Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
413 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 65 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 | #85284 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×C8×A4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #73111 | C2 | 1 | 2 | yes | maximal |
| 4 | #73110 | C2 | 1 | 2 | yes | maximal |
| 5 | #33208 | — | 3 | 1 | no | maximal, Sylow |
| 6 | C2×C8×A4 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C8×A4 | C22 | 1 | 1 | yes | characteristic |
| 8 | C42×A4 | C22 | 1 | 1 | yes | characteristic |
| 9 | C4⋊SL(2,ℤ/4) | C22 | 1 | 4 | yes | — |
| 10 | SL(2,ℤ/4)⋊C4 | — | 2 | 4 | no | — |
| 11 | #4683 | — | 4 | 1 | no | maximal |
| 12 | C2×#499 | — | 3 | 1 | no | — |
| 13 | C22×C4×C8 | D3 | 1 | 1 | yes | characteristic |
| 14 | #2851 | — | 3 | 2 | no | — |
| 15 | #1991 | — | 3 | 2 | no | — |
| 16 | #1942 | — | 3 | 2 | no | — |
| 17 | #1941 | — | 3 | 2 | no | — |
| 18 | #1846 | — | 3 | 1 | no | — |
| 19 | #1808 | — | 3 | 2 | no | — |
| 20 | #1807 | — | 3 | 2 | no | — |
| 21 | C2×C4×A4 | D4 | 1 | 2 | yes | — |
| 22 | C2×C4×A4 | C23 | 1 | 1 | yes | characteristic |
| 23 | C2×SL(2,ℤ/4) | — | 2 | 4 | no | — |
| 24 | #870 | — | 4 | 2 | no | — |
| 25 | C8×A4 | Q8 | 1 | 2 | yes | — |
| 26 | C8×A4 | Q8 | 1 | 2 | yes | — |
| 27 | C3×C4×C8 | — | 4 | 1 | no | — |
| 28 | #818 | — | 4 | 2 | no | — |
| 29 | #817 | — | 4 | 2 | no | — |
| 30 | C23×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 31 | C23×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 32 | C2×#125 | — | 3 | 2 | no | — |
| 33 | C22×C42 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | #499 | — | 6 | 1 | no | — |
| 35 | #499 | — | 6 | 1 | no | — |
| 36 | #499 | — | 6 | 1 | no | — |
| 37 | #499 | — | 6 | 1 | no | — |
| 38 | C2×#107 | — | 3 | 2 | no | — |
| 39 | C2×#107 | — | 3 | 2 | no | — |
| 40 | C2×#106 | — | 3 | 2 | no | — |
| 41 | C2×#106 | — | 3 | 2 | no | — |
| 42 | C2×C4×C8 | — | 3 | 1 | no | — |
| 43 | C2×C4×C8 | — | 3 | 1 | no | — |
| 44 | C2×C4×C8 | — | 3 | 1 | no | — |
| 45 | C2×C4×C8 | — | 3 | 1 | no | — |
| 46 | #389 | — | 3 | 4 | no | — |
| 47 | #387 | — | 6 | 4 | no | — |
| 48 | #385 | — | 6 | 4 | no | — |
| 49 | #382 | — | 3 | 2 | no | — |
| 50 | #380 | — | 3 | 4 | no | — |
| 51 | #340 | — | 3 | 4 | no | — |
| 52 | #340 | — | 3 | 4 | no | — |
| 53 | C22×A4 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C4×A4 | — | 2 | 2 | no | — |
| 55 | C4×A4 | C2×Q8 | 1 | 1 | yes | characteristic |
| 56 | C4×A4 | C2×Q8 | 1 | 1 | yes | characteristic |
| 57 | SL(2,ℤ/4) | — | 4 | 4 | no | — |
| 58 | C2×C3×C8 | — | 4 | 1 | no | — |
| 59 | C2×C3×C8 | — | 4 | 1 | no | — |
| 60 | C3×C42 | — | 4 | 1 | no | — |
| 61 | C4⋊Dic3 | — | 4 | 4 | no | — |
| 62 | Dic3⋊C4 | — | 8 | 4 | no | — |
| 63 | C23×C4 | D12 | 1 | 2 | yes | — |
| 64 | C23×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 65 | C22×C8 | — | 3 | 1 | no | — |
| 66 | C22×C8 | — | 3 | 1 | no | — |
| 67 | C22×C8 | — | 3 | 2 | no | — |
| 68 | C22×C8 | — | 3 | 2 | no | — |
| 69 | C22×C8 | — | 3 | 1 | no | — |
| 70 | C22×C8 | — | 3 | 1 | no | — |
| 71 | C22×C8 | Dic6 | 1 | 2 | yes | — |
| 72 | C22×C8 | Dic6 | 1 | 2 | yes | — |
| 73 | #125 | — | 6 | 4 | no | — |
| 74 | #125 | — | 6 | 2 | no | — |
| 75 | #125 | — | 6 | 2 | no | — |
| 76 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 77 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 78 | C2×C4⋊C4 | — | 6 | 4 | no | — |
| 79 | C2×C4⋊C4 | — | 6 | 4 | no | — |
| 80 | C2×C22⋊C4 | — | 6 | 4 | no | — |
| 81 | C2×C42 | — | 3 | 1 | no | — |
| 82 | C2×C42 | — | 3 | 2 | no | — |
| 83 | C2×C42 | — | 3 | 1 | no | — |
| 84 | #107 | — | 6 | 4 | no | — |
| 85 | #107 | — | 6 | 2 | no | — |
| 86 | #107 | — | 6 | 2 | no | — |
| 87 | #106 | — | 6 | 4 | no | — |
| 88 | #106 | — | 6 | 2 | no | — |
| 89 | #106 | — | 6 | 2 | no | — |
| 90 | C4×C8 | — | 3 | 1 | no | — |
| 91 | C4×C8 | — | 3 | 1 | no | — |
| 92 | C4×C8 | — | 3 | 1 | no | — |
| 93 | C4×C8 | — | 6 | 1 | no | — |
| 94 | C4×C8 | S4 | 1 | 1 | yes | characteristic |
| 95 | #95 | — | 6 | 4 | no | — |
| 96 | #95 | — | 6 | 4 | no | — |
| 97 | #95 | — | 6 | 4 | no | — |
| 98 | C2×A4 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 99 | C2×A4 | #135 | 1 | 2 | yes | — |
| 100 | C2×C3×C4 | — | 4 | 2 | no | — |
| 101 | C2×C3×C4 | — | 4 | 1 | no | — |
| 102 | C2×Dic3 | — | 8 | 4 | no | — |
| 103 | C3×C8 | — | 4 | 2 | no | — |
| 104 | C3×C8 | — | 4 | 2 | no | — |
| 105 | C24 | C2×D12 | 1 | 1 | yes | characteristic |
| 106 | C22×C4 | — | 2 | 2 | no | — |
| 107 | C22×C4 | — | 3 | 1 | no | — |
| 108 | C22×C4 | — | 3 | 1 | no | — |
| 109 | C22×C4 | — | 3 | 2 | no | — |
| 110 | C22×C4 | — | 3 | 1 | no | — |
| 111 | C22×C4 | — | 3 | 1 | no | — |
| 112 | C22×C4 | — | 3 | 2 | no | — |
| 113 | C22×C4 | — | 6 | 2 | no | — |
| 114 | C22×C4 | — | 6 | 4 | no | — |
| 115 | C22×C4 | — | 6 | 4 | no | — |
| 116 | C22×C4 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 117 | C22×C4 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 118 | C2×C8 | — | 3 | 1 | no | — |
| 119 | C2×C8 | — | 3 | 1 | no | — |
| 120 | C2×C8 | — | 3 | 2 | no | — |
| 121 | C2×C8 | — | 3 | 2 | no | — |
| 122 | C2×C8 | — | 3 | 2 | no | — |
| 123 | C2×C8 | — | 3 | 2 | no | — |
| 124 | C2×C8 | — | 3 | 2 | no | — |
| 125 | C2×C8 | — | 3 | 2 | no | — |
| 126 | C2×C8 | — | 6 | 1 | no | — |
| 127 | C2×C8 | — | 6 | 1 | no | — |
| 128 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 129 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 130 | C4⋊C4 | — | 6 | 4 | no | — |
| 131 | C4⋊C4 | — | 6 | 4 | no | — |
| 132 | C4⋊C4 | — | 6 | 4 | no | — |
| 133 | C4⋊C4 | — | 6 | 4 | no | — |
| 134 | C4⋊C4 | — | 12 | 8 | no | — |
| 135 | C4⋊C4 | — | 12 | 4 | no | — |
| 136 | C4⋊C4 | — | 12 | 4 | no | — |
| 137 | C22⋊C4 | — | 12 | 4 | no | — |
| 138 | C22⋊C4 | — | 12 | 4 | no | — |
| 139 | C42 | — | 3 | 2 | no | — |
| 140 | C42 | — | 3 | 1 | no | — |
| 141 | C42 | — | 6 | 1 | no | — |
| 142 | C42 | C2×S4 | 1 | 1 | yes | characteristic |
| 143 | C22×C3 | — | 4 | 1 | no | — |
| 144 | A4 | #499 | 1 | 1 | yes | characteristic, complemented |
| 145 | C3×C4 | — | 4 | 1 | no | — |
| 146 | C3×C4 | — | 4 | 1 | no | — |
| 147 | C3×C4 | — | 8 | 2 | no | — |
| 148 | Dic3 | — | 16 | 4 | no | — |
| 149 | C23 | — | 3 | 1 | no | — |
| 150 | C23 | — | 3 | 2 | no | — |
| 151 | C23 | — | 3 | 1 | no | — |
| 152 | C23 | #869 | 1 | 1 | yes | characteristic |
| 153 | C23 | #904 | 1 | 2 | yes | — |
| 154 | C2×C4 | — | 3 | 1 | no | — |
| 155 | C2×C4 | — | 3 | 1 | no | — |
| 156 | C2×C4 | — | 3 | 1 | no | — |
| 157 | C2×C4 | — | 3 | 1 | no | — |
| 158 | C2×C4 | — | 3 | 1 | no | — |
| 159 | C2×C4 | — | 3 | 1 | no | — |
| 160 | C2×C4 | — | 3 | 1 | no | — |
| 161 | C2×C4 | — | 3 | 4 | no | — |
| 162 | C2×C4 | — | 3 | 2 | no | — |
| 163 | C2×C4 | — | 6 | 2 | no | — |
| 164 | C2×C4 | — | 6 | 2 | no | — |
| 165 | C2×C4 | — | 6 | 1 | no | — |
| 166 | C2×C4 | — | 6 | 2 | no | — |
| 167 | C2×C4 | — | 12 | 4 | no | — |
| 168 | C2×C4 | — | 12 | 4 | no | — |
| 169 | C2×C4 | — | 12 | 4 | no | — |
| 170 | C2×C4 | — | 12 | 4 | no | — |
| 171 | C2×C4 | — | 12 | 4 | no | — |
| 172 | C2×C4 | — | 12 | 4 | no | — |
| 173 | C2×C4 | C4⋊S4 | 1 | 2 | yes | — |
| 174 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 175 | C8 | — | 3 | 2 | no | — |
| 176 | C8 | — | 3 | 2 | no | — |
| 177 | C8 | A4⋊Q8 | 1 | 2 | yes | — |
| 178 | C8 | A4⋊Q8 | 1 | 2 | yes | — |
| 179 | C2×C3 | — | 4 | 1 | no | — |
| 180 | C2×C3 | — | 4 | 2 | no | — |
| 181 | C22 | — | 3 | 1 | no | — |
| 182 | C22 | — | 3 | 2 | no | — |
| 183 | C22 | — | 3 | 1 | no | — |
| 184 | C22 | — | 3 | 1 | no | — |
| 185 | C22 | — | 3 | 2 | no | — |
| 186 | C22 | — | 3 | 2 | no | — |
| 187 | C22 | — | 6 | 1 | no | — |
| 188 | C22 | #4683 | 1 | 1 | yes | characteristic, complemented |
| 189 | C22 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| 190 | C4 | — | 2 | 2 | no | — |
| 191 | C4 | — | 3 | 1 | no | — |
| 192 | C4 | — | 3 | 1 | no | — |
| 193 | C4 | — | 6 | 2 | no | — |
| 194 | C4 | — | 24 | 4 | no | — |
| 195 | C4 | — | 24 | 4 | no | — |
| 196 | C4 | C2×A4⋊Q8 | 1 | 1 | yes | characteristic |
| 197 | C4 | C2×A4⋊Q8 | 1 | 1 | yes | characteristic |
| 198 | C3 | — | 4 | 1 | no | Sylow |
| 199 | C2 | — | 3 | 2 | no | — |
| 200 | C2 | — | 3 | 1 | no | — |
| 201 | C2 | — | 3 | 1 | no | — |
| 202 | C2 | #85283 | 1 | 1 | yes | characteristic |
| 203 | C2 | #85440 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 144. A4 | #499 | semidirect |
| 188. C22 | #4683 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #85284 | 2 |
| C4×C8×A4 | 2 |
| #73111 | 2 |
| #73110 | 2 |
| #33208 | 3 |
| #4683 | 4 |