Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
139 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 22 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 | D4×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | SL(2,𝔽3)⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #1627 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C22×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 6 | C2×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 7 | C2×#213 | — | 4 | 1 | no | maximal |
| 8 | C4×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 9 | D4×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #486 | — | 3 | 1 | no | — |
| 11 | #460 | — | 3 | 1 | no | — |
| 12 | #459 | — | 3 | 1 | no | — |
| 13 | #332 | — | 3 | 1 | no | — |
| 14 | Q8⋊C8 | — | 3 | 1 | no | — |
| 15 | D4⋊C8 | — | 3 | 1 | no | — |
| 16 | C2×C3×D4 | — | 4 | 1 | no | — |
| 17 | C2×D12 | — | 4 | 1 | no | — |
| 18 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 19 | C2×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 20 | GL(2,𝔽3) | — | 4 | 1 | no | — |
| 21 | #213 | — | 4 | 4 | no | — |
| 22 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 23 | C22×Q8 | — | 2 | 1 | no | — |
| 24 | C2×SD16 | — | 6 | 1 | no | — |
| 25 | C2×D8 | — | 6 | 1 | no | — |
| 26 | C4⋊Q8 | — | 3 | 1 | no | — |
| 27 | #127 | — | 3 | 1 | no | — |
| 28 | C22⋊Q8 | — | 6 | 1 | no | — |
| 29 | C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 30 | C4×D4 | — | 3 | 1 | no | — |
| 31 | C4⋊C8 | — | 3 | 1 | no | — |
| 32 | C4⋊C8 | — | 3 | 1 | no | — |
| 33 | #102 | — | 6 | 1 | no | — |
| 34 | #102 | — | 6 | 1 | no | — |
| 35 | #102 | — | 6 | 2 | no | — |
| 36 | C4×C8 | — | 3 | 1 | no | — |
| 37 | C23×C3 | — | 8 | 1 | no | — |
| 38 | C22×D3 | — | 8 | 1 | no | — |
| 39 | C3×D4 | — | 4 | 2 | no | — |
| 40 | C3×D4 | — | 8 | 1 | no | — |
| 41 | C2×C3×C4 | — | 4 | 1 | no | — |
| 42 | D12 | — | 4 | 2 | no | — |
| 43 | D12 | — | 8 | 1 | no | — |
| 44 | SL(2,𝔽3) | D8 | 1 | 1 | yes | characteristic, complemented |
| 45 | C3⋊C8 | — | 4 | 2 | no | — |
| 46 | C2×Q8 | — | 4 | 1 | no | — |
| 47 | C2×Q8 | — | 6 | 1 | no | — |
| 48 | C2×Q8 | — | 12 | 1 | no | — |
| 49 | C2×Q8 | #67 | 1 | 1 | yes | characteristic |
| 50 | C2×D4 | — | 6 | 1 | no | — |
| 51 | C2×D4 | — | 6 | 1 | no | — |
| 52 | C2×D4 | — | 6 | 1 | no | — |
| 53 | C2×D4 | S4 | 1 | 1 | yes | characteristic |
| 54 | C22×C4 | — | 6 | 1 | no | — |
| 55 | SD16 | — | 12 | 1 | no | — |
| 56 | SD16 | — | 12 | 1 | no | — |
| 57 | D8 | — | 12 | 2 | no | — |
| 58 | C2×C8 | — | 6 | 1 | no | — |
| 59 | C2×C8 | — | 6 | 1 | no | — |
| 60 | C2×C8 | — | 6 | 1 | no | — |
| 61 | C4⋊C4 | — | 3 | 1 | no | — |
| 62 | C4⋊C4 | — | 3 | 2 | no | — |
| 63 | C4⋊C4 | — | 3 | 1 | no | — |
| 64 | C22⋊C4 | — | 6 | 1 | no | — |
| 65 | C42 | — | 3 | 1 | no | — |
| 66 | C22×C3 | — | 4 | 1 | no | — |
| 67 | C22×C3 | — | 8 | 2 | no | — |
| 68 | C22×C3 | — | 16 | 1 | no | — |
| 69 | C22×C3 | — | 16 | 1 | no | — |
| 70 | C2×D3 | — | 8 | 2 | no | — |
| 71 | C2×D3 | — | 16 | 1 | no | — |
| 72 | C2×D3 | — | 16 | 1 | no | — |
| 73 | C3×C4 | — | 4 | 1 | no | — |
| 74 | C3×C4 | — | 4 | 1 | no | — |
| 75 | C23 | — | 2 | 1 | no | — |
| 76 | C23 | — | 12 | 1 | no | — |
| 77 | Q8 | — | 3 | 1 | no | — |
| 78 | Q8 | — | 12 | 1 | no | — |
| 79 | Q8 | — | 12 | 1 | no | — |
| 80 | Q8 | #213 | 1 | 1 | yes | characteristic, complemented |
| 81 | D4 | — | 2 | 1 | no | — |
| 82 | D4 | — | 12 | 1 | no | — |
| 83 | D4 | — | 12 | 1 | no | — |
| 84 | D4 | — | 12 | 2 | no | — |
| 85 | D4 | — | 12 | 1 | no | — |
| 86 | D4 | — | 12 | 1 | no | — |
| 87 | D4 | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 88 | C2×C4 | — | 3 | 1 | no | — |
| 89 | C2×C4 | — | 3 | 1 | no | — |
| 90 | C2×C4 | — | 6 | 1 | no | — |
| 91 | C2×C4 | — | 12 | 1 | no | — |
| 92 | C2×C4 | — | 12 | 1 | no | — |
| 93 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 94 | C8 | — | 12 | 1 | no | — |
| 95 | C8 | — | 12 | 1 | no | — |
| 96 | C8 | — | 12 | 1 | no | — |
| 97 | C2×C3 | — | 4 | 1 | no | — |
| 98 | C2×C3 | — | 4 | 1 | no | — |
| 99 | C2×C3 | — | 4 | 1 | no | — |
| 100 | C2×C3 | — | 16 | 2 | no | — |
| 101 | D3 | — | 16 | 2 | no | — |
| 102 | C22 | — | 2 | 2 | no | — |
| 103 | C22 | — | 4 | 1 | no | — |
| 104 | C22 | — | 4 | 1 | no | — |
| 105 | C22 | — | 24 | 1 | no | — |
| 106 | C22 | — | 24 | 1 | no | — |
| 107 | C22 | — | 24 | 1 | no | — |
| 108 | C22 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 109 | C4 | — | 3 | 1 | no | — |
| 110 | C4 | — | 3 | 1 | no | — |
| 111 | C4 | — | 6 | 1 | no | — |
| 112 | C4 | — | 12 | 1 | no | — |
| 113 | C4 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 114 | C4 | #983 | 1 | 1 | yes | characteristic |
| 115 | C3 | — | 4 | 1 | no | Sylow |
| 116 | C2 | — | 4 | 2 | no | — |
| 117 | C2 | — | 48 | 1 | no | — |
| 118 | C2 | #5415 | 1 | 1 | yes | characteristic |
| 119 | C2 | #5421 | 1 | 1 | yes | characteristic |
| 120 | C2 | #5431 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×SL(2,𝔽3) | C2 | semidirect |
| 2. C4⋊GL(2,𝔽3) | C2 | semidirect |
| 3. SL(2,𝔽3)⋊C8 | C2 | semidirect |
| 9. D4×Q8 | D3 | semidirect |
| 44. SL(2,𝔽3) | D8 | semidirect |
| 80. Q8 | #213 | semidirect |
| 87. D4 | GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×SL(2,𝔽3) | 2 |
| C4⋊GL(2,𝔽3) | 2 |
| SL(2,𝔽3)⋊C8 | 2 |
| #1627 | 3 |
| C2×#213 | 4 |