Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
568 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #82009 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #79180 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #71431 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #70656 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C3×#1384 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13309 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#904 | C22 | 1 | 2 | yes | — |
| 8 | C22×D3×C8 | C22 | 1 | 1 | yes | characteristic |
| 9 | #5524 | — | 2 | 2 | no | — |
| 10 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#427 | C22 | 1 | 2 | yes | — |
| 12 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5002 | — | 2 | 2 | no | — |
| 14 | #4909 | — | 2 | 2 | no | — |
| 15 | #4908 | — | 2 | 2 | no | — |
| 16 | #4904 | — | 2 | 2 | no | — |
| 17 | #4887 | — | 2 | 2 | no | — |
| 18 | #4883 | — | 2 | 2 | no | — |
| 19 | #4864 | — | 2 | 4 | no | — |
| 20 | (C2×D3)⋊C16 | — | 2 | 2 | no | — |
| 21 | #4488 | — | 2 | 4 | no | — |
| 22 | #3414 | — | 3 | 1 | no | — |
| 23 | #3050 | — | 3 | 2 | no | — |
| 24 | #2145 | — | 3 | 2 | no | — |
| 25 | #2112 | — | 3 | 1 | no | — |
| 26 | #1384 | D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C2×#235 | — | 2 | 2 | no | — |
| 28 | C22×C4×D3 | D4 | 1 | 1 | yes | characteristic |
| 29 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 30 | C3×#117 | — | 2 | 2 | no | — |
| 31 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×Dic12 | D4 | 1 | 2 | yes | complemented |
| 33 | #904 | — | 2 | 2 | no | — |
| 34 | #904 | — | 2 | 2 | no | — |
| 35 | #904 | — | 2 | 2 | no | — |
| 36 | #904 | — | 2 | 2 | no | — |
| 37 | C2×D24 | D4 | 1 | 2 | yes | complemented |
| 38 | C2×#204 | — | 2 | 2 | no | — |
| 39 | C2×D3×C8 | — | 2 | 2 | no | — |
| 40 | C2×D3×C8 | — | 2 | 2 | no | — |
| 41 | C2×D3×C8 | — | 2 | 1 | no | — |
| 42 | C2×D3×C8 | — | 2 | 1 | no | — |
| 43 | #897 | — | 2 | 4 | no | — |
| 44 | #895 | — | 2 | 4 | no | — |
| 45 | #886 | — | 4 | 2 | no | — |
| 46 | #882 | — | 4 | 2 | no | — |
| 47 | C4×D12 | — | 4 | 4 | no | — |
| 48 | C3×M32 | — | 2 | 2 | no | — |
| 49 | C2×C3×C16 | D4 | 1 | 2 | yes | complemented |
| 50 | C3×#107 | — | 2 | 2 | no | — |
| 51 | C3×#106 | D4 | 1 | 4 | yes | complemented |
| 52 | #812 | — | 2 | 2 | no | — |
| 53 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 54 | #810 | — | 2 | 4 | no | — |
| 55 | #809 | — | 2 | 4 | no | — |
| 56 | Dic24 | — | 2 | 2 | no | — |
| 57 | #800 | — | 2 | 4 | no | — |
| 58 | D48 | — | 2 | 2 | no | — |
| 59 | C2×#135 | — | 3 | 2 | no | — |
| 60 | C23×C8 | — | 3 | 1 | no | — |
| 61 | #525 | — | 6 | 2 | no | — |
| 62 | #510 | — | 6 | 2 | no | — |
| 63 | #509 | — | 6 | 2 | no | — |
| 64 | #508 | — | 6 | 2 | no | — |
| 65 | C2×M32 | — | 3 | 1 | no | — |
| 66 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 67 | #467 | — | 6 | 2 | no | — |
| 68 | #466 | — | 6 | 2 | no | — |
| 69 | #465 | — | 6 | 4 | no | — |
| 70 | #427 | C2×D3 | 1 | 2 | yes | — |
| 71 | #416 | — | 6 | 2 | no | — |
| 72 | #360 | — | 6 | 4 | no | — |
| 73 | C22⋊C16 | — | 6 | 2 | no | — |
| 74 | C23×D3 | D8 | 1 | 1 | yes | characteristic |
| 75 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C2×#67 | — | 4 | 2 | no | — |
| 77 | C22×Dic3 | D8 | 1 | 1 | yes | characteristic |
| 78 | #235 | — | 4 | 2 | no | — |
| 79 | #235 | — | 4 | 2 | no | — |
| 80 | #235 | — | 4 | 2 | no | — |
| 81 | #235 | — | 4 | 2 | no | — |
| 82 | C2×D12 | — | 2 | 2 | no | — |
| 83 | C2×D12 | — | 4 | 4 | no | — |
| 84 | C2×C4×D3 | — | 2 | 2 | no | — |
| 85 | C2×C4×D3 | — | 2 | 1 | no | — |
| 86 | C2×C4×D3 | — | 2 | 1 | no | — |
| 87 | C2×C4×D3 | — | 2 | 2 | no | — |
| 88 | C2×C4×D3 | — | 4 | 2 | no | — |
| 89 | C2×Dic6 | — | 2 | 2 | no | — |
| 90 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 92 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 93 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 94 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 95 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 96 | C3×C42 | — | 4 | 2 | no | — |
| 97 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 98 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 99 | C4⋊Dic3 | — | 4 | 4 | no | — |
| 100 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 101 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 102 | Dic12 | — | 2 | 2 | no | — |
| 103 | Dic12 | — | 2 | 2 | no | — |
| 104 | D24 | — | 2 | 2 | no | — |
| 105 | D24 | — | 2 | 2 | no | — |
| 106 | #204 | — | 4 | 2 | no | — |
| 107 | #204 | — | 4 | 2 | no | — |
| 108 | D3×C8 | — | 4 | 1 | no | — |
| 109 | D3×C8 | — | 4 | 2 | no | — |
| 110 | D3×C8 | — | 4 | 1 | no | — |
| 111 | C3×C16 | — | 2 | 2 | no | — |
| 112 | C3⋊C16 | — | 4 | 1 | no | — |
| 113 | C2×#41 | — | 6 | 2 | no | — |
| 114 | C23×C4 | — | 3 | 1 | no | — |
| 115 | #135 | — | 6 | 2 | no | — |
| 116 | #135 | — | 6 | 2 | no | — |
| 117 | #135 | — | 6 | 2 | no | — |
| 118 | #135 | — | 6 | 2 | no | — |
| 119 | C2×Q16 | — | 3 | 2 | no | — |
| 120 | C2×SD16 | — | 6 | 2 | no | — |
| 121 | C2×D8 | — | 3 | 2 | no | — |
| 122 | C22×C8 | — | 3 | 1 | no | — |
| 123 | C22×C8 | — | 6 | 1 | no | — |
| 124 | C22×C8 | — | 6 | 2 | no | — |
| 125 | C22×C8 | — | 6 | 2 | no | — |
| 126 | C22×C8 | — | 6 | 1 | no | — |
| 127 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 128 | #123 | — | 12 | 2 | no | — |
| 129 | C22⋊Q8 | — | 6 | 4 | no | — |
| 130 | #121 | — | 6 | 4 | no | — |
| 131 | #120 | — | 12 | 2 | no | — |
| 132 | C4×D4 | — | 12 | 4 | no | — |
| 133 | #117 | — | 2 | 2 | no | — |
| 134 | Q32 | — | 6 | 2 | no | — |
| 135 | SD32 | — | 6 | 4 | no | — |
| 136 | D16 | — | 6 | 2 | no | — |
| 137 | M32 | — | 2 | 2 | no | — |
| 138 | M32 | — | 6 | 2 | no | — |
| 139 | C2×C16 | — | 6 | 1 | no | — |
| 140 | C2×C16 | D12 | 1 | 2 | yes | complemented |
| 141 | #107 | — | 2 | 2 | no | — |
| 142 | #106 | D12 | 1 | 4 | yes | complemented |
| 143 | #103 | — | 6 | 4 | no | — |
| 144 | #102 | — | 6 | 4 | no | — |
| 145 | C23×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 146 | C22×D3 | — | 2 | 2 | no | — |
| 147 | C22×D3 | — | 2 | 1 | no | — |
| 148 | C22×D3 | — | 4 | 2 | no | — |
| 149 | C22×D3 | — | 4 | 2 | no | — |
| 150 | C2×C3×C4 | — | 4 | 4 | no | — |
| 151 | C2×C3×C4 | #120 | 1 | 2 | yes | — |
| 152 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 153 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 154 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 155 | #67 | — | 8 | 2 | no | — |
| 156 | #67 | — | 8 | 2 | no | — |
| 157 | C2×Dic3 | — | 2 | 2 | no | — |
| 158 | C2×Dic3 | — | 2 | 1 | no | — |
| 159 | C2×Dic3 | — | 4 | 2 | no | — |
| 160 | D12 | — | 4 | 2 | no | — |
| 161 | D12 | — | 4 | 2 | no | — |
| 162 | D12 | — | 8 | 8 | no | — |
| 163 | C4×D3 | — | 4 | 1 | no | — |
| 164 | C4×D3 | — | 4 | 2 | no | — |
| 165 | C4×D3 | — | 4 | 1 | no | — |
| 166 | C4×D3 | — | 8 | 2 | no | — |
| 167 | C4×D3 | — | 8 | 2 | no | — |
| 168 | Dic6 | — | 4 | 2 | no | — |
| 169 | Dic6 | — | 4 | 2 | no | — |
| 170 | C3×C8 | #120 | 1 | 1 | yes | characteristic |
| 171 | C3×C8 | #121 | 1 | 2 | yes | — |
| 172 | C3×C8 | #127 | 1 | 1 | yes | characteristic |
| 173 | C3⋊C8 | — | 4 | 1 | no | — |
| 174 | C24 | — | 3 | 1 | no | — |
| 175 | #41 | — | 12 | 2 | no | — |
| 176 | #41 | — | 12 | 2 | no | — |
| 177 | #41 | — | 12 | 2 | no | — |
| 178 | #41 | — | 12 | 2 | no | — |
| 179 | C2×Q8 | — | 6 | 2 | no | — |
| 180 | C2×D4 | — | 6 | 2 | no | — |
| 181 | C2×D4 | — | 12 | 4 | no | — |
| 182 | C2×D4 | — | 12 | 2 | no | — |
| 183 | C22×C4 | — | 3 | 1 | no | — |
| 184 | C22×C4 | — | 6 | 2 | no | — |
| 185 | C22×C4 | — | 6 | 1 | no | — |
| 186 | C22×C4 | — | 6 | 2 | no | — |
| 187 | C22×C4 | — | 6 | 1 | no | — |
| 188 | C22×C4 | — | 12 | 2 | no | — |
| 189 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 190 | Q16 | — | 6 | 2 | no | — |
| 191 | Q16 | — | 6 | 2 | no | — |
| 192 | SD16 | — | 12 | 2 | no | — |
| 193 | SD16 | — | 12 | 2 | no | — |
| 194 | D8 | — | 6 | 2 | no | — |
| 195 | D8 | — | 6 | 2 | no | — |
| 196 | C2×C8 | — | 6 | 2 | no | — |
| 197 | C2×C8 | — | 6 | 1 | no | — |
| 198 | C2×C8 | — | 12 | 1 | no | — |
| 199 | C2×C8 | — | 12 | 1 | no | — |
| 200 | C2×C8 | — | 12 | 2 | no | — |
| 201 | C2×C8 | C2×D12 | 1 | 1 | yes | characteristic |
| 202 | C2×C8 | C2×D12 | 1 | 2 | yes | — |
| 203 | C2×C8 | D3×D4 | 1 | 2 | yes | — |
| 204 | C2×C8 | #239 | 1 | 1 | yes | characteristic |
| 205 | C4⋊C4 | — | 2 | 4 | no | — |
| 206 | C4⋊C4 | — | 12 | 4 | no | — |
| 207 | C22⋊C4 | — | 4 | 2 | no | — |
| 208 | C22⋊C4 | — | 12 | 4 | no | — |
| 209 | C22⋊C4 | — | 12 | 4 | no | — |
| 210 | C42 | — | 4 | 2 | no | — |
| 211 | C16 | — | 2 | 2 | no | — |
| 212 | C16 | — | 12 | 1 | no | — |
| 213 | C22×C3 | — | 4 | 1 | no | — |
| 214 | C22×C3 | #447 | 1 | 2 | yes | — |
| 215 | C22×C3 | #466 | 1 | 1 | yes | characteristic |
| 216 | C2×D3 | — | 2 | 4 | no | — |
| 217 | C2×D3 | — | 2 | 2 | no | — |
| 218 | C2×D3 | — | 4 | 2 | no | — |
| 219 | C2×D3 | — | 4 | 1 | no | — |
| 220 | C2×D3 | — | 4 | 1 | no | — |
| 221 | C2×D3 | — | 8 | 4 | no | — |
| 222 | C2×D3 | — | 8 | 2 | no | — |
| 223 | C3×C4 | — | 8 | 4 | no | — |
| 224 | C3×C4 | #392 | 1 | 1 | yes | characteristic |
| 225 | C3×C4 | #449 | 1 | 2 | yes | — |
| 226 | C3×C4 | #469 | 1 | 1 | yes | characteristic |
| 227 | Dic3 | — | 4 | 1 | no | — |
| 228 | Dic3 | — | 8 | 2 | no | — |
| 229 | C23 | — | 6 | 1 | no | — |
| 230 | C23 | — | 6 | 2 | no | — |
| 231 | C23 | — | 12 | 2 | no | — |
| 232 | C23 | — | 12 | 2 | no | — |
| 233 | C23 | D3×D8 | 1 | 1 | yes | characteristic |
| 234 | Q8 | — | 12 | 2 | no | — |
| 235 | Q8 | — | 12 | 2 | no | — |
| 236 | D4 | — | 12 | 2 | no | — |
| 237 | D4 | — | 12 | 2 | no | — |
| 238 | D4 | — | 24 | 8 | no | — |
| 239 | D4 | — | 24 | 2 | no | — |
| 240 | D4 | — | 24 | 2 | no | — |
| 241 | C2×C4 | — | 4 | 4 | no | — |
| 242 | C2×C4 | — | 6 | 1 | no | — |
| 243 | C2×C4 | — | 6 | 2 | no | — |
| 244 | C2×C4 | — | 12 | 1 | no | — |
| 245 | C2×C4 | — | 12 | 1 | no | — |
| 246 | C2×C4 | — | 12 | 2 | no | — |
| 247 | C2×C4 | — | 12 | 2 | no | — |
| 248 | C2×C4 | — | 24 | 2 | no | — |
| 249 | C2×C4 | — | 24 | 2 | no | — |
| 250 | C2×C4 | #882 | 1 | 2 | yes | — |
| 251 | C2×C4 | #895 | 1 | 1 | yes | characteristic |
| 252 | C2×C4 | #914 | 1 | 2 | yes | — |
| 253 | C2×C4 | #919 | 1 | 1 | yes | characteristic |
| 254 | C8 | — | 12 | 1 | no | — |
| 255 | C8 | #874 | 1 | 1 | yes | characteristic |
| 256 | C8 | #882 | 1 | 1 | yes | characteristic |
| 257 | C8 | #895 | 1 | 2 | yes | — |
| 258 | C2×C3 | — | 2 | 2 | no | — |
| 259 | C2×C3 | — | 2 | 1 | no | — |
| 260 | C2×C3 | #1999 | 1 | 1 | yes | characteristic |
| 261 | D3 | — | 4 | 2 | no | — |
| 262 | D3 | — | 16 | 2 | no | — |
| 263 | C22 | — | 4 | 1 | no | — |
| 264 | C22 | — | 6 | 4 | no | — |
| 265 | C22 | — | 6 | 2 | no | — |
| 266 | C22 | — | 12 | 2 | no | — |
| 267 | C22 | — | 12 | 1 | no | — |
| 268 | C22 | — | 12 | 1 | no | — |
| 269 | C22 | — | 24 | 4 | no | — |
| 270 | C22 | — | 24 | 2 | no | — |
| 271 | C22 | #4773 | 1 | 2 | yes | — |
| 272 | C22 | #4883 | 1 | 1 | yes | characteristic |
| 273 | C4 | — | 8 | 4 | no | — |
| 274 | C4 | — | 12 | 1 | no | — |
| 275 | C4 | — | 24 | 2 | no | — |
| 276 | C4 | #4671 | 1 | 1 | yes | characteristic |
| 277 | C4 | #4810 | 1 | 2 | yes | — |
| 278 | C4 | #4865 | 1 | 1 | yes | characteristic |
| 279 | C3 | #13309 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 280 | C2 | — | 2 | 2 | no | — |
| 281 | C2 | — | 2 | 1 | no | — |
| 282 | C2 | — | 12 | 2 | no | — |
| 283 | C2 | — | 48 | 2 | no | — |
| 284 | C2 | #68886 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #82009 | C2 | semidirect |
| 2. #79180 | C2 | semidirect |
| 3. #71431 | C2 | semidirect |
| 4. #70656 | C2 | semidirect |
| 5. C3×#1384 | C2 | semidirect |
| 26. #1384 | D3 | semidirect |
| 32. C2×Dic12 | D4 | semidirect |
| 37. C2×D24 | D4 | semidirect |
| 49. C2×C3×C16 | D4 | semidirect |
| 51. C3×#106 | D4 | semidirect |
| 140. C2×C16 | D12 | semidirect |
| 142. #106 | D12 | semidirect |
| 279. C3 | #13309 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #82009 | 2 |
| #79180 | 2 |
| #71431 | 2 |
| #70656 | 2 |
| C3×#1384 | 2 |
| #13309 | 3 |