Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
199 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 36 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 | Q8×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5394 | C2 | 1 | 2 | yes | maximal |
| 3 | C4×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #5391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5389 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #3385 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | #946 | — | 4 | 1 | no | maximal |
| 9 | C4×SL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 10 | C4×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 11 | #859 | C22 | 1 | 1 | yes | characteristic |
| 12 | #859 | C22 | 1 | 2 | yes | — |
| 13 | Q82 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #554 | — | 3 | 1 | no | — |
| 15 | #501 | — | 3 | 2 | no | — |
| 16 | #500 | — | 3 | 1 | no | — |
| 17 | #479 | — | 3 | 2 | no | — |
| 18 | #477 | — | 3 | 1 | no | — |
| 19 | #476 | — | 3 | 1 | no | — |
| 20 | #474 | — | 3 | 2 | no | — |
| 21 | #446 | — | 3 | 1 | no | — |
| 22 | #440 | — | 3 | 2 | no | — |
| 23 | C4×SD16 | — | 3 | 1 | no | — |
| 24 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 25 | C2×C4×D3 | — | 4 | 1 | no | — |
| 26 | C2×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 27 | GL(2,𝔽3) | Q8 | 1 | 2 | yes | complemented |
| 28 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 29 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 30 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 31 | C2×SD16 | — | 3 | 1 | no | — |
| 32 | C4⋊Q8 | — | 3 | 2 | no | — |
| 33 | C4⋊Q8 | — | 3 | 1 | no | — |
| 34 | C4⋊Q8 | — | 3 | 1 | no | — |
| 35 | #125 | — | 3 | 2 | no | — |
| 36 | C22⋊Q8 | — | 6 | 2 | no | — |
| 37 | C22⋊Q8 | — | 6 | 1 | no | — |
| 38 | C4×Q8 | — | 3 | 1 | no | — |
| 39 | C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 40 | C4×Q8 | C2×D3 | 1 | 2 | yes | — |
| 41 | C4×D4 | — | 3 | 1 | no | — |
| 42 | C4×D4 | — | 3 | 2 | no | — |
| 43 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 44 | #107 | — | 3 | 2 | no | — |
| 45 | #107 | — | 3 | 2 | no | — |
| 46 | #107 | — | 3 | 2 | no | — |
| 47 | #106 | — | 3 | 1 | no | — |
| 48 | #106 | — | 3 | 2 | no | — |
| 49 | C4⋊C8 | — | 3 | 2 | no | — |
| 50 | C4⋊C8 | — | 3 | 1 | no | — |
| 51 | #103 | — | 3 | 1 | no | — |
| 52 | #103 | — | 3 | 2 | no | — |
| 53 | #102 | — | 3 | 1 | no | — |
| 54 | #102 | — | 3 | 2 | no | — |
| 55 | #97 | — | 3 | 2 | no | — |
| 56 | C4×C8 | — | 3 | 1 | no | — |
| 57 | C22×D3 | — | 4 | 1 | no | — |
| 58 | C3×Q8 | — | 8 | 2 | no | — |
| 59 | C2×C3×C4 | — | 4 | 1 | no | — |
| 60 | C2×C3×C4 | — | 4 | 2 | no | — |
| 61 | C2×Dic3 | — | 4 | 1 | no | — |
| 62 | C2×Dic3 | — | 4 | 2 | no | — |
| 63 | C4×D3 | — | 8 | 2 | no | — |
| 64 | SL(2,𝔽3) | C2×Q8 | 1 | 1 | yes | characteristic |
| 65 | C2×Q8 | — | 6 | 1 | no | — |
| 66 | C2×Q8 | — | 6 | 1 | no | — |
| 67 | C2×Q8 | S4 | 1 | 1 | yes | characteristic |
| 68 | C2×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 69 | C2×D4 | — | 3 | 1 | no | — |
| 70 | C22×C4 | — | 6 | 2 | no | — |
| 71 | C22×C4 | — | 6 | 1 | no | — |
| 72 | SD16 | — | 3 | 2 | no | — |
| 73 | SD16 | — | 3 | 2 | no | — |
| 74 | C2×C8 | — | 3 | 1 | no | — |
| 75 | C2×C8 | — | 3 | 1 | no | — |
| 76 | C2×C8 | — | 3 | 2 | no | — |
| 77 | C4⋊C4 | — | 3 | 1 | no | — |
| 78 | C4⋊C4 | — | 3 | 2 | no | — |
| 79 | C4⋊C4 | — | 3 | 1 | no | — |
| 80 | C4⋊C4 | — | 3 | 2 | no | — |
| 81 | C4⋊C4 | — | 3 | 1 | no | — |
| 82 | C4⋊C4 | — | 3 | 2 | no | — |
| 83 | C4⋊C4 | — | 6 | 2 | no | — |
| 84 | C4⋊C4 | — | 6 | 1 | no | — |
| 85 | C4⋊C4 | — | 6 | 2 | no | — |
| 86 | C22⋊C4 | — | 6 | 1 | no | — |
| 87 | C22⋊C4 | — | 6 | 2 | no | — |
| 88 | C42 | — | 3 | 1 | no | — |
| 89 | C42 | — | 3 | 2 | no | — |
| 90 | C22×C3 | — | 4 | 1 | no | — |
| 91 | C2×D3 | — | 4 | 2 | no | — |
| 92 | C2×D3 | — | 8 | 1 | no | — |
| 93 | C2×D3 | — | 8 | 1 | no | — |
| 94 | C3×C4 | — | 4 | 2 | no | — |
| 95 | C3×C4 | — | 8 | 2 | no | — |
| 96 | Dic3 | — | 8 | 1 | no | — |
| 97 | Dic3 | — | 8 | 2 | no | — |
| 98 | C23 | — | 6 | 1 | no | — |
| 99 | Q8 | — | 2 | 2 | no | — |
| 100 | Q8 | — | 3 | 1 | no | — |
| 101 | Q8 | — | 24 | 1 | no | — |
| 102 | Q8 | — | 24 | 1 | no | — |
| 103 | Q8 | D3×Q8 | 1 | 1 | yes | characteristic |
| 104 | D4 | — | 3 | 2 | no | — |
| 105 | D4 | — | 6 | 1 | no | — |
| 106 | C2×C4 | — | 3 | 1 | no | — |
| 107 | C2×C4 | — | 3 | 1 | no | — |
| 108 | C2×C4 | — | 3 | 2 | no | — |
| 109 | C2×C4 | — | 6 | 1 | no | — |
| 110 | C2×C4 | — | 6 | 2 | no | — |
| 111 | C2×C4 | — | 12 | 2 | no | — |
| 112 | C2×C4 | — | 24 | 2 | no | — |
| 113 | C2×C4 | C2×S4 | 1 | 2 | yes | — |
| 114 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 115 | C8 | — | 3 | 2 | no | — |
| 116 | C8 | — | 6 | 1 | no | — |
| 117 | C8 | — | 6 | 2 | no | — |
| 118 | C2×C3 | — | 4 | 1 | no | — |
| 119 | C2×C3 | — | 4 | 1 | no | — |
| 120 | C2×C3 | — | 4 | 1 | no | — |
| 121 | D3 | — | 8 | 2 | no | — |
| 122 | C22 | — | 6 | 2 | no | — |
| 123 | C22 | — | 12 | 1 | no | — |
| 124 | C22 | — | 12 | 1 | no | — |
| 125 | C22 | C22×S4 | 1 | 1 | yes | characteristic |
| 126 | C4 | — | 2 | 2 | no | — |
| 127 | C4 | — | 3 | 1 | no | — |
| 128 | C4 | — | 3 | 1 | no | — |
| 129 | C4 | — | 6 | 2 | no | — |
| 130 | C4 | — | 6 | 1 | no | — |
| 131 | C4 | — | 12 | 1 | no | — |
| 132 | C4 | — | 12 | 2 | no | — |
| 133 | C4 | #983 | 1 | 2 | yes | — |
| 134 | C3 | — | 4 | 1 | no | Sylow |
| 135 | C2 | — | 12 | 2 | no | — |
| 136 | C2 | C2×#983 | 1 | 1 | yes | characteristic |
| 137 | C2 | Q8×S4 | 1 | 1 | yes | characteristic |
| 138 | C2 | #5925 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×SL(2,𝔽3) | C2 | semidirect |
| 4. #5391 | C2 | semidirect |
| 5. #5389 | C2 | semidirect |
| 13. Q82 | D3 | semidirect |
| 27. GL(2,𝔽3) | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×SL(2,𝔽3) | 2 |
| #5394 | 2 |
| C4×GL(2,𝔽3) | 2 |
| #5391 | 2 |
| #5389 | 2 |
| #3385 | 3 |
| #946 | 4 |