Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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×#564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5700 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | #5699 | C2 | 1 | 10 | yes | maximal, complemented |
| 4 | #5691 | C2 | 1 | 10 | yes | maximal, complemented |
| 5 | #5688 | C2 | 1 | 5 | yes | maximal, complemented |
| 6 | #3531 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C3×#125 | C22 | 1 | 5 | yes | — |
| 8 | C3×#125 | C22 | 1 | 10 | yes | — |
| 9 | #898 | C22 | 1 | 20 | yes | — |
| 10 | #897 | C22 | 1 | 10 | yes | — |
| 11 | #896 | C22 | 1 | 20 | yes | — |
| 12 | #895 | C22 | 1 | 10 | yes | — |
| 13 | #894 | C22 | 1 | 20 | yes | — |
| 14 | #893 | C22 | 1 | 10 | yes | — |
| 15 | #892 | C22 | 1 | 10 | yes | — |
| 16 | #888 | C22 | 1 | 10 | yes | — |
| 17 | #887 | C22 | 1 | 10 | yes | — |
| 18 | #875 | C22 | 1 | 10 | yes | — |
| 19 | C42×D3 | C22 | 1 | 5 | yes | — |
| 20 | #870 | C22 | 1 | 5 | yes | — |
| 21 | #564 | D3 | 1 | 1 | yes | characteristic, complemented |
| 22 | #563 | — | 3 | 10 | no | — |
| 23 | #562 | — | 3 | 5 | no | — |
| 24 | #533 | — | 3 | 5 | no | — |
| 25 | #529 | — | 3 | 10 | no | — |
| 26 | C2×D12 | C23 | 1 | 10 | yes | — |
| 27 | C2×C4×D3 | C23 | 1 | 5 | yes | — |
| 28 | C2×C4×D3 | C23 | 1 | 10 | yes | — |
| 29 | C2×Dic6 | C23 | 1 | 10 | yes | — |
| 30 | C3×C4⋊C4 | C23 | 1 | 10 | yes | — |
| 31 | C3×C4⋊C4 | C23 | 1 | 10 | yes | — |
| 32 | C3×C4⋊C4 | C23 | 1 | 10 | yes | — |
| 33 | C3×C42 | C23 | 1 | 5 | yes | — |
| 34 | (C2×D3)⋊C4 | C23 | 1 | 20 | yes | — |
| 35 | (C2×D3)⋊C4 | C23 | 1 | 20 | yes | — |
| 36 | C4⋊Dic3 | C23 | 1 | 10 | yes | — |
| 37 | Dic3⋊C4 | C23 | 1 | 20 | yes | — |
| 38 | C4×Dic3 | C23 | 1 | 10 | yes | — |
| 39 | C4×Dic3 | C23 | 1 | 5 | yes | — |
| 40 | C4⋊Q8 | — | 3 | 10 | no | — |
| 41 | #126 | — | 3 | 20 | no | — |
| 42 | #125 | — | 3 | 5 | no | — |
| 43 | #125 | C2×D3 | 1 | 10 | yes | — |
| 44 | #125 | C2×D3 | 1 | 5 | yes | — |
| 45 | #124 | — | 3 | 10 | no | — |
| 46 | #123 | — | 3 | 20 | no | — |
| 47 | C22⋊Q8 | — | 3 | 10 | no | — |
| 48 | C22⋊Q8 | — | 3 | 20 | no | — |
| 49 | #121 | — | 3 | 10 | no | — |
| 50 | C4×Q8 | — | 3 | 10 | no | — |
| 51 | C4×D4 | — | 3 | 10 | no | — |
| 52 | #117 | — | 3 | 10 | no | — |
| 53 | C2×C42 | — | 3 | 5 | no | — |
| 54 | C22×D3 | C24 | 1 | 5 | yes | — |
| 55 | C2×C3×C4 | C24 | 1 | 10 | yes | — |
| 56 | C2×C3×C4 | C24 | 1 | 5 | yes | — |
| 57 | C2×Dic3 | C24 | 1 | 10 | yes | — |
| 58 | C2×Dic3 | C24 | 1 | 1 | yes | characteristic |
| 59 | D12 | — | 4 | 10 | no | — |
| 60 | C4×D3 | — | 2 | 10 | no | — |
| 61 | C4×D3 | — | 4 | 10 | no | — |
| 62 | Dic6 | — | 4 | 10 | no | — |
| 63 | C2×Q8 | — | 3 | 10 | no | — |
| 64 | C2×D4 | — | 3 | 10 | no | — |
| 65 | C22×C4 | — | 3 | 5 | no | — |
| 66 | C22×C4 | — | 3 | 10 | no | — |
| 67 | C4⋊C4 | — | 3 | 10 | no | — |
| 68 | C4⋊C4 | — | 3 | 20 | no | — |
| 69 | C4⋊C4 | C22×D3 | 1 | 10 | yes | — |
| 70 | C4⋊C4 | C22×D3 | 1 | 10 | yes | — |
| 71 | C4⋊C4 | C22×D3 | 1 | 10 | yes | — |
| 72 | C22⋊C4 | — | 3 | 20 | no | — |
| 73 | C22⋊C4 | — | 3 | 20 | no | — |
| 74 | C42 | — | 3 | 10 | no | — |
| 75 | C42 | — | 3 | 5 | no | — |
| 76 | C42 | C22×D3 | 1 | 5 | yes | — |
| 77 | C22×C3 | C25 | 1 | 1 | yes | characteristic |
| 78 | C2×D3 | — | 2 | 5 | no | — |
| 79 | C2×D3 | — | 2 | 10 | no | — |
| 80 | C3×C4 | — | 2 | 5 | no | — |
| 81 | C3×C4 | — | 2 | 10 | no | — |
| 82 | Dic3 | — | 2 | 10 | no | — |
| 83 | Dic3 | 2−1+4 | 1 | 2 | yes | — |
| 84 | C23 | — | 3 | 5 | no | — |
| 85 | Q8 | — | 12 | 10 | no | — |
| 86 | D4 | — | 12 | 10 | no | — |
| 87 | C2×C4 | — | 3 | 1 | no | — |
| 88 | C2×C4 | — | 3 | 10 | no | — |
| 89 | C2×C4 | — | 6 | 10 | no | — |
| 90 | C2×C4 | — | 12 | 10 | no | — |
| 91 | C2×C4 | C23×D3 | 1 | 5 | yes | — |
| 92 | C2×C4 | C23×D3 | 1 | 10 | yes | — |
| 93 | C2×C3 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 94 | C2×C3 | #585 | 1 | 2 | yes | — |
| 95 | D3 | — | 4 | 5 | no | — |
| 96 | C22 | — | 6 | 5 | no | — |
| 97 | C22 | — | 6 | 10 | no | — |
| 98 | C22 | C24×D3 | 1 | 1 | yes | characteristic |
| 99 | C4 | — | 2 | 5 | no | — |
| 100 | C4 | — | 2 | 10 | no | — |
| 101 | C4 | — | 3 | 2 | no | — |
| 102 | C4 | — | 6 | 10 | no | — |
| 103 | C3 | #3531 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 104 | C2 | — | 12 | 5 | no | — |
| 105 | C2 | D3×2−1+4 | 1 | 1 | yes | characteristic |
| 106 | C2 | #5968 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#564 | C2 |
| hypercenter | #564 | D3 |
| hypocenter | C3 | #3531 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#564 | C2 | semidirect |
| 2. #5700 | C2 | semidirect |
| 3. #5699 | C2 | semidirect |
| 4. #5691 | C2 | semidirect |
| 5. #5688 | C2 | semidirect |
| 21. #564 | D3 | semidirect |
| 103. C3 | #3531 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#564 | 2 |
| #5700 | 2 |
| #5699 | 2 |
| #5691 | 2 |
| #5688 | 2 |
| #3531 | 3 |