Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
130 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 130 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×D3×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3×S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊SL(2,ℤ/4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | GL(2,ℤ/4) | — | 3 | 1 | no | maximal |
| 5 | #937 | — | 3 | 1 | no | maximal |
| 6 | C2×C3×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | D3×A4 | — | 2 | 1 | no | — |
| 8 | C3×S4 | — | 2 | 1 | no | — |
| 9 | #625 | — | 4 | 1 | no | maximal |
| 10 | C23×D3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×A4 | — | 3 | 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 | SL(2,ℤ/4) | — | 3 | 1 | no | — |
| 17 | SL(2,ℤ/4) | — | 6 | 1 | no | — |
| 18 | #217 | — | 3 | 1 | no | — |
| 19 | (C2×D3)⋊C4 | — | 3 | 1 | no | — |
| 20 | (C2×D3)⋊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 | #120 | — | 9 | 1 | no | Sylow |
| 26 | C23×C3 | — | 3 | 1 | no | — |
| 27 | C23×C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | C22×D3 | — | 2 | 1 | no | — |
| 29 | C22×D3 | — | 3 | 1 | no | — |
| 30 | C22×D3 | — | 3 | 1 | no | — |
| 31 | C22×D3 | — | 6 | 1 | no | — |
| 32 | C2×A4 | — | 2 | 1 | no | — |
| 33 | C2×A4 | — | 6 | 1 | no | — |
| 34 | C2×A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 35 | S4 | — | 2 | 1 | no | — |
| 36 | C3×D4 | — | 6 | 1 | no | — |
| 37 | C3×D4 | — | 6 | 1 | no | — |
| 38 | C2×C3×C4 | — | 3 | 1 | no | — |
| 39 | #67 | — | 6 | 2 | no | — |
| 40 | #67 | — | 6 | 1 | no | — |
| 41 | #67 | — | 6 | 1 | no | — |
| 42 | #67 | — | 12 | 1 | no | — |
| 43 | C2×Dic3 | — | 3 | 1 | no | — |
| 44 | C2×Dic3 | — | 3 | 1 | no | — |
| 45 | C2×C32 | — | 4 | 1 | no | — |
| 46 | C3×D3 | — | 8 | 1 | no | — |
| 47 | C3×D3 | — | 8 | 1 | no | — |
| 48 | C24 | — | 3 | 1 | no | — |
| 49 | C2×D4 | — | 3 | 1 | no | — |
| 50 | C2×D4 | — | 9 | 1 | no | — |
| 51 | C2×D4 | — | 9 | 1 | no | — |
| 52 | C22⋊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 | — | 12 | 1 | no | — |
| 61 | C22×C3 | #67 | 1 | 1 | yes | characteristic, complemented |
| 62 | C2×D3 | — | 3 | 2 | no | — |
| 63 | C2×D3 | — | 3 | 1 | no | — |
| 64 | C2×D3 | — | 4 | 1 | no | — |
| 65 | C2×D3 | — | 6 | 1 | no | — |
| 66 | C2×D3 | — | 6 | 1 | no | — |
| 67 | C2×D3 | — | 6 | 1 | no | — |
| 68 | C2×D3 | S4 | 1 | 1 | yes | characteristic, complemented |
| 69 | A4 | — | 2 | 1 | no | — |
| 70 | A4 | #67 | 1 | 1 | yes | characteristic, complemented |
| 71 | C3×C4 | — | 6 | 1 | no | — |
| 72 | Dic3 | — | 6 | 1 | no | — |
| 73 | Dic3 | — | 6 | 1 | no | — |
| 74 | Dic3 | — | 12 | 1 | no | — |
| 75 | Dic3 | — | 24 | 1 | no | — |
| 76 | C32 | — | 4 | 1 | no | Sylow |
| 77 | C23 | — | 3 | 1 | no | — |
| 78 | C23 | — | 6 | 1 | no | — |
| 79 | C23 | — | 9 | 1 | no | — |
| 80 | C23 | — | 9 | 1 | no | — |
| 81 | C23 | — | 18 | 1 | no | — |
| 82 | C23 | D32 | 1 | 1 | yes | characteristic |
| 83 | D4 | — | 6 | 1 | no | — |
| 84 | D4 | — | 6 | 1 | no | — |
| 85 | D4 | — | 18 | 1 | no | — |
| 86 | D4 | — | 18 | 2 | no | — |
| 87 | D4 | — | 18 | 1 | no | — |
| 88 | C2×C4 | — | 3 | 1 | no | — |
| 89 | C2×C4 | — | 9 | 1 | no | — |
| 90 | C2×C4 | — | 9 | 1 | no | — |
| 91 | C2×C3 | — | 3 | 1 | no | — |
| 92 | C2×C3 | — | 3 | 1 | no | — |
| 93 | C2×C3 | — | 4 | 1 | no | — |
| 94 | C2×C3 | — | 8 | 1 | no | — |
| 95 | C2×C3 | — | 12 | 1 | no | — |
| 96 | C2×C3 | — | 24 | 1 | no | — |
| 97 | C2×C3 | C2×S4 | 1 | 1 | yes | characteristic |
| 98 | D3 | — | 2 | 1 | no | — |
| 99 | D3 | — | 6 | 1 | no | — |
| 100 | D3 | — | 8 | 1 | no | — |
| 101 | C22 | — | 3 | 1 | no | — |
| 102 | C22 | — | 3 | 1 | no | — |
| 103 | C22 | — | 3 | 1 | no | — |
| 104 | C22 | — | 6 | 1 | no | — |
| 105 | C22 | — | 6 | 1 | no | — |
| 106 | C22 | — | 6 | 1 | no | — |
| 107 | C22 | — | 9 | 2 | no | — |
| 108 | C22 | — | 9 | 1 | no | — |
| 109 | C22 | — | 18 | 1 | no | — |
| 110 | C22 | — | 18 | 1 | no | — |
| 111 | C22 | — | 18 | 1 | no | — |
| 112 | C22 | #625 | 1 | 1 | yes | characteristic, complemented |
| 113 | C4 | — | 6 | 1 | no | — |
| 114 | C4 | — | 18 | 1 | no | — |
| 115 | C4 | — | 18 | 1 | no | — |
| 116 | C3 | — | 4 | 1 | no | — |
| 117 | C3 | — | 8 | 1 | no | — |
| 118 | C3 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic, complemented |
| 119 | C2 | — | 3 | 1 | no | — |
| 120 | C2 | — | 3 | 1 | no | — |
| 121 | C2 | — | 6 | 1 | no | — |
| 122 | C2 | — | 12 | 1 | no | — |
| 123 | C2 | — | 18 | 1 | no | — |
| 124 | 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×D3×A4 | C2 | semidirect |
| 2. C2×C3×S4 | C2 | semidirect |
| 3. C3⋊SL(2,ℤ/4) | C2 | semidirect |
| 10. C23×D3 | D3 | semidirect |
| 12. C2×S4 | D3 | semidirect |
| 23. C3×A4 | D4 | semidirect |
| 61. C22×C3 | #67 | semidirect |
| 68. C2×D3 | S4 | semidirect |
| 70. A4 | #67 | semidirect |
| 112. C22 | #625 | semidirect |
| 118. C3 | GL(2,ℤ/4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×A4 | 2 |
| C2×C3×S4 | 2 |
| C3⋊SL(2,ℤ/4) | 2 |
| GL(2,ℤ/4) | 3 |
| #937 | 3 |
| #625 | 4 |