Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
406 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 59 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 | C3×Q8×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊(C3⋊S4) | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C4×C3⋊S4 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #5919 | — | 3 | 3 | no | maximal |
| 5 | #5817 | — | 3 | 1 | no | maximal |
| 6 | C2×C3⋊S4 | C22 | 1 | 3 | yes | — |
| 7 | #3832 | — | 4 | 1 | no | maximal |
| 8 | C3×C4×A4 | C22 | 1 | 3 | yes | — |
| 9 | C3⋊SL(2,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C3×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#239 | — | 3 | 1 | no | — |
| 12 | Q8×A4 | D3 | 1 | 3 | yes | complemented |
| 13 | C4⋊S4 | — | 3 | 9 | no | — |
| 14 | C4×S4 | — | 3 | 9 | no | — |
| 15 | #947 | — | 3 | 3 | no | — |
| 16 | #946 | — | 3 | 3 | no | — |
| 17 | Q8×Dic3 | — | 3 | 1 | no | — |
| 18 | #930 | — | 3 | 3 | no | — |
| 19 | C4×#67 | — | 3 | 3 | no | — |
| 20 | C2×C3×A4 | C23 | 1 | 1 | yes | characteristic |
| 21 | C3⋊S4 | — | 2 | 3 | no | — |
| 22 | C32×Q8 | — | 4 | 1 | no | — |
| 23 | C4⋊(C3⋊D3) | — | 4 | 3 | no | — |
| 24 | C4×C3⋊D3 | — | 4 | 3 | no | — |
| 25 | #548 | — | 9 | 1 | no | Sylow |
| 26 | C2×S4 | — | 3 | 9 | no | — |
| 27 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 28 | C2×C3×Q8 | — | 3 | 3 | no | — |
| 29 | C22×C3×C4 | C2×D3 | 1 | 3 | yes | — |
| 30 | C2×#67 | — | 3 | 3 | no | — |
| 31 | #239 | — | 6 | 3 | no | — |
| 32 | #239 | — | 6 | 1 | no | — |
| 33 | #239 | — | 12 | 3 | no | — |
| 34 | C2×D12 | — | 3 | 3 | no | — |
| 35 | C2×C4×D3 | — | 3 | 3 | no | — |
| 36 | C4×A4 | C2×D3 | 1 | 9 | yes | — |
| 37 | SL(2,ℤ/4) | — | 3 | 3 | no | — |
| 38 | #217 | — | 3 | 1 | no | — |
| 39 | (C2×D3)⋊C4 | — | 3 | 3 | no | — |
| 40 | (C2×D3)⋊C4 | — | 3 | 6 | no | — |
| 41 | C4⋊Dic3 | — | 3 | 3 | no | — |
| 42 | Dic3⋊C4 | — | 3 | 3 | no | — |
| 43 | C4×Dic3 | — | 3 | 3 | no | — |
| 44 | C2×C3⋊D3 | — | 4 | 3 | no | — |
| 45 | C3×A4 | #41 | 1 | 1 | yes | characteristic, complemented |
| 46 | C32×C4 | — | 4 | 3 | no | — |
| 47 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 48 | C2×#41 | — | 9 | 1 | no | — |
| 49 | C22×Q8 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 50 | #124 | — | 9 | 3 | no | — |
| 51 | C22⋊Q8 | — | 9 | 3 | no | — |
| 52 | #121 | — | 9 | 3 | no | — |
| 53 | C4×Q8 | — | 9 | 1 | no | — |
| 54 | C4×D4 | — | 9 | 3 | no | — |
| 55 | C23×C3 | C22×D3 | 1 | 1 | yes | characteristic |
| 56 | C22×D3 | — | 3 | 3 | no | — |
| 57 | C2×A4 | C22×D3 | 1 | 3 | yes | — |
| 58 | S4 | — | 6 | 9 | no | — |
| 59 | C3×Q8 | — | 3 | 3 | no | — |
| 60 | C3×Q8 | — | 4 | 3 | no | — |
| 61 | C3×Q8 | — | 6 | 1 | no | — |
| 62 | C3×Q8 | S4 | 1 | 1 | yes | characteristic, complemented |
| 63 | C2×C3×C4 | — | 3 | 3 | no | — |
| 64 | C2×C3×C4 | — | 3 | 3 | no | — |
| 65 | #67 | — | 6 | 3 | no | — |
| 66 | #67 | — | 6 | 3 | no | — |
| 67 | C2×Dic3 | — | 3 | 1 | no | — |
| 68 | C2×Dic3 | — | 3 | 3 | no | — |
| 69 | C2×Dic3 | — | 3 | 1 | no | — |
| 70 | D12 | — | 6 | 3 | no | — |
| 71 | D12 | — | 6 | 3 | no | — |
| 72 | D12 | — | 12 | 9 | no | — |
| 73 | C4×D3 | — | 6 | 3 | no | — |
| 74 | C4×D3 | — | 6 | 3 | no | — |
| 75 | C4×D3 | — | 12 | 9 | no | — |
| 76 | C2×C32 | — | 4 | 1 | no | — |
| 77 | C3⋊D3 | — | 8 | 3 | no | — |
| 78 | #41 | — | 18 | 3 | no | — |
| 79 | #41 | — | 18 | 1 | no | — |
| 80 | C2×Q8 | — | 3 | 1 | no | — |
| 81 | C2×Q8 | — | 3 | 3 | no | — |
| 82 | C2×D4 | — | 9 | 3 | no | — |
| 83 | C2×D4 | — | 9 | 3 | no | — |
| 84 | C22×C4 | — | 9 | 3 | no | — |
| 85 | C22×C4 | C2×C3⋊D3 | 1 | 3 | yes | — |
| 86 | C4⋊C4 | — | 9 | 3 | no | — |
| 87 | C4⋊C4 | — | 9 | 3 | no | — |
| 88 | C22⋊C4 | — | 9 | 3 | no | — |
| 89 | C22⋊C4 | — | 9 | 6 | no | — |
| 90 | C22⋊C4 | — | 9 | 1 | no | — |
| 91 | C42 | — | 9 | 3 | no | — |
| 92 | C22×C3 | — | 3 | 1 | no | — |
| 93 | C22×C3 | — | 3 | 1 | no | — |
| 94 | C22×C3 | #239 | 1 | 1 | yes | characteristic, complemented |
| 95 | C2×D3 | — | 6 | 3 | no | — |
| 96 | C2×D3 | — | 6 | 3 | no | — |
| 97 | C2×D3 | — | 6 | 3 | no | — |
| 98 | C2×D3 | — | 12 | 9 | no | — |
| 99 | A4 | #239 | 1 | 3 | yes | complemented |
| 100 | C3×C4 | — | 3 | 3 | no | — |
| 101 | C3×C4 | — | 4 | 9 | no | — |
| 102 | C3×C4 | C2×S4 | 1 | 3 | yes | — |
| 103 | Dic3 | — | 6 | 1 | no | — |
| 104 | Dic3 | — | 6 | 3 | no | — |
| 105 | Dic3 | — | 6 | 1 | no | — |
| 106 | Dic3 | — | 12 | 3 | no | — |
| 107 | C32 | — | 4 | 1 | no | Sylow |
| 108 | C23 | — | 9 | 3 | no | — |
| 109 | C23 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 110 | Q8 | — | 3 | 3 | no | — |
| 111 | Q8 | — | 6 | 1 | no | — |
| 112 | Q8 | C3⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 113 | D4 | — | 18 | 3 | no | — |
| 114 | D4 | — | 18 | 3 | no | — |
| 115 | D4 | — | 18 | 3 | no | — |
| 116 | D4 | — | 18 | 3 | no | — |
| 117 | C2×C4 | — | 3 | 3 | no | — |
| 118 | C2×C4 | — | 3 | 3 | no | — |
| 119 | C2×C4 | — | 9 | 1 | no | — |
| 120 | C2×C4 | — | 9 | 3 | no | — |
| 121 | C2×C4 | — | 9 | 1 | no | — |
| 122 | C2×C4 | — | 18 | 3 | no | — |
| 123 | C2×C4 | — | 18 | 3 | no | — |
| 124 | C2×C3 | — | 3 | 1 | no | — |
| 125 | C2×C3 | — | 3 | 1 | no | — |
| 126 | C2×C3 | — | 4 | 3 | no | — |
| 127 | C2×C3 | C22×S4 | 1 | 1 | yes | characteristic |
| 128 | D3 | — | 12 | 3 | no | — |
| 129 | D3 | — | 24 | 9 | no | — |
| 130 | C22 | — | 3 | 1 | no | — |
| 131 | C22 | — | 3 | 1 | no | — |
| 132 | C22 | — | 18 | 3 | no | — |
| 133 | C22 | — | 18 | 3 | no | — |
| 134 | C22 | — | 18 | 3 | no | — |
| 135 | C22 | #3832 | 1 | 1 | yes | characteristic, complemented |
| 136 | C4 | — | 3 | 3 | no | — |
| 137 | C4 | — | 18 | 3 | no | — |
| 138 | C4 | — | 18 | 1 | no | — |
| 139 | C4 | — | 18 | 1 | no | — |
| 140 | C4 | C2×C3⋊S4 | 1 | 3 | yes | — |
| 141 | C3 | — | 4 | 3 | no | — |
| 142 | C3 | #5919 | 1 | 1 | yes | characteristic, complemented |
| 143 | C2 | — | 3 | 1 | no | — |
| 144 | C2 | — | 3 | 1 | no | — |
| 145 | C2 | — | 36 | 3 | no | — |
| 146 | C2 | C22×C3⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊S4 |
| commutator, cosocle | C2×C3×A4 | C23 |
| Fitting | C22×C3×Q8 | D3 |
| socle | C23×C3 | C22×D3 |
| hypercenter | Q8 | C3⋊S4 |
| hypocenter | C3×A4 | #41 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×Q8×A4 | C2 | semidirect |
| 2. C4⋊(C3⋊S4) | C2 | semidirect |
| 3. C4×C3⋊S4 | C2 | semidirect |
| 10. C22×C3×Q8 | D3 | semidirect |
| 12. Q8×A4 | D3 | semidirect |
| 45. C3×A4 | #41 | semidirect |
| 49. C22×Q8 | C3⋊D3 | semidirect |
| 62. C3×Q8 | S4 | semidirect |
| 94. C22×C3 | #239 | semidirect |
| 99. A4 | #239 | semidirect |
| 112. Q8 | C3⋊S4 | semidirect |
| 135. C22 | #3832 | semidirect |
| 142. C3 | #5919 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q8×A4 | 2 |
| C4⋊(C3⋊S4) | 2 |
| C4×C3⋊S4 | 2 |
| #5919 | 3 |
| #5817 | 3 |
| #3832 | 4 |