← #10711296
Subgroups — #10711296
Order 768.
400 conjugacy classes of subgroups.
The subgroup lattice has 332 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#5196 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#2148 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72081 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#4511 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #70693 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #67820 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #67731 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #12825 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×#135 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#948 | — | 2 | 1 | no | — |
| 11 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C3×#108 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#837 | — | 2 | 1 | no | — |
| 14 | #5196 | — | 2 | 1 | no | — |
| 15 | #5196 | — | 2 | 1 | no | — |
| 16 | #5196 | C4 | 1 | 2 | yes | — |
| 17 | #5196 | C4 | 1 | 2 | yes | — |
| 18 | C2×#822 | — | 2 | 1 | no | — |
| 19 | #5118 | — | 2 | 1 | no | — |
| 20 | C2×#819 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#819 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 24 | C3×#362 | — | 2 | 1 | no | — |
| 25 | C3×#361 | — | 2 | 1 | no | — |
| 26 | #4565 | — | 2 | 1 | no | — |
| 27 | #4563 | — | 2 | 1 | no | — |
| 28 | #4511 | — | 2 | 1 | no | — |
| 29 | #4511 | — | 2 | 1 | no | — |
| 30 | C2×#443 | — | 3 | 1 | no | — |
| 31 | C2×#364 | — | 3 | 1 | no | — |
| 32 | #2148 | — | 3 | 1 | no | — |
| 33 | #2148 | D3 | 1 | 1 | yes | characteristic |
| 34 | #1848 | — | 3 | 1 | no | — |
| 35 | #1385 | — | 3 | 1 | no | — |
| 36 | #1385 | — | 3 | 1 | no | — |
| 37 | C2×C3×#41 | — | 2 | 1 | no | — |
| 38 | C3×#135 | — | 2 | 1 | no | — |
| 39 | C3×#135 | — | 2 | 1 | no | — |
| 40 | C3×#135 | C2×C4 | 1 | 2 | yes | — |
| 41 | C3×#135 | Q8 | 1 | 2 | yes | — |
| 42 | C2×C3×Q16 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×SD16 | — | 2 | 1 | no | — |
| 44 | C2×C3×D8 | D4 | 1 | 1 | yes | characteristic |
| 45 | C2×C3×M16 | — | 2 | 1 | no | — |
| 46 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 47 | #948 | — | 2 | 2 | no | — |
| 48 | #948 | — | 2 | 2 | no | — |
| 49 | #948 | — | 4 | 1 | no | — |
| 50 | #948 | — | 4 | 1 | no | — |
| 51 | C2×C4×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 52 | C2×#208 | — | 2 | 1 | no | — |
| 53 | C2×#208 | — | 2 | 1 | no | — |
| 54 | C2×#208 | — | 4 | 1 | no | — |
| 55 | C22×C3⋊C8 | — | 4 | 1 | no | — |
| 56 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 57 | C3×M32 | — | 2 | 1 | no | — |
| 58 | C3×M32 | — | 2 | 1 | no | — |
| 59 | C2×C3×C16 | — | 2 | 1 | no | — |
| 60 | C3×#108 | — | 2 | 1 | no | — |
| 61 | C3×#108 | — | 2 | 1 | no | — |
| 62 | #837 | — | 2 | 2 | no | — |
| 63 | #837 | — | 2 | 2 | no | — |
| 64 | #822 | — | 4 | 1 | no | — |
| 65 | #822 | — | 4 | 1 | no | — |
| 66 | #819 | — | 2 | 1 | no | — |
| 67 | #819 | — | 2 | 1 | no | — |
| 68 | #819 | — | 2 | 1 | no | — |
| 69 | #819 | C2×C4 | 1 | 2 | yes | — |
| 70 | Dic3⋊C8 | — | 4 | 2 | no | — |
| 71 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 72 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 73 | C8×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 74 | C8×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 75 | #812 | — | 2 | 1 | no | — |
| 76 | #812 | — | 2 | 1 | no | — |
| 77 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 78 | C2×#135 | C2×D3 | 1 | 1 | yes | characteristic |
| 79 | C2×#131 | — | 6 | 1 | no | — |
| 80 | C2×M32 | — | 3 | 1 | no | — |
| 81 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 82 | #443 | — | 3 | 2 | no | — |
| 83 | #443 | — | 3 | 2 | no | — |
| 84 | #443 | — | 6 | 1 | no | — |
| 85 | #443 | — | 6 | 1 | no | — |
| 86 | C2×#108 | — | 3 | 1 | no | — |
| 87 | C2×#108 | — | 3 | 1 | no | — |
| 88 | C2×#108 | — | 6 | 1 | no | — |
| 89 | C2×#108 | C2×D3 | 1 | 1 | yes | characteristic |
| 90 | #423 | — | 6 | 1 | no | — |
| 91 | C2×#104 | — | 6 | 1 | no | — |
| 92 | C2×C4×C8 | — | 3 | 1 | no | — |
| 93 | #364 | — | 6 | 1 | no | — |
| 94 | #364 | — | 6 | 1 | no | — |
| 95 | #362 | — | 2 | 1 | no | — |
| 96 | #362 | — | 6 | 1 | no | — |
| 97 | #361 | — | 2 | 1 | no | — |
| 98 | #361 | — | 6 | 1 | no | — |
| 99 | C3×#41 | — | 2 | 2 | no | — |
| 100 | C3×#41 | — | 2 | 2 | no | — |
| 101 | C3×#41 | — | 4 | 1 | no | — |
| 102 | C3×#41 | — | 4 | 1 | no | — |
| 103 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 104 | C2×C3×D4 | — | 2 | 1 | no | — |
| 105 | C2×C3×D4 | — | 4 | 1 | no | — |
| 106 | C22×C3×C4 | — | 4 | 1 | no | — |
| 107 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | C22×Dic3 | — | 2 | 1 | no | — |
| 109 | C3×Q16 | — | 2 | 1 | no | — |
| 110 | C3×Q16 | C4⋊C4 | 1 | 2 | yes | — |
| 111 | C3×SD16 | — | 2 | 2 | no | — |
| 112 | C3×SD16 | — | 2 | 2 | no | — |
| 113 | C3×D8 | — | 2 | 1 | no | — |
| 114 | C3×D8 | C4⋊C4 | 1 | 2 | yes | — |
| 115 | C3×M16 | — | 4 | 1 | no | — |
| 116 | C3×M16 | — | 4 | 1 | no | — |
| 117 | C2×C3×C8 | — | 4 | 1 | no | — |
| 118 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 119 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 120 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×C3×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 122 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 123 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 124 | C4×Dic3 | — | 2 | 1 | no | — |
| 125 | C4×Dic3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C4×Dic3 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 127 | #208 | — | 2 | 2 | no | — |
| 128 | #208 | — | 4 | 1 | no | — |
| 129 | #208 | — | 4 | 2 | no | — |
| 130 | #208 | — | 4 | 2 | no | — |
| 131 | #208 | — | 4 | 1 | no | — |
| 132 | #208 | — | 4 | 1 | no | — |
| 133 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 134 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 135 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 136 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 137 | C2×C3⋊C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 138 | C2×C3⋊C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 139 | C2×C3⋊C8 | D8 | 1 | 1 | yes | characteristic |
| 140 | C2×C3⋊C8 | D8 | 1 | 1 | yes | characteristic |
| 141 | C2×C3⋊C8 | SD16 | 1 | 1 | yes | characteristic |
| 142 | C2×C3⋊C8 | SD16 | 1 | 1 | yes | characteristic |
| 143 | C3×C16 | — | 4 | 1 | no | — |
| 144 | C3⋊C16 | — | 4 | 1 | no | — |
| 145 | C2×#41 | — | 2 | 1 | no | — |
| 146 | #135 | — | 2 | 1 | no | — |
| 147 | #135 | — | 2 | 1 | no | — |
| 148 | #135 | Dic6 | 1 | 2 | yes | — |
| 149 | #135 | C4×D3 | 1 | 2 | yes | — |
| 150 | C2×Q16 | #67 | 1 | 1 | yes | characteristic |
| 151 | C2×SD16 | — | 2 | 1 | no | — |
| 152 | C2×D8 | #67 | 1 | 1 | yes | characteristic |
| 153 | #131 | — | 6 | 2 | no | — |
| 154 | #131 | — | 6 | 2 | no | — |
| 155 | #131 | — | 12 | 1 | no | — |
| 156 | #131 | — | 12 | 1 | no | — |
| 157 | C2×M16 | — | 2 | 1 | no | — |
| 158 | C2×M16 | — | 6 | 1 | no | — |
| 159 | C2×M16 | — | 6 | 1 | no | — |
| 160 | C2×M16 | — | 12 | 1 | no | — |
| 161 | C22×C8 | — | 3 | 1 | no | — |
| 162 | C22×C8 | — | 12 | 1 | no | — |
| 163 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 164 | C2×C42 | — | 3 | 1 | no | — |
| 165 | M32 | — | 2 | 1 | no | — |
| 166 | M32 | — | 2 | 1 | no | — |
| 167 | M32 | — | 6 | 1 | no | — |
| 168 | M32 | — | 6 | 1 | no | — |
| 169 | C2×C16 | — | 2 | 1 | no | — |
| 170 | C2×C16 | — | 6 | 1 | no | — |
| 171 | #108 | — | 2 | 1 | no | — |
| 172 | #108 | — | 2 | 1 | no | — |
| 173 | #108 | — | 3 | 2 | no | — |
| 174 | #108 | — | 6 | 1 | no | — |
| 175 | #108 | — | 6 | 1 | no | — |
| 176 | #108 | — | 6 | 1 | no | — |
| 177 | #108 | — | 12 | 1 | no | — |
| 178 | #108 | — | 12 | 1 | no | — |
| 179 | C4⋊C8 | — | 12 | 2 | no | — |
| 180 | #104 | — | 6 | 2 | no | — |
| 181 | #104 | — | 6 | 2 | no | — |
| 182 | C4×C8 | — | 3 | 1 | no | — |
| 183 | C4×C8 | — | 3 | 1 | no | — |
| 184 | C4×C8 | — | 3 | 1 | no | — |
| 185 | C4×C8 | — | 3 | 1 | no | — |
| 186 | C23×C3 | — | 4 | 1 | no | — |
| 187 | C23×C3 | #137 | 1 | 1 | yes | characteristic |
| 188 | C3×Q8 | — | 2 | 2 | no | — |
| 189 | C3×Q8 | — | 4 | 1 | no | — |
| 190 | C3×D4 | — | 2 | 2 | no | — |
| 191 | C3×D4 | — | 4 | 2 | no | — |
| 192 | C3×D4 | — | 4 | 2 | no | — |
| 193 | C3×D4 | — | 4 | 1 | no | — |
| 194 | C2×C3×C4 | — | 4 | 1 | no | — |
| 195 | C2×C3×C4 | — | 4 | 2 | no | — |
| 196 | C2×C3×C4 | — | 4 | 2 | no | — |
| 197 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 198 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 199 | C2×C3×C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 200 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 201 | C2×C3×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 202 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 203 | C2×Dic3 | — | 2 | 1 | no | — |
| 204 | C2×Dic3 | — | 2 | 1 | no | — |
| 205 | C2×Dic3 | — | 4 | 2 | no | — |
| 206 | C3×C8 | — | 8 | 1 | no | — |
| 207 | C3×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 208 | C3×C8 | #117 | 1 | 1 | yes | characteristic |
| 209 | C3×C8 | #121 | 1 | 1 | yes | characteristic |
| 210 | C3×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 211 | C3⋊C8 | — | 4 | 2 | no | — |
| 212 | C3⋊C8 | — | 8 | 1 | no | — |
| 213 | C3⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 214 | C3⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 215 | C3⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 216 | C3⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 217 | #41 | — | 2 | 2 | no | — |
| 218 | #41 | — | 2 | 2 | no | — |
| 219 | #41 | — | 4 | 1 | no | — |
| 220 | #41 | — | 4 | 1 | no | — |
| 221 | C2×Q8 | — | 2 | 1 | no | — |
| 222 | C2×D4 | — | 2 | 1 | no | — |
| 223 | C2×D4 | — | 4 | 1 | no | — |
| 224 | C22×C4 | — | 4 | 1 | no | — |
| 225 | C22×C4 | — | 6 | 1 | no | — |
| 226 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 227 | Q16 | — | 2 | 1 | no | — |
| 228 | Q16 | Dic3⋊C4 | 1 | 2 | yes | — |
| 229 | SD16 | — | 2 | 2 | no | — |
| 230 | SD16 | — | 2 | 2 | no | — |
| 231 | D8 | — | 2 | 1 | no | — |
| 232 | D8 | Dic3⋊C4 | 1 | 2 | yes | — |
| 233 | M16 | — | 4 | 1 | no | — |
| 234 | M16 | — | 4 | 1 | no | — |
| 235 | M16 | — | 6 | 2 | no | — |
| 236 | M16 | — | 12 | 2 | no | — |
| 237 | M16 | — | 12 | 2 | no | — |
| 238 | M16 | — | 12 | 1 | no | — |
| 239 | M16 | — | 12 | 1 | no | — |
| 240 | M16 | — | 12 | 1 | no | — |
| 241 | C2×C8 | — | 3 | 1 | no | — |
| 242 | C2×C8 | — | 3 | 1 | no | — |
| 243 | C2×C8 | — | 3 | 1 | no | — |
| 244 | C2×C8 | — | 3 | 1 | no | — |
| 245 | C2×C8 | — | 3 | 1 | no | — |
| 246 | C2×C8 | — | 3 | 1 | no | — |
| 247 | C2×C8 | — | 4 | 1 | no | — |
| 248 | C2×C8 | — | 12 | 2 | no | — |
| 249 | C2×C8 | — | 12 | 2 | no | — |
| 250 | C2×C8 | — | 12 | 1 | no | — |
| 251 | C2×C8 | — | 12 | 1 | no | — |
| 252 | C2×C8 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 254 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 255 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 256 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 258 | C42 | — | 3 | 1 | no | — |
| 259 | C42 | — | 3 | 1 | no | — |
| 260 | C42 | — | 6 | 1 | no | — |
| 261 | C16 | — | 4 | 1 | no | — |
| 262 | C16 | — | 12 | 1 | no | — |
| 263 | C22×C3 | — | 4 | 1 | no | — |
| 264 | C22×C3 | — | 4 | 2 | no | — |
| 265 | C22×C3 | — | 8 | 2 | no | — |
| 266 | C22×C3 | #417 | 1 | 1 | yes | characteristic |
| 267 | C22×C3 | #461 | 1 | 1 | yes | characteristic |
| 268 | C22×C3 | #474 | 1 | 1 | yes | characteristic |
| 269 | C3×C4 | — | 4 | 2 | no | — |
| 270 | C3×C4 | #380 | 1 | 1 | yes | characteristic |
| 271 | C3×C4 | C2×#102 | 1 | 1 | yes | characteristic |
| 272 | C3×C4 | #459 | 1 | 1 | yes | characteristic |
| 273 | C3×C4 | #476 | 1 | 1 | yes | characteristic |
| 274 | Dic3 | — | 2 | 2 | no | — |
| 275 | Dic3 | — | 4 | 1 | no | — |
| 276 | C23 | — | 4 | 1 | no | — |
| 277 | C23 | #915 | 1 | 1 | yes | characteristic |
| 278 | Q8 | — | 2 | 2 | no | — |
| 279 | Q8 | — | 4 | 1 | no | — |
| 280 | D4 | — | 2 | 2 | no | — |
| 281 | D4 | — | 4 | 2 | no | — |
| 282 | D4 | — | 4 | 2 | no | — |
| 283 | D4 | — | 4 | 1 | no | — |
| 284 | C2×C4 | — | 4 | 1 | no | — |
| 285 | C2×C4 | — | 4 | 2 | no | — |
| 286 | C2×C4 | — | 4 | 2 | no | — |
| 287 | C2×C4 | — | 6 | 1 | no | — |
| 288 | C2×C4 | — | 6 | 1 | no | — |
| 289 | C2×C4 | — | 12 | 2 | no | — |
| 290 | C2×C4 | #878 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | D3×D8 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | #911 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | D3×SD16 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #939 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 3 | 1 | no | — |
| 297 | C8 | — | 3 | 1 | no | — |
| 298 | C8 | — | 3 | 1 | no | — |
| 299 | C8 | — | 3 | 1 | no | — |
| 300 | C8 | — | 8 | 1 | no | — |
| 301 | C8 | — | 12 | 2 | no | — |
| 302 | C8 | — | 24 | 1 | no | — |
| 303 | C8 | #877 | 1 | 1 | yes | characteristic |
| 304 | C8 | #878 | 1 | 1 | yes | characteristic |
| 305 | C8 | C2×Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 306 | C8 | #939 | 1 | 1 | yes | characteristic |
| 307 | C2×C3 | — | 2 | 1 | no | — |
| 308 | C2×C3 | — | 2 | 1 | no | — |
| 309 | C2×C3 | — | 2 | 1 | no | — |
| 310 | C2×C3 | — | 8 | 2 | no | — |
| 311 | C2×C3 | #1802 | 1 | 1 | yes | characteristic |
| 312 | C22 | — | 4 | 1 | no | — |
| 313 | C22 | — | 4 | 2 | no | — |
| 314 | C22 | — | 8 | 2 | no | — |
| 315 | C22 | #4759 | 1 | 1 | yes | characteristic |
| 316 | C22 | #4771 | 1 | 1 | yes | characteristic |
| 317 | C22 | #5162 | 1 | 1 | yes | characteristic |
| 318 | C4 | — | 4 | 2 | no | — |
| 319 | C4 | — | 6 | 2 | no | — |
| 320 | C4 | — | 12 | 1 | no | — |
| 321 | C4 | #4761 | 1 | 1 | yes | characteristic |
| 322 | C4 | D3×#102 | 1 | 1 | yes | characteristic |
| 323 | C4 | #4942 | 1 | 1 | yes | characteristic |
| 324 | C4 | #5150 | 1 | 1 | yes | characteristic |
| 325 | C3 | #12825 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 326 | C2 | — | 2 | 1 | no | — |
| 327 | C2 | — | 2 | 1 | no | — |
| 328 | C2 | — | 2 | 1 | no | — |
| 329 | C2 | — | 8 | 2 | no | — |
| 330 | C2 | #70153 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta