Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1402 conjugacy classes of subgroups; only the 275 normal subgroups are shown.
This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 275 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×#3458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #82269 | C2 | 1 | 3 | yes | maximal |
| 3 | #71897 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #71592 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #23469 | — | 3 | 1 | no | maximal, Sylow |
| 6 | C23×D12 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3×#544 | — | 2 | 6 | no | — |
| 8 | C3×#543 | C22 | 1 | 4 | yes | — |
| 9 | C3×#533 | — | 2 | 6 | no | — |
| 10 | C2×C3×#117 | C22 | 1 | 3 | yes | — |
| 11 | C2×#926 | C22 | 1 | 3 | yes | — |
| 12 | #4988 | — | 2 | 6 | no | — |
| 13 | #4955 | C22 | 1 | 12 | yes | — |
| 14 | #4939 | — | 2 | 6 | no | — |
| 15 | #4526 | C22 | 1 | 12 | yes | — |
| 16 | #3458 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | #2309 | — | 3 | 3 | no | — |
| 18 | #2029 | — | 3 | 8 | no | — |
| 19 | #1740 | — | 3 | 3 | no | — |
| 20 | C24×D3 | C23 | 1 | 2 | yes | — |
| 21 | C23×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 22 | C22×D12 | — | 2 | 3 | no | — |
| 23 | C22×D12 | — | 2 | 3 | no | — |
| 24 | C22×D12 | — | 2 | 6 | no | — |
| 25 | C22×D12 | Q8 | 1 | 4 | yes | complemented |
| 26 | C3×C4⋊Q8 | — | 2 | 12 | no | — |
| 27 | C3×#125 | — | 2 | 12 | no | — |
| 28 | C3×C22⋊Q8 | — | 2 | 24 | no | — |
| 29 | C3×C4×Q8 | — | 2 | 8 | no | — |
| 30 | C3×#117 | — | 2 | 6 | no | — |
| 31 | C3×#117 | — | 2 | 6 | no | — |
| 32 | C2×C3×C4⋊C4 | D4 | 1 | 6 | yes | — |
| 33 | C2×C3×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 34 | C2×C3×C42 | D4 | 1 | 6 | yes | — |
| 35 | C2×#217 | C23 | 1 | 6 | yes | — |
| 36 | C2×(C2×D3)⋊C4 | — | 2 | 24 | no | — |
| 37 | #926 | — | 2 | 12 | no | — |
| 38 | C2×C4⋊Dic3 | — | 2 | 3 | no | — |
| 39 | C2×C4×Dic3 | — | 2 | 3 | no | — |
| 40 | C23×D4 | — | 3 | 1 | no | — |
| 41 | #544 | — | 2 | 6 | no | — |
| 42 | #543 | C2×D3 | 1 | 4 | yes | — |
| 43 | #533 | — | 2 | 6 | no | — |
| 44 | C2×#117 | — | 3 | 3 | no | — |
| 45 | C2×#117 | C2×D3 | 1 | 3 | yes | — |
| 46 | #390 | — | 6 | 6 | no | — |
| 47 | #390 | — | 6 | 6 | no | — |
| 48 | #379 | — | 3 | 12 | no | — |
| 49 | #342 | — | 3 | 12 | no | — |
| 50 | C24×C3 | C24 | 1 | 1 | yes | characteristic |
| 51 | C23×D3 | — | 2 | 6 | no | — |
| 52 | C23×D3 | — | 2 | 6 | no | — |
| 53 | C23×D3 | — | 8 | 4 | no | — |
| 54 | C23×D3 | C2×Q8 | 1 | 4 | yes | — |
| 55 | C2×C3×Q8 | — | 2 | 8 | no | — |
| 56 | C22×C3×C4 | C2×D4 | 1 | 12 | yes | — |
| 57 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 58 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 59 | C22×C3×C4 | C2×Q8 | 1 | 2 | yes | — |
| 60 | C22×C3×C4 | #41 | 1 | 6 | yes | — |
| 61 | C22×Dic3 | — | 2 | 6 | no | — |
| 62 | C2×D12 | — | 2 | 12 | no | — |
| 63 | C2×D12 | — | 2 | 12 | no | — |
| 64 | C2×D12 | — | 4 | 1 | no | — |
| 65 | C2×D12 | — | 4 | 3 | no | — |
| 66 | C2×D12 | — | 4 | 6 | no | — |
| 67 | C2×D12 | — | 4 | 6 | no | — |
| 68 | C3×C4⋊C4 | — | 2 | 24 | no | — |
| 69 | C3×C4⋊C4 | — | 2 | 12 | no | — |
| 70 | C3×C22⋊C4 | — | 2 | 12 | no | — |
| 71 | C3×C42 | — | 2 | 6 | no | — |
| 72 | C3×C42 | — | 2 | 6 | no | — |
| 73 | #217 | — | 2 | 12 | no | — |
| 74 | (C2×D3)⋊C4 | — | 8 | 24 | no | — |
| 75 | C4⋊Dic3 | — | 4 | 6 | no | — |
| 76 | C4×Dic3 | — | 4 | 6 | no | — |
| 77 | C25 | — | 3 | 2 | no | — |
| 78 | C22×D4 | — | 3 | 4 | no | — |
| 79 | C22×D4 | — | 6 | 6 | no | — |
| 80 | C22×D4 | — | 6 | 3 | no | — |
| 81 | C22×D4 | — | 6 | 3 | no | — |
| 82 | C23×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 83 | C4⋊Q8 | — | 2 | 12 | no | — |
| 84 | #125 | — | 2 | 12 | no | — |
| 85 | C22⋊Q8 | — | 2 | 24 | no | — |
| 86 | C4×Q8 | — | 2 | 8 | no | — |
| 87 | #117 | — | 2 | 6 | no | — |
| 88 | #117 | — | 2 | 6 | no | — |
| 89 | #117 | — | 6 | 12 | no | — |
| 90 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 91 | C2×C4⋊C4 | #67 | 1 | 6 | yes | — |
| 92 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 93 | C2×C22⋊C4 | — | 6 | 24 | no | — |
| 94 | C2×C22⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 95 | C2×C42 | — | 6 | 3 | no | — |
| 96 | C2×C42 | #67 | 1 | 6 | yes | — |
| 97 | C23×C3 | — | 4 | 2 | no | — |
| 98 | C23×C3 | C22×D4 | 1 | 3 | yes | — |
| 99 | C23×C3 | C22×Q8 | 1 | 1 | yes | characteristic |
| 100 | C23×C3 | C2×#41 | 1 | 3 | yes | — |
| 101 | C22×D3 | — | 2 | 12 | no | — |
| 102 | C22×D3 | — | 4 | 24 | no | — |
| 103 | C22×D3 | — | 4 | 2 | no | — |
| 104 | C22×D3 | — | 4 | 6 | no | — |
| 105 | C22×D3 | — | 8 | 4 | no | — |
| 106 | C22×D3 | — | 8 | 12 | no | — |
| 107 | C3×Q8 | — | 8 | 8 | no | — |
| 108 | C2×C3×C4 | — | 2 | 12 | no | — |
| 109 | C2×C3×C4 | — | 4 | 12 | no | — |
| 110 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 111 | C2×C3×C4 | #120 | 1 | 3 | yes | — |
| 112 | C2×C3×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 113 | C2×C3×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 114 | C2×C3×C4 | #124 | 1 | 6 | yes | — |
| 115 | C2×C3×C4 | #124 | 1 | 6 | yes | — |
| 116 | C2×Dic3 | — | 4 | 12 | no | — |
| 117 | C2×Dic3 | — | 4 | 6 | no | — |
| 118 | D12 | — | 4 | 4 | no | — |
| 119 | D12 | — | 4 | 12 | no | — |
| 120 | C24 | — | 3 | 4 | no | — |
| 121 | C24 | — | 6 | 6 | no | — |
| 122 | C24 | — | 6 | 6 | no | — |
| 123 | C24 | — | 24 | 4 | no | — |
| 124 | C24 | C23×D3 | 1 | 1 | yes | characteristic |
| 125 | C2×Q8 | — | 2 | 8 | no | — |
| 126 | C2×D4 | — | 6 | 12 | no | — |
| 127 | C2×D4 | — | 6 | 12 | no | — |
| 128 | C2×D4 | — | 12 | 6 | no | — |
| 129 | C2×D4 | — | 12 | 6 | no | — |
| 130 | C2×D4 | — | 12 | 1 | no | — |
| 131 | C2×D4 | — | 12 | 3 | no | — |
| 132 | C22×C4 | — | 6 | 6 | no | — |
| 133 | C22×C4 | D3×Q8 | 1 | 2 | yes | — |
| 134 | C22×C4 | #239 | 1 | 6 | yes | — |
| 135 | C22×C4 | C2×#67 | 1 | 12 | yes | — |
| 136 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 137 | C22×C4 | C2×#67 | 1 | 3 | yes | — |
| 138 | C4⋊C4 | — | 2 | 24 | no | — |
| 139 | C4⋊C4 | — | 2 | 12 | no | — |
| 140 | C4⋊C4 | — | 12 | 6 | no | — |
| 141 | C22⋊C4 | — | 2 | 12 | no | — |
| 142 | C22⋊C4 | — | 6 | 12 | no | — |
| 143 | C22⋊C4 | — | 24 | 24 | no | — |
| 144 | C42 | — | 2 | 6 | no | — |
| 145 | C42 | — | 2 | 6 | no | — |
| 146 | C42 | — | 12 | 6 | no | — |
| 147 | C22×C3 | — | 2 | 6 | no | — |
| 148 | C22×C3 | — | 4 | 1 | no | — |
| 149 | C22×C3 | — | 4 | 3 | no | — |
| 150 | C22×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 151 | C22×C3 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 152 | C22×C3 | C2×#124 | 1 | 3 | yes | — |
| 153 | C2×D3 | — | 4 | 24 | no | — |
| 154 | C2×D3 | — | 4 | 4 | no | — |
| 155 | C2×D3 | — | 8 | 12 | no | — |
| 156 | C2×D3 | — | 8 | 4 | no | — |
| 157 | C3×C4 | — | 4 | 12 | no | — |
| 158 | C3×C4 | #393 | 1 | 2 | yes | — |
| 159 | C3×C4 | #393 | 1 | 6 | yes | — |
| 160 | Dic3 | — | 8 | 6 | no | — |
| 161 | C23 | — | 4 | 2 | no | — |
| 162 | C23 | — | 6 | 12 | no | — |
| 163 | C23 | — | 12 | 24 | no | — |
| 164 | C23 | — | 12 | 6 | no | — |
| 165 | C23 | — | 12 | 2 | no | — |
| 166 | C23 | — | 24 | 4 | no | — |
| 167 | C23 | — | 24 | 12 | no | — |
| 168 | C23 | C2×D3×Q8 | 1 | 1 | yes | characteristic |
| 169 | C23 | C2×#239 | 1 | 3 | yes | — |
| 170 | C23 | C22×#67 | 1 | 3 | yes | — |
| 171 | Q8 | — | 8 | 8 | no | — |
| 172 | D4 | — | 12 | 12 | no | — |
| 173 | D4 | — | 12 | 4 | no | — |
| 174 | C2×C4 | — | 2 | 12 | no | — |
| 175 | C2×C4 | — | 4 | 12 | no | — |
| 176 | C2×C4 | — | 12 | 12 | no | — |
| 177 | C2×C4 | — | 12 | 6 | no | — |
| 178 | C2×C4 | #946 | 1 | 6 | yes | — |
| 179 | C2×C4 | #946 | 1 | 6 | yes | — |
| 180 | C2×C4 | #947 | 1 | 6 | yes | — |
| 181 | C2×C4 | #947 | 1 | 6 | yes | — |
| 182 | C2×C4 | #953 | 1 | 1 | yes | characteristic |
| 183 | C2×C4 | #953 | 1 | 3 | yes | — |
| 184 | C2×C3 | — | 2 | 3 | no | — |
| 185 | C2×C3 | — | 2 | 3 | no | — |
| 186 | C2×C3 | C2×#393 | 1 | 1 | yes | characteristic |
| 187 | C2×C3 | #3027 | 1 | 2 | yes | — |
| 188 | D3 | — | 8 | 4 | no | — |
| 189 | C22 | — | 2 | 6 | no | — |
| 190 | C22 | — | 4 | 1 | no | — |
| 191 | C22 | — | 4 | 3 | no | — |
| 192 | C22 | — | 12 | 24 | no | — |
| 193 | C22 | — | 12 | 4 | no | — |
| 194 | C22 | — | 24 | 12 | no | — |
| 195 | C22 | — | 24 | 4 | no | — |
| 196 | C22 | C2×#946 | 1 | 3 | yes | — |
| 197 | C22 | C2×#947 | 1 | 3 | yes | — |
| 198 | C22 | C2×#953 | 1 | 1 | yes | characteristic |
| 199 | C4 | — | 4 | 12 | no | — |
| 200 | C4 | — | 24 | 6 | no | — |
| 201 | C4 | #5231 | 1 | 6 | yes | — |
| 202 | C4 | #5231 | 1 | 2 | yes | — |
| 203 | C3 | #23469 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 204 | C2 | — | 2 | 3 | no | — |
| 205 | C2 | — | 2 | 3 | no | — |
| 206 | C2 | — | 24 | 4 | no | — |
| 207 | C2 | C2×#5231 | 1 | 1 | yes | characteristic |
| 208 | C2 | #84126 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3458 | C2 | semidirect |
| 3. #71897 | C2 | semidirect |
| 4. #71592 | C2 | semidirect |
| 16. #3458 | D3 | semidirect |
| 25. C22×D12 | Q8 | semidirect |
| 203. C3 | #23469 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3458 | 2 |
| #82269 | 2 |
| #71897 | 2 |
| #71592 | 2 |
| #23469 | 3 |