Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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 | 2 | yes | maximal, complemented |
| 2 | #78499 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #71448 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #69845 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×#1375 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13340 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#904 | — | 2 | 1 | no | — |
| 8 | C2×#904 | C22 | 1 | 2 | yes | — |
| 9 | C2×#904 | C22 | 1 | 1 | yes | characteristic |
| 10 | #5477 | — | 2 | 1 | no | — |
| 11 | C2×C3×M32 | C22 | 1 | 2 | yes | — |
| 12 | C2×C3×#97 | C22 | 1 | 1 | yes | characteristic |
| 13 | #5110 | — | 2 | 1 | no | — |
| 14 | C2×#819 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#805 | — | 2 | 1 | no | — |
| 16 | #4909 | — | 2 | 2 | no | — |
| 17 | #4909 | — | 2 | 2 | no | — |
| 18 | #4908 | — | 2 | 2 | no | — |
| 19 | #4908 | — | 2 | 2 | no | — |
| 20 | #4904 | — | 2 | 4 | no | — |
| 21 | #4715 | — | 2 | 2 | no | — |
| 22 | #4712 | — | 2 | 2 | no | — |
| 23 | #4510 | — | 2 | 4 | no | — |
| 24 | #3414 | — | 3 | 2 | no | — |
| 25 | #3150 | — | 3 | 1 | no | — |
| 26 | #2143 | — | 3 | 2 | no | — |
| 27 | #1975 | — | 3 | 1 | no | — |
| 28 | #1375 | D3 | 1 | 1 | yes | characteristic, complemented |
| 29 | C2×#235 | — | 2 | 1 | no | — |
| 30 | C2×#235 | — | 2 | 2 | no | — |
| 31 | C22×C3×C8 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 34 | #930 | — | 4 | 2 | no | — |
| 35 | C2×#208 | — | 2 | 1 | no | — |
| 36 | C2×Dic12 | — | 2 | 1 | no | — |
| 37 | C2×Dic12 | D4 | 1 | 2 | yes | — |
| 38 | C2×Dic12 | D4 | 1 | 1 | yes | characteristic, complemented |
| 39 | #904 | — | 2 | 2 | no | — |
| 40 | #904 | — | 2 | 2 | no | — |
| 41 | #904 | — | 2 | 2 | no | — |
| 42 | #904 | — | 2 | 2 | no | — |
| 43 | #904 | — | 2 | 2 | no | — |
| 44 | #904 | — | 2 | 2 | no | — |
| 45 | #904 | — | 4 | 2 | no | — |
| 46 | #904 | — | 4 | 2 | no | — |
| 47 | C2×D24 | — | 2 | 1 | no | — |
| 48 | C2×D24 | D4 | 1 | 2 | yes | — |
| 49 | C2×D24 | D4 | 1 | 1 | yes | characteristic, complemented |
| 50 | C2×#204 | — | 2 | 2 | no | — |
| 51 | C2×#204 | — | 2 | 2 | no | — |
| 52 | C2×#204 | — | 2 | 1 | no | — |
| 53 | #875 | — | 2 | 2 | no | — |
| 54 | #874 | — | 2 | 1 | no | — |
| 55 | C4×D12 | — | 4 | 2 | no | — |
| 56 | #869 | — | 2 | 1 | no | — |
| 57 | C3×M32 | — | 2 | 2 | no | — |
| 58 | C3×M32 | — | 2 | 2 | no | — |
| 59 | C2×C3×C16 | D4 | 1 | 4 | yes | complemented |
| 60 | C3×#97 | — | 2 | 2 | no | — |
| 61 | (C2×D3)⋊C8 | — | 4 | 2 | no | — |
| 62 | #819 | — | 2 | 2 | no | — |
| 63 | #805 | — | 4 | 2 | no | — |
| 64 | Dic24 | — | 2 | 4 | no | — |
| 65 | #800 | — | 2 | 4 | no | — |
| 66 | #800 | — | 2 | 4 | no | — |
| 67 | D48 | — | 2 | 4 | no | — |
| 68 | C2×#135 | — | 3 | 2 | no | — |
| 69 | C2×#135 | — | 3 | 1 | no | — |
| 70 | C2×#135 | — | 6 | 1 | no | — |
| 71 | #532 | — | 6 | 1 | no | — |
| 72 | #510 | — | 6 | 2 | no | — |
| 73 | #510 | — | 6 | 2 | no | — |
| 74 | #509 | — | 6 | 2 | no | — |
| 75 | #509 | — | 6 | 2 | no | — |
| 76 | #508 | — | 6 | 4 | no | — |
| 77 | C2×M32 | C2×D3 | 1 | 2 | yes | — |
| 78 | #497 | — | 6 | 2 | no | — |
| 79 | #496 | — | 6 | 2 | no | — |
| 80 | C2×#108 | — | 3 | 1 | no | — |
| 81 | C2×#104 | — | 6 | 1 | no | — |
| 82 | #408 | — | 6 | 1 | no | — |
| 83 | C2×#97 | C2×D3 | 1 | 1 | yes | characteristic |
| 84 | #359 | — | 6 | 4 | no | — |
| 85 | C22×C3×C4 | — | 2 | 1 | no | — |
| 86 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×#67 | — | 4 | 2 | no | — |
| 88 | C2×#67 | — | 4 | 1 | no | — |
| 89 | #235 | — | 4 | 1 | no | — |
| 90 | #235 | — | 4 | 2 | no | — |
| 91 | #235 | — | 4 | 1 | no | — |
| 92 | #235 | — | 4 | 2 | no | — |
| 93 | #235 | — | 4 | 1 | no | — |
| 94 | #235 | — | 4 | 2 | no | — |
| 95 | #235 | — | 4 | 2 | no | — |
| 96 | #235 | — | 4 | 1 | no | — |
| 97 | C2×D12 | — | 2 | 2 | no | — |
| 98 | C2×D12 | — | 2 | 1 | no | — |
| 99 | C2×D12 | — | 4 | 2 | no | — |
| 100 | C2×C4×D3 | — | 4 | 1 | no | — |
| 101 | C2×C4×D3 | — | 4 | 2 | no | — |
| 102 | C2×Dic6 | — | 2 | 2 | no | — |
| 103 | C2×Dic6 | — | 2 | 1 | no | — |
| 104 | C2×C3×C8 | — | 2 | 1 | no | — |
| 105 | C2×C3×C8 | — | 2 | 1 | no | — |
| 106 | C2×C3×C8 | D8 | 1 | 2 | yes | — |
| 107 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 108 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 110 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 111 | C3×C42 | — | 2 | 1 | no | — |
| 112 | C3×C42 | D8 | 1 | 2 | yes | — |
| 113 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 114 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 115 | #208 | — | 4 | 1 | no | — |
| 116 | #208 | — | 4 | 1 | no | — |
| 117 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 118 | Dic12 | — | 2 | 2 | no | — |
| 119 | Dic12 | — | 2 | 2 | no | — |
| 120 | Dic12 | — | 2 | 2 | no | — |
| 121 | Dic12 | — | 4 | 2 | no | — |
| 122 | D24 | — | 2 | 2 | no | — |
| 123 | D24 | — | 2 | 2 | no | — |
| 124 | D24 | — | 2 | 2 | no | — |
| 125 | D24 | — | 4 | 2 | no | — |
| 126 | #204 | — | 4 | 4 | no | — |
| 127 | #204 | — | 4 | 2 | no | — |
| 128 | #204 | — | 4 | 2 | no | — |
| 129 | #204 | — | 4 | 2 | no | — |
| 130 | C3×C16 | — | 2 | 4 | no | — |
| 131 | C2×#41 | — | 6 | 2 | no | — |
| 132 | C2×#41 | — | 6 | 1 | no | — |
| 133 | #135 | — | 6 | 2 | no | — |
| 134 | #135 | — | 6 | 2 | no | — |
| 135 | #135 | — | 6 | 2 | no | — |
| 136 | #135 | — | 6 | 2 | no | — |
| 137 | #135 | — | 6 | 2 | no | — |
| 138 | #135 | — | 6 | 2 | no | — |
| 139 | #135 | — | 12 | 2 | no | — |
| 140 | #135 | — | 12 | 2 | no | — |
| 141 | C2×Q16 | — | 3 | 2 | no | — |
| 142 | C2×Q16 | — | 3 | 1 | no | — |
| 143 | C2×Q16 | — | 6 | 1 | no | — |
| 144 | C2×SD16 | — | 6 | 2 | no | — |
| 145 | C2×SD16 | — | 6 | 2 | no | — |
| 146 | C2×SD16 | — | 6 | 1 | no | — |
| 147 | C2×D8 | — | 3 | 2 | no | — |
| 148 | C2×D8 | — | 3 | 1 | no | — |
| 149 | C2×D8 | — | 6 | 1 | no | — |
| 150 | C2×M16 | — | 6 | 1 | no | — |
| 151 | C22×C8 | D12 | 1 | 1 | yes | characteristic |
| 152 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 153 | C4⋊Q8 | — | 6 | 1 | no | — |
| 154 | #127 | — | 6 | 1 | no | — |
| 155 | #124 | — | 6 | 2 | no | — |
| 156 | #121 | — | 12 | 2 | no | — |
| 157 | C4×D4 | — | 12 | 2 | no | — |
| 158 | C2×C42 | D12 | 1 | 1 | yes | characteristic |
| 159 | Q32 | — | 6 | 4 | no | — |
| 160 | SD32 | — | 6 | 4 | no | — |
| 161 | SD32 | — | 6 | 4 | no | — |
| 162 | D16 | — | 6 | 4 | no | — |
| 163 | M32 | — | 2 | 2 | no | — |
| 164 | M32 | — | 2 | 2 | no | — |
| 165 | C2×C16 | D12 | 1 | 4 | yes | complemented |
| 166 | #108 | — | 6 | 2 | no | — |
| 167 | #104 | — | 12 | 2 | no | — |
| 168 | C22⋊C8 | — | 12 | 2 | no | — |
| 169 | #97 | — | 2 | 2 | no | — |
| 170 | C23×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 171 | C22×D3 | — | 4 | 2 | no | — |
| 172 | C22×D3 | — | 4 | 1 | no | — |
| 173 | C2×C3×C4 | — | 2 | 2 | no | — |
| 174 | C2×C3×C4 | — | 8 | 1 | no | — |
| 175 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 176 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 177 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 179 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 180 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 181 | #67 | — | 8 | 1 | no | — |
| 182 | #67 | — | 8 | 2 | no | — |
| 183 | #67 | — | 8 | 2 | no | — |
| 184 | #67 | — | 8 | 1 | no | — |
| 185 | C2×Dic3 | — | 4 | 2 | no | — |
| 186 | C2×Dic3 | — | 4 | 1 | no | — |
| 187 | D12 | — | 4 | 2 | no | — |
| 188 | D12 | — | 4 | 1 | no | — |
| 189 | D12 | — | 4 | 2 | no | — |
| 190 | D12 | — | 4 | 1 | no | — |
| 191 | D12 | — | 8 | 4 | no | — |
| 192 | C4×D3 | — | 8 | 1 | no | — |
| 193 | C4×D3 | — | 8 | 2 | no | — |
| 194 | C4×D3 | — | 8 | 1 | no | — |
| 195 | C4×D3 | — | 8 | 2 | no | — |
| 196 | Dic6 | — | 4 | 2 | no | — |
| 197 | Dic6 | — | 4 | 1 | no | — |
| 198 | Dic6 | — | 4 | 2 | no | — |
| 199 | Dic6 | — | 4 | 1 | no | — |
| 200 | C3×C8 | — | 2 | 2 | no | — |
| 201 | C3×C8 | #120 | 1 | 2 | yes | — |
| 202 | C3×C8 | #121 | 1 | 2 | yes | — |
| 203 | C3⋊C8 | — | 8 | 1 | no | — |
| 204 | #41 | — | 12 | 2 | no | — |
| 205 | #41 | — | 12 | 1 | no | — |
| 206 | #41 | — | 12 | 1 | no | — |
| 207 | #41 | — | 12 | 2 | no | — |
| 208 | #41 | — | 12 | 1 | no | — |
| 209 | #41 | — | 12 | 2 | no | — |
| 210 | #41 | — | 12 | 2 | no | — |
| 211 | #41 | — | 12 | 1 | no | — |
| 212 | C2×Q8 | — | 6 | 2 | no | — |
| 213 | C2×Q8 | — | 6 | 1 | no | — |
| 214 | C2×D4 | — | 6 | 2 | no | — |
| 215 | C2×D4 | — | 6 | 1 | no | — |
| 216 | C2×D4 | — | 12 | 2 | no | — |
| 217 | C2×D4 | — | 12 | 2 | no | — |
| 218 | C2×D4 | — | 12 | 1 | no | — |
| 219 | C22×C4 | — | 2 | 1 | no | — |
| 220 | C22×C4 | — | 12 | 1 | no | — |
| 221 | C22×C4 | — | 12 | 2 | no | — |
| 222 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 223 | Q16 | — | 6 | 2 | no | — |
| 224 | Q16 | — | 6 | 2 | no | — |
| 225 | Q16 | — | 6 | 2 | no | — |
| 226 | Q16 | — | 12 | 2 | no | — |
| 227 | SD16 | — | 12 | 4 | no | — |
| 228 | SD16 | — | 12 | 2 | no | — |
| 229 | SD16 | — | 12 | 2 | no | — |
| 230 | SD16 | — | 12 | 2 | no | — |
| 231 | D8 | — | 6 | 2 | no | — |
| 232 | D8 | — | 6 | 2 | no | — |
| 233 | D8 | — | 6 | 2 | no | — |
| 234 | D8 | — | 12 | 2 | no | — |
| 235 | M16 | — | 12 | 1 | no | — |
| 236 | M16 | — | 12 | 1 | no | — |
| 237 | C2×C8 | — | 2 | 1 | no | — |
| 238 | C2×C8 | — | 2 | 1 | no | — |
| 239 | C2×C8 | — | 12 | 1 | no | — |
| 240 | C2×C8 | D24 | 1 | 2 | yes | — |
| 241 | C2×C8 | C2×D12 | 1 | 2 | yes | — |
| 242 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | D3×D4 | 1 | 2 | yes | — |
| 244 | C2×C8 | #239 | 1 | 1 | yes | characteristic |
| 245 | C4⋊C4 | — | 12 | 2 | no | — |
| 246 | C22⋊C4 | — | 12 | 4 | no | — |
| 247 | C42 | — | 2 | 1 | no | — |
| 248 | C42 | D24 | 1 | 2 | yes | — |
| 249 | C16 | — | 2 | 4 | no | — |
| 250 | C22×C3 | — | 4 | 1 | no | — |
| 251 | C22×C3 | #447 | 1 | 1 | yes | characteristic |
| 252 | C22×C3 | #459 | 1 | 1 | yes | characteristic |
| 253 | C22×C3 | #496 | 1 | 1 | yes | characteristic |
| 254 | C2×D3 | — | 8 | 4 | no | — |
| 255 | C2×D3 | — | 8 | 2 | no | — |
| 256 | C2×D3 | — | 8 | 1 | no | — |
| 257 | C2×D3 | — | 8 | 2 | no | — |
| 258 | C3×C4 | — | 4 | 2 | no | — |
| 259 | C3×C4 | #392 | 1 | 1 | yes | characteristic |
| 260 | C3×C4 | #447 | 1 | 1 | yes | characteristic |
| 261 | C3×C4 | #459 | 1 | 1 | yes | characteristic |
| 262 | C3×C4 | #493 | 1 | 1 | yes | characteristic |
| 263 | Dic3 | — | 8 | 2 | no | — |
| 264 | Dic3 | — | 8 | 1 | no | — |
| 265 | C23 | — | 12 | 2 | no | — |
| 266 | C23 | — | 12 | 1 | no | — |
| 267 | C23 | #908 | 1 | 1 | yes | characteristic |
| 268 | Q8 | — | 12 | 2 | no | — |
| 269 | Q8 | — | 12 | 1 | no | — |
| 270 | Q8 | — | 12 | 2 | no | — |
| 271 | Q8 | — | 12 | 1 | no | — |
| 272 | D4 | — | 12 | 1 | no | — |
| 273 | D4 | — | 12 | 2 | no | — |
| 274 | D4 | — | 12 | 2 | no | — |
| 275 | D4 | — | 12 | 1 | no | — |
| 276 | D4 | — | 24 | 4 | no | — |
| 277 | D4 | — | 24 | 2 | no | — |
| 278 | D4 | — | 24 | 1 | no | — |
| 279 | D4 | — | 24 | 1 | no | — |
| 280 | D4 | — | 24 | 2 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 8 | 1 | no | — |
| 283 | C2×C4 | — | 12 | 2 | no | — |
| 284 | C2×C4 | — | 12 | 1 | no | — |
| 285 | C2×C4 | — | 24 | 1 | no | — |
| 286 | C2×C4 | — | 24 | 2 | no | — |
| 287 | C2×C4 | — | 24 | 1 | no | — |
| 288 | C2×C4 | — | 24 | 2 | no | — |
| 289 | C2×C4 | #874 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #882 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | #895 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | C2×D24 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C2×D24 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | #908 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 2 | 2 | no | — |
| 296 | C8 | — | 24 | 1 | no | — |
| 297 | C8 | #882 | 1 | 2 | yes | — |
| 298 | C8 | #895 | 1 | 2 | yes | — |
| 299 | C2×C3 | — | 2 | 1 | no | — |
| 300 | C2×C3 | — | 2 | 1 | no | — |
| 301 | C2×C3 | — | 2 | 1 | no | — |
| 302 | C2×C3 | #2011 | 1 | 1 | yes | characteristic |
| 303 | D3 | — | 16 | 2 | no | — |
| 304 | D3 | — | 16 | 1 | no | — |
| 305 | C22 | — | 4 | 1 | no | — |
| 306 | C22 | — | 24 | 4 | no | — |
| 307 | C22 | — | 24 | 2 | no | — |
| 308 | C22 | — | 24 | 1 | no | — |
| 309 | C22 | — | 24 | 2 | no | — |
| 310 | C22 | #4712 | 1 | 1 | yes | characteristic |
| 311 | C22 | #4732 | 1 | 1 | yes | characteristic |
| 312 | C22 | #4843 | 1 | 1 | yes | characteristic |
| 313 | C4 | — | 4 | 2 | no | — |
| 314 | C4 | — | 24 | 2 | no | — |
| 315 | C4 | — | 24 | 1 | no | — |
| 316 | C4 | #4671 | 1 | 1 | yes | characteristic |
| 317 | C4 | #4695 | 1 | 1 | yes | characteristic |
| 318 | C4 | #4732 | 1 | 1 | yes | characteristic |
| 319 | C4 | #4843 | 1 | 1 | yes | characteristic |
| 320 | C3 | #13340 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 321 | C2 | — | 2 | 1 | no | — |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 2 | 1 | no | — |
| 324 | C2 | — | 48 | 2 | no | — |
| 325 | C2 | — | 48 | 1 | no | — |
| 326 | C2 | #68674 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #82009 | C2 | semidirect |
| 2. #78499 | C2 | semidirect |
| 3. #71448 | C2 | semidirect |
| 4. #69845 | C2 | semidirect |
| 5. C3×#1375 | C2 | semidirect |
| 28. #1375 | D3 | semidirect |
| 38. C2×Dic12 | D4 | semidirect |
| 49. C2×D24 | D4 | semidirect |
| 59. C2×C3×C16 | D4 | semidirect |
| 165. C2×C16 | D12 | semidirect |
| 320. C3 | #13340 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #82009 | 2 |
| #78499 | 2 |
| #71448 | 2 |
| #69845 | 2 |
| C3×#1375 | 2 |
| #13340 | 3 |