $C_6:D_4^2$ x 3, $C_4:D_4\times D_6$ x 3, $D_{12}:C_2^4$ x 2, $D_{12}:C_2^4$ x 2, $D_{12}:C_2^4$ x 2, $C_2^5:D_6$ x 2, $C_2^5:D_6$ x 2, $(C_2\times D_{12}):C_2^3$ x 2, $C_2^2.D_4\times D_6$ x 2, $C_6:D_4^2$ x 2, $C_2^5:D_6$ x 2, $C_6:D_4^2$ x 2, $C_4\times D_4\times D_6$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $D_6\times C_2^5$, $C_{12}.C_2^5$, $C_{12}:C_2^5$, $D_6:C_2^5$, $C_2^4\times D_{12}$, $C_{12}:C_2^5$, $D_4^2:C_6$, $C_6\times D_4:D_4$, $C_6\times D_4^2$, $C_4^2:C_2^2\times C_6$, $C_6\times C_2^3:D_4$, $C_4^2:C_2^2\times C_6$, $C_2^6:C_6$, $C_4^2:C_2^2\times C_6$, $C_2^5:D_6$, $(C_6\times D_4):D_4$, $(C_6\times D_4):D_4$, $(C_2^2\times C_{12}):D_4$, $(C_2^2\times C_{12}):D_4$, $C_2^5:D_6$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^5.D_6$, $D_4:(S_3\times D_4)$, $S_3\times D_4:D_4$, $D_4^2:S_3$, $S_3\times D_4^2$, $(C_2\times D_{12}):D_4$, $C_4^2:C_2^2\times S_3$, $C_2^3:(S_3\times D_4)$, $S_3\times C_2^3:D_4$, $C_6:D_4^2$, $C_2\times C_4^2:D_6$, $C_4:D_4\times D_6$, $C_2\times D_{12}:D_4$, $C_2\times C_4^2:D_6$, $C_2\times C_4^2:D_6$, $(C_2\times D_{12}):D_4$, $C_4^2:C_2^2\times S_3$, $(C_2\times D_{12}):C_2^3$, $(C_4\times D_6):C_2^3$, $C_2\times D_{12}:D_4$, $C_2^2:Q_8\times D_6$, $(C_4\times D_6):D_4$, $C_2\times D_{12}:D_4$, $(C_4\times D_6):C_2^3$, $(C_4\times D_6):D_4$, $(C_2\times D_{12}):C_2^3$, $(C_4\times D_6):D_4$, $C_2^5:D_6$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^5.D_6$, $(C_2\times D_4):D_{12}$, $C_4^2:C_2^2\times S_3$, $C_2\times D_{12}:D_4$, $C_2\times D_4:D_{12}$, $C_2\times C_4^2:D_6$, $C_2\times C_4^2:D_6$, $C_2\times C_4^2:D_6$, $C_2\times C_4^2:D_6$, $C_2\times C_4^2:D_6$, $C_2\times C_4^2:D_6$, $C_2^5.D_6$, $C_2^4:D_{12}$, $C_2^5.D_6$

Order 256: $C_4^2:C_2^4$ x 11, $D_4:C_2^5$, $D_4\times C_2^5$, $C_4^2:C_2^4$, $C_2^2\times D_4^2$, $C_4^2:C_2^4$, $C_2^5:D_4$, $C_4^2:C_2^4$, $C_2^5:D_4$, $C_4^2:C_2^4$

Order 192: $C_{12}:C_2^4$ x 44, $D_6\times C_2^4$ x 12, $D_6:C_2^4$ x 12, $C_{12}:C_2^4$ x 11, $C_2^4:D_6$ x 11, $C_2^3:D_{12}$ x 11, $C_2^4.D_6$ x 11, $C_{12}:C_2^4$ x 9, $C_2^3\times D_{12}$ x 9, $C_2^4:D_6$ x 7, $D_{12}:C_2^3$ x 6, $C_2^5:C_6$ x 5, $C_2^4:D_6$ x 5, $D_{12}:C_2^3$ x 4, $D_{12}:C_2^3$ x 4, $C_{12}.C_2^4$ x 4, $D_{12}:C_2^3$ x 4, $C_2^4.D_6$ x 4, $C_2^4.D_6$ x 4, $C_2\times C_{12}:D_4$ x 3, $C_2^4.D_6$ x 3, $C_2^4:D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3.D_{12}$ x 3, $D_{12}:D_4$ x 3, $S_3\times C_4:D_4$ x 3, $C_2\times C_4:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_{12}.C_2^4$ x 2, $C_{12}.C_2^4$ x 2, $D_{12}:C_2^3$ x 2, $(C_2^3\times C_{12}):C_2$ x 2, $C_2\times D_4\times C_{12}$ x 2, $C_2^4:C_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $(C_2\times D_6):D_4$ x 2, $S_3\times C_2^2.D_4$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $D_4\times D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_2\times D_6.D_4$ x 2, $C_2\times D_{12}:C_4$ x 2, $C_4:C_4\times D_6$ x 2, $C_2^3.D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_4^2:D_6$ x 2, $C_2^5\times C_6$, $C_2^4\times C_{12}$, $C_6.C_2^5$, $D_{12}:C_2^3$, $C_{12}.C_2^4$, $C_3\times D_4:D_4$, $C_3\times D_4^2$, $C_3\times C_4^2:C_2^2$, $C_3\times C_2^3:D_4$, $C_2\times C_{12}:D_4$, $C_2\times C_4^2:C_6$, $C_3\times C_4^2:C_2^2$, $C_2^2:Q_8\times C_6$, $C_3\times C_4^2:C_2^2$, $C_2\times C_4^2:C_6$, $(C_2\times C_{12}):D_4$, $(C_2\times D_4):D_6$, $(C_2\times C_{12}):D_4$, $C_2\times C_{12}.D_4$, $C_2\times D_6:Q_8$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_3:D_4^2$, $C_4^2:D_6$, $C_4^2:D_6$, $D_{12}:D_4$, $C_4^2:D_6$, $C_4^2:D_6$, $(C_4\times D_6):C_2^2$, $(D_4\times D_6):C_2$, $D_{12}:D_4$, $(C_4\times S_3):D_4$, $S_3\times C_2^2:Q_8$, $D_{12}:D_4$, $(C_2\times D_4):D_6$, $(C_4\times S_3):D_4$, $(C_4\times S_3):D_4$, $(C_2\times D_4):D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $D_4:D_{12}$, $D_{12}:D_4$, $C_4^2:D_6$, $C_4^2:D_6$, $C_4^2:D_6$, $C_4^2:D_6$, $C_4^2:D_6$, $C_4^2:D_6$, $C_2\times D_6:Q_8$, $C_2\times D_6:Q_8$, $(C_4\times D_6):C_4$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_4^2:D_6$, $C_4^2:D_6$, $C_4^2:D_6$, $D_6\times C_4^2$

Order 128: $D_4\times C_2^4$ x 16, $C_2^4:D_4$ x 14, $C_4^2:C_2^3$ x 13, $C_2\times D_4^2$ x 13, $D_4^2:C_2$ x 10, $C_2^4:D_4$ x 10, $C_4^2:C_2^3$ x 10, $C_4^2:C_2^3$ x 9, $D_4:C_2^4$ x 8, $C_4^2:C_2^3$ x 6, $C_2^4:D_4$ x 3, $D_4:C_2^4$ x 2, $C_2^4.D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^5:C_4$ x 2, $C_2^7$, $C_2^5\times C_4$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_2^4:Q_8$, $C_4^2:C_2^3$

Order 96: $D_4\times D_6$ x 53, $C_2^3\times D_6$ x 48, $C_2^3:D_6$ x 34, $C_{12}:C_2^3$ x 31, $C_{12}:C_2^3$ x 23, $C_2^2\times D_{12}$ x 23, $C_2^2.D_{12}$ x 11, $C_2^2:D_{12}$ x 9, $D_6.D_4$ x 9, $D_6:D_4$ x 9, $C_2^4\times C_6$ x 8, $C_2^3\times C_{12}$ x 8, $C_6.C_2^4$ x 8, $D_4:D_6$ x 8, $D_{12}:C_2^2$ x 8, $C_2^3:C_{12}$ x 7, $C_2^3.D_6$ x 7, $D_6:D_4$ x 4, $C_6.C_2^4$ x 4, $Q_8:D_6$ x 4, $C_2^4:C_6$ x 4, $C_2^4:S_3$ x 4, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 3, $D_4:D_6$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_2^3.D_6$ x 3, $C_4:D_{12}$ x 3, $D_6.D_4$ x 2, $C_2^2.D_{12}$ x 2, $D_6.D_4$ x 2, $C_4\times D_{12}$ x 2, $D_4:D_6$ x 2, $D_4:D_6$ x 2, $Q_8\times D_6$ x 2, $C_6.C_2^4$ x 2, $(C_2^2\times C_{12}):C_2$ x 2, $D_4\times C_{12}$ x 2, $C_2\times C_4:C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3.D_6$ x 2, $C_2^3.D_6$ x 2, $C_2^2.D_{12}$ x 2, $C_6:C_4^2$ x 2, $D_6.D_4$ x 2, $D_{12}:C_4$ x 2, $(C_4\times S_3):C_4$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^3.D_6$, $C_4^2:S_3$, $C_4:D_{12}$, $C_4^2:S_3$, $S_3\times C_4^2$, $C_6.C_2^4$, $C_6.C_2^4$, $C_{12}:D_4$, $C_4^2:C_6$, $C_{12}.D_4$, $C_4^2:C_6$, $C_2\times C_4\times C_{12}$, $C_{12}.D_4$, $D_6:Q_8$, $C_{12}.D_4$, $C_2^3.D_6$, $C_{12}.D_4$, $D_6:Q_8$, $D_6:Q_8$

Order 64: $D_4\times C_2^3$ x 74, $C_2^3:D_4$ x 39, $C_2^3:D_4$ x 23, $C_2^4:C_4$ x 18, $D_4:C_2^3$ x 16, $C_4^2:C_2^2$ x 16, $C_2^3.D_4$ x 14, $C_2^6$ x 13, $D_4:C_2^3$ x 13, $C_2^4\times C_4$ x 13, $D_4:D_4$ x 12, $D_4^2$ x 12, $C_2^3:D_4$ x 9, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 8, $C_2^3:Q_8$ x 7, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 5, $C_2^3.D_4$ x 2, $Q_8\times C_2^3$, $C_2^2\times C_4^2$

Order 48: $C_2^2\times D_6$ x 89, $C_6:D_4$ x 36, $C_4\times D_6$ x 33, $C_2^3\times C_6$ x 24, $C_6\times D_4$ x 23, $C_2\times D_{12}$ x 23, $S_3\times D_4$ x 21, $C_2^2\times C_{12}$ x 17, $C_6.C_2^3$ x 17, $D_6:C_4$ x 8, $C_2^2:C_{12}$ x 5, $C_6.D_4$ x 5, $D_4:S_3$ x 4, $D_{12}:C_2$ x 4, $C_6:Q_8$ x 4, $C_6.D_4$ x 3, $D_4:C_6$ x 2, $C_6\times Q_8$ x 2, $D_{12}:C_2$ x 2, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_{12}:C_4$ x 2, $S_3\times Q_8$, $C_4\times C_{12}$

Order 32: $C_2^2\times D_4$ x 133, $C_2^5$ x 56, $C_2^3\times C_4$ x 47, $C_2^3:C_4$ x 34, $D_4:C_2^2$ x 26, $C_2^2\wr C_2$ x 26, $C_4:D_4$ x 20, $C_4\times D_4$ x 14, $C_2^2.D_4$ x 12, $C_2^2.D_4$ x 9, $D_4:C_2^2$ x 6, $C_2^2:Q_8$ x 6, $C_2^2\times Q_8$ x 5, $C_4:D_4$ x 5, $C_4^2:C_2$ x 5, $C_4^2:C_2$ x 5, $C_2\times C_4^2$ x 4

Order 24: $C_2\times D_6$ x 81, $C_2^2\times C_6$ x 34, $C_2\times C_{12}$ x 14, $C_6:C_4$ x 14, $C_3:D_4$ x 13, $C_4\times S_3$ x 12, $D_{12}$ x 8, $C_3\times D_4$ x 8, $C_3:Q_8$ x 2, $C_3\times Q_8$

Order 16: $C_2^4$ x 113, $C_2\times D_4$ x 103, $C_2^2\times C_4$ x 67, $C_2^2:C_4$ x 18, $D_4:C_2$ x 12, $C_4:C_4$ x 7, $C_2\times Q_8$ x 7, $C_4^2$ x 3

Order 12: $D_6$ x 35, $C_2\times C_6$ x 24, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 8: $C_2^3$ x 115, $C_2\times C_4$ x 40, $D_4$ x 29, $Q_8$ x 3

Order 6: $C_6$ x 8, $S_3$ x 6

Order 4: $C_2^2$ x 59, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 14

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 6 or order up to 8 are shown.

Order 1536: $C_2\times D_4^2:D_6$ ($C_1$)

Order 768: $D_4^2:D_6$ ($C_2$) x 64, $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$) x 8, $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$) x 8, $C_2\times (C_2\times D_4):C_2^2\times S_3$ ($C_2$) x 8, $D_6\times D_4^2$ ($C_2$) x 8, $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$) x 4, $C_3:(C_2^3.C_2^5)$ ($C_2$) x 4, $C_2\times C_2^4:C_2^2\times S_3$ ($C_2$) x 4, $C_2^6:D_6$ ($C_2$) x 4, $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$) x 2, $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$) x 2, $C_2\times (C_2^2\times D_4):C_2\times S_3$ ($C_2$) x 2, $C_4^2:C_2^2\times D_6$ ($C_2$) x 2, $C_2\times D_4^2:C_6$ ($C_2$), $D_{12}:C_2^5$ ($C_2$), $C_{12}:C_2^6$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_2^3:(C_2\times C_4)\times S_3$ ($C_2$)

Order 384: $D_4:(S_3\times D_4)$ ($C_2^2$) x 256, $S_3\times D_4:D_4$ ($C_2^2$) x 256, $D_4^2:S_3$ ($C_2^2$) x 256, $S_3\times D_4^2$ ($C_2^2$) x 256, $C_2^5:D_6$ ($C_2^2$) x 196, $C_2^3:(S_3\times D_4)$ ($C_2^2$) x 128, $S_3\times C_2^3:D_4$ ($C_2^2$) x 128, $C_2^5:D_6$ ($C_2^2$) x 128, $C_{12}:C_2^5$ ($C_2^2$) x 72, $(C_2\times D_{12}):D_4$ ($C_2^2$) x 64, $C_4^2:C_2^2\times S_3$ ($C_2^2$) x 64, $(C_2\times D_{12}):D_4$ ($C_2^2$) x 64, $C_4^2:C_2^2\times S_3$ ($C_2^2$) x 64, $D_{12}:C_2^4$ ($C_2^2$) x 32, $D_4^2:C_6$ ($C_2^2$) x 32, $(C_6\times D_4):D_4$ ($C_2^2$) x 32, $C_2^5:D_6$ ($C_2^2$) x 32, $C_6:D_4^2$ ($C_2^2$) x 32, $C_6:D_4^2$ ($C_2^2$) x 32, $C_6:D_4^2$ ($C_2^2$) x 32, $(C_2\times D_4):D_{12}$ ($C_2^2$) x 32, $C_4^2:C_2^2\times S_3$ ($C_2^2$) x 32, $C_4:D_4\times D_6$ ($C_2^2$) x 28, $(C_2\times D_{12}):C_2^3$ ($C_2^2$) x 16, $(C_2\times D_{12}):C_2^3$ ($C_2^2$) x 16, $(C_4\times D_6):C_2^3$ ($C_2^2$) x 16, $(C_2\times D_{12}):C_2^3$ ($C_2^2$) x 16, $C_2^5.D_6$ ($C_2^2$) x 16, $C_2^5:D_6$ ($C_2^2$) x 16, $C_6:D_4^2$ ($C_2^2$) x 16, $C_4\times D_4\times D_6$ ($C_2^2$) x 16, $C_2^5.D_6$ ($C_2^2$) x 16, $C_2^5:D_6$ ($C_2^2$) x 12, $C_2^2.D_4\times D_6$ ($C_2^2$) x 12, $C_2^4:D_{12}$ ($C_2^2$) x 12, $C_2^5.D_6$ ($C_2^2$) x 12, $D_{12}:C_2^4$ ($C_2^2$) x 9, $C_6\times D_4:D_4$ ($C_2^2$) x 8, $C_6\times D_4^2$ ($C_2^2$) x 8, $(C_6\times D_4):D_4$ ($C_2^2$) x 8, $C_2^5:D_6$ ($C_2^2$) x 8, $C_2^5.D_6$ ($C_2^2$) x 8, $C_6:D_4^2$ ($C_2^2$) x 8, $C_2\times D_{12}:D_4$ ($C_2^2$) x 8, $C_2\times D_{12}:D_4$ ($C_2^2$) x 8, $C_2\times D_{12}:D_4$ ($C_2^2$) x 8, $(C_4\times D_6):D_4$ ($C_2^2$) x 8, $C_2^5.D_6$ ($C_2^2$) x 8, $C_2\times D_{12}:D_4$ ($C_2^2$) x 8, $C_2\times D_4:D_{12}$ ($C_2^2$) x 8, $C_2\times C_4^2:D_6$ ($C_2^2$) x 8, $C_2\times C_4^2:D_6$ ($C_2^2$) x 8, $D_{12}:C_2^4$ ($C_2^2$) x 6, $D_{12}:C_2^4$ ($C_2^2$) x 6, $C_6\times C_2^3:D_4$ ($C_2^2$) x 4, $C_2^6:C_6$ ($C_2^2$) x 4, $C_2^5:D_6$ ($C_2^2$) x 4, $C_2\times C_4^2:D_6$ ($C_2^2$) x 4, $C_4:D_4\times D_6$ ($C_2^2$) x 4, $C_2\times C_4^2:D_6$ ($C_2^2$) x 4, $C_2\times C_4^2:D_6$ ($C_2^2$) x 4, $(C_4\times D_6):C_2^3$ ($C_2^2$) x 4, $C_2^2:Q_8\times D_6$ ($C_2^2$) x 4, $(C_4\times D_6):D_4$ ($C_2^2$) x 4, $(C_4\times D_6):D_4$ ($C_2^2$) x 4, $C_2^5.D_6$ ($C_2^2$) x 4, $C_2^4:D_{12}$ ($C_2^2$) x 4, $C_2^5.D_6$ ($C_2^2$) x 4, $D_6\times C_2^5$ ($C_2^2$) x 2, $D_6:C_2^5$ ($C_2^2$) x 2, $C_4^2:C_2^2\times C_6$ ($C_2^2$) x 2, $C_4^2:C_2^2\times C_6$ ($C_2^2$) x 2, $(C_2^2\times C_{12}):D_4$ ($C_2^2$) x 2, $(C_2^2\times C_{12}):D_4$ ($C_2^2$) x 2, $C_2^5.D_6$ ($C_2^2$) x 2, $C_2^5.D_6$ ($C_2^2$) x 2, $C_2\times C_4^2:D_6$ ($C_2^2$) x 2, $C_2\times C_4^2:D_6$ ($C_2^2$) x 2, $C_2\times C_4^2:D_6$ ($C_2^2$) x 2, $C_2\times C_4^2:D_6$ ($C_2^2$) x 2, $C_{12}.C_2^5$ ($C_2^2$), $C_{12}:C_2^5$ ($C_2^2$), $C_2^4\times D_{12}$ ($C_2^2$), $C_{12}:C_2^5$ ($C_2^2$), $C_4^2:C_2^2\times C_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$)

Order 256: $C_4^2:C_2^4$ ($S_3$)

Order 192: $C_2^4:D_6$ ($C_2^3$) x 1088, $C_{12}:C_2^4$ ($C_2^3$) x 806, $C_2^4:D_6$ ($C_2^3$) x 512, $D_{12}:D_4$ ($C_2^3$) x 512, $C_2^4:D_6$ ($C_2^3$) x 512, $S_3\times C_4:D_4$ ($C_2^3$) x 448, $C_2^4:D_6$ ($C_2^3$) x 300, $C_2^3:D_{12}$ ($C_2^3$) x 300, $C_2^4.D_6$ ($C_2^3$) x 284, $(D_4\times D_6):C_2$ ($C_2^3$) x 256, $(C_2\times D_6):D_4$ ($C_2^3$) x 256, $(C_2\times D_4):D_6$ ($C_2^3$) x 256, $(C_2\times D_4):D_6$ ($C_2^3$) x 256, $C_2^4.D_6$ ($C_2^3$) x 256, $C_2^4:D_6$ ($C_2^3$) x 256, $D_4\times D_{12}$ ($C_2^3$) x 256, $C_4^2:D_6$ ($C_2^3$) x 256, $C_2^4.D_6$ ($C_2^3$) x 256, $S_3\times C_2^2.D_4$ ($C_2^3$) x 192, $D_{12}:C_2^3$ ($C_2^3$) x 144, $C_3\times D_4:D_4$ ($C_2^3$) x 128, $C_3\times D_4^2$ ($C_2^3$) x 128, $(C_2\times D_4):D_6$ ($C_2^3$) x 128, $C_2^4.D_6$ ($C_2^3$) x 128, $C_2^4:D_6$ ($C_2^3$) x 128, $C_3:D_4^2$ ($C_2^3$) x 128, $D_{12}:D_4$ ($C_2^3$) x 128, $D_{12}:D_4$ ($C_2^3$) x 128, $D_{12}:D_4$ ($C_2^3$) x 128, $(C_4\times S_3):D_4$ ($C_2^3$) x 128, $C_2^4.D_6$ ($C_2^3$) x 128, $D_4:D_{12}$ ($C_2^3$) x 128, $D_{12}:D_4$ ($C_2^3$) x 128, $C_4^2:D_6$ ($C_2^3$) x 128, $C_4^2:D_6$ ($C_2^3$) x 128, $C_2^5:C_6$ ($C_2^3$) x 100, $C_2^4:D_6$ ($C_2^3$) x 100, $D_{12}:C_2^3$ ($C_2^3$) x 96, $D_{12}:C_2^3$ ($C_2^3$) x 96, $D_6:C_2^4$ ($C_2^3$) x 88, $C_3\times C_2^3:D_4$ ($C_2^3$) x 64, $C_4^2:D_6$ ($C_2^3$) x 64, $C_4^2:D_6$ ($C_2^3$) x 64, $C_4^2:D_6$ ($C_2^3$) x 64, $C_4^2:D_6$ ($C_2^3$) x 64, $(C_4\times D_6):C_2^2$ ($C_2^3$) x 64, $(C_4\times S_3):D_4$ ($C_2^3$) x 64, $S_3\times C_2^2:Q_8$ ($C_2^3$) x 64, $(C_4\times S_3):D_4$ ($C_2^3$) x 64, $C_2^4.D_6$ ($C_2^3$) x 64, $C_2^3:D_{12}$ ($C_2^3$) x 64, $C_2^4.D_6$ ($C_2^3$) x 64, $D_6\times C_2^4$ ($C_2^3$) x 63, $C_2^4.D_6$ ($C_2^3$) x 56, $C_2^4.D_6$ ($C_2^3$) x 56, $C_{12}:C_2^4$ ($C_2^3$) x 44, $C_2^3\times D_{12}$ ($C_2^3$) x 44, $C_{12}:C_2^4$ ($C_2^3$) x 44, $C_{12}:C_2^4$ ($D_4$) x 32, $C_3\times C_4^2:C_2^2$ ($C_2^3$) x 32, $C_3\times C_4^2:C_2^2$ ($C_2^3$) x 32, $(C_2\times C_{12}):D_4$ ($C_2^3$) x 32, $(C_2\times C_{12}):D_4$ ($C_2^3$) x 32, $C_2^4.D_6$ ($C_2^3$) x 32, $C_2^4.D_6$ ($C_2^3$) x 32, $C_2^4.D_6$ ($C_2^3$) x 32, $C_4^2:D_6$ ($C_2^3$) x 32, $C_4^2:D_6$ ($C_2^3$) x 32, $C_4^2:D_6$ ($C_2^3$) x 32, $C_4^2:D_6$ ($C_2^3$) x 32, $C_2^4.D_6$ ($C_2^3$) x 32, $C_2\times C_{12}:D_4$ ($C_2^3$) x 28, $C_2^4.D_6$ ($C_2^3$) x 28, $C_2^3:D_{12}$ ($C_2^3$) x 28, $C_2\times C_4:D_{12}$ ($C_2^3$) x 28, $C_2^4.D_6$ ($C_2^3$) x 24, $C_2^3.D_{12}$ ($C_2^3$) x 24, $C_2\times D_6.D_4$ ($C_2^3$) x 24, $C_2^4.D_6$ ($C_2^3$) x 24, $C_{12}.C_2^4$ ($C_2^3$) x 18, $D_{12}:C_2^3$ ($C_2^3$) x 18, $C_{12}.C_2^4$ ($C_2^3$) x 16, $C_3\times C_4^2:C_2^2$ ($C_2^3$) x 16, $C_2\times D_4\times C_{12}$ ($C_2^3$) x 16, $C_2^4.D_6$ ($C_2^3$) x 16, $C_2^4.D_6$ ($C_2^3$) x 16, $C_2\times D_{12}:C_4$ ($C_2^3$) x 16, $C_2^4.D_6$ ($C_2^3$) x 16, $C_4^2:D_6$ ($C_2^3$) x 16, $(C_2^3\times C_{12}):C_2$ ($C_2^3$) x 12, $C_2^4:C_{12}$ ($C_2^3$) x 12, $C_2^4.D_6$ ($C_2^3$) x 12, $C_2^4.D_6$ ($C_2^3$) x 12, $C_2^4.D_6$ ($C_2^3$) x 12, $C_4:C_4\times D_6$ ($C_2^3$) x 12, $C_2^3.D_{12}$ ($C_2^3$) x 12, $C_2\times D_6:Q_8$ ($C_2^3$) x 8, $C_2^4.D_6$ ($C_2^3$) x 8, $C_{12}.C_2^4$ ($C_2^3$) x 6, $D_{12}:C_2^3$ ($C_2^3$) x 6, $C_2\times C_{12}:D_4$ ($C_2^3$) x 4, $C_2\times C_4^2:C_6$ ($C_2^3$) x 4, $C_2^2:Q_8\times C_6$ ($C_2^3$) x 4, $C_2\times C_{12}.D_4$ ($C_2^3$) x 4, $C_2\times D_6:Q_8$ ($C_2^3$) x 4, $C_2^4.D_6$ ($C_2^3$) x 4, $C_2^4.D_6$ ($C_2^3$) x 4, $C_2\times D_6:Q_8$ ($C_2^3$) x 4, $(C_4\times D_6):C_4$ ($C_2^3$) x 4, $C_2^4.D_6$ ($C_2^3$) x 4, $C_4^2:D_6$ ($C_2^3$) x 4, $C_4^2:D_6$ ($C_2^3$) x 4, $C_4^2:D_6$ ($C_2^3$) x 4, $D_6\times C_4^2$ ($C_2^3$) x 4, $C_2^5\times C_6$ ($C_2^3$) x 2, $C_{12}.C_2^4$ ($C_2^3$) x 2, $C_2\times C_4^2:C_6$ ($C_2^3$) x 2, $C_2^4.D_6$ ($C_2^3$) x 2, $C_2^4\times C_{12}$ ($C_2^3$), $C_6.C_2^5$ ($C_2^3$)

Order 128: $D_4^2:C_2$ ($D_6$) x 32, $C_4^2:C_2^3$ ($D_6$) x 8, $C_2\times D_4^2$ ($D_6$) x 8, $C_2^4:D_4$ ($D_6$) x 4, $C_2^4:D_4$ ($D_6$) x 4, $C_4^2:C_2^3$ ($D_6$) x 2, $C_4^2:C_2^3$ ($D_6$) x 2, $D_4:C_2^4$ ($D_6$), $D_4\times C_2^4$ ($D_6$), $C_4^2:C_2^3$ ($D_6$)

Order 96: $D_4\times D_6$ ($C_2^4$) x 1328, $C_2^2:D_{12}$ ($C_2^4$) x 864, $D_6:D_4$ ($C_2^4$) x 864, $D_6.D_4$ ($C_2^4$) x 736, $C_2^3:D_6$ ($C_2^4$) x 556, $D_6:D_4$ ($C_2^4$) x 448, $C_2^3.D_6$ ($C_2^4$) x 448, $D_4\times D_6$ ($C_2\times D_4$) x 384, $C_2^3\times D_6$ ($C_2^4$) x 314, $C_2^4:C_6$ ($C_2^4$) x 288, $C_2^4:S_3$ ($C_2^4$) x 288, $C_{12}:C_2^3$ ($C_2^4$) x 284, $C_2^2.D_{12}$ ($C_2^4$) x 280, $C_{12}:C_2^3$ ($C_2^4$) x 278, $C_2^2\times D_{12}$ ($C_2^4$) x 278, $C_2^3.D_6$ ($C_2^4$) x 256, $C_{12}:D_4$ ($C_2^4$) x 256, $C_{12}:D_4$ ($C_2^4$) x 224, $C_{12}:D_4$ ($C_2^4$) x 224, $C_{12}:D_4$ ($C_2^4$) x 224, $C_4:D_{12}$ ($C_2^4$) x 224, $D_6.D_4$ ($C_2^4$) x 192, $C_{12}:D_4$ ($C_2^4$) x 192, $D_6.D_4$ ($C_2^4$) x 192, $D_4:D_6$ ($C_2^4$) x 144, $D_{12}:C_2^2$ ($C_2^4$) x 144, $C_2^3:C_{12}$ ($C_2^4$) x 140, $C_2^3.D_6$ ($C_2^4$) x 140, $C_2^3.D_6$ ($C_2^4$) x 128, $C_4\times D_{12}$ ($C_2^4$) x 128, $D_4\times C_{12}$ ($C_2^4$) x 128, $C_2^3.D_6$ ($C_2^4$) x 128, $D_{12}:C_4$ ($C_2^4$) x 128, $D_6.D_4$ ($C_2^4$) x 96, $C_2^2.D_{12}$ ($C_2^4$) x 96, $(C_2^2\times C_{12}):C_2$ ($C_2^4$) x 96, $C_2^3.D_6$ ($C_2^4$) x 96, $C_2^3.D_6$ ($C_2^4$) x 96, $C_2^3.D_6$ ($C_2^4$) x 64, $D_6:Q_8$ ($C_2^4$) x 64, $C_6.C_2^4$ ($C_2^4$) x 48, $Q_8:D_6$ ($C_2^4$) x 48, $C_2^4\times C_6$ ($C_2^4$) x 33, $(C_4\times S_3):C_4$ ($C_2^4$) x 32, $C_2^3.D_6$ ($C_2^4$) x 32, $C_4^2:S_3$ ($C_2^4$) x 32, $C_4:D_{12}$ ($C_2^4$) x 32, $C_4^2:S_3$ ($C_2^4$) x 32, $S_3\times C_4^2$ ($C_2^4$) x 32, $C_2^3\times D_6$ ($C_2\times D_4$) x 32, $C_2^3:D_6$ ($C_2\times D_4$) x 32, $C_{12}:D_4$ ($C_2^4$) x 32, $C_4^2:C_6$ ($C_2^4$) x 32, $C_{12}.D_4$ ($C_2^4$) x 32, $C_{12}.D_4$ ($C_2^4$) x 32, $D_6:Q_8$ ($C_2^4$) x 32, $C_{12}.D_4$ ($C_2^4$) x 32, $C_{12}.D_4$ ($C_2^4$) x 32, $D_6:Q_8$ ($C_2^4$) x 32, $C_2^3\times C_{12}$ ($C_2^4$) x 30, $C_6.C_2^4$ ($C_2^4$) x 30, $C_2^3.D_6$ ($C_2^4$) x 24, $C_{12}:C_2^3$ ($C_2\times D_4$) x 16, $Q_8\times D_6$ ($C_2^4$) x 16, $C_2^2\times D_{12}$ ($C_2\times D_4$) x 16, $C_{12}:C_2^3$ ($C_2\times D_4$) x 16, $C_4^2:C_6$ ($C_2^4$) x 16, $C_2^3.D_6$ ($C_2^4$) x 16, $C_2\times C_4:C_{12}$ ($C_2^4$) x 12, $C_2^2.D_{12}$ ($C_2^4$) x 12, $C_6:C_4^2$ ($C_2^4$) x 12, $C_6.C_2^4$ ($C_2^4$) x 6, $C_2\times C_4\times C_{12}$ ($C_2^4$) x 4, $C_6.C_2^4$ ($C_2^4$) x 2

Order 64: $D_4:D_4$ ($C_2\times D_6$) x 128, $D_4^2$ ($C_2\times D_6$) x 128, $C_2^3:D_4$ ($C_2\times D_6$) x 100, $C_2^3:D_4$ ($C_2\times D_6$) x 64, $D_4\times C_2^3$ ($C_2\times D_6$) x 44, $C_4^2:C_2^2$ ($C_2\times D_6$) x 32, $C_4^2:C_2^2$ ($C_2\times D_6$) x 32, $C_2^3:D_4$ ($C_2\times D_6$) x 28, $D_4:C_2^3$ ($C_2\times D_6$) x 16, $C_4^2:C_2^2$ ($C_2\times D_6$) x 16, $C_4^2:C_2^2$ ($C_2\times D_6$) x 16, $C_2^3.D_4$ ($C_2\times D_6$) x 12, $C_2^4:C_4$ ($C_2\times D_6$) x 12, $D_4:C_2^3$ ($C_2\times D_6$) x 6, $C_4^2:C_2^2$ ($C_2\times D_6$) x 4, $C_4^2:C_2^2$ ($C_2\times D_6$) x 4, $C_2^3:Q_8$ ($C_2\times D_6$) x 4, $C_2^6$ ($C_2\times D_6$) x 2, $C_4^2:C_2^2$ ($C_2\times D_6$) x 2, $C_2^4\times C_4$ ($C_2\times D_6$)

Order 48: $S_3\times D_4$ ($C_2^2\times D_4$) x 512, $C_6:D_4$ ($C_2^5$) x 432, $D_6:C_4$ ($C_2^5$) x 352, $C_2^2\times D_6$ ($C_2^5$) x 280, $C_4\times D_6$ ($C_2^5$) x 240, $C_2^2\times D_6$ ($C_2^2\times D_4$) x 216, $C_6\times D_4$ ($C_2^5$) x 216, $C_2\times D_{12}$ ($C_2^5$) x 216, $C_6:D_4$ ($C_2^2\times D_4$) x 192, $C_2^2:C_{12}$ ($C_2^5$) x 176, $C_6.D_4$ ($C_2^5$) x 176, $C_2^2\times C_{12}$ ($C_2^5$) x 104, $C_6.C_2^3$ ($C_2^5$) x 104, $C_6\times D_4$ ($C_2^2\times D_4$) x 96, $C_2\times D_{12}$ ($C_2^2\times D_4$) x 96, $C_4\times D_6$ ($C_2^2\times D_4$) x 96, $C_6.D_4$ ($C_2^5$) x 96, $C_2^3\times C_6$ ($C_2^5$) x 83, $C_4:C_{12}$ ($C_2^5$) x 48, $C_{12}:C_4$ ($C_2^5$) x 48, $C_{12}:C_4$ ($C_2^5$) x 48, $C_6:Q_8$ ($C_2^5$) x 24, $C_2^3\times C_6$ ($C_2^2\times D_4$) x 16, $C_2^2\times D_6$ ($D_4:C_2^2$) x 16, $C_4\times C_{12}$ ($C_2^5$) x 16, $C_6\times Q_8$ ($C_2^5$) x 8, $C_2^2\times C_{12}$ ($C_2^2\times D_4$) x 8, $C_6.C_2^3$ ($C_2^2\times D_4$) x 8

Order 32: $C_2^2\wr C_2$ ($C_2^2\times D_6$) x 288, $C_2^2\times D_4$ ($C_2^2\times D_6$) x 278, $C_4:D_4$ ($C_2^2\times D_6$) x 224, $C_2^3:C_4$ ($C_2^2\times D_6$) x 140, $C_4\times D_4$ ($C_2^2\times D_6$) x 128, $C_2^2.D_4$ ($C_2^2\times D_6$) x 96, $D_4:C_2^2$ ($C_2^2\times D_6$) x 48, $C_2^5$ ($C_2^2\times D_6$) x 33, $C_4:D_4$ ($C_2^2\times D_6$) x 32, $C_4^2:C_2$ ($C_2^2\times D_6$) x 32, $C_2^2:Q_8$ ($C_2^2\times D_6$) x 32, $C_2^3\times C_4$ ($C_2^2\times D_6$) x 30, $C_2^2\times D_4$ ($S_3\times D_4$) x 16, $C_4^2:C_2$ ($C_2^2\times D_6$) x 16, $C_2^2.D_4$ ($C_2^2\times D_6$) x 12, $C_2\times C_4^2$ ($C_2^2\times D_6$) x 4, $C_2^2\times Q_8$ ($C_2^2\times D_6$) x 2

Order 24: $C_2\times D_6$ ($D_4\times C_2^3$) x 200, $C_3:D_4$ ($D_4\times C_2^3$) x 128, $D_{12}$ ($D_4\times C_2^3$) x 64, $C_4\times S_3$ ($D_4\times C_2^3$) x 64, $C_3\times D_4$ ($D_4\times C_2^3$) x 64, $C_2^2\times C_6$ ($D_4\times C_2^3$) x 52, $C_2\times D_6$ ($D_4:C_2^3$) x 48, $C_2\times C_{12}$ ($C_2^6$) x 32, $C_6:C_4$ ($C_2^6$) x 32, $C_2\times D_6$ ($C_2^6$) x 32, $C_2^2\times C_6$ ($C_2^6$) x 31, $C_2\times C_{12}$ ($D_4\times C_2^3$) x 24, $C_6:C_4$ ($D_4\times C_2^3$) x 24, $C_2^2\times C_6$ ($D_4:C_2^3$) x 8

Order 16: $C_2\times D_4$ ($C_2^3\times D_6$) x 216, $C_2^2:C_4$ ($C_2^3\times D_6$) x 176, $C_2^2\times C_4$ ($C_2^3\times D_6$) x 104, $C_2\times D_4$ ($D_4\times D_6$) x 96, $C_2^4$ ($C_2^3\times D_6$) x 83, $C_4:C_4$ ($C_2^3\times D_6$) x 48, $C_4^2$ ($C_2^3\times D_6$) x 16, $C_2^4$ ($D_4\times D_6$) x 16, $C_2\times Q_8$ ($C_2^3\times D_6$) x 8, $C_2^2\times C_4$ ($D_4\times D_6$) x 8

Order 12: $D_6$ ($D_4\times C_2^4$) x 24, $C_2\times C_6$ ($D_4\times C_2^4$) x 22, $D_6$ ($D_4:C_2^4$) x 16, $D_6$ ($D_4^2:C_2$) x 16, $C_2\times C_6$ ($D_4:C_2^4$) x 12, $C_{12}$ ($D_4\times C_2^4$) x 8, $C_3:C_4$ ($D_4\times C_2^4$) x 8, $C_2\times C_6$ ($C_2^7$)

Order 8: $D_4$ ($C_{12}:C_2^4$) x 64, $C_2^3$ ($C_{12}:C_2^4$) x 52, $C_2\times C_4$ ($D_6\times C_2^4$) x 32, $C_2^3$ ($D_6\times C_2^4$) x 31, $C_2\times C_4$ ($C_{12}:C_2^4$) x 24, $C_2^3$ ($D_{12}:C_2^3$) x 8

Order 6: $S_3$ ($C_4^2:C_2^4$) x 8, $C_6$ ($C_4^2:C_2^4$) x 4, $C_6$ ($D_4:C_2^5$) x 2, $C_6$ ($D_4\times C_2^5$)

Order 4: $C_2^2$ ($C_{12}:C_2^5$) x 22, $C_2^2$ ($D_{12}:C_2^4$) x 12, $C_4$ ($C_{12}:C_2^5$) x 8, $C_2^2$ ($D_6\times C_2^5$)

Order 3: $C_3$ ($C_4^2:C_2^5$)

Order 2: $C_2$ ($D_4^2:D_6$) x 4, $C_2$ ($D_{12}:C_2^5$) x 2, $C_2$ ($C_{12}:C_2^6$)

Order 1: $C_1$ ($C_2\times D_4^2:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

All normal subgroups of index up to 6 or order up to 8 are shown.

Order 1536: $C_2\times D_4^2:D_6$ ($C_1$)

Order 768: $C_2\times D_4^2:C_6$ ($C_2$), $D_{12}:C_2^5$ ($C_2$), $C_{12}:C_2^6$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $C_3:(C_2^3.C_2^5)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2$), $D_4^2:D_6$ ($C_2$), $C_2\times (C_2\times D_4):C_2^2\times S_3$ ($C_2$), $D_6\times D_4^2$ ($C_2$), $C_2\times (C_2^2\times D_4):C_2\times S_3$ ($C_2$), $C_2\times C_2^4:C_2^2\times S_3$ ($C_2$), $C_4^2:C_2^2\times D_6$ ($C_2$), $C_2^6:D_6$ ($C_2$), $C_2\times C_2^3:(C_2\times C_4)\times S_3$ ($C_2$)

Order 384: $C_{12}:C_2^5$ ($C_2^2$) x 12, $C_2^5:D_6$ ($C_2^2$) x 8, $D_{12}:C_2^4$ ($C_2^2$) x 3, $C_6:D_4^2$ ($C_2^2$) x 3, $C_6:D_4^2$ ($C_2^2$) x 3, $C_4:D_4\times D_6$ ($C_2^2$) x 3, $D_{12}:C_2^4$ ($C_2^2$) x 2, $D_{12}:C_2^4$ ($C_2^2$) x 2, $C_2^5:D_6$ ($C_2^2$) x 2, $C_2^5:D_6$ ($C_2^2$) x 2, $(C_2\times D_{12}):C_2^3$ ($C_2^2$) x 2, $C_2^2.D_4\times D_6$ ($C_2^2$) x 2, $C_6:D_4^2$ ($C_2^2$) x 2, $C_2^5:D_6$ ($C_2^2$) x 2, $C_6:D_4^2$ ($C_2^2$) x 2, $C_4\times D_4\times D_6$ ($C_2^2$) x 2, $C_2^4:D_{12}$ ($C_2^2$) x 2, $C_2^5.D_6$ ($C_2^2$) x 2, $D_6\times C_2^5$ ($C_2^2$), $C_{12}.C_2^5$ ($C_2^2$), $C_{12}:C_2^5$ ($C_2^2$), $D_6:C_2^5$ ($C_2^2$), $D_{12}:C_2^4$ ($C_2^2$), $C_2^4\times D_{12}$ ($C_2^2$), $C_{12}:C_2^5$ ($C_2^2$), $D_4^2:C_6$ ($C_2^2$), $C_6\times D_4:D_4$ ($C_2^2$), $C_6\times D_4^2$ ($C_2^2$), $C_4^2:C_2^2\times C_6$ ($C_2^2$), $C_6\times C_2^3:D_4$ ($C_2^2$), $C_4^2:C_2^2\times C_6$ ($C_2^2$), $C_2^6:C_6$ ($C_2^2$), $C_4^2:C_2^2\times C_6$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $(C_6\times D_4):D_4$ ($C_2^2$), $(C_6\times D_4):D_4$ ($C_2^2$), $(C_2^2\times C_{12}):D_4$ ($C_2^2$), $(C_2^2\times C_{12}):D_4$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $D_4:(S_3\times D_4)$ ($C_2^2$), $S_3\times D_4:D_4$ ($C_2^2$), $D_4^2:S_3$ ($C_2^2$), $S_3\times D_4^2$ ($C_2^2$), $(C_2\times D_{12}):D_4$ ($C_2^2$), $C_4^2:C_2^2\times S_3$ ($C_2^2$), $C_2^3:(S_3\times D_4)$ ($C_2^2$), $S_3\times C_2^3:D_4$ ($C_2^2$), $C_6:D_4^2$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_4:D_4\times D_6$ ($C_2^2$), $C_2\times D_{12}:D_4$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $(C_2\times D_{12}):D_4$ ($C_2^2$), $C_4^2:C_2^2\times S_3$ ($C_2^2$), $(C_2\times D_{12}):C_2^3$ ($C_2^2$), $(C_4\times D_6):C_2^3$ ($C_2^2$), $C_2\times D_{12}:D_4$ ($C_2^2$), $C_2^2:Q_8\times D_6$ ($C_2^2$), $(C_4\times D_6):D_4$ ($C_2^2$), $C_2\times D_{12}:D_4$ ($C_2^2$), $(C_4\times D_6):C_2^3$ ($C_2^2$), $(C_4\times D_6):D_4$ ($C_2^2$), $(C_2\times D_{12}):C_2^3$ ($C_2^2$), $(C_4\times D_6):D_4$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $(C_2\times D_4):D_{12}$ ($C_2^2$), $C_4^2:C_2^2\times S_3$ ($C_2^2$), $C_2\times D_{12}:D_4$ ($C_2^2$), $C_2\times D_4:D_{12}$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_2\times C_4^2:D_6$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $C_2^4:D_{12}$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$)

Order 256: $C_4^2:C_2^4$ ($S_3$)

Order 192: $C_{12}:C_2^4$ ($C_2^3$) x 32, $D_6:C_2^4$ ($C_2^3$) x 11, $C_2^4:D_6$ ($C_2^3$) x 11, $C_2^3:D_{12}$ ($C_2^3$) x 11, $C_2^4.D_6$ ($C_2^3$) x 11, $D_6\times C_2^4$ ($C_2^3$) x 10, $C_{12}:C_2^4$ ($C_2^3$) x 10, $C_{12}:C_2^4$ ($C_2^3$) x 8, $C_2^3\times D_{12}$ ($C_2^3$) x 8, $C_2^4:D_6$ ($C_2^3$) x 7, $C_2^5:C_6$ ($C_2^3$) x 5, $C_2^4:D_6$ ($C_2^3$) x 5, $C_{12}.C_2^4$ ($C_2^3$) x 4, $D_{12}:C_2^3$ ($C_2^3$) x 4, $C_2^4.D_6$ ($C_2^3$) x 4, $C_2^4.D_6$ ($C_2^3$) x 4, $D_{12}:C_2^3$ ($C_2^3$) x 3, $C_2\times C_{12}:D_4$ ($C_2^3$) x 3, $C_2^4.D_6$ ($C_2^3$) x 3, $C_2^4:D_6$ ($C_2^3$) x 3, $C_2^4.D_6$ ($C_2^3$) x 3, $C_2^3:D_{12}$ ($C_2^3$) x 3, $C_2^4.D_6$ ($C_2^3$) x 3, $C_2^3.D_{12}$ ($C_2^3$) x 3, $D_{12}:D_4$ ($C_2^3$) x 3, $S_3\times C_4:D_4$ ($C_2^3$) x 3, $C_2\times C_4:D_{12}$ ($C_2^3$) x 3, $C_2^4.D_6$ ($C_2^3$) x 3, $C_{12}.C_2^4$ ($C_2^3$) x 2, $D_{12}:C_2^3$ ($C_2^3$) x 2, $D_{12}:C_2^3$ ($C_2^3$) x 2, $D_{12}:C_2^3$ ($C_2^3$) x 2, $(C_2^3\times C_{12}):C_2$ ($C_2^3$) x 2, $C_2\times D_4\times C_{12}$ ($C_2^3$) x 2, $C_2^4:C_{12}$ ($C_2^3$) x 2, $C_2^4.D_6$ ($C_2^3$) x 2, $C_2^4:D_6$ ($C_2^3$) x 2, $C_2^4.D_6$ ($C_2^3$) x 2, $C_2^4.D_6$ ($C_2^3$) x 2, $C_2^4.D_6$ ($C_2^3$) x 2, $(C_2\times D_6):D_4$ ($C_2^3$) x 2, $S_3\times C_2^2.D_4$ ($C_2^3$) x 2, $C_2^4:D_6$ ($C_2^3$) x 2, $C_2^4:D_6$ ($C_2^3$) x 2, $D_4\times D_{12}$ ($C_2^3$) x 2, $C_4^2:D_6$ ($C_2^3$) x 2, $C_2\times D_6.D_4$ ($C_2^3$) x 2, $C_2\times D_{12}:C_4$ ($C_2^3$) x 2, $C_4:C_4\times D_6$ ($C_2^3$) x 2, $C_2^3.D_{12}$ ($C_2^3$) x 2, $C_2^4.D_6$ ($C_2^3$) x 2, $C_4^2:D_6$ ($C_2^3$) x 2, $C_2^5\times C_6$ ($C_2^3$), $C_{12}.C_2^4$ ($C_2^3$), $C_2^4\times C_{12}$ ($C_2^3$), $C_6.C_2^5$ ($C_2^3$), $C_{12}.C_2^4$ ($C_2^3$), $C_{12}:C_2^4$ ($D_4$), $C_3\times D_4:D_4$ ($C_2^3$), $C_3\times D_4^2$ ($C_2^3$), $C_3\times C_4^2:C_2^2$ ($C_2^3$), $C_3\times C_2^3:D_4$ ($C_2^3$), $C_2\times C_{12}:D_4$ ($C_2^3$), $C_2\times C_4^2:C_6$ ($C_2^3$), $C_3\times C_4^2:C_2^2$ ($C_2^3$), $C_2^2:Q_8\times C_6$ ($C_2^3$), $C_3\times C_4^2:C_2^2$ ($C_2^3$), $C_2\times C_4^2:C_6$ ($C_2^3$), $(C_2\times C_{12}):D_4$ ($C_2^3$), $(C_2\times D_4):D_6$ ($C_2^3$), $(C_2\times C_{12}):D_4$ ($C_2^3$), $C_2\times C_{12}.D_4$ ($C_2^3$), $C_2\times D_6:Q_8$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_3:D_4^2$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $D_{12}:D_4$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $(C_4\times D_6):C_2^2$ ($C_2^3$), $(D_4\times D_6):C_2$ ($C_2^3$), $D_{12}:D_4$ ($C_2^3$), $(C_4\times S_3):D_4$ ($C_2^3$), $S_3\times C_2^2:Q_8$ ($C_2^3$), $D_{12}:D_4$ ($C_2^3$), $(C_2\times D_4):D_6$ ($C_2^3$), $(C_4\times S_3):D_4$ ($C_2^3$), $(C_4\times S_3):D_4$ ($C_2^3$), $(C_2\times D_4):D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $D_4:D_{12}$ ($C_2^3$), $D_{12}:D_4$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_2\times D_6:Q_8$ ($C_2^3$), $C_2\times D_6:Q_8$ ($C_2^3$), $(C_4\times D_6):C_4$ ($C_2^3$), $C_2^3:D_{12}$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $C_4^2:D_6$ ($C_2^3$), $D_6\times C_4^2$ ($C_2^3$)

Order 128: $D_4:C_2^4$ ($D_6$), $D_4\times C_2^4$ ($D_6$), $D_4^2:C_2$ ($D_6$), $C_4^2:C_2^3$ ($D_6$), $C_2\times D_4^2$ ($D_6$), $C_4^2:C_2^3$ ($D_6$), $C_2^4:D_4$ ($D_6$), $C_4^2:C_2^3$ ($D_6$), $C_2^4:D_4$ ($D_6$), $C_4^2:C_2^3$ ($D_6$)

Order 96: $C_2^3\times D_6$ ($C_2^4$) x 25, $C_2^3:D_6$ ($C_2^4$) x 23, $D_4\times D_6$ ($C_2^4$) x 21, $C_{12}:C_2^3$ ($C_2^4$) x 21, $C_{12}:C_2^3$ ($C_2^4$) x 15, $C_2^2\times D_{12}$ ($C_2^4$) x 15, $C_2^2.D_{12}$ ($C_2^4$) x 11, $C_2^2:D_{12}$ ($C_2^4$) x 9, $D_6.D_4$ ($C_2^4$) x 9, $D_6:D_4$ ($C_2^4$) x 9, $C_2^3\times C_{12}$ ($C_2^4$) x 7, $C_6.C_2^4$ ($C_2^4$) x 7, $C_2^3:C_{12}$ ($C_2^4$) x 7, $C_2^3.D_6$ ($C_2^4$) x 7, $C_2^4\times C_6$ ($C_2^4$) x 6, $D_6:D_4$ ($C_2^4$) x 4, $D_4:D_6$ ($C_2^4$) x 4, $D_{12}:C_2^2$ ($C_2^4$) x 4, $C_2^4:C_6$ ($C_2^4$) x 4, $C_2^4:S_3$ ($C_2^4$) x 4, $C_2^3.D_6$ ($C_2^4$) x 4, $C_2^3.D_6$ ($C_2^4$) x 3, $D_4\times D_6$ ($C_2\times D_4$) x 3, $C_{12}:D_4$ ($C_2^4$) x 3, $C_{12}:D_4$ ($C_2^4$) x 3, $C_{12}:D_4$ ($C_2^4$) x 3, $C_{12}:D_4$ ($C_2^4$) x 3, $C_{12}:D_4$ ($C_2^4$) x 3, $C_2^3.D_6$ ($C_2^4$) x 3, $C_4:D_{12}$ ($C_2^4$) x 3, $D_6.D_4$ ($C_2^4$) x 2, $C_2^2.D_{12}$ ($C_2^4$) x 2, $D_6.D_4$ ($C_2^4$) x 2, $C_4\times D_{12}$ ($C_2^4$) x 2, $C_6.C_2^4$ ($C_2^4$) x 2, $Q_8:D_6$ ($C_2^4$) x 2, $C_6.C_2^4$ ($C_2^4$) x 2, $(C_2^2\times C_{12}):C_2$ ($C_2^4$) x 2, $D_4\times C_{12}$ ($C_2^4$) x 2, $C_2\times C_4:C_{12}$ ($C_2^4$) x 2, $C_2^3.D_6$ ($C_2^4$) x 2, $C_2^3.D_6$ ($C_2^4$) x 2, $C_2^3.D_6$ ($C_2^4$) x 2, $C_2^2.D_{12}$ ($C_2^4$) x 2, $C_6:C_4^2$ ($C_2^4$) x 2, $D_6.D_4$ ($C_2^4$) x 2, $D_{12}:C_4$ ($C_2^4$) x 2, $(C_4\times S_3):C_4$ ($C_2^4$), $C_2^3.D_6$ ($C_2^4$), $C_2^3.D_6$ ($C_2^4$), $C_2^3.D_6$ ($C_2^4$), $C_4^2:S_3$ ($C_2^4$), $C_4:D_{12}$ ($C_2^4$), $C_4^2:S_3$ ($C_2^4$), $S_3\times C_4^2$ ($C_2^4$), $C_2^3\times D_6$ ($C_2\times D_4$), $C_6.C_2^4$ ($C_2^4$), $C_{12}:C_2^3$ ($C_2\times D_4$), $C_2^3:D_6$ ($C_2\times D_4$), $Q_8\times D_6$ ($C_2^4$), $C_2^2\times D_{12}$ ($C_2\times D_4$), $C_{12}:C_2^3$ ($C_2\times D_4$), $C_{12}:D_4$ ($C_2^4$), $C_4^2:C_6$ ($C_2^4$), $C_{12}.D_4$ ($C_2^4$), $C_4^2:C_6$ ($C_2^4$), $C_2\times C_4\times C_{12}$ ($C_2^4$), $C_{12}.D_4$ ($C_2^4$), $D_6:Q_8$ ($C_2^4$), $C_{12}.D_4$ ($C_2^4$), $C_2^3.D_6$ ($C_2^4$), $C_{12}.D_4$ ($C_2^4$), $D_6:Q_8$ ($C_2^4$), $D_6:Q_8$ ($C_2^4$)

Order 64: $D_4\times C_2^3$ ($C_2\times D_6$) x 8, $C_2^3:D_4$ ($C_2\times D_6$) x 5, $C_2^3:D_4$ ($C_2\times D_6$) x 3, $D_4:C_2^3$ ($C_2\times D_6$) x 2, $C_2^3.D_4$ ($C_2\times D_6$) x 2, $C_4^2:C_2^2$ ($C_2\times D_6$) x 2, $C_2^4:C_4$ ($C_2\times D_6$) x 2, $C_2^6$ ($C_2\times D_6$), $D_4:C_2^3$ ($C_2\times D_6$), $C_2^4\times C_4$ ($C_2\times D_6$), $D_4:D_4$ ($C_2\times D_6$), $D_4^2$ ($C_2\times D_6$), $C_4^2:C_2^2$ ($C_2\times D_6$), $C_2^3:D_4$ ($C_2\times D_6$), $C_4^2:C_2^2$ ($C_2\times D_6$), $C_4^2:C_2^2$ ($C_2\times D_6$), $C_4^2:C_2^2$ ($C_2\times D_6$), $C_2^3:Q_8$ ($C_2\times D_6$), $C_4^2:C_2^2$ ($C_2\times D_6$), $C_4^2:C_2^2$ ($C_2\times D_6$)

Order 48: $C_2^2\times D_6$ ($C_2^5$) x 22, $C_6:D_4$ ($C_2^5$) x 13, $C_4\times D_6$ ($C_2^5$) x 12, $C_2^3\times C_6$ ($C_2^5$) x 10, $C_2^2\times C_{12}$ ($C_2^5$) x 10, $C_6.C_2^3$ ($C_2^5$) x 10, $C_6\times D_4$ ($C_2^5$) x 8, $C_2\times D_{12}$ ($C_2^5$) x 8, $D_6:C_4$ ($C_2^5$) x 8, $C_2^2\times D_6$ ($C_2^2\times D_4$) x 5, $C_2^2:C_{12}$ ($C_2^5$) x 5, $C_6.D_4$ ($C_2^5$) x 5, $C_4\times D_6$ ($C_2^2\times D_4$) x 3, $C_6.D_4$ ($C_2^5$) x 3, $C_6\times D_4$ ($C_2^2\times D_4$) x 2, $C_6:D_4$ ($C_2^2\times D_4$) x 2, $S_3\times D_4$ ($C_2^2\times D_4$) x 2, $C_2\times D_{12}$ ($C_2^2\times D_4$) x 2, $C_6:Q_8$ ($C_2^5$) x 2, $C_4:C_{12}$ ($C_2^5$) x 2, $C_{12}:C_4$ ($C_2^5$) x 2, $C_{12}:C_4$ ($C_2^5$) x 2, $C_2^3\times C_6$ ($C_2^2\times D_4$), $C_2^2\times D_6$ ($D_4:C_2^2$), $C_6\times Q_8$ ($C_2^5$), $C_2^2\times C_{12}$ ($C_2^2\times D_4$), $C_6.C_2^3$ ($C_2^2\times D_4$), $C_4\times C_{12}$ ($C_2^5$)

Order 32: $C_2^2\times D_4$ ($C_2^2\times D_6$) x 15, $C_2^3\times C_4$ ($C_2^2\times D_6$) x 7, $C_2^3:C_4$ ($C_2^2\times D_6$) x 7, $C_2^5$ ($C_2^2\times D_6$) x 6, $C_2^2\wr C_2$ ($C_2^2\times D_6$) x 4, $C_4:D_4$ ($C_2^2\times D_6$) x 3, $D_4:C_2^2$ ($C_2^2\times D_6$) x 2, $C_2^2.D_4$ ($C_2^2\times D_6$) x 2, $C_4\times D_4$ ($C_2^2\times D_6$) x 2, $C_2^2.D_4$ ($C_2^2\times D_6$) x 2, $C_2^2\times Q_8$ ($C_2^2\times D_6$), $C_2^2\times D_4$ ($S_3\times D_4$), $C_4:D_4$ ($C_2^2\times D_6$), $C_4^2:C_2$ ($C_2^2\times D_6$), $C_2^2:Q_8$ ($C_2^2\times D_6$), $C_4^2:C_2$ ($C_2^2\times D_6$), $C_2\times C_4^2$ ($C_2^2\times D_6$)

Order 24: $C_2\times D_6$ ($D_4\times C_2^3$) x 7, $C_2^2\times C_6$ ($C_2^6$) x 6, $C_2\times D_6$ ($C_2^6$) x 6, $C_2\times C_{12}$ ($C_2^6$) x 4, $C_6:C_4$ ($C_2^6$) x 4, $C_2^2\times C_6$ ($D_4\times C_2^3$) x 3, $C_2\times D_6$ ($D_4:C_2^3$) x 3, $C_2\times C_{12}$ ($D_4\times C_2^3$) x 2, $C_6:C_4$ ($D_4\times C_2^3$) x 2, $C_4\times S_3$ ($D_4\times C_2^3$) x 2, $C_3:D_4$ ($D_4\times C_2^3$), $D_{12}$ ($D_4\times C_2^3$), $C_2^2\times C_6$ ($D_4:C_2^3$), $C_3\times D_4$ ($D_4\times C_2^3$)

Order 16: $C_2^4$ ($C_2^3\times D_6$) x 10, $C_2^2\times C_4$ ($C_2^3\times D_6$) x 10, $C_2\times D_4$ ($C_2^3\times D_6$) x 8, $C_2^2:C_4$ ($C_2^3\times D_6$) x 5, $C_4:C_4$ ($C_2^3\times D_6$) x 2, $C_2\times D_4$ ($D_4\times D_6$) x 2, $C_4^2$ ($C_2^3\times D_6$), $C_2^4$ ($D_4\times D_6$), $C_2\times Q_8$ ($C_2^3\times D_6$), $C_2^2\times C_4$ ($D_4\times D_6$)

Order 12: $C_2\times C_6$ ($D_4\times C_2^4$) x 3, $D_6$ ($D_4\times C_2^4$) x 3, $C_2\times C_6$ ($D_4:C_2^4$) x 2, $D_6$ ($D_4:C_2^4$) x 2, $C_2\times C_6$ ($C_2^7$), $D_6$ ($D_4^2:C_2$), $C_{12}$ ($D_4\times C_2^4$), $C_3:C_4$ ($D_4\times C_2^4$)

Order 8: $C_2^3$ ($D_6\times C_2^4$) x 6, $C_2\times C_4$ ($D_6\times C_2^4$) x 4, $C_2^3$ ($C_{12}:C_2^4$) x 3, $C_2\times C_4$ ($C_{12}:C_2^4$) x 2, $C_2^3$ ($D_{12}:C_2^3$), $D_4$ ($C_{12}:C_2^4$)

Order 6: $C_6$ ($D_4:C_2^5$), $C_6$ ($D_4\times C_2^5$), $C_6$ ($C_4^2:C_2^4$), $S_3$ ($C_4^2:C_2^4$)

Order 4: $C_2^2$ ($C_{12}:C_2^5$) x 3, $C_2^2$ ($D_{12}:C_2^4$) x 2, $C_2^2$ ($D_6\times C_2^5$), $C_4$ ($C_{12}:C_2^5$)

Order 3: $C_3$ ($C_4^2:C_2^5$)

Order 2: $C_2$ ($D_{12}:C_2^5$), $C_2$ ($C_{12}:C_2^6$), $C_2$ ($D_4^2:D_6$)

Order 1: $C_1$ ($C_2\times D_4^2:D_6$)

Series

Derived series

$C_2\times D_4^2:D_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_2\times D_4^2:D_6$

$\rhd$

$C_2\times C_2^3:(C_2\times C_4)\times S_3$

$\rhd$

$C_4^2:C_2^2\times C_6$

$\rhd$

$C_4^2:C_2^3$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^2\times D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times D_4^2:D_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_4^2:C_2^4$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $264 \times 264$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.