Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
406 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 77 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 | #79321 | C2 | 1 | 3 | yes | maximal |
| 2 | #70837 | C2 | 1 | 3 | yes | maximal |
| 3 | C3×#1308 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10211 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #5470 | C22 | 1 | 3 | yes | — |
| 6 | C3×#432 | C22 | 1 | 3 | yes | — |
| 7 | #4937 | C22 | 1 | 1 | yes | characteristic |
| 8 | #4851 | — | 2 | 3 | no | — |
| 9 | #4850 | — | 2 | 3 | no | — |
| 10 | #4723 | — | 2 | 6 | no | — |
| 11 | #4703 | — | 2 | 3 | no | — |
| 12 | #4702 | — | 2 | 3 | no | — |
| 13 | #4467 | — | 2 | 6 | no | — |
| 14 | #3237 | — | 3 | 3 | no | — |
| 15 | #1515 | — | 3 | 3 | no | — |
| 16 | #1308 | D3 | 1 | 1 | yes | characteristic |
| 17 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 18 | #926 | — | 2 | 3 | no | — |
| 19 | #924 | — | 2 | 3 | no | — |
| 20 | #924 | — | 2 | 3 | no | — |
| 21 | #870 | — | 2 | 3 | no | — |
| 22 | #869 | D4 | 1 | 3 | yes | — |
| 23 | #869 | D4 | 1 | 3 | yes | — |
| 24 | C4×Dic6 | — | 2 | 3 | no | — |
| 25 | C4×Dic6 | — | 2 | 3 | no | — |
| 26 | C3×C4⋊C8 | — | 2 | 3 | no | — |
| 27 | C3×C22⋊C8 | D4 | 1 | 6 | yes | — |
| 28 | C3×#97 | — | 2 | 3 | no | — |
| 29 | C3⋊#95 | — | 2 | 3 | no | — |
| 30 | #818 | — | 2 | 6 | no | — |
| 31 | #817 | — | 2 | 6 | no | — |
| 32 | #816 | — | 2 | 6 | no | — |
| 33 | #816 | — | 2 | 6 | no | — |
| 34 | #533 | — | 3 | 3 | no | — |
| 35 | #501 | — | 6 | 3 | no | — |
| 36 | #491 | — | 6 | 3 | no | — |
| 37 | #485 | — | 6 | 6 | no | — |
| 38 | #477 | — | 6 | 3 | no | — |
| 39 | #475 | — | 6 | 3 | no | — |
| 40 | #432 | C2×D3 | 1 | 3 | yes | — |
| 41 | #383 | — | 3 | 1 | no | — |
| 42 | #328 | — | 6 | 6 | no | — |
| 43 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 44 | C22×Dic3 | — | 4 | 1 | no | — |
| 45 | C2×Dic6 | — | 2 | 3 | no | — |
| 46 | C2×Dic6 | — | 2 | 3 | no | — |
| 47 | C2×C3×C8 | — | 2 | 6 | no | — |
| 48 | C3×C42 | C2×D4 | 1 | 3 | yes | — |
| 49 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 50 | #217 | — | 4 | 3 | no | — |
| 51 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 52 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 53 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 54 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 55 | Dic3⋊C4 | — | 4 | 3 | no | — |
| 56 | Dic3⋊C4 | — | 4 | 3 | no | — |
| 57 | C4×Dic3 | — | 4 | 3 | no | — |
| 58 | C4⋊Q8 | — | 3 | 3 | no | — |
| 59 | C4⋊Q8 | — | 3 | 3 | no | — |
| 60 | #125 | — | 6 | 3 | no | — |
| 61 | C22⋊Q8 | — | 6 | 3 | no | — |
| 62 | C22⋊Q8 | — | 6 | 3 | no | — |
| 63 | C4×Q8 | — | 6 | 3 | no | — |
| 64 | C4×Q8 | — | 6 | 3 | no | — |
| 65 | #117 | — | 6 | 3 | no | — |
| 66 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 67 | #107 | — | 6 | 6 | no | — |
| 68 | #106 | — | 6 | 6 | no | — |
| 69 | C4⋊C8 | — | 2 | 3 | no | — |
| 70 | #103 | — | 6 | 6 | no | — |
| 71 | #103 | — | 6 | 6 | no | — |
| 72 | C22⋊C8 | D12 | 1 | 6 | yes | — |
| 73 | #97 | — | 2 | 3 | no | — |
| 74 | #95 | — | 6 | 3 | no | — |
| 75 | C23×C3 | #127 | 1 | 1 | yes | characteristic |
| 76 | C2×C3×C4 | — | 4 | 3 | no | — |
| 77 | C2×C3×C4 | #120 | 1 | 3 | yes | — |
| 78 | C2×C3×C4 | #121 | 1 | 3 | yes | — |
| 79 | C2×Dic3 | — | 4 | 3 | no | — |
| 80 | C2×Dic3 | — | 4 | 3 | no | — |
| 81 | C2×Dic3 | — | 8 | 3 | no | — |
| 82 | Dic6 | — | 4 | 6 | no | — |
| 83 | Dic6 | — | 8 | 3 | no | — |
| 84 | C3×C8 | — | 4 | 6 | no | — |
| 85 | C2×Q8 | — | 6 | 3 | no | — |
| 86 | C2×Q8 | — | 6 | 3 | no | — |
| 87 | C22×C4 | — | 12 | 1 | no | — |
| 88 | C22×C4 | C2×D12 | 1 | 3 | yes | — |
| 89 | C2×C8 | — | 2 | 6 | no | — |
| 90 | C4⋊C4 | — | 6 | 3 | no | — |
| 91 | C4⋊C4 | — | 6 | 3 | no | — |
| 92 | C4⋊C4 | — | 6 | 3 | no | — |
| 93 | C4⋊C4 | — | 6 | 3 | no | — |
| 94 | C4⋊C4 | — | 12 | 3 | no | — |
| 95 | C4⋊C4 | — | 12 | 3 | no | — |
| 96 | C22⋊C4 | — | 12 | 3 | no | — |
| 97 | C42 | — | 12 | 3 | no | — |
| 98 | C42 | D3×D4 | 1 | 3 | yes | — |
| 99 | C42 | #239 | 1 | 1 | yes | characteristic |
| 100 | C22×C3 | — | 2 | 3 | no | — |
| 101 | C22×C3 | #392 | 1 | 1 | yes | characteristic |
| 102 | C3×C4 | — | 2 | 3 | no | — |
| 103 | C3×C4 | — | 2 | 3 | no | — |
| 104 | Dic3 | — | 8 | 3 | no | — |
| 105 | Dic3 | — | 8 | 3 | no | — |
| 106 | Dic3 | — | 16 | 1 | no | — |
| 107 | C23 | #874 | 1 | 1 | yes | characteristic |
| 108 | Q8 | — | 12 | 6 | no | — |
| 109 | Q8 | — | 24 | 3 | no | — |
| 110 | C2×C4 | — | 4 | 3 | no | — |
| 111 | C2×C4 | — | 12 | 3 | no | — |
| 112 | C2×C4 | — | 12 | 3 | no | — |
| 113 | C2×C4 | — | 24 | 3 | no | — |
| 114 | C2×C4 | #882 | 1 | 3 | yes | — |
| 115 | C2×C4 | #895 | 1 | 3 | yes | — |
| 116 | C8 | — | 4 | 6 | no | — |
| 117 | C2×C3 | — | 4 | 1 | no | — |
| 118 | C2×C3 | #2009 | 1 | 3 | yes | — |
| 119 | C22 | — | 2 | 3 | no | — |
| 120 | C22 | #4671 | 1 | 1 | yes | characteristic |
| 121 | C4 | — | 2 | 3 | no | — |
| 122 | C4 | — | 2 | 3 | no | — |
| 123 | C4 | — | 24 | 3 | no | — |
| 124 | C4 | — | 24 | 3 | no | — |
| 125 | C4 | — | 48 | 1 | no | — |
| 126 | C3 | #10211 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 127 | C2 | — | 4 | 1 | no | — |
| 128 | C2 | #69031 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 126. C3 | #10211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #79321 | 2 |
| #70837 | 2 |
| C3×#1308 | 2 |
| #10211 | 3 |