Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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×#1304 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊#1304 | C2 | 1 | 3 | yes | maximal |
| 3 | C3⋊#1304 | C2 | 1 | 3 | yes | maximal |
| 4 | #10690 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C3×#379 | C22 | 1 | 3 | yes | — |
| 6 | #5247 | C22 | 1 | 1 | yes | characteristic |
| 7 | #5247 | C22 | 1 | 3 | yes | — |
| 8 | #1304 | — | 3 | 3 | no | — |
| 9 | #1304 | — | 3 | 3 | no | — |
| 10 | #1304 | D3 | 1 | 1 | yes | characteristic |
| 11 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×C3×C22⋊C4 | — | 2 | 3 | no | — |
| 13 | C2×C3×C22⋊C4 | — | 2 | 3 | no | — |
| 14 | C2×C3×C22⋊C4 | — | 2 | 3 | no | — |
| 15 | C2×#217 | — | 2 | 3 | no | — |
| 16 | C2×#217 | — | 2 | 3 | no | — |
| 17 | C2×#217 | — | 2 | 3 | no | — |
| 18 | C2×#217 | — | 2 | 3 | no | — |
| 19 | #379 | — | 3 | 1 | no | — |
| 20 | #379 | — | 3 | 3 | no | — |
| 21 | #379 | C2×D3 | 1 | 3 | yes | — |
| 22 | C24×C3 | — | 8 | 3 | no | — |
| 23 | C24×C3 | #41 | 1 | 3 | yes | — |
| 24 | C24×C3 | #41 | 1 | 3 | yes | — |
| 25 | C24×C3 | #41 | 1 | 1 | yes | characteristic |
| 26 | C22×C3×C4 | — | 4 | 3 | no | — |
| 27 | C22×Dic3 | — | 4 | 1 | no | — |
| 28 | C22×Dic3 | — | 4 | 3 | no | — |
| 29 | C3×C22⋊C4 | — | 8 | 3 | no | — |
| 30 | C3×C22⋊C4 | — | 8 | 3 | no | — |
| 31 | C3×C22⋊C4 | — | 8 | 3 | no | — |
| 32 | #217 | — | 8 | 3 | no | — |
| 33 | #217 | — | 8 | 3 | no | — |
| 34 | #217 | — | 8 | 3 | no | — |
| 35 | #217 | — | 8 | 3 | no | — |
| 36 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 37 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 38 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 39 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 40 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 41 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 42 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 43 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 44 | C23×C3 | — | 4 | 3 | no | — |
| 45 | C23×C3 | — | 4 | 3 | no | — |
| 46 | C23×C3 | — | 4 | 3 | no | — |
| 47 | C23×C3 | — | 4 | 3 | no | — |
| 48 | C23×C3 | — | 4 | 3 | no | — |
| 49 | C23×C3 | — | 4 | 3 | no | — |
| 50 | C23×C3 | — | 4 | 3 | no | — |
| 51 | C23×C3 | — | 8 | 8 | no | — |
| 52 | C23×C3 | #126 | 1 | 1 | yes | characteristic |
| 53 | C23×C3 | #126 | 1 | 3 | yes | — |
| 54 | C23×C3 | #126 | 1 | 3 | yes | — |
| 55 | C2×C3×C4 | — | 8 | 3 | no | — |
| 56 | C2×C3×C4 | — | 8 | 3 | no | — |
| 57 | C2×C3×C4 | — | 8 | 3 | no | — |
| 58 | C2×Dic3 | — | 8 | 3 | no | — |
| 59 | C2×Dic3 | — | 8 | 3 | no | — |
| 60 | C2×Dic3 | — | 8 | 3 | no | — |
| 61 | C2×Dic3 | — | 8 | 3 | no | — |
| 62 | C24 | — | 8 | 3 | no | — |
| 63 | C24 | #235 | 1 | 3 | yes | — |
| 64 | C24 | #237 | 1 | 3 | yes | — |
| 65 | C24 | #239 | 1 | 1 | yes | characteristic |
| 66 | C22×C4 | — | 4 | 3 | no | — |
| 67 | C22×C4 | — | 12 | 1 | no | — |
| 68 | C22×C4 | — | 12 | 3 | no | — |
| 69 | C22⋊C4 | — | 8 | 3 | no | — |
| 70 | C22⋊C4 | — | 8 | 3 | no | — |
| 71 | C22⋊C4 | — | 8 | 3 | no | — |
| 72 | C22⋊C4 | — | 24 | 3 | no | — |
| 73 | C22⋊C4 | — | 24 | 3 | no | — |
| 74 | C22⋊C4 | — | 24 | 3 | no | — |
| 75 | C22⋊C4 | — | 24 | 3 | no | — |
| 76 | C22×C3 | — | 2 | 3 | no | — |
| 77 | C22×C3 | — | 2 | 3 | no | — |
| 78 | C22×C3 | — | 2 | 3 | no | — |
| 79 | C22×C3 | — | 2 | 3 | no | — |
| 80 | C22×C3 | — | 2 | 3 | no | — |
| 81 | C22×C3 | — | 2 | 3 | no | — |
| 82 | C22×C3 | — | 2 | 3 | no | — |
| 83 | C22×C3 | — | 8 | 2 | no | — |
| 84 | C22×C3 | — | 8 | 6 | no | — |
| 85 | C22×C3 | — | 8 | 6 | no | — |
| 86 | C22×C3 | #401 | 1 | 1 | yes | characteristic |
| 87 | C3×C4 | — | 16 | 3 | no | — |
| 88 | Dic3 | — | 16 | 1 | no | — |
| 89 | Dic3 | — | 16 | 3 | no | — |
| 90 | C23 | — | 4 | 3 | no | — |
| 91 | C23 | — | 4 | 3 | no | — |
| 92 | C23 | — | 4 | 3 | no | — |
| 93 | C23 | — | 4 | 3 | no | — |
| 94 | C23 | — | 4 | 3 | no | — |
| 95 | C23 | — | 4 | 3 | no | — |
| 96 | C23 | — | 4 | 3 | no | — |
| 97 | C23 | — | 8 | 8 | no | — |
| 98 | C23 | #876 | 1 | 1 | yes | characteristic |
| 99 | C23 | #879 | 1 | 3 | yes | — |
| 100 | C23 | #898 | 1 | 3 | yes | — |
| 101 | C2×C4 | — | 8 | 3 | no | — |
| 102 | C2×C4 | — | 8 | 3 | no | — |
| 103 | C2×C4 | — | 8 | 3 | no | — |
| 104 | C2×C4 | — | 24 | 3 | no | — |
| 105 | C2×C4 | — | 24 | 3 | no | — |
| 106 | C2×C4 | — | 24 | 3 | no | — |
| 107 | C2×C4 | — | 24 | 3 | no | — |
| 108 | C2×C3 | — | 4 | 1 | no | — |
| 109 | C2×C3 | — | 4 | 3 | no | — |
| 110 | C2×C3 | — | 4 | 3 | no | — |
| 111 | C2×C3 | #2104 | 1 | 3 | yes | — |
| 112 | C22 | — | 2 | 3 | no | — |
| 113 | C22 | — | 2 | 3 | no | — |
| 114 | C22 | — | 2 | 3 | no | — |
| 115 | C22 | — | 2 | 3 | no | — |
| 116 | C22 | — | 2 | 3 | no | — |
| 117 | C22 | — | 2 | 3 | no | — |
| 118 | C22 | — | 2 | 3 | no | — |
| 119 | C22 | — | 8 | 6 | no | — |
| 120 | C22 | — | 8 | 6 | no | — |
| 121 | C22 | — | 8 | 2 | no | — |
| 122 | C22 | C3⋊#401 | 1 | 1 | yes | characteristic |
| 123 | C4 | — | 16 | 3 | no | — |
| 124 | C4 | — | 48 | 1 | no | — |
| 125 | C4 | — | 48 | 3 | no | — |
| 126 | C3 | #10690 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 127 | C2 | — | 4 | 3 | no | — |
| 128 | C2 | — | 4 | 1 | no | — |
| 129 | C2 | — | 4 | 3 | no | — |
| 130 | C2 | C3⋊#2104 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 126. C3 | #10690 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#1304 | 2 |
| C3⋊#1304 | 2 |
| #10690 | 3 |