Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
132 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 25 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 | #3811 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#624 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C22⋊#157 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #3773 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | D32⋊C4 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C22×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×D32 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#157 | C22 | 1 | 2 | yes | — |
| 9 | C3×#67 | — | 2 | 1 | no | — |
| 10 | C2×C3×Dic3 | — | 2 | 1 | no | — |
| 11 | #626 | — | 2 | 1 | no | — |
| 12 | #624 | — | 2 | 1 | no | — |
| 13 | #624 | C22 | 1 | 2 | yes | — |
| 14 | #624 | C22 | 1 | 1 | yes | characteristic |
| 15 | D3×D4 | — | 6 | 1 | no | — |
| 16 | C2×C4×D3 | — | 6 | 1 | no | — |
| 17 | C22×C32 | D4 | 1 | 1 | yes | characteristic |
| 18 | C2×C3⋊D3 | — | 2 | 2 | no | — |
| 19 | C2×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×C3⋊D3 | C23 | 1 | 1 | yes | characteristic |
| 21 | C2×C3×D3 | — | 2 | 1 | no | — |
| 22 | D32 | — | 2 | 2 | no | — |
| 23 | #157 | — | 2 | 2 | no | — |
| 24 | C3×Dic3 | — | 2 | 2 | no | — |
| 25 | C3×Dic3 | — | 2 | 1 | no | — |
| 26 | #123 | — | 9 | 1 | no | maximal, Sylow |
| 27 | C22×D3 | — | 6 | 1 | no | — |
| 28 | C22×D3 | — | 6 | 1 | no | — |
| 29 | C22×D3 | — | 6 | 1 | no | — |
| 30 | C3×D4 | — | 6 | 1 | no | — |
| 31 | C2×C3×C4 | — | 6 | 1 | no | — |
| 32 | #67 | — | 2 | 1 | no | — |
| 33 | #67 | — | 6 | 1 | no | — |
| 34 | C2×Dic3 | — | 2 | 1 | no | — |
| 35 | D12 | — | 6 | 1 | no | — |
| 36 | C4×D3 | — | 6 | 2 | no | — |
| 37 | C4×D3 | — | 6 | 1 | no | — |
| 38 | C4×D3 | — | 6 | 2 | no | — |
| 39 | C2×C32 | — | 2 | 1 | no | — |
| 40 | C2×C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 41 | C3⋊D3 | — | 2 | 1 | no | — |
| 42 | C3⋊D3 | #41 | 1 | 2 | yes | — |
| 43 | C3×D3 | — | 4 | 1 | no | — |
| 44 | C2×D4 | — | 9 | 1 | no | — |
| 45 | C22×C4 | — | 9 | 1 | no | — |
| 46 | C4⋊C4 | — | 9 | 2 | no | — |
| 47 | C22⋊C4 | — | 9 | 2 | no | — |
| 48 | C22⋊C4 | — | 9 | 1 | no | — |
| 49 | C22×C3 | — | 2 | 1 | no | — |
| 50 | C22×C3 | — | 2 | 1 | no | — |
| 51 | C22×C3 | — | 6 | 1 | no | — |
| 52 | C2×D3 | — | 2 | 1 | no | — |
| 53 | C2×D3 | — | 6 | 1 | no | — |
| 54 | C2×D3 | — | 6 | 1 | no | — |
| 55 | C2×D3 | — | 6 | 1 | no | — |
| 56 | C2×D3 | — | 6 | 1 | no | — |
| 57 | C2×D3 | — | 6 | 4 | no | — |
| 58 | C2×D3 | — | 12 | 2 | no | — |
| 59 | C2×D3 | — | 12 | 2 | no | — |
| 60 | C3×C4 | — | 6 | 2 | no | — |
| 61 | C3×C4 | — | 6 | 1 | no | — |
| 62 | Dic3 | — | 2 | 2 | no | — |
| 63 | Dic3 | — | 2 | 1 | no | — |
| 64 | C32 | #123 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 65 | C23 | — | 9 | 1 | no | — |
| 66 | C23 | — | 9 | 1 | no | — |
| 67 | D4 | — | 6 | 1 | no | — |
| 68 | D4 | — | 18 | 1 | no | — |
| 69 | C2×C4 | — | 6 | 1 | no | — |
| 70 | C2×C4 | — | 9 | 2 | no | — |
| 71 | C2×C4 | — | 9 | 1 | no | — |
| 72 | C2×C4 | — | 9 | 2 | no | — |
| 73 | C2×C4 | — | 18 | 1 | no | — |
| 74 | C2×C3 | — | 2 | 2 | no | — |
| 75 | C2×C3 | — | 2 | 1 | no | — |
| 76 | C2×C3 | — | 2 | 1 | no | — |
| 77 | C2×C3 | — | 4 | 1 | no | — |
| 78 | C2×C3 | — | 12 | 1 | no | — |
| 79 | D3 | — | 4 | 1 | no | — |
| 80 | D3 | — | 6 | 2 | no | — |
| 81 | D3 | — | 6 | 2 | no | — |
| 82 | D3 | — | 6 | 2 | no | — |
| 83 | D3 | — | 12 | 1 | no | — |
| 84 | C22 | — | 6 | 1 | no | — |
| 85 | C22 | — | 9 | 1 | no | — |
| 86 | C22 | — | 9 | 1 | no | — |
| 87 | C22 | — | 18 | 2 | no | — |
| 88 | C22 | — | 18 | 2 | no | — |
| 89 | C22 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 90 | C4 | — | 6 | 2 | no | — |
| 91 | C4 | — | 6 | 1 | no | — |
| 92 | C4 | — | 18 | 2 | no | — |
| 93 | C3 | — | 2 | 1 | no | — |
| 94 | C3 | — | 2 | 1 | no | — |
| 95 | C2 | — | 2 | 1 | no | — |
| 96 | C2 | — | 9 | 2 | no | — |
| 97 | C2 | — | 12 | 1 | no | — |
| 98 | C2 | — | 18 | 1 | no | — |
| 99 | C2 | C2×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#624 | C2 | semidirect |
| 3. C22⋊#157 | C2 | semidirect |
| 4. #3773 | C2 | semidirect |
| 5. D32⋊C4 | C2 | semidirect |
| 64. C32 | #123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3811 | 2 |
| C2×#624 | 2 |
| C22⋊#157 | 2 |
| #3773 | 2 |
| D32⋊C4 | 2 |
| #123 | 9 |