Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
2234 conjugacy classes of subgroups; only the 275 normal subgroups are shown.
This group has 2234 conjugacy classes of subgroups — too many to draw. Only its 275 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×A42 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(C3⋊#1020) | — | 3 | 2 | no | maximal |
| 3 | C3⋊(A4⋊S4) | — | 3 | 36 | no | maximal |
| 4 | #11724473 | — | 3 | 2 | no | maximal |
| 5 | C3⋊(C3⋊(C3⋊S4)) | — | 4 | 2 | no | maximal |
| 6 | C32×C22⋊A4 | D3 | 1 | 2 | yes | complemented |
| 7 | C22×C32×A4 | D3 | 1 | 2 | yes | complemented |
| 8 | C2×C3⋊(C3⋊S4) | — | 3 | 2 | no | — |
| 9 | C3×A42 | D3 | 1 | 36 | yes | complemented |
| 10 | C3⋊(C3⋊SL(2,ℤ/4)) | — | 3 | 2 | no | — |
| 11 | C33×A4 | — | 4 | 2 | no | — |
| 12 | C3⋊#1020 | — | 9 | 24 | no | — |
| 13 | A4⋊S4 | — | 9 | 81 | no | — |
| 14 | #64269 | — | 9 | 24 | no | — |
| 15 | #64162 | — | 9 | 1 | no | — |
| 16 | C2×C32×A4 | — | 3 | 2 | no | — |
| 17 | C3⋊(C3⋊S4) | — | 6 | 2 | no | — |
| 18 | C3⋊(C3⋊S4) | — | 12 | 4 | no | — |
| 19 | C3⋊(C3⋊S4) | — | 12 | 72 | no | — |
| 20 | C3⋊(C3⋊S4) | — | 12 | 2 | no | — |
| 21 | #6312 | — | 12 | 2 | no | — |
| 22 | C3⋊(C3⋊(C3⋊D3)) | — | 16 | 1 | no | — |
| 23 | C24×C32 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C3×C22⋊A4 | C3⋊D3 | 1 | 24 | yes | complemented |
| 25 | C22×C3×A4 | C3⋊D3 | 1 | 24 | yes | complemented |
| 26 | C2×C3⋊S4 | — | 9 | 24 | no | — |
| 27 | A42 | C3⋊D3 | 1 | 81 | yes | complemented |
| 28 | C2×#638 | — | 9 | 2 | no | — |
| 29 | C2×#638 | — | 9 | 1 | no | — |
| 30 | C3⋊SL(2,ℤ/4) | — | 9 | 24 | no | — |
| 31 | #3757 | — | 9 | 2 | no | — |
| 32 | #3757 | — | 9 | 1 | no | — |
| 33 | C22×C33 | — | 4 | 2 | no | — |
| 34 | C2×C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 35 | C32×A4 | — | 4 | 4 | no | — |
| 36 | C32×A4 | — | 4 | 72 | no | — |
| 37 | C32×A4 | — | 12 | 2 | no | — |
| 38 | C32×A4 | S4 | 1 | 2 | yes | complemented |
| 39 | C3⋊(C3⋊Dic3) | — | 12 | 2 | no | — |
| 40 | #1020 | — | 27 | 18 | no | — |
| 41 | GL(2,ℤ/4) | — | 27 | 18 | no | — |
| 42 | #953 | — | 27 | 4 | no | — |
| 43 | C34 | — | 16 | 1 | no | Sylow |
| 44 | C23×C32 | — | 3 | 2 | no | — |
| 45 | C23×C32 | — | 9 | 1 | no | — |
| 46 | C22×C3⋊D3 | — | 9 | 1 | no | — |
| 47 | C2×C3×A4 | — | 3 | 24 | no | — |
| 48 | C3⋊S4 | — | 18 | 24 | no | — |
| 49 | C3⋊S4 | — | 36 | 48 | no | — |
| 50 | C3⋊S4 | — | 36 | 24 | no | — |
| 51 | C3⋊S4 | — | 36 | 162 | no | — |
| 52 | #638 | — | 18 | 2 | no | — |
| 53 | #638 | — | 18 | 2 | no | — |
| 54 | #638 | — | 18 | 2 | no | — |
| 55 | #638 | — | 36 | 24 | no | — |
| 56 | C2×C3⋊Dic3 | — | 9 | 2 | no | — |
| 57 | C2×C3⋊Dic3 | — | 9 | 1 | no | — |
| 58 | C2×C33 | — | 12 | 2 | no | — |
| 59 | C3⋊(C3⋊D3) | — | 24 | 2 | no | — |
| 60 | C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 61 | C3⋊(C3⋊D3) | — | 48 | 36 | no | — |
| 62 | C24×C3 | C3⋊(C3⋊D3) | 1 | 4 | yes | complemented |
| 63 | C22⋊A4 | C3⋊(C3⋊D3) | 1 | 18 | yes | complemented |
| 64 | C22×A4 | C3⋊(C3⋊D3) | 1 | 18 | yes | complemented |
| 65 | C2×S4 | — | 27 | 18 | no | — |
| 66 | C2×#67 | — | 27 | 8 | no | — |
| 67 | C2×#67 | — | 27 | 4 | no | — |
| 68 | SL(2,ℤ/4) | — | 27 | 18 | no | — |
| 69 | #217 | — | 27 | 8 | no | — |
| 70 | #217 | — | 27 | 4 | no | — |
| 71 | C22×C32 | — | 3 | 2 | no | — |
| 72 | C22×C32 | — | 4 | 24 | no | — |
| 73 | C22×C32 | — | 9 | 1 | no | — |
| 74 | C22×C32 | — | 9 | 2 | no | — |
| 75 | C22×C32 | C3⋊S4 | 1 | 2 | yes | complemented |
| 76 | C2×C3⋊D3 | — | 18 | 2 | no | — |
| 77 | C2×C3⋊D3 | — | 18 | 1 | no | — |
| 78 | C2×C3⋊D3 | — | 36 | 24 | no | — |
| 79 | C3×A4 | — | 4 | 48 | no | — |
| 80 | C3×A4 | — | 4 | 162 | no | — |
| 81 | C3×A4 | — | 12 | 24 | no | — |
| 82 | C3×A4 | C3⋊S4 | 1 | 24 | yes | complemented |
| 83 | C3⋊Dic3 | — | 18 | 2 | no | — |
| 84 | C3⋊Dic3 | — | 18 | 1 | no | — |
| 85 | C3⋊Dic3 | — | 36 | 24 | no | — |
| 86 | #120 | — | 81 | 1 | no | Sylow |
| 87 | C33 | — | 4 | 2 | no | — |
| 88 | C33 | — | 16 | 36 | no | — |
| 89 | C33 | — | 16 | 2 | no | — |
| 90 | C23×C3 | — | 3 | 8 | no | — |
| 91 | C23×C3 | — | 9 | 4 | no | — |
| 92 | C22×D3 | — | 27 | 4 | no | — |
| 93 | C2×A4 | — | 3 | 18 | no | — |
| 94 | S4 | — | 54 | 18 | no | — |
| 95 | S4 | — | 108 | 36 | no | — |
| 96 | S4 | — | 108 | 18 | no | — |
| 97 | #67 | — | 54 | 8 | no | — |
| 98 | #67 | — | 54 | 8 | no | — |
| 99 | #67 | — | 54 | 8 | no | — |
| 100 | #67 | — | 108 | 18 | no | — |
| 101 | C2×Dic3 | — | 27 | 8 | no | — |
| 102 | C2×Dic3 | — | 27 | 4 | no | — |
| 103 | C2×C32 | — | 3 | 2 | no | — |
| 104 | C2×C32 | — | 9 | 1 | no | — |
| 105 | C2×C32 | — | 12 | 24 | no | — |
| 106 | C3⋊D3 | — | 36 | 1 | no | — |
| 107 | C3⋊D3 | — | 72 | 24 | no | — |
| 108 | C3⋊D3 | — | 144 | 24 | no | — |
| 109 | C3⋊D3 | — | 144 | 81 | no | — |
| 110 | C24 | C3⋊(C3⋊(C3⋊D3)) | 1 | 1 | yes | characteristic, complemented |
| 111 | C2×D4 | — | 81 | 2 | no | — |
| 112 | C2×D4 | — | 81 | 1 | no | — |
| 113 | C22⋊C4 | — | 81 | 2 | no | — |
| 114 | C22⋊C4 | — | 81 | 1 | no | — |
| 115 | C22×C3 | — | 3 | 8 | no | — |
| 116 | C22×C3 | — | 4 | 18 | no | — |
| 117 | C22×C3 | — | 9 | 4 | no | — |
| 118 | C22×C3 | — | 9 | 8 | no | — |
| 119 | C22×C3 | C3⋊(C3⋊S4) | 1 | 8 | yes | complemented |
| 120 | C2×D3 | — | 54 | 8 | no | — |
| 121 | C2×D3 | — | 54 | 4 | no | — |
| 122 | C2×D3 | — | 108 | 18 | no | — |
| 123 | A4 | — | 4 | 36 | no | — |
| 124 | A4 | — | 12 | 18 | no | — |
| 125 | A4 | C3⋊(C3⋊S4) | 1 | 18 | yes | complemented |
| 126 | Dic3 | — | 54 | 8 | no | — |
| 127 | Dic3 | — | 54 | 4 | no | — |
| 128 | Dic3 | — | 108 | 18 | no | — |
| 129 | C32 | — | 4 | 24 | no | — |
| 130 | C32 | — | 16 | 81 | no | — |
| 131 | C32 | — | 16 | 24 | no | — |
| 132 | C32 | A4⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 133 | C23 | — | 3 | 2 | no | — |
| 134 | C23 | — | 9 | 1 | no | — |
| 135 | C23 | — | 81 | 1 | no | — |
| 136 | D4 | — | 162 | 2 | no | — |
| 137 | D4 | — | 162 | 2 | no | — |
| 138 | D4 | — | 162 | 2 | no | — |
| 139 | C2×C4 | — | 81 | 2 | no | — |
| 140 | C2×C4 | — | 81 | 1 | no | — |
| 141 | C2×C3 | — | 3 | 8 | no | — |
| 142 | C2×C3 | — | 9 | 4 | no | — |
| 143 | C2×C3 | — | 12 | 18 | no | — |
| 144 | D3 | — | 108 | 4 | no | — |
| 145 | D3 | — | 432 | 18 | no | — |
| 146 | D3 | — | 216 | 18 | no | — |
| 147 | C22 | — | 3 | 2 | no | — |
| 148 | C22 | — | 9 | 1 | no | — |
| 149 | C22 | — | 9 | 2 | no | — |
| 150 | C22 | — | 162 | 2 | no | — |
| 151 | C22 | — | 162 | 1 | no | — |
| 152 | C22 | C3⋊(C3⋊(C3⋊S4)) | 1 | 2 | yes | complemented |
| 153 | C4 | — | 162 | 2 | no | — |
| 154 | C4 | — | 162 | 1 | no | — |
| 155 | C3 | — | 4 | 18 | no | — |
| 156 | C3 | — | 16 | 18 | no | — |
| 157 | C3 | C3⋊(A4⋊S4) | 1 | 4 | yes | complemented |
| 158 | C2 | — | 3 | 2 | no | — |
| 159 | C2 | — | 9 | 1 | no | — |
| 160 | C2 | — | 324 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×A42 | C2 |
| Fitting, socle | C24×C32 | C3⋊D3 |
| LCS | G ▷ C32×A42 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×A42 | C2 | semidirect |
| 6. C32×C22⋊A4 | D3 | semidirect |
| 7. C22×C32×A4 | D3 | semidirect |
| 9. C3×A42 | D3 | semidirect |
| 23. C24×C32 | C3⋊D3 | semidirect |
| 24. C3×C22⋊A4 | C3⋊D3 | semidirect |
| 25. C22×C3×A4 | C3⋊D3 | semidirect |
| 27. A42 | C3⋊D3 | semidirect |
| 38. C32×A4 | S4 | semidirect |
| 62. C24×C3 | C3⋊(C3⋊D3) | semidirect |
| 63. C22⋊A4 | C3⋊(C3⋊D3) | semidirect |
| 64. C22×A4 | C3⋊(C3⋊D3) | semidirect |
| 75. C22×C32 | C3⋊S4 | semidirect |
| 82. C3×A4 | C3⋊S4 | semidirect |
| 110. C24 | C3⋊(C3⋊(C3⋊D3)) | semidirect |
| 119. C22×C3 | C3⋊(C3⋊S4) | semidirect |
| 125. A4 | C3⋊(C3⋊S4) | semidirect |
| 132. C32 | A4⋊S4 | semidirect |
| 152. C22 | C3⋊(C3⋊(C3⋊S4)) | semidirect |
| 157. C3 | C3⋊(A4⋊S4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×A42 | 2 |
| C3⋊(C3⋊#1020) | 3 |
| C3⋊(A4⋊S4) | 3 |
| #11724473 | 3 |
| C3⋊(C3⋊(C3⋊S4)) | 4 |