Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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 | C3×#585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #5825 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #5243 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #5240 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #3028 | — | 3 | 1 | no | maximal, Sylow |
| 6 | C3×2−1+4 | — | 2 | 3 | no | — |
| 7 | C3×H(5,𝔽2) | — | 2 | 3 | no | — |
| 8 | C3×H(5,𝔽2) | C22 | 1 | 4 | yes | — |
| 9 | C2×C3×#41 | — | 2 | 6 | no | — |
| 10 | C2×C3×#41 | C22 | 1 | 3 | yes | — |
| 11 | C22×Dic6 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | #944 | — | 2 | 3 | no | — |
| 13 | #936 | — | 2 | 3 | no | — |
| 14 | #935 | C22 | 1 | 12 | yes | complemented |
| 15 | #926 | C22 | 1 | 3 | yes | complemented |
| 16 | #924 | — | 2 | 6 | no | — |
| 17 | #834 | C22 | 1 | 12 | yes | complemented |
| 18 | #585 | D3 | 1 | 1 | yes | characteristic |
| 19 | #536 | — | 3 | 3 | no | — |
| 20 | #458 | — | 3 | 8 | no | — |
| 21 | #410 | — | 3 | 3 | no | — |
| 22 | C3×#41 | — | 2 | 4 | no | — |
| 23 | C3×#41 | — | 2 | 12 | no | — |
| 24 | C3×#41 | — | 2 | 6 | no | — |
| 25 | C3×#41 | — | 2 | 12 | no | — |
| 26 | C3×#41 | — | 2 | 6 | no | — |
| 27 | C2×C3×Q8 | — | 2 | 6 | no | — |
| 28 | C2×C3×Q8 | D4 | 1 | 3 | yes | — |
| 29 | C2×C3×D4 | — | 2 | 6 | no | — |
| 30 | C2×C3×D4 | — | 2 | 12 | no | — |
| 31 | C2×C3×D4 | D4 | 1 | 3 | yes | — |
| 32 | C2×C3×D4 | C23 | 1 | 6 | yes | — |
| 33 | C22×C3×C4 | — | 2 | 4 | no | — |
| 34 | C22×C3×C4 | D4 | 1 | 6 | yes | — |
| 35 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 36 | C22×Dic3 | C23 | 1 | 2 | yes | — |
| 37 | C2×Dic6 | — | 2 | 3 | no | — |
| 38 | C2×Dic6 | — | 2 | 3 | no | — |
| 39 | #217 | — | 2 | 12 | no | — |
| 40 | #217 | C23 | 1 | 6 | yes | — |
| 41 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 42 | Dic3⋊C4 | — | 2 | 12 | no | — |
| 43 | C4×Dic3 | — | 2 | 3 | no | — |
| 44 | 2−1+4 | — | 2 | 3 | no | — |
| 45 | H(5,𝔽2) | — | 2 | 3 | no | — |
| 46 | H(5,𝔽2) | C2×D3 | 1 | 4 | yes | — |
| 47 | C2×#41 | — | 2 | 6 | no | — |
| 48 | C2×#41 | C2×D3 | 1 | 3 | yes | — |
| 49 | C22×Q8 | — | 3 | 1 | no | — |
| 50 | C4⋊Q8 | — | 6 | 3 | no | — |
| 51 | #124 | — | 6 | 3 | no | — |
| 52 | #123 | — | 3 | 12 | no | — |
| 53 | C22⋊Q8 | — | 6 | 6 | no | — |
| 54 | #117 | — | 3 | 3 | no | — |
| 55 | #99 | — | 3 | 12 | no | — |
| 56 | C23×C3 | — | 2 | 4 | no | — |
| 57 | C23×C3 | C2×D4 | 1 | 6 | yes | — |
| 58 | C23×C3 | C24 | 1 | 1 | yes | characteristic |
| 59 | C3×Q8 | — | 2 | 4 | no | — |
| 60 | C3×Q8 | — | 2 | 6 | no | — |
| 61 | C3×D4 | — | 2 | 12 | no | — |
| 62 | C3×D4 | — | 2 | 12 | no | — |
| 63 | C3×D4 | — | 2 | 6 | no | — |
| 64 | C2×C3×C4 | — | 2 | 12 | no | — |
| 65 | C2×C3×C4 | — | 2 | 6 | no | — |
| 66 | C2×C3×C4 | — | 2 | 6 | no | — |
| 67 | C2×C3×C4 | C2×D4 | 1 | 6 | yes | — |
| 68 | C2×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 69 | C2×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 70 | C2×Dic3 | — | 2 | 6 | no | — |
| 71 | C2×Dic3 | — | 2 | 6 | no | — |
| 72 | Dic6 | — | 4 | 1 | no | — |
| 73 | Dic6 | — | 4 | 3 | no | — |
| 74 | #41 | — | 2 | 4 | no | — |
| 75 | #41 | — | 2 | 12 | no | — |
| 76 | #41 | — | 2 | 6 | no | — |
| 77 | #41 | — | 2 | 12 | no | — |
| 78 | #41 | — | 2 | 6 | no | — |
| 79 | C2×Q8 | — | 2 | 6 | no | — |
| 80 | C2×Q8 | — | 6 | 3 | no | — |
| 81 | C2×Q8 | — | 6 | 3 | no | — |
| 82 | C2×Q8 | #67 | 1 | 3 | yes | — |
| 83 | C2×D4 | — | 2 | 6 | no | — |
| 84 | C2×D4 | — | 2 | 12 | no | — |
| 85 | C2×D4 | #67 | 1 | 3 | yes | — |
| 86 | C2×D4 | C22×D3 | 1 | 6 | yes | — |
| 87 | C22×C4 | — | 2 | 4 | no | — |
| 88 | C22×C4 | — | 3 | 2 | no | — |
| 89 | C22×C4 | #67 | 1 | 6 | yes | — |
| 90 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 91 | C4⋊C4 | — | 6 | 3 | no | — |
| 92 | C4⋊C4 | — | 6 | 12 | no | — |
| 93 | C22⋊C4 | — | 3 | 6 | no | — |
| 94 | C22⋊C4 | — | 6 | 12 | no | — |
| 95 | C42 | — | 6 | 3 | no | — |
| 96 | C22×C3 | — | 2 | 6 | no | — |
| 97 | C22×C3 | — | 4 | 1 | no | — |
| 98 | C22×C3 | — | 4 | 6 | no | — |
| 99 | C22×C3 | — | 8 | 4 | no | — |
| 100 | C22×C3 | C22×D4 | 1 | 3 | yes | — |
| 101 | C3×C4 | — | 2 | 6 | no | — |
| 102 | C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 103 | C3×C4 | #120 | 1 | 3 | yes | — |
| 104 | Dic3 | — | 4 | 6 | no | — |
| 105 | Dic3 | — | 4 | 2 | no | — |
| 106 | C23 | — | 2 | 4 | no | — |
| 107 | C23 | C2×#67 | 1 | 6 | yes | — |
| 108 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 109 | Q8 | — | 2 | 4 | no | — |
| 110 | Q8 | — | 2 | 6 | no | — |
| 111 | Q8 | — | 12 | 1 | no | — |
| 112 | Q8 | — | 12 | 3 | no | — |
| 113 | D4 | — | 2 | 12 | no | — |
| 114 | D4 | — | 2 | 12 | no | — |
| 115 | D4 | — | 2 | 6 | no | — |
| 116 | C2×C4 | — | 2 | 12 | no | — |
| 117 | C2×C4 | — | 2 | 6 | no | — |
| 118 | C2×C4 | — | 2 | 6 | no | — |
| 119 | C2×C4 | — | 6 | 6 | no | — |
| 120 | C2×C4 | — | 6 | 6 | no | — |
| 121 | C2×C4 | C2×#67 | 1 | 6 | yes | — |
| 122 | C2×C4 | C2×#67 | 1 | 3 | yes | — |
| 123 | C2×C4 | C2×#67 | 1 | 3 | yes | — |
| 124 | C2×C3 | — | 2 | 3 | no | — |
| 125 | C2×C3 | — | 4 | 6 | no | — |
| 126 | C2×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 127 | C22 | — | 2 | 6 | no | — |
| 128 | C22 | — | 4 | 1 | no | — |
| 129 | C22 | — | 4 | 6 | no | — |
| 130 | C22 | — | 8 | 4 | no | — |
| 131 | C22 | C22×#67 | 1 | 3 | yes | — |
| 132 | C4 | — | 2 | 6 | no | — |
| 133 | C4 | — | 12 | 6 | no | — |
| 134 | C4 | — | 12 | 2 | no | — |
| 135 | C4 | #953 | 1 | 1 | yes | characteristic |
| 136 | C4 | #953 | 1 | 3 | yes | — |
| 137 | C3 | #3028 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | C2 | — | 2 | 3 | no | — |
| 139 | C2 | — | 4 | 6 | no | — |
| 140 | C2 | C2×#953 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5825 | C2 | semidirect |
| 3. #5243 | C2 | semidirect |
| 4. #5240 | C2 | semidirect |
| 11. C22×Dic6 | C22 | semidirect |
| 14. #935 | C22 | semidirect |
| 15. #926 | C22 | semidirect |
| 17. #834 | C22 | semidirect |
| 137. C3 | #3028 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585 | 2 |
| #5825 | 2 |
| #5243 | 2 |
| #5240 | 2 |
| #3028 | 3 |