Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
164 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 25 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 | #71262 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#1410 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #67788 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #7527 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C3×#492 | C22 | 1 | 1 | yes | characteristic |
| 6 | #5088 | — | 2 | 1 | no | — |
| 7 | #5073 | — | 2 | 1 | no | — |
| 8 | #4695 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3×#356 | — | 2 | 1 | no | — |
| 10 | #4545 | — | 2 | 1 | no | — |
| 11 | #4455 | C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | #1721 | — | 3 | 1 | no | — |
| 13 | #1410 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #1409 | — | 3 | 1 | no | — |
| 15 | C2×C3×SD16 | — | 4 | 1 | no | — |
| 16 | C3×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 17 | C3×#127 | D4 | 1 | 1 | yes | characteristic |
| 18 | C2×#213 | — | 4 | 2 | no | — |
| 19 | C2×D24 | — | 4 | 1 | no | — |
| 20 | #874 | — | 2 | 1 | no | — |
| 21 | C3×#101 | — | 4 | 1 | no | — |
| 22 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | #834 | — | 4 | 1 | no | — |
| 24 | #809 | — | 4 | 2 | no | — |
| 25 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 26 | #493 | — | 3 | 1 | no | — |
| 27 | #493 | — | 6 | 1 | no | — |
| 28 | #492 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | #486 | — | 6 | 1 | no | — |
| 30 | #356 | — | 2 | 1 | no | — |
| 31 | #353 | — | 6 | 1 | no | — |
| 32 | #322 | — | 3 | 1 | no | — |
| 33 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 34 | C2×C3×D4 | — | 2 | 1 | no | — |
| 35 | C2×C3×D4 | — | 4 | 1 | no | — |
| 36 | C2×D12 | — | 4 | 1 | no | — |
| 37 | C2×D12 | — | 4 | 1 | no | — |
| 38 | C2×D12 | — | 4 | 1 | no | — |
| 39 | C3×SD16 | — | 8 | 2 | no | — |
| 40 | C3×M16 | — | 8 | 1 | no | — |
| 41 | C2×C3×C8 | — | 2 | 1 | no | — |
| 42 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 43 | C3×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 44 | #217 | — | 8 | 1 | no | — |
| 45 | #213 | — | 8 | 4 | no | — |
| 46 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 47 | D24 | — | 8 | 2 | no | — |
| 48 | C2×SD16 | — | 4 | 1 | no | — |
| 49 | C2×D8 | — | 12 | 1 | no | — |
| 50 | C2×D8 | — | 12 | 2 | no | — |
| 51 | C4⋊Q8 | #67 | 1 | 1 | yes | characteristic |
| 52 | #127 | — | 6 | 1 | no | — |
| 53 | #127 | D12 | 1 | 1 | yes | characteristic |
| 54 | #102 | — | 12 | 2 | no | — |
| 55 | #101 | — | 4 | 1 | no | — |
| 56 | #99 | — | 12 | 1 | no | — |
| 57 | C4×C8 | — | 6 | 1 | no | — |
| 58 | C4×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 59 | C23×C3 | — | 4 | 1 | no | — |
| 60 | C22×D3 | — | 8 | 1 | no | — |
| 61 | C3×Q8 | — | 8 | 1 | no | — |
| 62 | C3×D4 | — | 8 | 1 | no | — |
| 63 | C3×D4 | — | 8 | 2 | no | — |
| 64 | C2×C3×C4 | — | 2 | 1 | no | — |
| 65 | C2×C3×C4 | — | 4 | 1 | no | — |
| 66 | C2×C3×C4 | #99 | 1 | 1 | yes | characteristic |
| 67 | C2×Dic3 | — | 16 | 1 | no | — |
| 68 | D12 | — | 8 | 2 | no | — |
| 69 | D12 | — | 8 | 2 | no | — |
| 70 | D12 | — | 8 | 2 | no | — |
| 71 | C3×C8 | — | 2 | 2 | no | — |
| 72 | C3×C8 | — | 16 | 1 | no | — |
| 73 | C3⋊C8 | — | 4 | 2 | no | — |
| 74 | C2×Q8 | — | 2 | 1 | no | — |
| 75 | C2×D4 | — | 2 | 1 | no | — |
| 76 | C2×D4 | — | 4 | 1 | no | — |
| 77 | C2×D4 | — | 12 | 1 | no | — |
| 78 | C2×D4 | — | 12 | 1 | no | — |
| 79 | C2×D4 | — | 12 | 1 | no | — |
| 80 | SD16 | — | 8 | 2 | no | — |
| 81 | D8 | — | 24 | 2 | no | — |
| 82 | D8 | — | 24 | 4 | no | — |
| 83 | M16 | — | 8 | 1 | no | — |
| 84 | C2×C8 | — | 2 | 1 | no | — |
| 85 | C2×C8 | — | 6 | 2 | no | — |
| 86 | C4⋊C4 | — | 4 | 1 | no | — |
| 87 | C22⋊C4 | — | 24 | 1 | no | — |
| 88 | C42 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 89 | C22×C3 | — | 8 | 1 | no | — |
| 90 | C22×C3 | — | 16 | 1 | no | — |
| 91 | C22×C3 | #353 | 1 | 1 | yes | characteristic |
| 92 | C2×D3 | — | 16 | 1 | no | — |
| 93 | C2×D3 | — | 16 | 1 | no | — |
| 94 | C2×D3 | — | 16 | 1 | no | — |
| 95 | C3×C4 | — | 2 | 2 | no | — |
| 96 | C3×C4 | — | 2 | 1 | no | — |
| 97 | C3×C4 | — | 8 | 1 | no | — |
| 98 | Dic3 | — | 32 | 1 | no | — |
| 99 | C23 | — | 4 | 1 | no | — |
| 100 | C23 | — | 24 | 1 | no | — |
| 101 | Q8 | — | 8 | 1 | no | — |
| 102 | D4 | — | 8 | 1 | no | — |
| 103 | D4 | — | 8 | 2 | no | — |
| 104 | D4 | — | 24 | 2 | no | — |
| 105 | D4 | — | 24 | 2 | no | — |
| 106 | D4 | — | 24 | 2 | no | — |
| 107 | C2×C4 | — | 2 | 1 | no | — |
| 108 | C2×C4 | — | 4 | 1 | no | — |
| 109 | C2×C4 | — | 48 | 1 | no | — |
| 110 | C2×C4 | #806 | 1 | 1 | yes | characteristic |
| 111 | C8 | — | 2 | 2 | no | — |
| 112 | C8 | — | 12 | 2 | no | — |
| 113 | C8 | — | 16 | 1 | no | — |
| 114 | C2×C3 | — | 2 | 1 | no | — |
| 115 | C2×C3 | — | 16 | 1 | no | — |
| 116 | C2×C3 | #1408 | 1 | 1 | yes | characteristic |
| 117 | D3 | — | 32 | 1 | no | — |
| 118 | C22 | — | 8 | 1 | no | — |
| 119 | C22 | — | 16 | 1 | no | — |
| 120 | C22 | — | 48 | 1 | no | — |
| 121 | C22 | — | 48 | 1 | no | — |
| 122 | C22 | — | 48 | 1 | no | — |
| 123 | C22 | #4474 | 1 | 1 | yes | characteristic |
| 124 | C4 | — | 2 | 2 | no | — |
| 125 | C4 | — | 2 | 1 | no | — |
| 126 | C4 | — | 8 | 1 | no | — |
| 127 | C4 | — | 96 | 1 | no | — |
| 128 | C3 | #7527 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 129 | C2 | — | 2 | 1 | no | — |
| 130 | C2 | — | 16 | 1 | no | — |
| 131 | C2 | — | 96 | 1 | no | — |
| 132 | C2 | #67541 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#1410 | C2 | semidirect |
| 3. #67788 | C2 | semidirect |
| 8. #4695 | C4 | semidirect |
| 11. #4455 | C4 | semidirect |
| 13. #1410 | D3 | semidirect |
| 128. C3 | #7527 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #71262 | 2 |
| C3×#1410 | 2 |
| #67788 | 2 |
| #7527 | 3 |