Abstract group 12130560.c: $\PGL(4,3)$

• Label: 12130560.c
• Order: \( 2^{8} \cdot 3^{6} \cdot 5 \cdot 13 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $40$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$\PGL(4,3)$
Order:	\(12130560\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 5 \cdot 13 \)
Exponent:	\(4680\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Automorphism group:	$\PSL(4,3).C_2^2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \)
Composition factors:	$C_2$, $\PSL(4,3)$
Derived length:	$1$

This group is nonabelian, almost simple, and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	13	20	24	26	40
Elements	1	8451	82160	636012	303264	1375920	1566864	449280	303264	842400	1866240	606528	1010880	1866240	1213056	12130560
Conjugacy classes	1	3	3	4	1	5	6	1	1	2	4	2	2	4	4	43
Divisions	1	3	3	4	1	5	4	1	1	2	1	1	1	1	1	30
Autjugacy classes	1	3	3	4	1	5	4	1	1	2	2	1	1	2	2	33

Dimension	1	39	52	90	104	130	260	351	390	416	468	520	640	729	780	832	936	1040	1170	1664	2080	2560
Irr. complex chars.	2	2	3	2	0	1	2	2	2	8	3	1	8	2	2	0	0	2	1	0	0	0	43
Irr. rational chars.	2	2	1	2	1	1	2	2	2	2	1	1	0	2	2	1	1	0	1	1	1	2	30

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	39	39	39
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PGL(4,3)$, $\PGammaL(4,3)$
Permutation group:	Degree $40$ $\langle(1,31,17,6,25,19,21,39,36,35,12,24,34,16,22,20,7,28,18,38,13,26,9,23,8,3,29,10,27,32,11,4,30,33,37,15,40,14,5,2) \!\cdots\! \rangle$
Transitive group:	40T177552				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\GL(4,3)$.

Homology

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

Subgroups

There are 51404899 subgroups in 1228 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PGL(4,3)$
Commutator:	$G' \simeq$ $\PSL(4,3)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PGL(4,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PGL(4,3)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PGL(4,3)$
Socle:	$\operatorname{soc} \simeq$ $\PSL(4,3)$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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 12130560: $\PGL(4,3)$

Order 6065280: $\PSL(4,3)$

Order 303264: $C_3^3:C_2.\SL(3,3)$ x 2

Order 151632: $C_3^3.\SL(3,3)$ x 2

Order 93312: $C_3^4.(Q_8.A_4).D_6$

Order 51840: $\SU(4,2).C_2$

Order 46656: $C_3^4.(Q_8.A_4).C_6$ x 2, $C_3^4.Q_8.C_3:S_4$

Order 31104: $C_3^4.Q_8.D_6.C_2^2$ x 2

Order 25920: $\SU(4,2)$

Order 23328: $C_3^4.Q_8.S_3^2$ x 2, $C_3^3:C_3^2.Q_8.D_6$, $C_3^4.(Q_8.A_4).C_3$

Order 15552: $C_3^4.Q_8.D_6.C_2$ x 2, $C_3^4.Q_8.C_3.D_4$ x 2, $C_3^4.Q_8.D_6.C_2$ x 2, $C_3^4.(\SD_{16}.A_4)$ x 2, $C_3^4.Q_8.D_6.C_2$ x 2, $C_3^4.Q_8.D_{12}$ x 2, $C_3^4.Q_8.D_6.C_2$ x 2, $C_3^4.Q_8.S_4$

Order 11664: $C_3^4.Q_8.C_3.C_6$ x 2, $C_3^4.Q_8.C_3.S_3$ x 2, $C_3^4.(C_3\times Q_8).C_6$ x 2, $C_3^3.\AGL(2,3)$, $C_3^3.\AGL(2,3)$, $C_3^3:C_3^2.Q_8.C_6$

Order 11232: $C_2\times \SL(3,3)$

Order 10368: $C_3^4.D_4:C_2.C_2^3$

Order 7776: $C_3^4.(C_4.S_4)$ x 2, $C_3^4.Q_8.D_6$ x 2, $C_3^4.C_{12}.C_2^3$ x 2, $C_3^4.Q_8.D_6$ x 2, $C_3^4.Q_8.D_6$ x 2, $C_3^4.Q_8.D_6$ x 2, $C_3^4.(Q_8.A_4)$ x 2, $C_3^4.Q_8.C_{12}$ x 2, $C_3^4.(D_4.A_4)$ x 2, $C_3^4.Q_8.D_6$ x 2, $C_3^3.S_3\wr C_2.C_2^2$, $C_3^4.Q_8.A_4$

Order 5832: $C_3^4.Q_8.C_3^2$ x 2, $C_3^3.S_3^3$, $C_3^3.\ASL(2,3)$

Order 5760: $C_4.A_6.C_2^2$

Order 5616: $\SL(3,3)$

Order 5184: $C_3^4.D_4.C_2^3$ x 2, $C_3^4.(C_4.C_2^4)$ x 2, $C_3^4.D_4:D_4$ x 2, $C_3^4.D_8:C_2.C_2$ x 2, $C_3^4.D_4:C_2.C_2^2$ x 2, $C_3^4.\SD_{16}:C_2:C_2$ x 2, $C_3^4.\OD_{16}:C_2.C_2$ x 2, $C_3^4.C_4:\SD_{16}$

Order 3888: $C_3^4.Q_8.C_6$ x 2, $C_3^4.C_8:S_3$ x 2, $C_3^4.Q_8.S_3$ x 2, $C_3^4:C_3.\SD_{16}$ x 2, $C_3^3.(C_2\times F_9)$ x 2, $C_3^4.Q_8.S_3$ x 2, $C_3^4.Q_8:S_3$ x 2, $C_3^4.D_6.C_2^2$ x 2, $C_3^4.Q_8.S_3$ x 2, $C_3^2.S_3^3.C_2$ x 2, $C_3^4.Q_8.S_3$ x 2, $C_3^4.Q_8.C_6$ x 2, $C_3^4.C_{12}.C_2^2$ x 2, $C_3^3.S_3\wr C_2.C_2$, $C_3^3.S_3\wr C_2.C_2$, $C_3^3.S_3\wr C_2.C_2$, $\He_3:(C_3:S_3).C_2^3$, $C_3^3.S_3^2.C_2^2$, $C_3^3.(S_3\times S_4)$, $C_3^3.(C_2\times F_9)$, $C_3^3.S_3\wr C_2.C_2$

Order 2916: $C_3^4.(C_6\times S_3)$ x 2, $C_3^4:C_3.D_6$ x 2, $C_3.C_3^4:C_6.C_2$, $C_3^4:C_3.D_6$, $C_3^4:C_3.D_6$

Order 2880: $C_4.A_6.C_2$, $C_4.A_6.C_2$, $C_4.A_6.C_2$

Order 2592: $C_3^4.D_4.C_2^2$ x 4, $C_3^4.\OD_{16}:C_2$ x 4, $C_3^4.C_4\wr C_2$ x 4, $C_3^4.D_8:C_2$ x 4, $\ASL(2,3).D_6$ x 2, $C_3^4.\SD_{16}.C_2$ x 2, $C_3^4.(\OD_{16}.C_2)$ x 2, $C_3^4.\SD_{16}:C_2$ x 2, $C_3^4.\SD_{16}.C_2$ x 2, $C_3^4.Q_8.C_2^2$ x 2, $C_3^4.\SD_{16}.C_2$ x 2, $C_3^4.Q_8.C_2^2$ x 2, $C_3^4.(C_2\times \SD_{16})$ x 2, $C_3^4.\OD_{16}.C_2$ x 2, $C_3^4.D_4.C_2^2$ x 2, $C_3^4.\OD_{16}.C_2$, $C_3^4.(C_2^3.C_4)$, $C_3^4.(C_4\times C_8)$, $C_3^4.Q_8:C_2^2$, $C_3^4.Q_8.C_2^2$, $C_3^4.C_4:D_4$, $C_3^4.C_4:Q_8$

Order 2304: $\SL(2,3):A_4.D_4$

Order 2106: $C_3^3:C_{13}.C_6$ x 2

Order 1944: $C_3^4:\SL(2,3)$ x 2, $C_3^2.S_3^3$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^4:D_{12}$ x 2, $C_3^3:\SOPlus(4,2)$ x 2, $C_3^3:\SOPlus(4,2)$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^3:F_9$ x 2, $C_3^4:C_{24}$ x 2, $C_3^4:\SL(2,3)$ x 2, $C_3^3:(S_3\times A_4)$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:S_3^2.C_2$, $C_3^3:F_9$, $C_3^3:F_9$

Order 1920: $C_2^4.S_5$

Order 1458: $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:(C_3\times C_6)$, $C_3^4:C_3:S_3$, $C_3^4:C_3:S_3$

Order 1440: $C_2\times S_6$, $S_6:C_2$, $C_4\times A_6$, $A_6:C_4$

Order 1296: $C_3^4:D_4:C_2$ x 8, $C_3^4:\SD_{16}$ x 8, $C_3^4:\SD_{16}$ x 6, $C_3^4:\OD_{16}$ x 6, $C_3^4:(C_2\times D_4)$ x 5, $C_3^4:(C_2\times Q_8)$ x 4, $C_3^4:Q_{16}$ x 4, $C_3^4:D_8$ x 4, $C_3^2:(C_2\times F_9)$ x 4, $S_3\wr S_3$ x 3, $S_3\times \PU(3,2)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^3:(S_3\times D_4)$, $C_3^2\wr C_2.Q_8$, $C_3^4:C_4^2$, $\He_3:\GL(2,3)$

Order 1152: $\SL(2,3):A_4.C_4$, $A_4^2.D_4$, $C_2^3:A_4.D_6$, $Q_8.A_4.D_6$

Order 1053: $C_3^3:C_{13}:C_3$ x 2

Order 972: $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:C_3:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3\wr A_4$, $C_3^3:S_3^2$, $C_3^3:C_3^2:C_4$, $C_3^3:C_3^2:C_4$, $C_3^3:S_3^2$

Order 960: $C_2^4:A_5$

Order 864: $F_9:D_6$ x 2, $C_2\times C_3^2:\GL(2,3)$ x 2

Order 729: $C_3^4:C_3^2$

Order 720: $S_6$ x 2, $C_2\times A_6$, $A_6.C_2$, $\PGL(2,9)$

Order 702: $F_{27}$ x 2

Order 648: $C_3^4:D_4$ x 10, $C_3^4:Q_8$ x 6, $C_3^4:(C_2\times C_4)$ x 6, $C_3^2:F_9$ x 4, $C_3^3:S_4$ x 4, $C_3:S_3^3$ x 3, $S_3\wr C_3$ x 3, $C_3^3:S_4$ x 3, $C_3^2:F_9$ x 2, $C_3^3:\SL(2,3)$ x 2, $C_3.S_3^3$ x 2, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $\Unitary(3,2)$

Order 576: $\POPlus(4,3)$, $S_4^2$, $A_4^2:C_4$, $F_9:D_4$, $\SOPlus(4,3)$, $\OmegaPlus(4,3):C_2$, $\GL(2,3):A_4$

Order 486: $C_3^3:(C_3\times C_6)$ x 3, $C_3^4:S_3$ x 2, $C_3\wr C_3:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^3:(C_3\times C_6)$ x 2, $C_3\wr C_3:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $(C_3\times \He_3):S_3$, $C_3^3:(C_3\times S_3)$, $(C_3\times \He_3):S_3$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:S_3$

Order 480: $C_4:S_5$

Order 432: $C_3^2:\GL(2,3)$ x 6, $S_3^3:C_2$ x 3, $C_3^3:\SD_{16}$ x 3, $S_3\times \PSU(3,2)$ x 2, $F_9:S_3$ x 2, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2, $S_3\times F_9$ x 2, $C_2\times \PU(3,2)$ x 2, $\He_3:\SD_{16}$

Order 384: $C_2\wr S_4$, $\GL(2,3):D_4$

Order 360: $A_6$ x 2

Order 351: $C_3^3:C_{13}$ x 2

Order 324: $C_3^2:S_3^2$ x 5, $C_3^2:S_3^2$ x 4, $C_3^4:C_4$ x 4, $C_3^3:A_4$ x 4, $C_3^3:D_6$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$ x 3, $C_3\wr C_2^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 2, $C_3^4:C_4$, $C_3^3:C_{12}$

Order 320: $C_2^4:F_5$

Order 288: $F_9:C_2^2$ x 6, $C_4:\PSU(3,2)$, $C_4:\SOPlus(4,2)$, $C_4\times F_9$, $\OmegaPlus(4,3)$, $S_3\times \GL(2,3)$, $\PSOPlus(4,3)$, $A_4\wr C_2$, $A_4\times S_4$

Order 256: $D_4^2.C_2^2$

Order 243: $C_3^3:C_3^2$ x 2, $C_3^4:C_3$ x 2, $C_3^3:C_3^2$, $\He_3:C_3^2$, $C_3^4:C_3$

Order 240: $C_2\times S_5$ x 2, $C_4\times A_5$

Order 216: $S_3^3$ x 5, $S_3^2:S_3$ x 5, $S_3^2:C_6$ x 4, $\PU(3,2)$ x 4, $C_3^3:Q_8$ x 3, $C_3^2:D_{12}$ x 3, $C_3^2:C_4\times S_3$ x 3, $C_3:F_9$ x 3, $\He_3:C_8$ x 2, $C_3^3:Q_8$ x 2, $C_3\times F_9$ x 2, $\SU(3,2)$, $\He_3:D_4$, $C_6.S_3^2$

Order 192: $D_4.S_4$, $\SL(2,3):D_4$, $C_8.S_4$, $\GL(2,3):C_4$, $C_2^3:S_4$, $C_2\wr A_4$, $Q_8:S_4$, $C_2^3:S_4$, $\GL(2,3):C_2^2$, $\GL(2,3):C_2^2$, $D_4\times S_4$, $\SL(2,3).D_4$

Order 162: $C_3^3:C_6$ x 9, $C_3\wr S_3$ x 6, $C_3^3:S_3$ x 5, $C_3^3:C_6$ x 5, $C_3^2\wr C_2$ x 4, $C_3^3:S_3$ x 4, $S_3\times \He_3$ x 3, $C_3^3:S_3$ x 3, $C_3^3:C_6$ x 3, $C_3^3:S_3$, $D_9:C_3^2$, $\He_3:S_3$

Order 160: $C_{40}:C_4$, $C_2^4:D_5$

Order 144: $F_9:C_2$ x 17, $S_3^2:C_2^2$ x 6, $C_2\times \PSU(3,2)$ x 4, $C_2\times F_9$ x 4, $A_4^2$, $S_3\times S_4$, $C_{12}:D_6$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $S_3\times \SL(2,3)$, $C_3:\GL(2,3)$, $C_3\times \GL(2,3)$, $C_2.\PSU(3,2)$

Order 128: $C_2\wr D_4$ x 2, $C_4^2.D_4$ x 2, $D_4:\SD_{16}$, $D_8:D_4$, $(C_4\times C_8):C_4$

Order 120: $S_5$ x 3, $C_2\times A_5$

Order 108: $C_3\times S_3^2$ x 12, $C_3^2:D_6$ x 9, $C_3:S_3^2$ x 7, $C_3:S_3^2$ x 6, $C_3^3:C_4$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 2, $\He_3:C_4$ x 2, $C_3^2:D_6$

Order 96: $C_2^2:S_4$ x 2, $C_2^2\times S_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_8.A_4$, $C_2^3:A_4$, $\Unitary(2,3)$, $Q_8:A_4$, $D_4.A_4$, $\SL(2,3):C_2^2$, $D_4\times A_4$, $\GL(2,3):C_2$, $\GL(2,3):C_2$, $C_4.S_4$, $\GL(2,3):C_2$, $C_2\times \GL(2,3)$, $C_4:S_4$, $C_4\times S_4$, $S_3\times \SD_{16}$

Order 81: $C_3\wr C_3$ x 6, $C_3\times \He_3$ x 6, $\He_3:C_3$ x 2, $C_3^4$, $C_9:C_3^2$

Order 80: $C_{20}:C_4$ x 2, $C_2^4:C_5$, $C_8\times D_5$

Order 78: $C_{13}:C_6$

Order 72: $\SOPlus(4,2)$ x 18, $\PSU(3,2)$ x 9, $F_9$ x 9, $C_2\times C_3^2:C_4$ x 7, $S_3\times D_6$ x 6, $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$, $C_{12}:S_3$, $C_3\times D_{12}$, $C_3\times \SL(2,3)$, $D_6:S_3$

Order 64: $C_4^2:C_4$ x 2, $C_2\wr C_4$ x 2, $D_8:C_2^2$ x 2, $C_8.D_4$ x 2, $C_2\wr C_2^2$ x 2, $D_4:D_4$ x 2, $D_4:C_8$, $D_4^2$, $C_8:D_4$, $D_4:Q_8$, $D_4.D_4$, $D_8:C_4$, $C_4.D_8$

Order 60: $A_5$ x 2

Order 54: $C_3^2:C_6$ x 15, $S_3\times C_3^2$ x 13, $C_3^2:C_6$ x 12, $C_3^2:S_3$ x 9, $C_3^2:S_3$ x 4, $C_9:C_6$ x 3, $C_3\times D_9$, $C_2\times \He_3$

Order 48: $C_2\times S_4$ x 6, $\GL(2,3)$ x 3, $C_2^2\times A_4$ x 2, $S_3\times D_4$ x 2, $C_{24}:C_2$, $C_2^2:A_4$, $S_3\times Q_8$, $S_3\times C_8$, $\SL(2,3):C_2$, $C_2\times \SL(2,3)$, $C_4\times A_4$, $A_4:C_4$, $C_2.S_4$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$, $C_{40}$, $C_5:C_8$

Order 39: $C_{13}:C_3$

Order 36: $S_3^2$ x 22, $C_3^2:C_4$ x 15, $C_6\times S_3$ x 7, $C_6:S_3$ x 4, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3\times A_4$

Order 32: $C_2^3:C_4$ x 3, $Q_{16}:C_2$ x 3, $D_8:C_2$ x 3, $C_2\times \SD_{16}$ x 3, $C_2^2\wr C_2$ x 3, $\OD_{16}:C_2$ x 2, $D_4:C_2^2$ x 2, $D_8:C_2$ x 2, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_4:C_4$, $C_4.D_4$, $C_4.C_2^3$, $C_2^2\times D_4$, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $C_4:D_4$, $C_4\times D_4$, $C_8.C_4$, $C_8:C_4$, $C_4:C_8$

Order 27: $\He_3$ x 10, $C_3^3$ x 8, $C_9:C_3$ x 3, $C_3\times C_9$

Order 26: $C_{26}$

Order 24: $S_4$ x 6, $C_3:D_4$ x 3, $C_2\times D_6$ x 3, $C_2\times A_4$ x 3, $D_{12}$ x 2, $C_4\times S_3$ x 2, $\SL(2,3)$ x 2, $C_3\times D_4$ x 2, $C_3:Q_8$, $C_{24}$, $C_3\times Q_8$, $C_3:C_8$

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 21, $C_3:S_3$ x 13, $C_3\times C_6$ x 5, $D_9$

Order 16: $C_2\times D_4$ x 8, $\SD_{16}$ x 7, $C_2^2:C_4$ x 4, $D_4:C_2$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $Q_{16}$ x 2, $D_8$ x 2, $C_4:C_4$ x 2, $C_2^4$ x 2, $C_2\times Q_8$ x 2, $C_4^2$, $C_2^2\times C_4$

Order 13: $C_{13}$

Order 12: $D_6$ x 10, $C_2\times C_6$ x 3, $A_4$ x 3, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 10, $C_9$

Order 8: $D_4$ x 9, $C_2\times C_4$ x 6, $C_2^3$ x 4, $C_8$ x 4, $Q_8$ x 3

Order 6: $S_3$ x 8, $C_6$ x 5

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12130560: $\PGL(4,3)$

Order 6065280: $\PSL(4,3)$

Order 303264: $C_3^3:C_2.\SL(3,3)$

Order 151632: $C_3^3.\SL(3,3)$

Order 93312: $C_3^4.(Q_8.A_4).D_6$

Order 51840: $\SU(4,2).C_2$

Order 46656: $C_3^4.Q_8.C_3:S_4$, $C_3^4.(Q_8.A_4).C_6$

Order 31104: $C_3^4.Q_8.D_6.C_2^2$

Order 25920: $\SU(4,2)$

Order 23328: $C_3^3:C_3^2.Q_8.D_6$, $C_3^4.(Q_8.A_4).C_3$, $C_3^4.Q_8.S_3^2$

Order 15552: $C_3^4.Q_8.D_6.C_2$, $C_3^4.Q_8.C_3.D_4$, $C_3^4.Q_8.D_6.C_2$, $C_3^4.(\SD_{16}.A_4)$, $C_3^4.Q_8.D_6.C_2$, $C_3^4.Q_8.D_{12}$, $C_3^4.Q_8.D_6.C_2$, $C_3^4.Q_8.S_4$

Order 11664: $C_3^3.\AGL(2,3)$, $C_3^4.Q_8.C_3.C_6$, $C_3^3.\AGL(2,3)$, $C_3^3:C_3^2.Q_8.C_6$, $C_3^4.Q_8.C_3.S_3$, $C_3^4.(C_3\times Q_8).C_6$

Order 11232: $C_2\times \SL(3,3)$

Order 10368: $C_3^4.D_4:C_2.C_2^3$

Order 7776: $C_3^4.(C_4.S_4)$, $C_3^4.Q_8.D_6$, $C_3^4.C_{12}.C_2^3$, $C_3^3.S_3\wr C_2.C_2^2$, $C_3^4.Q_8.D_6$, $C_3^4.Q_8.D_6$, $C_3^4.Q_8.D_6$, $C_3^4.Q_8.A_4$, $C_3^4.(Q_8.A_4)$, $C_3^4.Q_8.C_{12}$, $C_3^4.(D_4.A_4)$, $C_3^4.Q_8.D_6$

Order 5832: $C_3^3.S_3^3$, $C_3^3.\ASL(2,3)$, $C_3^4.Q_8.C_3^2$

Order 5760: $C_4.A_6.C_2^2$

Order 5616: $\SL(3,3)$

Order 5184: $C_3^4.D_4:D_4$ x 2, $C_3^4.D_4.C_2^3$, $C_3^4.(C_4.C_2^4)$, $C_3^4.D_8:C_2.C_2$, $C_3^4.C_4:\SD_{16}$, $C_3^4.D_4:C_2.C_2^2$, $C_3^4.\SD_{16}:C_2:C_2$, $C_3^4.\OD_{16}:C_2.C_2$

Order 3888: $C_3^4.Q_8.C_6$, $C_3^4.C_8:S_3$, $C_3^4.Q_8.S_3$, $C_3^3.S_3\wr C_2.C_2$, $C_3^3.S_3\wr C_2.C_2$, $C_3^3.S_3\wr C_2.C_2$, $C_3^4:C_3.\SD_{16}$, $\He_3:(C_3:S_3).C_2^3$, $C_3^3.(C_2\times F_9)$, $C_3^4.Q_8.S_3$, $C_3^4.Q_8:S_3$, $C_3^4.D_6.C_2^2$, $C_3^3.S_3^2.C_2^2$, $C_3^4.Q_8.S_3$, $C_3^3.(S_3\times S_4)$, $C_3^2.S_3^3.C_2$, $C_3^4.Q_8.S_3$, $C_3^4.Q_8.C_6$, $C_3^3.(C_2\times F_9)$, $C_3^3.S_3\wr C_2.C_2$, $C_3^4.C_{12}.C_2^2$

Order 2916: $C_3.C_3^4:C_6.C_2$, $C_3^4:C_3.D_6$, $C_3^4:C_3.D_6$, $C_3^4.(C_6\times S_3)$, $C_3^4:C_3.D_6$

Order 2880: $C_4.A_6.C_2$, $C_4.A_6.C_2$, $C_4.A_6.C_2$

Order 2592: $C_3^4.D_4.C_2^2$ x 2, $C_3^4.\OD_{16}:C_2$ x 2, $C_3^4.C_4\wr C_2$ x 2, $C_3^4.D_8:C_2$ x 2, $\ASL(2,3).D_6$, $C_3^4.\SD_{16}.C_2$, $C_3^4.(\OD_{16}.C_2)$, $C_3^4.\OD_{16}.C_2$, $C_3^4.(C_2^3.C_4)$, $C_3^4.\SD_{16}:C_2$, $C_3^4.\SD_{16}.C_2$, $C_3^4.Q_8.C_2^2$, $C_3^4.\SD_{16}.C_2$, $C_3^4.Q_8.C_2^2$, $C_3^4.(C_2\times \SD_{16})$, $C_3^4.(C_4\times C_8)$, $C_3^4.\OD_{16}.C_2$, $C_3^4.Q_8:C_2^2$, $C_3^4.Q_8.C_2^2$, $C_3^4.C_4:D_4$, $C_3^4.D_4.C_2^2$, $C_3^4.C_4:Q_8$

Order 2304: $\SL(2,3):A_4.D_4$

Order 2106: $C_3^3:C_{13}.C_6$

Order 1944: $C_3^3:(S_3\times A_4)$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:\SL(2,3)$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:S_3^2.C_2$, $C_3^3:F_9$, $C_3^3:F_9$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^3:\PSU(3,2)$, $C_3^4:D_{12}$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$, $C_3^3:F_9$, $C_3^4:C_{24}$, $C_3^4:\SL(2,3)$

Order 1920: $C_2^4.S_5$

Order 1458: $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times C_6)$, $C_3^4:C_3:S_3$, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$

Order 1440: $C_2\times S_6$, $S_6:C_2$, $C_4\times A_6$, $A_6:C_4$

Order 1296: $C_3^4:D_4:C_2$ x 4, $C_3^4:(C_2\times D_4)$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 3, $C_3^4:\OD_{16}$ x 3, $C_3^4:(C_2\times Q_8)$ x 2, $C_3^4:Q_{16}$ x 2, $C_3^4:D_8$ x 2, $C_3^2:(C_2\times F_9)$ x 2, $S_3\wr S_3$ x 2, $C_3^3:(S_3\times D_4)$, $C_3^2\wr C_2.Q_8$, $C_3^4:C_4^2$, $S_3\times \PU(3,2)$, $C_3^3:\GL(2,3)$, $C_3^3:\GL(2,3)$, $\He_3:\GL(2,3)$

Order 1152: $\SL(2,3):A_4.C_4$, $A_4^2.D_4$, $C_2^3:A_4.D_6$, $Q_8.A_4.D_6$

Order 1053: $C_3^3:C_{13}:C_3$

Order 972: $C_3^3:S_3^2$ x 2, $C_3\wr A_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_3^2:C_4$, $C_3^3:C_3^2:C_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$, $C_3^4:C_3:C_4$, $C_3^4:C_{12}$

Order 960: $C_2^4:A_5$

Order 864: $F_9:D_6$, $C_2\times C_3^2:\GL(2,3)$

Order 729: $C_3^4:C_3^2$

Order 720: $S_6$ x 2, $C_2\times A_6$, $A_6.C_2$, $\PGL(2,9)$

Order 702: $F_{27}$

Order 648: $C_3^4:D_4$ x 5, $C_3^4:(C_2\times C_4)$ x 4, $C_3:S_3^3$ x 3, $C_3^4:Q_8$ x 3, $C_3^3:S_4$ x 3, $C_3^2:F_9$ x 2, $S_3\wr C_3$ x 2, $C_3^3:S_4$ x 2, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $C_3^2:F_9$, $C_3^3:\SL(2,3)$, $C_3.S_3^3$, $\Unitary(3,2)$

Order 576: $\POPlus(4,3)$, $S_4^2$, $A_4^2:C_4$, $F_9:D_4$, $\SOPlus(4,3)$, $\OmegaPlus(4,3):C_2$, $\GL(2,3):A_4$

Order 486: $C_3^3:(C_3\times C_6)$ x 2, $(C_3\times \He_3):S_3$, $C_3^3:(C_3\times S_3)$, $C_3^4:S_3$, $(C_3\times \He_3):S_3$, $C_3\wr C_3:C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^3:(C_3\times C_6)$, $C_3\wr C_3:C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 480: $C_4:S_5$

Order 432: $C_3^2:\GL(2,3)$ x 3, $S_3^3:C_2$ x 2, $C_3^3:\SD_{16}$ x 2, $S_3\times \PSU(3,2)$, $F_9:S_3$, $C_3^3:\SD_{16}$, $F_9:C_6$, $S_3\times F_9$, $C_2\times \PU(3,2)$, $\He_3:\SD_{16}$

Order 384: $C_2\wr S_4$, $\GL(2,3):D_4$

Order 360: $A_6$ x 2

Order 351: $C_3^3:C_{13}$

Order 324: $C_3^2:S_3^2$ x 3, $C_3\wr C_2^2$ x 3, $C_3^3:A_4$ x 3, $C_3^2:S_3^2$ x 3, $C_3^3:D_6$ x 2, $C_3^4:C_4$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$ x 2, $C_3^2:S_3^2$ x 2, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^2:S_3^2$

Order 320: $C_2^4:F_5$

Order 288: $F_9:C_2^2$ x 3, $C_4:\PSU(3,2)$, $C_4:\SOPlus(4,2)$, $C_4\times F_9$, $\OmegaPlus(4,3)$, $S_3\times \GL(2,3)$, $\PSOPlus(4,3)$, $A_4\wr C_2$, $A_4\times S_4$

Order 256: $D_4^2.C_2^2$

Order 243: $C_3^3:C_3^2$, $\He_3:C_3^2$, $C_3^3:C_3^2$, $C_3^4:C_3$, $C_3^4:C_3$

Order 240: $C_2\times S_5$ x 2, $C_4\times A_5$

Order 216: $S_3^3$ x 4, $S_3^2:S_3$ x 4, $S_3^2:C_6$ x 3, $\He_3:C_8$ x 2, $C_3^3:Q_8$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_3:F_9$ x 2, $\PU(3,2)$ x 2, $\SU(3,2)$, $\He_3:D_4$, $C_3^3:Q_8$, $C_3\times F_9$, $C_6.S_3^2$

Order 192: $D_4.S_4$, $\SL(2,3):D_4$, $C_8.S_4$, $\GL(2,3):C_4$, $C_2^3:S_4$, $C_2\wr A_4$, $Q_8:S_4$, $C_2^3:S_4$, $\GL(2,3):C_2^2$, $\GL(2,3):C_2^2$, $D_4\times S_4$, $\SL(2,3).D_4$

Order 162: $C_3^3:C_6$ x 5, $C_3\wr S_3$ x 4, $C_3^2\wr C_2$ x 3, $C_3^3:S_3$ x 3, $C_3^3:S_3$ x 3, $C_3^3:C_6$ x 3, $S_3\times \He_3$ x 2, $C_3^3:S_3$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $D_9:C_3^2$, $\He_3:S_3$

Order 160: $C_{40}:C_4$, $C_2^4:D_5$

Order 144: $F_9:C_2$ x 9, $S_3^2:C_2^2$ x 5, $C_2\times \PSU(3,2)$ x 2, $C_2\times F_9$ x 2, $A_4^2$, $S_3\times S_4$, $C_{12}:D_6$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $S_3\times \SL(2,3)$, $C_3:\GL(2,3)$, $C_3\times \GL(2,3)$, $C_2.\PSU(3,2)$

Order 128: $C_2\wr D_4$ x 2, $C_4^2.D_4$ x 2, $D_4:\SD_{16}$, $D_8:D_4$, $(C_4\times C_8):C_4$

Order 120: $S_5$ x 3, $C_2\times A_5$

Order 108: $C_3\times S_3^2$ x 9, $C_3^2:D_6$ x 6, $C_3:S_3^2$ x 5, $C_3:S_3^2$ x 5, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 3, $\He_3:C_4$ x 2, $C_3^2:D_6$, $C_3^2:D_6$

Order 96: $C_2^2:S_4$ x 2, $C_2^2\times S_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_8.A_4$, $C_2^3:A_4$, $\Unitary(2,3)$, $Q_8:A_4$, $D_4.A_4$, $\SL(2,3):C_2^2$, $D_4\times A_4$, $\GL(2,3):C_2$, $\GL(2,3):C_2$, $C_4.S_4$, $\GL(2,3):C_2$, $C_2\times \GL(2,3)$, $C_4:S_4$, $C_4\times S_4$, $S_3\times \SD_{16}$

Order 81: $C_3\wr C_3$ x 4, $C_3\times \He_3$ x 4, $\He_3:C_3$ x 2, $C_3^4$, $C_9:C_3^2$

Order 80: $C_{20}:C_4$ x 2, $C_2^4:C_5$, $C_8\times D_5$

Order 78: $C_{13}:C_6$

Order 72: $\SOPlus(4,2)$ x 11, $S_3\times D_6$ x 5, $C_2\times C_3^2:C_4$ x 5, $\PSU(3,2)$ x 5, $F_9$ x 5, $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$, $C_{12}:S_3$, $C_3\times D_{12}$, $C_3\times \SL(2,3)$, $D_6:S_3$

Order 64: $C_4^2:C_4$ x 2, $C_2\wr C_4$ x 2, $D_8:C_2^2$ x 2, $C_8.D_4$ x 2, $C_2\wr C_2^2$ x 2, $D_4:D_4$ x 2, $D_4:C_8$, $D_4^2$, $C_8:D_4$, $D_4:Q_8$, $D_4.D_4$, $D_8:C_4$, $C_4.D_8$

Order 60: $A_5$ x 2

Order 54: $C_3^2:C_6$ x 9, $S_3\times C_3^2$ x 9, $C_3^2:C_6$ x 8, $C_3^2:S_3$ x 7, $C_3^2:S_3$ x 3, $C_9:C_6$ x 2, $C_3\times D_9$, $C_2\times \He_3$

Order 48: $C_2\times S_4$ x 6, $\GL(2,3)$ x 3, $C_2^2\times A_4$ x 2, $S_3\times D_4$ x 2, $C_{24}:C_2$, $C_2^2:A_4$, $S_3\times Q_8$, $S_3\times C_8$, $\SL(2,3):C_2$, $C_2\times \SL(2,3)$, $C_4\times A_4$, $A_4:C_4$, $C_2.S_4$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$, $C_{40}$, $C_5:C_8$

Order 39: $C_{13}:C_3$

Order 36: $S_3^2$ x 17, $C_3^2:C_4$ x 10, $C_6\times S_3$ x 6, $C_6:S_3$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3\times A_4$

Order 32: $C_2^3:C_4$ x 3, $Q_{16}:C_2$ x 3, $D_8:C_2$ x 3, $C_2\times \SD_{16}$ x 3, $C_2^2\wr C_2$ x 3, $\OD_{16}:C_2$ x 2, $D_4:C_2^2$ x 2, $D_8:C_2$ x 2, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_4:C_4$, $C_4.D_4$, $C_4.C_2^3$, $C_2^2\times D_4$, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $C_4:D_4$, $C_4\times D_4$, $C_8.C_4$, $C_8:C_4$, $C_4:C_8$

Order 27: $\He_3$ x 7, $C_3^3$ x 6, $C_9:C_3$ x 2, $C_3\times C_9$

Order 26: $C_{26}$

Order 24: $S_4$ x 6, $C_3:D_4$ x 3, $C_2\times D_6$ x 3, $C_2\times A_4$ x 3, $D_{12}$ x 2, $C_4\times S_3$ x 2, $\SL(2,3)$ x 2, $C_3\times D_4$ x 2, $C_3:Q_8$, $C_{24}$, $C_3\times Q_8$, $C_3:C_8$

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 16, $C_3:S_3$ x 10, $C_3\times C_6$ x 4, $D_9$

Order 16: $C_2\times D_4$ x 8, $\SD_{16}$ x 7, $C_2^2:C_4$ x 4, $D_4:C_2$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $Q_{16}$ x 2, $D_8$ x 2, $C_4:C_4$ x 2, $C_2^4$ x 2, $C_2\times Q_8$ x 2, $C_4^2$, $C_2^2\times C_4$

Order 13: $C_{13}$

Order 12: $D_6$ x 9, $C_2\times C_6$ x 3, $A_4$ x 3, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 8, $C_9$

Order 8: $D_4$ x 9, $C_2\times C_4$ x 6, $C_2^3$ x 4, $C_8$ x 4, $Q_8$ x 3

Order 6: $S_3$ x 7, $C_6$ x 5

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12130560: $\PGL(4,3)$ ($C_1$)

Order 6065280: $\PSL(4,3)$ ($C_2$)

Order 1: $C_1$ ($\PGL(4,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12130560: $\PGL(4,3)$ ($C_1$)

Order 6065280: $\PSL(4,3)$ ($C_2$)

Order 1: $C_1$ ($\PGL(4,3)$)

Series

Derived series	$\PGL(4,3)$	$\rhd$	$\PSL(4,3)$
Chief series	$\PGL(4,3)$	$\rhd$	$\PSL(4,3)$	$\rhd$	$C_1$
Lower central series	$\PGL(4,3)$	$\rhd$	$\PSL(4,3)$
Upper central series	$C_1$

Character theory

Complex character table

See the $43 \times 43$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.