Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
122 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 19 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 | C2×C3⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3×S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Dic3×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4⋊S4 | — | 3 | 1 | no | maximal |
| 5 | #939 | — | 3 | 1 | no | maximal |
| 6 | C2×C3×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊S4 | — | 2 | 1 | no | — |
| 8 | C3×S4 | — | 2 | 1 | no | — |
| 9 | #626 | — | 4 | 1 | no | maximal |
| 10 | C2×S4 | — | 3 | 1 | no | — |
| 11 | C2×S4 | — | 6 | 1 | no | — |
| 12 | C2×S4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×C3×D4 | — | 3 | 1 | no | — |
| 14 | C2×#67 | — | 3 | 1 | no | — |
| 15 | C2×#67 | — | 3 | 1 | no | — |
| 16 | C22×Dic3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | C4×A4 | — | 3 | 1 | no | — |
| 18 | #217 | — | 3 | 1 | no | — |
| 19 | (C2×D3)⋊C4 | — | 3 | 1 | no | — |
| 20 | Dic3⋊C4 | — | 3 | 1 | no | — |
| 21 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 22 | C2×C3×D3 | — | 4 | 1 | no | — |
| 23 | C3×A4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 24 | C3×Dic3 | — | 4 | 1 | no | — |
| 25 | #121 | — | 9 | 1 | no | Sylow |
| 26 | C23×C3 | — | 3 | 1 | no | — |
| 27 | C23×C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | C22×D3 | — | 3 | 1 | no | — |
| 29 | C2×A4 | — | 2 | 1 | no | — |
| 30 | C2×A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 31 | S4 | — | 2 | 1 | no | — |
| 32 | S4 | — | 6 | 2 | no | — |
| 33 | S4 | — | 6 | 1 | no | — |
| 34 | C3×D4 | — | 6 | 1 | no | — |
| 35 | C3×D4 | — | 6 | 1 | no | — |
| 36 | C2×C3×C4 | — | 3 | 1 | no | — |
| 37 | #67 | — | 6 | 1 | no | — |
| 38 | #67 | — | 6 | 1 | no | — |
| 39 | #67 | — | 6 | 1 | no | — |
| 40 | #67 | — | 6 | 1 | no | — |
| 41 | C2×Dic3 | — | 3 | 1 | no | — |
| 42 | C2×Dic3 | — | 3 | 1 | no | — |
| 43 | C2×Dic3 | — | 3 | 1 | no | — |
| 44 | D12 | — | 12 | 1 | no | — |
| 45 | C2×C32 | — | 4 | 1 | no | — |
| 46 | C3⋊D3 | — | 8 | 1 | no | — |
| 47 | C3×D3 | — | 8 | 1 | no | — |
| 48 | C2×D4 | — | 3 | 1 | no | — |
| 49 | C2×D4 | — | 9 | 1 | no | — |
| 50 | C2×D4 | — | 9 | 1 | no | — |
| 51 | C22×C4 | — | 3 | 1 | no | — |
| 52 | C4⋊C4 | — | 9 | 1 | no | — |
| 53 | C22⋊C4 | — | 9 | 1 | no | — |
| 54 | C22⋊C4 | — | 9 | 1 | no | — |
| 55 | C22×C3 | — | 3 | 1 | no | — |
| 56 | C22×C3 | — | 3 | 1 | no | — |
| 57 | C22×C3 | — | 6 | 1 | no | — |
| 58 | C22×C3 | — | 6 | 1 | no | — |
| 59 | C22×C3 | — | 6 | 1 | no | — |
| 60 | C22×C3 | D12 | 1 | 1 | yes | characteristic, complemented |
| 61 | C2×D3 | — | 4 | 1 | no | — |
| 62 | C2×D3 | — | 6 | 1 | no | — |
| 63 | C2×D3 | — | 6 | 1 | no | — |
| 64 | C2×D3 | — | 6 | 1 | no | — |
| 65 | C2×D3 | — | 12 | 1 | no | — |
| 66 | C2×D3 | — | 24 | 1 | no | — |
| 67 | A4 | — | 2 | 1 | no | — |
| 68 | A4 | #67 | 1 | 1 | yes | characteristic, complemented |
| 69 | C3×C4 | — | 6 | 1 | no | — |
| 70 | C3×C4 | — | 12 | 1 | no | — |
| 71 | Dic3 | — | 3 | 1 | no | — |
| 72 | Dic3 | — | 6 | 1 | no | — |
| 73 | Dic3 | S4 | 1 | 1 | yes | characteristic, complemented |
| 74 | C32 | — | 4 | 1 | no | Sylow |
| 75 | C23 | — | 3 | 1 | no | — |
| 76 | C23 | — | 9 | 1 | no | — |
| 77 | C23 | D32 | 1 | 1 | yes | characteristic |
| 78 | D4 | — | 6 | 1 | no | — |
| 79 | D4 | — | 6 | 1 | no | — |
| 80 | D4 | — | 18 | 1 | no | — |
| 81 | D4 | — | 18 | 1 | no | — |
| 82 | D4 | — | 18 | 1 | no | — |
| 83 | D4 | — | 18 | 1 | no | — |
| 84 | C2×C4 | — | 3 | 1 | no | — |
| 85 | C2×C4 | — | 9 | 1 | no | — |
| 86 | C2×C4 | — | 9 | 1 | no | — |
| 87 | C2×C4 | — | 9 | 1 | no | — |
| 88 | C2×C3 | — | 3 | 1 | no | — |
| 89 | C2×C3 | — | 3 | 1 | no | — |
| 90 | C2×C3 | — | 4 | 1 | no | — |
| 91 | C2×C3 | — | 8 | 1 | no | — |
| 92 | C2×C3 | — | 12 | 1 | no | — |
| 93 | C2×C3 | C2×S4 | 1 | 1 | yes | characteristic |
| 94 | D3 | — | 8 | 1 | no | — |
| 95 | D3 | — | 12 | 1 | no | — |
| 96 | D3 | — | 24 | 1 | no | — |
| 97 | D3 | — | 24 | 2 | no | — |
| 98 | C22 | — | 3 | 1 | no | — |
| 99 | C22 | — | 3 | 1 | no | — |
| 100 | C22 | — | 6 | 1 | no | — |
| 101 | C22 | — | 6 | 1 | no | — |
| 102 | C22 | — | 6 | 1 | no | — |
| 103 | C22 | — | 18 | 1 | no | — |
| 104 | C22 | — | 18 | 1 | no | — |
| 105 | C22 | — | 18 | 1 | no | — |
| 106 | C22 | #626 | 1 | 1 | yes | characteristic, complemented |
| 107 | C4 | — | 3 | 1 | no | — |
| 108 | C4 | — | 6 | 1 | no | — |
| 109 | C4 | — | 9 | 1 | no | — |
| 110 | C4 | — | 18 | 1 | no | — |
| 111 | C3 | — | 4 | 1 | no | — |
| 112 | C3 | — | 8 | 1 | no | — |
| 113 | C3 | C4⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 114 | C2 | — | 3 | 1 | no | — |
| 115 | C2 | — | 3 | 1 | no | — |
| 116 | C2 | — | 12 | 1 | no | — |
| 117 | C2 | — | 36 | 1 | no | — |
| 118 | C2 | D3×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S4 |
| commutator, cosocle | C2×C3×A4 | C22 |
| Fitting, socle | C23×C3 | C2×D3 |
| hypocenter | C3×A4 | D4 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3⋊S4 | C2 | semidirect |
| 2. C2×C3×S4 | C2 | semidirect |
| 3. Dic3×A4 | C2 | semidirect |
| 12. C2×S4 | D3 | semidirect |
| 16. C22×Dic3 | D3 | semidirect |
| 23. C3×A4 | D4 | semidirect |
| 60. C22×C3 | D12 | semidirect |
| 68. A4 | #67 | semidirect |
| 73. Dic3 | S4 | semidirect |
| 106. C22 | #626 | semidirect |
| 113. C3 | C4⋊S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊S4 | 2 |
| C2×C3×S4 | 2 |
| Dic3×A4 | 2 |
| C4⋊S4 | 3 |
| #939 | 3 |
| #626 | 4 |