Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
574 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 105 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 | #79268 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #78499 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #70790 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #69822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×#1281 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10225 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#904 | — | 2 | 2 | no | — |
| 8 | #5477 | C22 | 1 | 2 | yes | — |
| 9 | C2×#869 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#431 | C22 | 1 | 2 | yes | — |
| 11 | C2×C3×#97 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#816 | — | 2 | 2 | no | — |
| 13 | #4936 | C22 | 1 | 1 | yes | characteristic |
| 14 | #4846 | — | 2 | 4 | no | — |
| 15 | #4736 | — | 2 | 4 | no | — |
| 16 | #4715 | — | 2 | 2 | no | — |
| 17 | #4712 | — | 2 | 2 | no | — |
| 18 | #4694 | — | 2 | 2 | no | — |
| 19 | #4682 | — | 2 | 2 | no | — |
| 20 | #4470 | — | 2 | 4 | no | — |
| 21 | #3226 | — | 3 | 2 | no | — |
| 22 | #3150 | — | 3 | 1 | no | — |
| 23 | #1960 | — | 3 | 1 | no | — |
| 24 | #1517 | — | 3 | 2 | no | — |
| 25 | #1281 | D3 | 1 | 1 | yes | characteristic, complemented |
| 26 | C2×#235 | — | 2 | 2 | no | — |
| 27 | C22×Dic6 | — | 2 | 1 | no | — |
| 28 | C22×C3×C8 | D4 | 1 | 2 | yes | — |
| 29 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 30 | #930 | — | 2 | 2 | no | — |
| 31 | #930 | — | 2 | 2 | no | — |
| 32 | C2×C4⋊Dic3 | — | 2 | 2 | no | — |
| 33 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 34 | C2×Dic12 | — | 2 | 2 | no | — |
| 35 | #904 | — | 4 | 4 | no | — |
| 36 | #904 | — | 4 | 4 | no | — |
| 37 | C2×D24 | — | 2 | 2 | no | — |
| 38 | C2×#204 | — | 2 | 4 | no | — |
| 39 | #875 | — | 2 | 2 | no | — |
| 40 | #874 | D4 | 1 | 2 | yes | — |
| 41 | C4×D12 | — | 2 | 2 | no | — |
| 42 | C4×D12 | — | 2 | 2 | no | — |
| 43 | #869 | — | 2 | 2 | no | — |
| 44 | #869 | — | 2 | 1 | no | — |
| 45 | #869 | D4 | 1 | 2 | yes | — |
| 46 | #869 | D4 | 1 | 2 | yes | — |
| 47 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 48 | C3×C22⋊C8 | D4 | 1 | 4 | yes | — |
| 49 | C3×#97 | — | 2 | 2 | no | — |
| 50 | C3×C4×C8 | — | 2 | 2 | no | — |
| 51 | C3⋊#95 | — | 2 | 2 | no | — |
| 52 | #821 | — | 2 | 8 | no | — |
| 53 | #818 | — | 2 | 8 | no | — |
| 54 | #816 | — | 4 | 4 | no | — |
| 55 | C2×#135 | — | 6 | 2 | no | — |
| 56 | #532 | — | 3 | 2 | no | — |
| 57 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 58 | #500 | — | 6 | 2 | no | — |
| 59 | #497 | — | 6 | 2 | no | — |
| 60 | #496 | — | 6 | 2 | no | — |
| 61 | #486 | — | 6 | 2 | no | — |
| 62 | #480 | — | 6 | 4 | no | — |
| 63 | #474 | — | 6 | 4 | no | — |
| 64 | #431 | C2×D3 | 1 | 2 | yes | — |
| 65 | C2×#103 | — | 6 | 2 | no | — |
| 66 | C2×#97 | C2×D3 | 1 | 1 | yes | characteristic |
| 67 | #382 | — | 3 | 1 | no | — |
| 68 | #327 | — | 6 | 4 | no | — |
| 69 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 70 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C2×#67 | — | 4 | 2 | no | — |
| 72 | C22×Dic3 | — | 4 | 1 | no | — |
| 73 | C22×Dic3 | — | 4 | 1 | no | — |
| 74 | #235 | — | 4 | 4 | no | — |
| 75 | #235 | — | 4 | 4 | no | — |
| 76 | C2×D12 | — | 2 | 2 | no | — |
| 77 | C2×D12 | — | 2 | 2 | no | — |
| 78 | C2×D12 | — | 2 | 2 | no | — |
| 79 | C2×C4×D3 | — | 4 | 2 | no | — |
| 80 | C2×Dic6 | — | 2 | 2 | no | — |
| 81 | C2×Dic6 | — | 2 | 2 | no | — |
| 82 | C2×Dic6 | — | 4 | 2 | no | — |
| 83 | C2×Dic6 | — | 4 | 2 | no | — |
| 84 | C2×Dic6 | — | 4 | 1 | no | — |
| 85 | C2×C3×C8 | — | 2 | 2 | no | — |
| 86 | C2×C3×C8 | — | 2 | 2 | no | — |
| 87 | C2×C3×C8 | — | 2 | 4 | no | — |
| 88 | C2×C3×C8 | D8 | 1 | 4 | yes | — |
| 89 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 90 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 92 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 93 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 94 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 95 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 96 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 97 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 98 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 99 | Dic12 | — | 4 | 4 | no | — |
| 100 | D24 | — | 4 | 4 | no | — |
| 101 | #204 | — | 4 | 8 | no | — |
| 102 | C2×#41 | — | 6 | 2 | no | — |
| 103 | C22×Q8 | — | 6 | 1 | no | — |
| 104 | #135 | — | 12 | 4 | no | — |
| 105 | #135 | — | 12 | 4 | no | — |
| 106 | C2×Q16 | — | 6 | 2 | no | — |
| 107 | C2×SD16 | — | 6 | 4 | no | — |
| 108 | C2×D8 | — | 6 | 2 | no | — |
| 109 | C22×C8 | D12 | 1 | 2 | yes | — |
| 110 | C4⋊Q8 | — | 3 | 2 | no | — |
| 111 | C4⋊Q8 | — | 3 | 2 | no | — |
| 112 | C4⋊Q8 | — | 6 | 2 | no | — |
| 113 | C4⋊Q8 | — | 6 | 1 | no | — |
| 114 | #127 | — | 3 | 2 | no | — |
| 115 | #124 | — | 6 | 2 | no | — |
| 116 | #121 | — | 6 | 2 | no | — |
| 117 | #121 | — | 6 | 2 | no | — |
| 118 | C4×D4 | — | 6 | 2 | no | — |
| 119 | C4×D4 | — | 6 | 2 | no | — |
| 120 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 121 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 122 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 123 | #107 | — | 6 | 8 | no | — |
| 124 | C4⋊C8 | — | 2 | 2 | no | — |
| 125 | #103 | — | 12 | 4 | no | — |
| 126 | #102 | — | 6 | 8 | no | — |
| 127 | C22⋊C8 | D12 | 1 | 4 | yes | — |
| 128 | #97 | — | 2 | 2 | no | — |
| 129 | C4×C8 | — | 2 | 2 | no | — |
| 130 | #95 | — | 6 | 2 | no | — |
| 131 | C23×C3 | #127 | 1 | 1 | yes | characteristic |
| 132 | C22×D3 | — | 4 | 2 | no | — |
| 133 | C2×C3×C4 | — | 4 | 2 | no | — |
| 134 | C2×C3×C4 | #120 | 1 | 2 | yes | — |
| 135 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 136 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 137 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 138 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 139 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 140 | #67 | — | 8 | 2 | no | — |
| 141 | #67 | — | 8 | 2 | no | — |
| 142 | C2×Dic3 | — | 4 | 2 | no | — |
| 143 | C2×Dic3 | — | 4 | 2 | no | — |
| 144 | C2×Dic3 | — | 8 | 1 | no | — |
| 145 | C2×Dic3 | — | 8 | 1 | no | — |
| 146 | C2×Dic3 | — | 8 | 2 | no | — |
| 147 | C2×Dic3 | — | 8 | 1 | no | — |
| 148 | D12 | — | 4 | 4 | no | — |
| 149 | D12 | — | 4 | 4 | no | — |
| 150 | D12 | — | 8 | 2 | no | — |
| 151 | C4×D3 | — | 8 | 4 | no | — |
| 152 | Dic6 | — | 4 | 4 | no | — |
| 153 | Dic6 | — | 4 | 4 | no | — |
| 154 | Dic6 | — | 8 | 2 | no | — |
| 155 | C3×C8 | — | 2 | 4 | no | — |
| 156 | C3×C8 | — | 4 | 4 | no | — |
| 157 | #41 | — | 12 | 4 | no | — |
| 158 | #41 | — | 12 | 4 | no | — |
| 159 | C2×Q8 | — | 6 | 2 | no | — |
| 160 | C2×Q8 | — | 6 | 2 | no | — |
| 161 | C2×Q8 | — | 12 | 1 | no | — |
| 162 | C2×Q8 | — | 12 | 2 | no | — |
| 163 | C2×Q8 | — | 12 | 2 | no | — |
| 164 | C2×D4 | — | 6 | 2 | no | — |
| 165 | C2×D4 | — | 6 | 2 | no | — |
| 166 | C2×D4 | — | 6 | 2 | no | — |
| 167 | C2×D4 | — | 12 | 2 | no | — |
| 168 | C22×C4 | — | 12 | 1 | no | — |
| 169 | C22×C4 | — | 12 | 2 | no | — |
| 170 | C22×C4 | — | 12 | 1 | no | — |
| 171 | C22×C4 | C2×D12 | 1 | 2 | yes | — |
| 172 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 173 | Q16 | — | 12 | 4 | no | — |
| 174 | SD16 | — | 12 | 8 | no | — |
| 175 | D8 | — | 12 | 4 | no | — |
| 176 | C2×C8 | — | 2 | 2 | no | — |
| 177 | C2×C8 | — | 2 | 2 | no | — |
| 178 | C2×C8 | — | 2 | 4 | no | — |
| 179 | C2×C8 | D24 | 1 | 4 | yes | — |
| 180 | C4⋊C4 | — | 6 | 4 | no | — |
| 181 | C4⋊C4 | — | 6 | 2 | no | — |
| 182 | C4⋊C4 | — | 6 | 2 | no | — |
| 183 | C4⋊C4 | — | 12 | 2 | no | — |
| 184 | C4⋊C4 | — | 12 | 2 | no | — |
| 185 | C22⋊C4 | — | 12 | 2 | no | — |
| 186 | C22⋊C4 | — | 12 | 2 | no | — |
| 187 | C42 | D3×D4 | 1 | 2 | yes | — |
| 188 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 189 | C42 | #239 | 1 | 1 | yes | characteristic |
| 190 | C22×C3 | — | 2 | 2 | no | — |
| 191 | C22×C3 | #392 | 1 | 1 | yes | characteristic |
| 192 | C22×C3 | #493 | 1 | 1 | yes | characteristic |
| 193 | C22×C3 | #496 | 1 | 1 | yes | characteristic |
| 194 | C2×D3 | — | 8 | 2 | no | — |
| 195 | C2×D3 | — | 8 | 2 | no | — |
| 196 | C2×D3 | — | 8 | 2 | no | — |
| 197 | C3×C4 | — | 2 | 2 | no | — |
| 198 | C3×C4 | — | 2 | 2 | no | — |
| 199 | C3×C4 | #447 | 1 | 2 | yes | — |
| 200 | C3×C4 | #459 | 1 | 2 | yes | — |
| 201 | Dic3 | — | 8 | 2 | no | — |
| 202 | Dic3 | — | 8 | 2 | no | — |
| 203 | Dic3 | — | 16 | 1 | no | — |
| 204 | C23 | — | 12 | 2 | no | — |
| 205 | C23 | #874 | 1 | 1 | yes | characteristic |
| 206 | Q8 | — | 12 | 4 | no | — |
| 207 | Q8 | — | 12 | 4 | no | — |
| 208 | Q8 | — | 24 | 2 | no | — |
| 209 | D4 | — | 12 | 4 | no | — |
| 210 | D4 | — | 12 | 4 | no | — |
| 211 | D4 | — | 24 | 2 | no | — |
| 212 | D4 | — | 24 | 2 | no | — |
| 213 | D4 | — | 24 | 2 | no | — |
| 214 | C2×C4 | — | 4 | 2 | no | — |
| 215 | C2×C4 | — | 12 | 2 | no | — |
| 216 | C2×C4 | — | 12 | 2 | no | — |
| 217 | C2×C4 | — | 24 | 1 | no | — |
| 218 | C2×C4 | — | 24 | 1 | no | — |
| 219 | C2×C4 | — | 24 | 4 | no | — |
| 220 | C2×C4 | — | 24 | 2 | no | — |
| 221 | C2×C4 | — | 24 | 1 | no | — |
| 222 | C2×C4 | #882 | 1 | 2 | yes | — |
| 223 | C2×C4 | #882 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #895 | 1 | 2 | yes | — |
| 225 | C2×C4 | #895 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | C2×D24 | 1 | 2 | yes | — |
| 227 | C2×C4 | #908 | 1 | 2 | yes | — |
| 228 | C8 | — | 2 | 4 | no | — |
| 229 | C8 | — | 4 | 4 | no | — |
| 230 | C2×C3 | — | 2 | 1 | no | — |
| 231 | C2×C3 | — | 2 | 1 | no | — |
| 232 | C2×C3 | #2009 | 1 | 2 | yes | — |
| 233 | C2×C3 | #2011 | 1 | 1 | yes | characteristic |
| 234 | D3 | — | 16 | 2 | no | — |
| 235 | C22 | — | 2 | 2 | no | — |
| 236 | C22 | — | 24 | 2 | no | — |
| 237 | C22 | — | 24 | 2 | no | — |
| 238 | C22 | — | 24 | 2 | no | — |
| 239 | C22 | #4671 | 1 | 1 | yes | characteristic |
| 240 | C22 | #4695 | 1 | 1 | yes | characteristic |
| 241 | C22 | #4712 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 2 | 2 | no | — |
| 243 | C4 | — | 2 | 2 | no | — |
| 244 | C4 | — | 24 | 2 | no | — |
| 245 | C4 | — | 24 | 2 | no | — |
| 246 | C4 | — | 48 | 1 | no | — |
| 247 | C4 | #4732 | 1 | 2 | yes | — |
| 248 | C4 | #4843 | 1 | 2 | yes | — |
| 249 | C3 | #10225 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 250 | C2 | — | 2 | 1 | no | — |
| 251 | C2 | — | 2 | 1 | no | — |
| 252 | C2 | — | 48 | 2 | no | — |
| 253 | C2 | #68674 | 1 | 1 | yes | characteristic |
| 254 | C2 | #69031 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #79268 | C2 | semidirect |
| 3. #70790 | C2 | semidirect |
| 4. #69822 | C2 | semidirect |
| 5. C3×#1281 | C2 | semidirect |
| 25. #1281 | D3 | semidirect |
| 249. C3 | #10225 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #79268 | 2 |
| #78499 | 2 |
| #70790 | 2 |
| #69822 | 2 |
| C3×#1281 | 2 |
| #10225 | 3 |