Abstract group 1536.15184889: $(C_2\times C_{12}).D_4^2$

• Label: 1536.15184889
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $4$

Group information

Description:	$(C_2\times C_{12}).D_4^2$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3:(C_2^5.C_2^6.C_2^4)$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	215	2	424	142	384	368	1536
Conjugacy classes	1	27	1	27	23	10	31	120
Divisions	1	27	1	22	19	5	15	90
Autjugacy classes	1	15	1	15	13	3	11	59

Dimension	1	2	4	8	16
Irr. complex chars.	32	48	30	9	1	120
Irr. rational chars.	16	28	30	14	2	90

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$4$
Inequivalent generating quadruples:	$16773120$

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:	${\langle a, b, c, d, e, f \mid a^{4}=b^{2}=c^{4}=d^{4}=e^{2}=f^{6}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2,4,6)(3,5,7,8)(9,10)(11,12), (10,12), (1,3)(4,7)(6,8)(9,11)(10,12)(14,15) \!\cdots\! \rangle$
Direct product:	$D_4$ $\, \times\, $ $((C_2^2\times D_6):C_4)$
Semidirect product:	$(C_2^5.D_6)$ $\,\rtimes\,$ $C_4$	$(C_6.D_4^2)$ $\,\rtimes\,$ $C_4$	$(C_6.D_4^2)$ $\,\rtimes\,$ $C_4$	$(C_6.D_4^2)$ $\,\rtimes\,$ $C_4$	all 53
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_6:D_4^2)$ . $C_2$	$(C_2\times D_4^2)$ . $D_6$	$(C_2^5:C_6)$ . $D_4$	$C_2^5$ . $(C_6:D_4)$	all 118

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 42154 subgroups in 5286 conjugacy classes, 339 normal (139 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^4.D_{12}$
Commutator:	$G' \simeq$ $C_2^3\times C_6$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times D_4$	$G/\Phi \simeq$ $C_2^2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_2^3:C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2\times C_{12}).D_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^5:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^2\times D_4).C_2^4$
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: $(C_2\times C_{12}).D_4^2$

Order 768: $(C_2^3\times C_{12}).D_4$ x 2, $(C_2^3\times C_{12}).D_4$ x 2, $(D_6\times C_2^4):C_4$ x 2, $(D_6\times C_2^4):C_4$ x 2, $(C_2^3\times D_{12}):C_4$, $(C_2^3\times D_{12}):C_4$, $C_{12}:C_2\wr C_4$, $C_{12}:C_2\wr C_4$, $D_6:D_4^2$, $C_{12}.D_4^2$, $D_4\times C_2^3:C_{12}$

Order 512: $(C_2^2\times D_4).C_2^4$

Order 384: $(C_2^3\times D_6):C_4$ x 20, $(C_6\times D_4).D_4$ x 4, $C_2^4:(C_2\times C_{12})$ x 2, $C_2^5:C_{12}$ x 2, $(C_6\times D_4).D_4$ x 2, $(C_2^2\times C_{12}).D_4$ x 2, $C_2^5:D_6$ x 2, $C_2^5:C_{12}$ x 2, $C_2^5:D_6$ x 2, $C_2^2\times C_{12}.D_4$ x 2, $C_6.D_4^2$ x 2, $C_2^5.D_6$ x 2, $C_6.D_4^2$ x 2, $C_2^5:D_6$ x 2, $(C_2^2\times D_4):C_{12}$, $(C_2\times C_4^2):C_{12}$, $C_4\times C_2^3:C_{12}$, $(C_4\times C_{12}).D_4$, $(C_4\times C_{12}).D_4$, $C_4\times C_{12}.D_4$, $C_{12}:C_2^5$, $C_6\times D_4^2$, $C_6:D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $(C_4\times D_6):D_4$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $C_4\times D_6:D_4$, $D_4\times C_3:\OD_{16}$

Order 256: $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^6:C_4$ x 2, $C_2^6:C_4$ x 2, $(D_4\times C_2^3):C_4$, $(D_4\times C_2^3):C_4$, $C_4:C_2\wr C_4$, $C_2^4:C_4^2$, $C_2^2:D_4^2$, $C_4.D_4^2$, $C_2^2.D_4^2$

Order 192: $C_2^4:D_6$ x 29, $(C_2^2\times D_6):C_4$ x 22, $C_{12}:C_2^4$ x 16, $C_2^4:C_{12}$ x 14, $C_2\times C_{12}.D_4$ x 14, $C_2^4:D_6$ x 8, $C_2^3.D_{12}$ x 7, $C_2^4.D_6$ x 6, $C_2^4.D_6$ x 5, $C_{12}:C_2^4$ x 4, $D_6:C_2^4$ x 4, $C_3\times D_4^2$ x 4, $C_2\times C_{12}:D_4$ x 4, $C_2^5:C_6$ x 4, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2\times D_4\times C_{12}$ x 3, $(C_2\times C_4).D_{12}$ x 2, $(C_2\times C_{12}):D_4$ x 2, $C_2^4:C_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^3.D_{12}$ x 2, $(C_2\times C_6):\OD_{16}$ x 2, $C_4^2.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^3.D_{12}$ x 2, $D_6\times C_2^4$ x 2, $(C_2^2\times C_{12}):C_4$ 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, $C_2^3.D_{12}$ x 2, $C_{12}.C_2^4$ x 2, $(C_2\times C_{12}):D_4$, $(C_2^2\times C_{12}):C_4$, $C_4\times C_2^2:C_{12}$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^4.D_6$, $C_{12}:\OD_{16}$, $C_4^2.D_6$, $(C_2\times C_4):D_{12}$, $C_6.(C_4\times D_4)$, $(C_2\times D_{12}):C_4$, $(C_4\times D_6):C_4$, $C_2^4.D_6$, $(C_2\times C_4):D_{12}$, $D_6:C_4^2$, $C_{12}:\OD_{16}$, $C_{12}:\OD_{16}$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2\times C_{12}:D_4$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 128: $C_2^5:C_4$ x 20, $C_2^3.C_4^2$ x 4, $C_2^4:D_4$ x 4, $C_2^4:D_4$ x 4, $C_2.D_4^2$ x 3, $\OD_{16}:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^5.C_4$ x 2, $C_2^5:C_4$ x 2, $C_2\times D_4^2$ x 2, $C_2^4:D_4$ x 2, $\OD_{16}:C_2^3$ x 2, $C_2^5:C_4$ x 2, $C_2^3\wr C_2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $\OD_{16}:D_4$, $C_2^4.C_2^3$, $C_4^2.D_4$, $C_2^4.C_2^3$, $C_4^2.D_4$, $C_2^3:C_4^2$, $D_4\times C_2^4$, $D_4\times \OD_{16}$, $C_2^4:D_4$, $C_2^3.C_2^4$

Order 96: $D_4\times D_6$ x 116, $D_6:D_4$ x 56, $C_{12}:C_2^3$ x 46, $C_2^3:D_6$ x 45, $C_2^3\times D_6$ x 31, $C_2^3:C_{12}$ x 24, $C_2^2.D_{12}$ x 23, $C_2^3.D_6$ x 19, $C_2^3:C_{12}$ x 15, $C_{12}.D_4$ x 15, $C_6:\OD_{16}$ x 12, $D_4\times C_{12}$ x 10, $C_2^2\times D_{12}$ x 8, $C_{12}:C_2^3$ x 8, $C_{12}:D_4$ x 8, $C_2^4:C_6$ x 8, $C_2^4:S_3$ x 8, $C_2^3.D_6$ x 8, $C_6.C_4^2$ x 6, $C_2^3\times C_{12}$ x 6, $C_{12}.D_4$ x 4, $C_2^4\times C_6$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_2^3.D_6$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_6.C_2^4$ x 3, $C_2\times C_4:C_{12}$ x 3, $C_6.C_4^2$ x 2, $C_{12}:D_4$ x 2, $C_2\times C_4\times C_{12}$ x 2, $C_{12}.C_2^3$ x 2, $C_{12}:C_8$ x 2, $C_{12}:C_8$, $C_2^2.D_{12}$, $C_2^3.D_6$, $C_6:C_4^2$, $C_6.C_4^2$

Order 64: $C_2^3:D_4$ x 35, $D_4\times C_2^3$ x 25, $C_2\wr C_4$ x 22, $\OD_{16}:C_2^2$ x 14, $C_2^4:C_4$ x 14, $C_2^3.D_4$ x 14, $D_4^2$ x 12, $C_2^4:C_4$ x 11, $C_2^3:D_4$ x 9, $C_2^3:D_4$ x 8, $C_2^4:C_4$ x 6, $C_4^2:C_2^2$ x 5, $C_2^3.C_2^3$ x 4, $C_2^3.D_4$ x 3, $C_2^3.D_4$ x 3, $C_2^2:C_4^2$ x 3, $C_2^2:\OD_{16}$ x 2, $C_2^3.D_4$ x 2, $C_2^6$ x 2, $C_2^2\times \OD_{16}$ x 2, $\OD_{16}:C_4$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $C_8:D_4$ x 2, $C_4\times \OD_{16}$, $C_2^4\times C_4$, $C_8:D_4$, $C_8\times D_4$, $C_4:\OD_{16}$

Order 48: $C_2^2\times D_6$ x 206, $C_6:D_4$ x 140, $S_3\times D_4$ x 128, $C_6\times D_4$ x 108, $C_2^3\times C_6$ x 40, $C_2^2\times C_{12}$ x 39, $C_2\times D_{12}$ x 30, $C_4\times D_6$ x 30, $C_2^2:C_{12}$ x 29, $D_6:C_4$ x 28, $C_6.D_4$ x 24, $C_6.C_2^3$ x 18, $C_3:\OD_{16}$ x 13, $C_6:C_8$ x 8, $C_4:C_{12}$ x 5, $C_4\times C_{12}$ x 4, $C_{12}:C_4$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2

Order 32: $C_2^2\times D_4$ x 215, $C_2^2\wr C_2$ x 72, $C_2^3:C_4$ x 66, $C_2^5$ x 35, $C_4:D_4$ x 24, $C_4\times D_4$ x 18, $C_2^3\times C_4$ x 17, $\OD_{16}:C_2$ x 15, $C_2^3:C_4$ x 15, $C_2\times \OD_{16}$ x 12, $C_2.C_4^2$ x 8, $C_4:D_4$ x 6, $C_2^2.D_4$ x 5, $C_2^2:C_8$ x 4, $C_2\times C_4^2$ x 3, $C_2^2\times C_8$ x 2, $C_4:C_8$ x 2, $C_8:C_4$, $C_4\times C_8$

Order 24: $C_2\times D_6$ x 350, $C_2^2\times C_6$ x 113, $C_3:D_4$ x 68, $C_2\times C_{12}$ x 56, $C_3\times D_4$ x 49, $C_6:C_4$ x 29, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_3:C_8$ x 5

Order 16: $C_2\times D_4$ x 406, $C_2^4$ x 246, $C_2^2\times C_4$ x 87, $C_2^2:C_4$ x 81, $\OD_{16}$ x 13, $C_4:C_4$ x 9, $C_2\times C_8$ x 8, $C_4^2$ x 6

Order 12: $D_6$ x 132, $C_2\times C_6$ x 90, $C_{12}$ x 15, $C_3:C_4$ x 7

Order 8: $C_2^3$ x 463, $D_4$ x 133, $C_2\times C_4$ x 101, $C_8$ x 5

Order 6: $C_6$ x 19, $S_3$ x 8

Order 4: $C_2^2$ x 222, $C_4$ x 22

Order 3: $C_3$

Order 2: $C_2$ x 27

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $(C_2\times C_{12}).D_4^2$

Order 768: $(C_2^3\times D_{12}):C_4$, $(C_2^3\times D_{12}):C_4$, $(C_2^3\times C_{12}).D_4$, $(C_2^3\times C_{12}).D_4$, $C_{12}:C_2\wr C_4$, $(D_6\times C_2^4):C_4$, $C_{12}:C_2\wr C_4$, $D_6:D_4^2$, $(D_6\times C_2^4):C_4$, $C_{12}.D_4^2$, $D_4\times C_2^3:C_{12}$

Order 512: $(C_2^2\times D_4).C_2^4$

Order 384: $(C_2^3\times D_6):C_4$ x 8, $(C_6\times D_4).D_4$ x 3, $(C_2^2\times D_4):C_{12}$, $C_2^4:(C_2\times C_{12})$, $(C_2\times C_4^2):C_{12}$, $C_2^5:C_{12}$, $C_4\times C_2^3:C_{12}$, $(C_6\times D_4).D_4$, $(C_4\times C_{12}).D_4$, $(C_2^2\times C_{12}).D_4$, $(C_4\times C_{12}).D_4$, $C_4\times C_{12}.D_4$, $C_{12}:C_2^5$, $C_6\times D_4^2$, $C_6:D_4^2$, $C_2^5:D_6$, $C_2^5:C_{12}$, $C_6.D_4^2$, $C_2^5:D_6$, $C_6.D_4^2$, $C_2^2\times C_{12}.D_4$, $C_6.D_4^2$, $(C_4\times D_6):D_4$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $C_2^5.D_6$, $C_6.D_4^2$, $C_2^5:D_6$, $C_6.D_4^2$, $C_6.D_4^2$, $C_4\times D_6:D_4$, $D_4\times C_3:\OD_{16}$

Order 256: $(D_4\times C_2^3):C_4$, $(D_4\times C_2^3):C_4$, $C_2^5.D_4$, $C_2^5.D_4$, $C_4:C_2\wr C_4$, $C_2^6:C_4$, $C_2^4:C_4^2$, $C_2^2:D_4^2$, $C_2^6:C_4$, $C_4.D_4^2$, $C_2^2.D_4^2$

Order 192: $C_2^4:D_6$ x 13, $C_2^4:C_{12}$ x 7, $C_2\times C_{12}.D_4$ x 7, $(C_2^2\times D_6):C_4$ x 7, $C_{12}:C_2^4$ x 7, $C_2^3.D_{12}$ x 4, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_{12}:C_2^4$ x 3, $C_3\times D_4^2$ x 3, $C_2\times C_{12}:D_4$ x 3, $C_2\times D_4\times C_{12}$ x 3, $(C_2\times C_4).D_{12}$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $D_6:C_2^4$ x 2, $(C_2^2\times C_{12}):C_4$ x 2, $C_2^5:C_6$ x 2, $(C_2\times C_{12}):D_4$, $(C_2\times C_{12}):D_4$, $(C_2^2\times C_{12}):C_4$, $C_2^4:C_{12}$, $C_4\times C_2^2:C_{12}$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $C_2^4.D_6$, $(C_2\times C_6):\OD_{16}$, $C_{12}:\OD_{16}$, $C_4^2.D_6$, $C_4^2.D_6$, $(C_2\times C_4):D_{12}$, $C_6.(C_4\times D_4)$, $(C_2\times D_{12}):C_4$, $(C_4\times D_6):C_4$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^4.D_6$, $(C_2\times C_4):D_{12}$, $D_6:C_4^2$, $C_{12}:\OD_{16}$, $C_{12}:\OD_{16}$, $D_6\times C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2\times C_{12}:D_4$, $C_2^4:C_{12}$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_{12}.C_2^4$

Order 128: $C_2^5:C_4$ x 8, $C_2^3.C_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2\times D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^4:D_4$ x 2, $C_2.D_4^2$ x 2, $C_2^4:D_4$ x 2, $\OD_{16}:D_4$, $C_2^4.C_2^3$, $\OD_{16}:D_4$, $C_2^4.D_4$, $C_4^2.D_4$, $C_2^4.C_2^3$, $C_2^5.C_4$, $C_2^5:C_4$, $C_4^2.D_4$, $C_2^3:C_4^2$, $D_4\times C_2^4$, $C_2^4:D_4$, $D_4\times \OD_{16}$, $\OD_{16}:C_2^3$, $C_2^5:C_4$, $C_2^4:D_4$, $C_2^3\wr C_2$, $C_2^3.C_2^4$

Order 96: $C_{12}:C_2^3$ x 25, $D_4\times D_6$ x 19, $D_6:D_4$ x 19, $C_2^3:D_6$ x 14, $C_2^3:C_{12}$ x 13, $C_2^3\times D_6$ x 11, $C_2^2.D_{12}$ x 11, $C_2^3.D_6$ x 10, $C_2^3:C_{12}$ x 8, $C_{12}.D_4$ x 8, $C_{12}:D_4$ x 6, $D_4\times C_{12}$ x 6, $C_6:\OD_{16}$ x 6, $C_6.C_4^2$ x 5, $C_2^2\times D_{12}$ x 5, $C_{12}:C_2^3$ x 5, $C_2^3\times C_{12}$ x 4, $C_2^4:C_6$ x 4, $C_2\times C_4:C_{12}$ x 3, $C_6.C_4^2$ x 2, $C_{12}.D_4$ x 2, $C_2^4\times C_6$ x 2, $C_6.C_2^4$ x 2, $C_{12}:D_4$ x 2, $C_2\times C_4\times C_{12}$ x 2, $C_{12}:C_8$ x 2, $C_{12}:C_8$, $C_2^4:S_3$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}:D_4$, $C_2^2.D_{12}$, $C_2^3.D_6$, $C_6:C_4^2$, $C_{12}.C_2^3$, $C_6.C_4^2$

Order 64: $C_2^3:D_4$ x 16, $D_4\times C_2^3$ x 13, $C_2^3.D_4$ x 8, $\OD_{16}:C_2^2$ x 7, $C_2^4:C_4$ x 7, $C_2\wr C_4$ x 7, $C_2^4:C_4$ x 6, $C_2^3:D_4$ x 6, $C_2^3:D_4$ x 5, $C_4^2:C_2^2$ x 5, $C_2^3.C_2^3$ x 4, $D_4^2$ x 4, $C_2^3.D_4$ x 3, $C_2^2:C_4^2$ x 3, $C_2^4:C_4$ x 3, $C_2^3.D_4$ x 2, $C_2^3.D_4$ x 2, $\OD_{16}:C_4$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2:\OD_{16}$, $C_4\times \OD_{16}$, $C_2^6$, $C_2^4\times C_4$, $C_2^2\times \OD_{16}$, $C_8:D_4$, $C_8:D_4$, $C_8\times D_4$, $C_4:\OD_{16}$

Order 48: $C_6\times D_4$ x 50, $C_2^2\times D_6$ x 44, $C_6:D_4$ x 30, $C_2^2\times C_{12}$ x 23, $C_2^3\times C_6$ x 19, $C_2^2:C_{12}$ x 15, $C_4\times D_6$ x 11, $D_6:C_4$ x 11, $C_6.D_4$ x 10, $C_6.C_2^3$ x 9, $C_2\times D_{12}$ x 9, $S_3\times D_4$ x 7, $C_3:\OD_{16}$ x 7, $C_6:C_8$ x 4, $C_4:C_{12}$ x 4, $C_4\times C_{12}$ x 3, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 32: $C_2^2\times D_4$ x 63, $C_2^3:C_4$ x 34, $C_2^2\wr C_2$ x 24, $C_2^5$ x 13, $C_2^3\times C_4$ x 11, $C_4:D_4$ x 9, $\OD_{16}:C_2$ x 8, $C_2^3:C_4$ x 8, $C_4\times D_4$ x 8, $C_2.C_4^2$ x 7, $C_2\times \OD_{16}$ x 6, $C_2^2.D_4$ x 5, $C_4:D_4$ x 3, $C_2\times C_4^2$ x 3, $C_2^2:C_8$ x 2, $C_4:C_8$ x 2, $C_8:C_4$, $C_2^2\times C_8$, $C_4\times C_8$

Order 24: $C_2\times D_6$ x 57, $C_2^2\times C_6$ x 51, $C_2\times C_{12}$ x 34, $C_3\times D_4$ x 22, $C_6:C_4$ x 15, $C_3:D_4$ x 10, $D_{12}$ x 3, $C_4\times S_3$ x 3, $C_3:C_8$ x 3

Order 16: $C_2\times D_4$ x 96, $C_2^4$ x 63, $C_2^2\times C_4$ x 43, $C_2^2:C_4$ x 36, $\OD_{16}$ x 7, $C_4:C_4$ x 6, $C_2\times C_8$ x 4, $C_4^2$ x 4

Order 12: $C_2\times C_6$ x 45, $D_6$ x 25, $C_{12}$ x 11, $C_3:C_4$ x 4

Order 8: $C_2^3$ x 108, $C_2\times C_4$ x 52, $D_4$ x 35, $C_8$ x 3

Order 6: $C_6$ x 13, $S_3$ x 2

Order 4: $C_2^2$ x 70, $C_4$ x 15

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $(C_2\times C_{12}).D_4^2$ ($C_1$)

Order 768: $(C_2^3\times C_{12}).D_4$ ($C_2$) x 2, $(C_2^3\times C_{12}).D_4$ ($C_2$) x 2, $(D_6\times C_2^4):C_4$ ($C_2$) x 2, $(D_6\times C_2^4):C_4$ ($C_2$) x 2, $(C_2^3\times D_{12}):C_4$ ($C_2$), $(C_2^3\times D_{12}):C_4$ ($C_2$), $C_{12}:C_2\wr C_4$ ($C_2$), $C_{12}:C_2\wr C_4$ ($C_2$), $D_6:D_4^2$ ($C_2$), $C_{12}.D_4^2$ ($C_2$), $D_4\times C_2^3:C_{12}$ ($C_2$)

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

Order 256: $C_2^2.D_4^2$ ($S_3$)

Order 192: $C_2^4:D_6$ ($C_2\times C_4$) x 6, $C_2^4:C_{12}$ ($C_2^3$) x 4, $C_2\times C_{12}.D_4$ ($C_2^3$) x 4, $(C_2^2\times D_6):C_4$ ($D_4$) x 4, $C_2^3.D_{12}$ ($C_2\times C_4$) x 3, $C_2\times C_{12}:D_4$ ($D_4$) x 3, $C_2^4:D_6$ ($C_2^3$) x 3, $C_2^4.D_6$ ($C_2\times C_4$) x 2, $C_2^4.D_6$ ($C_2\times C_4$) x 2, $D_6\times C_2^4$ ($C_2\times C_4$) x 2, $C_{12}:C_2^4$ ($C_2^3$) x 2, $D_6:C_2^4$ ($C_2\times C_4$) x 2, $C_2^5:C_6$ ($D_4$) x 2, $C_2^5:C_6$ ($C_2\times C_4$) x 2, $C_2^4.D_6$ ($C_2\times C_4$) x 2, $C_2^4.D_6$ ($C_2^3$), $C_2^3.D_{12}$ ($C_2\times C_4$), $C_2^4.D_6$ ($C_2\times C_4$), $C_2^4.D_6$ ($C_2\times C_4$), $C_{12}:C_2^4$ ($D_4$), $C_{12}:C_2^4$ ($C_2\times C_4$), $C_2^3\times D_{12}$ ($C_2\times C_4$), $C_{12}:C_2^4$ ($C_2\times C_4$), $C_2\times C_{12}:D_4$ ($D_4$), $C_2\times C_{12}:D_4$ ($C_2\times C_4$), $C_2\times D_4\times C_{12}$ ($C_2^3$), $C_2\times D_4\times C_{12}$ ($D_4$)

Order 128: $C_2^5:C_4$ ($D_6$) x 2, $C_2^5:C_4$ ($D_6$) x 2, $C_2^4.C_2^3$ ($D_6$), $C_2^3:C_4^2$ ($D_6$), $C_2\times D_4^2$ ($D_6$)

Order 96: $C_{12}:C_2^3$ ($C_2^2:C_4$) x 9, $C_2^3:C_{12}$ ($C_2^2:C_4$) x 6, $C_{12}:C_2^3$ ($C_2\times D_4$) x 4, $C_2^3.D_6$ ($C_2^2\times C_4$) x 4, $C_2^3\times D_6$ ($C_2^2\times C_4$) x 3, $C_2^3\times C_{12}$ ($C_2\times D_4$) x 3, $C_2^3:C_{12}$ ($C_2\times D_4$) x 2, $C_{12}.D_4$ ($C_2\times D_4$) x 2, $C_2^4\times C_6$ ($C_2\times D_4$) x 2, $C_2^4\times C_6$ ($C_2^2\times C_4$) x 2, $C_{12}:C_2^3$ ($C_2^2\times C_4$) x 2, $D_6:D_4$ ($D_4:C_2$) x 2, $D_6:D_4$ ($C_2\times D_4$) x 2, $C_{12}:C_2^3$ ($C_2^4$), $C_2^3\times C_{12}$ ($C_2^2\times C_4$), $C_6.C_2^4$ ($C_2^2\times C_4$), $C_2\times C_4:C_{12}$ ($C_2^2:C_4$), $C_2\times C_4:C_{12}$ ($C_2\times D_4$), $C_2^3:C_{12}$ ($C_2\times D_4$), $C_2^3:C_{12}$ ($C_2^2\times C_4$), $C_2\times C_4\times C_{12}$ ($C_2\times D_4$)

Order 64: $C_2^4:C_4$ ($C_2\times D_6$) x 4, $D_4\times C_2^3$ ($C_2\times D_6$) x 2, $C_2^3:D_4$ ($D_{12}$) x 2, $C_2^3:D_4$ ($C_3:D_4$) x 2, $C_2^3:D_4$ ($C_4\times S_3$) x 2, $D_4\times C_2^3$ ($C_3:D_4$), $D_4\times C_2^3$ ($C_4\times S_3$), $C_4^2:C_2^2$ ($D_{12}$), $C_2^3:D_4$ ($C_3:D_4$), $C_2^3:D_4$ ($C_4\times S_3$), $C_4^2:C_2^2$ ($D_{12}$), $C_4^2:C_2^2$ ($C_2\times D_6$)

Order 48: $C_2^3\times C_6$ ($C_2^3:C_4$) x 6, $C_2^2\times C_{12}$ ($C_2^3:C_4$) x 6, $C_6\times D_4$ ($C_2^3:C_4$) x 4, $C_2^2\times D_6$ ($C_4\times D_4$) x 2, $C_6\times D_4$ ($D_4:C_2^2$) x 2, $C_6.D_4$ ($C_4\times D_4$) x 2, $C_2^3\times C_6$ ($C_2^2\times D_4$), $C_2^3\times C_6$ ($C_2^3\times C_4$), $C_6\times D_4$ ($D_4:C_2^2$), $C_6\times D_4$ ($C_2^2\times D_4$), $C_2^2\times C_{12}$ ($C_2^2\times D_4$)

Order 32: $C_2^2\times D_4$ ($D_6:C_4$) x 9, $C_2^3:C_4$ ($D_6:C_4$) x 6, $C_2^3:C_4$ ($S_3\times D_4$) x 2, $C_2^5$ ($C_6:D_4$) x 2, $C_2^5$ ($C_4\times D_6$) x 2, $C_2^2\times D_4$ ($C_6:D_4$) x 2, $C_2^2\times D_4$ ($C_2\times D_{12}$) x 2, $C_2^2\times D_4$ ($C_4\times D_6$) x 2, $C_2^3\times C_4$ ($C_2\times D_{12}$) x 2, $C_2^2\times D_4$ ($C_2^2\times D_6$), $C_2^3\times C_4$ ($C_6:D_4$), $C_2^3\times C_4$ ($C_4\times D_6$), $C_2^2.D_4$ ($C_2\times D_{12}$), $C_2^2.D_4$ ($D_6:C_4$), $C_2^3:C_4$ ($C_6:D_4$), $C_2^3:C_4$ ($C_4\times D_6$), $C_2\times C_4^2$ ($C_2\times D_{12}$)

Order 24: $C_2^2\times C_6$ ($C_2^4:C_4$) x 4, $C_3\times D_4$ ($C_2\wr C_4$) x 4, $C_2\times C_{12}$ ($C_2^4:C_4$) x 2, $C_2\times C_{12}$ ($D_4:D_4$), $C_2\times C_{12}$ ($D_4^2$), $C_2^2\times C_6$ ($D_4:D_4$), $C_2^2\times C_6$ ($D_4^2$), $C_2^2\times C_6$ ($C_4^2:C_2^2$), $C_2^2\times C_6$ ($C_4^2:C_2^2$), $C_2^2\times C_6$ ($C_2^4:C_4$)

Order 16: $C_2^4$ ($C_2^2.D_{12}$) x 6, $C_2^2\times C_4$ ($C_2^2.D_{12}$) x 6, $C_2\times D_4$ ($C_2^2.D_{12}$) x 4, $C_2^4$ ($C_2^3:D_6$), $C_2^4$ ($C_{12}:C_2^3$), $C_2\times D_4$ ($D_4:D_6$), $C_2\times D_4$ ($D_4:D_6$), $C_2\times D_4$ ($D_4:D_6$), $C_2\times D_4$ ($D_4\times D_6$), $C_2^2\times C_4$ ($C_2^2\times D_{12}$)

Order 12: $C_2\times C_6$ ($C_2^5:C_4$) x 4, $C_{12}$ ($C_2^5:C_4$) x 2, $C_2\times C_6$ ($C_2^3.C_2^4$), $C_2\times C_6$ ($C_2^5:C_4$), $C_2\times C_6$ ($C_2.D_4^2$)

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

Order 6: $C_6$ ($(D_4\times C_2^3):C_4$), $C_6$ ($C_2^6:C_4$), $C_6$ ($C_2^2.D_4^2$)

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

Order 3: $C_3$ ($C_2^4.C_2^3.C_2^2$)

Order 2: $C_2$ ($C_3:(C_2^3.C_2^4.C_2)$), $C_2$ ($(D_6\times C_2^4):C_4$), $C_2$ ($D_4\times C_2^2.D_{12}$)

Order 1: $C_1$ ($(C_2\times C_{12}).D_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $(C_2\times C_{12}).D_4^2$ ($C_1$)

Order 768: $(C_2^3\times D_{12}):C_4$ ($C_2$), $(C_2^3\times D_{12}):C_4$ ($C_2$), $(C_2^3\times C_{12}).D_4$ ($C_2$), $(C_2^3\times C_{12}).D_4$ ($C_2$), $C_{12}:C_2\wr C_4$ ($C_2$), $(D_6\times C_2^4):C_4$ ($C_2$), $C_{12}:C_2\wr C_4$ ($C_2$), $D_6:D_4^2$ ($C_2$), $(D_6\times C_2^4):C_4$ ($C_2$), $C_{12}.D_4^2$ ($C_2$), $D_4\times C_2^3:C_{12}$ ($C_2$)

Order 384: $(C_2^3\times D_6):C_4$ ($C_2^2$) x 7, $(C_6\times D_4).D_4$ ($C_2^2$) x 3, $(C_2\times C_4^2):C_{12}$ ($C_2^2$), $C_2^5:C_{12}$ ($C_2^2$), $C_4\times C_2^3:C_{12}$ ($C_2^2$), $(C_6\times D_4).D_4$ ($C_2^2$), $(C_4\times C_{12}).D_4$ ($C_2^2$), $C_4\times C_{12}.D_4$ ($C_2^2$), $C_{12}:C_2^5$ ($C_4$), $C_6\times D_4^2$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $C_2^5:C_{12}$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $C_6.D_4^2$ ($C_4$), $C_2^2\times C_{12}.D_4$ ($C_2^2$), $(C_4\times D_6):D_4$ ($C_4$), $C_6.D_4^2$ ($C_4$), $C_2^5.D_6$ ($C_4$), $C_6.D_4^2$ ($C_4$), $C_6.D_4^2$ ($C_2^2$), $C_4\times D_6:D_4$ ($C_2^2$)

Order 256: $C_2^2.D_4^2$ ($S_3$)

Order 192: $C_2^4:C_{12}$ ($C_2^3$) x 3, $C_2\times C_{12}.D_4$ ($C_2^3$) x 3, $C_2^4:D_6$ ($C_2\times C_4$) x 3, $C_2^3.D_{12}$ ($C_2\times C_4$) x 2, $C_2\times C_{12}:D_4$ ($D_4$) x 2, $C_2^4:D_6$ ($C_2^3$) x 2, $C_2^4.D_6$ ($C_2\times C_4$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2\times C_4$), $C_2^3.D_{12}$ ($C_2\times C_4$), $C_2^4.D_6$ ($C_2\times C_4$), $C_2^4.D_6$ ($C_2\times C_4$), $(C_2^2\times D_6):C_4$ ($D_4$), $D_6\times C_2^4$ ($C_2\times C_4$), $C_{12}:C_2^4$ ($C_2^3$), $C_{12}:C_2^4$ ($D_4$), $C_{12}:C_2^4$ ($C_2\times C_4$), $D_6:C_2^4$ ($C_2\times C_4$), $C_2^3\times D_{12}$ ($C_2\times C_4$), $C_{12}:C_2^4$ ($C_2\times C_4$), $C_2\times C_{12}:D_4$ ($D_4$), $C_2\times C_{12}:D_4$ ($C_2\times C_4$), $C_2^5:C_6$ ($D_4$), $C_2^5:C_6$ ($C_2\times C_4$), $C_2\times D_4\times C_{12}$ ($C_2^3$), $C_2\times D_4\times C_{12}$ ($D_4$), $C_2^4.D_6$ ($C_2\times C_4$)

Order 128: $C_2^4.C_2^3$ ($D_6$), $C_2^5:C_4$ ($D_6$), $C_2^3:C_4^2$ ($D_6$), $C_2\times D_4^2$ ($D_6$), $C_2^5:C_4$ ($D_6$)

Order 96: $C_{12}:C_2^3$ ($C_2^2:C_4$) x 6, $C_2^3:C_{12}$ ($C_2^2:C_4$) x 3, $C_2^3.D_6$ ($C_2^2\times C_4$) x 3, $C_2^3\times D_6$ ($C_2^2\times C_4$) x 2, $C_{12}:C_2^3$ ($C_2\times D_4$) x 2, $C_2^3\times C_{12}$ ($C_2\times D_4$) x 2, $C_2^3:C_{12}$ ($C_2\times D_4$), $C_{12}.D_4$ ($C_2\times D_4$), $C_2^4\times C_6$ ($C_2\times D_4$), $C_2^4\times C_6$ ($C_2^2\times C_4$), $C_{12}:C_2^3$ ($C_2^4$), $C_{12}:C_2^3$ ($C_2^2\times C_4$), $C_2^3\times C_{12}$ ($C_2^2\times C_4$), $C_6.C_2^4$ ($C_2^2\times C_4$), $C_2\times C_4:C_{12}$ ($C_2^2:C_4$), $C_2\times C_4:C_{12}$ ($C_2\times D_4$), $C_2^3:C_{12}$ ($C_2\times D_4$), $C_2^3:C_{12}$ ($C_2^2\times C_4$), $C_2\times C_4\times C_{12}$ ($C_2\times D_4$), $D_6:D_4$ ($D_4:C_2$), $D_6:D_4$ ($C_2\times D_4$)

Order 64: $C_2^4:C_4$ ($C_2\times D_6$) x 3, $D_4\times C_2^3$ ($C_3:D_4$), $D_4\times C_2^3$ ($C_4\times S_3$), $D_4\times C_2^3$ ($C_2\times D_6$), $C_4^2:C_2^2$ ($D_{12}$), $C_2^3:D_4$ ($C_3:D_4$), $C_2^3:D_4$ ($D_{12}$), $C_2^3:D_4$ ($C_4\times S_3$), $C_2^3:D_4$ ($C_3:D_4$), $C_2^3:D_4$ ($C_4\times S_3$), $C_4^2:C_2^2$ ($D_{12}$), $C_4^2:C_2^2$ ($C_2\times D_6$)

Order 48: $C_2^2\times C_{12}$ ($C_2^3:C_4$) x 5, $C_2^3\times C_6$ ($C_2^3:C_4$) x 3, $C_6\times D_4$ ($C_2^3:C_4$) x 3, $C_6\times D_4$ ($D_4:C_2^2$) x 2, $C_2^3\times C_6$ ($C_2^2\times D_4$), $C_2^3\times C_6$ ($C_2^3\times C_4$), $C_2^2\times D_6$ ($C_4\times D_4$), $C_6\times D_4$ ($D_4:C_2^2$), $C_6\times D_4$ ($C_2^2\times D_4$), $C_2^2\times C_{12}$ ($C_2^2\times D_4$), $C_6.D_4$ ($C_4\times D_4$)

Order 32: $C_2^2\times D_4$ ($D_6:C_4$) x 6, $C_2^3:C_4$ ($D_6:C_4$) x 3, $C_2^3:C_4$ ($S_3\times D_4$), $C_2^5$ ($C_6:D_4$), $C_2^5$ ($C_4\times D_6$), $C_2^2\times D_4$ ($C_2^2\times D_6$), $C_2^2\times D_4$ ($C_6:D_4$), $C_2^2\times D_4$ ($C_2\times D_{12}$), $C_2^2\times D_4$ ($C_4\times D_6$), $C_2^3\times C_4$ ($C_6:D_4$), $C_2^3\times C_4$ ($C_2\times D_{12}$), $C_2^3\times C_4$ ($C_4\times D_6$), $C_2^2.D_4$ ($C_2\times D_{12}$), $C_2^2.D_4$ ($D_6:C_4$), $C_2^3:C_4$ ($C_6:D_4$), $C_2^3:C_4$ ($C_4\times D_6$), $C_2\times C_4^2$ ($C_2\times D_{12}$)

Order 24: $C_2\times C_{12}$ ($C_2^4:C_4$) x 2, $C_2^2\times C_6$ ($C_2^4:C_4$) x 2, $C_2\times C_{12}$ ($D_4:D_4$), $C_2\times C_{12}$ ($D_4^2$), $C_2^2\times C_6$ ($D_4:D_4$), $C_2^2\times C_6$ ($D_4^2$), $C_2^2\times C_6$ ($C_4^2:C_2^2$), $C_2^2\times C_6$ ($C_4^2:C_2^2$), $C_2^2\times C_6$ ($C_2^4:C_4$), $C_3\times D_4$ ($C_2\wr C_4$)

Order 16: $C_2^2\times C_4$ ($C_2^2.D_{12}$) x 5, $C_2^4$ ($C_2^2.D_{12}$) x 3, $C_2\times D_4$ ($C_2^2.D_{12}$) x 3, $C_2^4$ ($C_2^3:D_6$), $C_2^4$ ($C_{12}:C_2^3$), $C_2\times D_4$ ($D_4:D_6$), $C_2\times D_4$ ($D_4:D_6$), $C_2\times D_4$ ($D_4:D_6$), $C_2\times D_4$ ($D_4\times D_6$), $C_2^2\times C_4$ ($C_2^2\times D_{12}$)

Order 12: $C_2\times C_6$ ($C_2^5:C_4$), $C_2\times C_6$ ($C_2^3.C_2^4$), $C_2\times C_6$ ($C_2^5:C_4$), $C_2\times C_6$ ($C_2.D_4^2$), $C_{12}$ ($C_2^5:C_4$)

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

Order 6: $C_6$ ($(D_4\times C_2^3):C_4$), $C_6$ ($C_2^6:C_4$), $C_6$ ($C_2^2.D_4^2$)

Order 4: $C_2^2$ ($(C_2^3\times D_6):C_4$), $C_2^2$ ($(D_4\times D_6):C_4$), $C_2^2$ ($C_6.D_4^2$), $C_2^2$ ($C_2^4.D_{12}$), $C_4$ ($(C_2^3\times D_6):C_4$)

Order 3: $C_3$ ($C_2^4.C_2^3.C_2^2$)

Order 2: $C_2$ ($C_3:(C_2^3.C_2^4.C_2)$), $C_2$ ($(D_6\times C_2^4):C_4$), $C_2$ ($D_4\times C_2^2.D_{12}$)

Order 1: $C_1$ ($(C_2\times C_{12}).D_4^2$)

Series

Derived series

$(C_2\times C_{12}).D_4^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_1$

Chief series

$(C_2\times C_{12}).D_4^2$

$\rhd$

$D_4\times C_2^3:C_{12}$

$\rhd$

$C_2^5:C_{12}$

$\rhd$

$C_2^4:C_{12}$

$\rhd$

$C_{12}:C_2^3$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_{12}).D_4^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2\times D_4$

$\lhd$

$C_2\times D_4^2$

$\lhd$

$C_2^2.D_4^2$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.