Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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 | #5713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5073 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #4848 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×#331 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #4484 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #1620 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C3×#127 | C22 | 1 | 1 | yes | characteristic |
| 8 | #936 | — | 2 | 2 | no | — |
| 9 | #935 | — | 2 | 2 | no | — |
| 10 | D4×Dic3 | — | 2 | 2 | no | — |
| 11 | C2×#213 | — | 2 | 2 | no | — |
| 12 | C2×#204 | — | 2 | 2 | no | — |
| 13 | #874 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×Dic6 | C22 | 1 | 2 | yes | — |
| 15 | C3×C4⋊C8 | C22 | 1 | 2 | yes | — |
| 16 | #821 | — | 2 | 2 | no | — |
| 17 | C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 18 | #559 | — | 3 | 1 | no | — |
| 19 | #493 | — | 3 | 1 | no | — |
| 20 | #460 | — | 3 | 2 | no | — |
| 21 | #331 | D3 | 1 | 1 | yes | characteristic, complemented |
| 22 | Q8⋊C8 | — | 3 | 2 | no | — |
| 23 | C2×C3×D4 | — | 2 | 2 | no | — |
| 24 | C2×C3×D4 | D4 | 1 | 2 | yes | — |
| 25 | C22×Dic3 | — | 2 | 2 | no | — |
| 26 | C2×D12 | — | 2 | 2 | no | — |
| 27 | C2×D12 | — | 2 | 1 | no | — |
| 28 | C2×Dic6 | D4 | 1 | 2 | yes | — |
| 29 | C2×C3×C8 | — | 2 | 2 | no | — |
| 30 | C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 31 | #217 | — | 2 | 4 | no | — |
| 32 | #217 | — | 2 | 2 | no | — |
| 33 | #213 | — | 4 | 4 | no | — |
| 34 | C4⋊Dic3 | D4 | 1 | 2 | yes | — |
| 35 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 36 | C4×Dic3 | — | 2 | 2 | no | — |
| 37 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 38 | #204 | — | 4 | 4 | no | — |
| 39 | C2×SD16 | — | 6 | 2 | no | — |
| 40 | C2×D8 | — | 6 | 2 | no | — |
| 41 | #127 | — | 3 | 1 | no | — |
| 42 | #127 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | #124 | — | 6 | 2 | no | — |
| 44 | #123 | — | 6 | 2 | no | — |
| 45 | C4×Q8 | — | 3 | 2 | no | — |
| 46 | C4×D4 | — | 6 | 2 | no | — |
| 47 | C4⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 48 | #102 | — | 6 | 2 | no | — |
| 49 | C4×C8 | — | 3 | 1 | no | — |
| 50 | C23×C3 | — | 2 | 2 | no | — |
| 51 | C22×D3 | — | 4 | 1 | no | — |
| 52 | C3×D4 | — | 4 | 2 | no | — |
| 53 | C3×D4 | — | 4 | 2 | no | — |
| 54 | C3×D4 | — | 4 | 2 | no | — |
| 55 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 56 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 57 | C2×Dic3 | — | 2 | 2 | no | — |
| 58 | C2×Dic3 | — | 2 | 2 | no | — |
| 59 | C2×Dic3 | — | 8 | 2 | no | — |
| 60 | D12 | — | 4 | 2 | no | — |
| 61 | D12 | — | 4 | 2 | no | — |
| 62 | D12 | — | 4 | 2 | no | — |
| 63 | Dic6 | — | 2 | 2 | no | — |
| 64 | Dic6 | — | 2 | 2 | no | — |
| 65 | C3×C8 | — | 4 | 2 | no | — |
| 66 | C3⋊C8 | — | 2 | 2 | no | — |
| 67 | C2×Q8 | — | 3 | 2 | no | — |
| 68 | C2×D4 | — | 2 | 2 | no | — |
| 69 | C2×D4 | — | 6 | 1 | no | — |
| 70 | C2×D4 | — | 6 | 2 | no | — |
| 71 | C2×D4 | D12 | 1 | 2 | yes | — |
| 72 | C22×C4 | — | 6 | 2 | no | — |
| 73 | SD16 | — | 12 | 4 | no | — |
| 74 | D8 | — | 12 | 4 | no | — |
| 75 | C2×C8 | — | 2 | 2 | no | — |
| 76 | C2×C8 | — | 6 | 1 | no | — |
| 77 | C4⋊C4 | — | 3 | 2 | no | — |
| 78 | C4⋊C4 | — | 6 | 2 | no | — |
| 79 | C22⋊C4 | — | 6 | 4 | no | — |
| 80 | C22⋊C4 | — | 6 | 2 | no | — |
| 81 | C42 | — | 6 | 2 | no | — |
| 82 | C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 83 | C22×C3 | — | 4 | 4 | no | — |
| 84 | C22×C3 | — | 4 | 2 | no | — |
| 85 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 86 | C2×D3 | — | 8 | 2 | no | — |
| 87 | C2×D3 | — | 8 | 1 | no | — |
| 88 | C3×C4 | — | 2 | 1 | no | — |
| 89 | C3×C4 | #135 | 1 | 2 | yes | — |
| 90 | C3×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 91 | Dic3 | — | 4 | 2 | no | — |
| 92 | Dic3 | — | 4 | 2 | no | — |
| 93 | C23 | — | 2 | 2 | no | — |
| 94 | C23 | — | 12 | 1 | no | — |
| 95 | Q8 | — | 6 | 2 | no | — |
| 96 | Q8 | — | 6 | 2 | no | — |
| 97 | D4 | — | 4 | 2 | no | — |
| 98 | D4 | — | 4 | 2 | no | — |
| 99 | D4 | — | 4 | 2 | no | — |
| 100 | D4 | — | 12 | 2 | no | — |
| 101 | D4 | — | 12 | 2 | no | — |
| 102 | D4 | — | 12 | 2 | no | — |
| 103 | C2×C4 | — | 6 | 2 | no | — |
| 104 | C2×C4 | — | 6 | 2 | no | — |
| 105 | C2×C4 | — | 24 | 2 | no | — |
| 106 | C2×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 107 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 108 | C8 | — | 4 | 2 | no | — |
| 109 | C8 | — | 6 | 2 | no | — |
| 110 | C2×C3 | — | 8 | 2 | no | — |
| 111 | C2×C3 | #449 | 1 | 2 | yes | — |
| 112 | C2×C3 | #453 | 1 | 1 | yes | characteristic |
| 113 | D3 | — | 16 | 1 | no | — |
| 114 | C22 | — | 4 | 2 | no | — |
| 115 | C22 | — | 4 | 4 | no | — |
| 116 | C22 | — | 24 | 1 | no | — |
| 117 | C22 | — | 24 | 2 | no | — |
| 118 | C22 | #882 | 1 | 1 | yes | characteristic |
| 119 | C4 | — | 2 | 1 | no | — |
| 120 | C4 | — | 12 | 2 | no | — |
| 121 | C4 | — | 12 | 2 | no | — |
| 122 | C4 | #911 | 1 | 2 | yes | — |
| 123 | C4 | #916 | 1 | 2 | yes | — |
| 124 | C3 | #1620 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 125 | C2 | — | 8 | 2 | no | — |
| 126 | C2 | — | 48 | 1 | no | — |
| 127 | C2 | #4747 | 1 | 1 | yes | characteristic |
| 128 | C2 | #4781 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5713 | C2 | semidirect |
| 3. #4848 | C2 | semidirect |
| 4. C3×#331 | C2 | semidirect |
| 5. #4484 | C2 | semidirect |
| 21. #331 | D3 | semidirect |
| 124. C3 | #1620 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5713 | 2 |
| #5073 | 2 |
| #4848 | 2 |
| C3×#331 | 2 |
| #4484 | 2 |
| #1620 | 3 |