Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
335 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 44 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×C4⋊S4 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×C8×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5402 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C2×#466 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C22×S4 | — | 2 | 2 | no | — |
| 6 | C2×C4×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | C4⋊S4 | — | 2 | 2 | no | — |
| 8 | C4⋊S4 | C22 | 1 | 4 | yes | complemented |
| 9 | C2×D24 | — | 4 | 1 | no | maximal |
| 10 | C8×A4 | C22 | 1 | 2 | yes | complemented |
| 11 | C22×D8 | — | 3 | 1 | no | — |
| 12 | C23×C8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#121 | — | 3 | 2 | no | — |
| 14 | #466 | — | 3 | 4 | no | — |
| 15 | #466 | — | 3 | 4 | no | — |
| 16 | C2×#107 | — | 3 | 1 | no | — |
| 17 | C2×#102 | — | 3 | 2 | no | — |
| 18 | C22×A4 | D4 | 1 | 1 | yes | characteristic |
| 19 | C2×S4 | — | 2 | 4 | no | — |
| 20 | C2×S4 | — | 4 | 4 | no | — |
| 21 | C2×D12 | — | 4 | 2 | no | — |
| 22 | C4×A4 | D4 | 1 | 1 | yes | characteristic |
| 23 | C4×A4 | C23 | 1 | 1 | yes | characteristic |
| 24 | C2×C3×C8 | — | 4 | 1 | no | — |
| 25 | D24 | — | 4 | 4 | no | — |
| 26 | C22×D4 | — | 3 | 2 | no | — |
| 27 | C22×D4 | — | 6 | 2 | no | — |
| 28 | C23×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C2×D8 | — | 3 | 4 | no | — |
| 30 | C2×D8 | — | 6 | 1 | no | — |
| 31 | C2×D8 | — | 6 | 2 | no | — |
| 32 | C2×D8 | — | 6 | 1 | no | — |
| 33 | C22×C8 | — | 3 | 1 | no | — |
| 34 | C22×C8 | — | 3 | 1 | no | — |
| 35 | C22×C8 | — | 3 | 2 | no | — |
| 36 | C22×C8 | C2×D3 | 1 | 2 | yes | complemented |
| 37 | #121 | — | 3 | 4 | no | — |
| 38 | #121 | — | 3 | 4 | no | — |
| 39 | #121 | — | 6 | 2 | no | — |
| 40 | #121 | — | 6 | 2 | no | — |
| 41 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 42 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 43 | #107 | — | 3 | 4 | no | — |
| 44 | #102 | — | 3 | 8 | no | — |
| 45 | C22×D3 | — | 8 | 2 | no | — |
| 46 | C2×A4 | D8 | 1 | 2 | yes | complemented |
| 47 | C2×A4 | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | S4 | — | 4 | 4 | no | — |
| 49 | C2×C3×C4 | — | 4 | 1 | no | — |
| 50 | D12 | — | 4 | 4 | no | — |
| 51 | D12 | — | 8 | 2 | no | — |
| 52 | C3×C8 | — | 4 | 2 | no | — |
| 53 | C24 | — | 6 | 2 | no | — |
| 54 | C24 | D12 | 1 | 1 | yes | characteristic |
| 55 | C2×D4 | — | 3 | 4 | no | — |
| 56 | C2×D4 | — | 6 | 2 | no | — |
| 57 | C2×D4 | — | 6 | 2 | no | — |
| 58 | C2×D4 | — | 6 | 2 | no | — |
| 59 | C2×D4 | — | 6 | 2 | no | — |
| 60 | C2×D4 | — | 6 | 4 | no | — |
| 61 | C2×D4 | — | 6 | 4 | no | — |
| 62 | C2×D4 | — | 6 | 2 | no | — |
| 63 | C2×D4 | — | 12 | 4 | no | — |
| 64 | C2×D4 | — | 12 | 4 | no | — |
| 65 | C22×C4 | — | 3 | 1 | no | — |
| 66 | C22×C4 | — | 3 | 1 | no | — |
| 67 | C22×C4 | — | 3 | 1 | no | — |
| 68 | C22×C4 | — | 3 | 1 | no | — |
| 69 | C22×C4 | — | 6 | 2 | no | — |
| 70 | C22×C4 | D12 | 1 | 1 | yes | characteristic |
| 71 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 72 | D8 | — | 6 | 4 | no | — |
| 73 | D8 | — | 6 | 4 | no | — |
| 74 | C2×C8 | — | 3 | 1 | no | — |
| 75 | C2×C8 | — | 3 | 2 | no | — |
| 76 | C2×C8 | — | 3 | 2 | no | — |
| 77 | C2×C8 | — | 3 | 2 | no | — |
| 78 | C2×C8 | — | 6 | 1 | no | — |
| 79 | C2×C8 | S4 | 1 | 1 | yes | characteristic, complemented |
| 80 | C4⋊C4 | — | 3 | 4 | no | — |
| 81 | C4⋊C4 | — | 6 | 2 | no | — |
| 82 | C22⋊C4 | — | 6 | 4 | no | — |
| 83 | C22⋊C4 | — | 6 | 4 | no | — |
| 84 | C22×C3 | — | 4 | 1 | no | — |
| 85 | C2×D3 | — | 8 | 4 | no | — |
| 86 | C2×D3 | — | 16 | 4 | no | — |
| 87 | A4 | C2×D8 | 1 | 1 | yes | characteristic, complemented |
| 88 | C3×C4 | — | 4 | 1 | no | — |
| 89 | C3×C4 | — | 4 | 1 | no | — |
| 90 | C23 | — | 3 | 1 | no | — |
| 91 | C23 | — | 3 | 1 | no | — |
| 92 | C23 | — | 3 | 2 | no | — |
| 93 | C23 | — | 6 | 4 | no | — |
| 94 | C23 | — | 12 | 2 | no | — |
| 95 | C23 | — | 12 | 4 | no | — |
| 96 | C23 | — | 12 | 4 | no | — |
| 97 | C23 | — | 12 | 2 | no | — |
| 98 | C23 | D24 | 1 | 2 | yes | complemented |
| 99 | C23 | C2×D12 | 1 | 1 | yes | characteristic |
| 100 | D4 | — | 6 | 4 | no | — |
| 101 | D4 | — | 6 | 4 | no | — |
| 102 | D4 | — | 12 | 2 | no | — |
| 103 | D4 | — | 12 | 8 | no | — |
| 104 | D4 | — | 12 | 4 | no | — |
| 105 | D4 | — | 12 | 4 | no | — |
| 106 | D4 | — | 12 | 2 | no | — |
| 107 | C2×C4 | — | 3 | 1 | no | — |
| 108 | C2×C4 | — | 3 | 1 | no | — |
| 109 | C2×C4 | — | 3 | 1 | no | — |
| 110 | C2×C4 | — | 3 | 1 | no | — |
| 111 | C2×C4 | — | 3 | 1 | no | — |
| 112 | C2×C4 | — | 3 | 1 | no | — |
| 113 | C2×C4 | — | 3 | 1 | no | — |
| 114 | C2×C4 | — | 6 | 1 | no | — |
| 115 | C2×C4 | — | 6 | 4 | no | — |
| 116 | C2×C4 | — | 12 | 4 | no | — |
| 117 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 118 | C8 | — | 3 | 2 | no | — |
| 119 | C8 | C2×S4 | 1 | 2 | yes | complemented |
| 120 | C2×C3 | — | 4 | 1 | no | — |
| 121 | C2×C3 | — | 4 | 2 | no | — |
| 122 | D3 | — | 16 | 4 | no | — |
| 123 | C22 | — | 3 | 2 | no | — |
| 124 | C22 | — | 3 | 1 | no | — |
| 125 | C22 | — | 3 | 1 | no | — |
| 126 | C22 | — | 3 | 1 | no | — |
| 127 | C22 | — | 3 | 2 | no | — |
| 128 | C22 | — | 3 | 2 | no | — |
| 129 | C22 | — | 6 | 1 | no | — |
| 130 | C22 | — | 12 | 4 | no | — |
| 131 | C22 | — | 12 | 4 | no | — |
| 132 | C22 | — | 12 | 4 | no | — |
| 133 | C22 | — | 24 | 4 | no | — |
| 134 | C22 | — | 24 | 4 | no | — |
| 135 | C22 | C2×D24 | 1 | 1 | yes | characteristic, complemented |
| 136 | C22 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 137 | C4 | — | 3 | 1 | no | — |
| 138 | C4 | — | 3 | 1 | no | — |
| 139 | C4 | — | 12 | 4 | no | — |
| 140 | C4 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 141 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 142 | C3 | — | 4 | 1 | no | Sylow |
| 143 | C2 | — | 3 | 1 | no | — |
| 144 | C2 | — | 3 | 1 | no | — |
| 145 | C2 | — | 3 | 2 | no | — |
| 146 | C2 | — | 24 | 4 | no | — |
| 147 | C2 | #5402 | 1 | 2 | yes | direct product factor |
| 148 | C2 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C4⋊S4 |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C23×C8 | D3 |
| socle | C24 | D12 |
| hypercenter | C2×C8 | S4 |
| hypocenter | A4 | C2×D8 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4⋊S4 | C2 | semidirect |
| 2. C2×C8×A4 | C2 | semidirect |
| 3. #5402 | C2 | direct (N × H) |
| 8. C4⋊S4 | C22 | semidirect |
| 10. C8×A4 | C22 | semidirect |
| 12. C23×C8 | D3 | semidirect |
| 36. C22×C8 | C2×D3 | semidirect |
| 46. C2×A4 | D8 | semidirect |
| 79. C2×C8 | S4 | semidirect |
| 87. A4 | C2×D8 | semidirect |
| 98. C23 | D24 | semidirect |
| 119. C8 | C2×S4 | semidirect |
| 135. C22 | C2×D24 | semidirect |
| 147. C2 | #5402 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊S4 | 2 |
| C2×C8×A4 | 2 |
| #5402 | 2 |
| C2×#466 | 3 |
| C2×D24 | 4 |