← #10691722
Subgroups — #10691722
Order 768.
428 conjugacy classes of subgroups.
The subgroup lattice has 342 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #83223 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #79008 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C3×#332 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #71789 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #70438 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #67707 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67706 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #9272 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 10 | #5470 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#836 | — | 2 | 1 | no | — |
| 14 | C2×#836 | — | 2 | 1 | no | — |
| 15 | #5102 | — | 2 | 1 | no | — |
| 16 | #5101 | — | 2 | 1 | no | — |
| 17 | #5094 | — | 2 | 2 | no | — |
| 18 | #5093 | — | 2 | 1 | no | — |
| 19 | #5080 | — | 2 | 1 | no | — |
| 20 | #4971 | C4 | 1 | 4 | yes | — |
| 21 | #4925 | C22 | 1 | 1 | yes | characteristic |
| 22 | #4924 | C22 | 1 | 1 | yes | characteristic |
| 23 | C4×#208 | C22 | 1 | 1 | yes | characteristic |
| 24 | #4831 | — | 2 | 1 | no | — |
| 25 | #4830 | — | 2 | 1 | no | — |
| 26 | #4828 | — | 2 | 2 | no | — |
| 27 | C3×#332 | — | 2 | 1 | no | — |
| 28 | C3×#332 | — | 2 | 1 | no | — |
| 29 | #4482 | — | 2 | 1 | no | — |
| 30 | #4481 | — | 2 | 1 | no | — |
| 31 | #3229 | — | 3 | 1 | no | — |
| 32 | #2962 | — | 3 | 1 | no | — |
| 33 | #1966 | — | 3 | 1 | no | — |
| 34 | #1550 | — | 3 | 1 | no | — |
| 35 | C2×#332 | D3 | 1 | 1 | yes | characteristic |
| 36 | #1301 | — | 3 | 1 | no | — |
| 37 | #1283 | — | 3 | 1 | no | — |
| 38 | C22×C3×Q8 | — | 2 | 1 | no | — |
| 39 | C22×C3×C8 | — | 2 | 1 | no | — |
| 40 | C22×C3×C8 | — | 2 | 1 | no | — |
| 41 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 42 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 43 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 44 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 45 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 46 | C2×C3×C4⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 47 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 48 | #926 | — | 2 | 1 | no | — |
| 49 | #924 | — | 2 | 1 | no | — |
| 50 | #924 | C2×C4 | 1 | 2 | yes | — |
| 51 | C2×#208 | — | 2 | 1 | no | — |
| 52 | C2×#208 | D4 | 1 | 1 | yes | characteristic |
| 53 | C2×#208 | D4 | 1 | 1 | yes | characteristic |
| 54 | #870 | — | 2 | 1 | no | — |
| 55 | #869 | D4 | 1 | 1 | yes | characteristic |
| 56 | #869 | D4 | 1 | 1 | yes | characteristic |
| 57 | C4×Dic6 | — | 2 | 1 | no | — |
| 58 | C4×Dic6 | D4 | 1 | 2 | yes | — |
| 59 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 60 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 61 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 62 | C3×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 63 | C3×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 64 | #836 | — | 4 | 2 | no | — |
| 65 | #836 | — | 4 | 2 | no | — |
| 66 | #830 | C2×C4 | 1 | 2 | yes | — |
| 67 | #815 | — | 2 | 2 | no | — |
| 68 | Dic3⋊C8 | — | 2 | 2 | no | — |
| 69 | Dic3⋊C8 | — | 2 | 2 | no | — |
| 70 | C8×Dic3 | — | 2 | 2 | no | — |
| 71 | #810 | — | 2 | 4 | no | — |
| 72 | #808 | — | 2 | 4 | no | — |
| 73 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 74 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 75 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 76 | #803 | — | 2 | 1 | no | — |
| 77 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 78 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 79 | #533 | — | 3 | 1 | no | — |
| 80 | C2×C4⋊Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 81 | #498 | — | 6 | 1 | no | — |
| 82 | #487 | — | 6 | 1 | no | — |
| 83 | #483 | — | 3 | 4 | no | — |
| 84 | #475 | — | 6 | 2 | no | — |
| 85 | #446 | — | 6 | 2 | no | — |
| 86 | #446 | — | 6 | 1 | no | — |
| 87 | C8×Q8 | — | 6 | 1 | no | — |
| 88 | #431 | — | 3 | 1 | no | — |
| 89 | #424 | — | 6 | 1 | no | — |
| 90 | #423 | — | 3 | 1 | no | — |
| 91 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 92 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 93 | C2×#101 | — | 6 | 1 | no | — |
| 94 | C2×#101 | — | 6 | 1 | no | — |
| 95 | #405 | — | 6 | 1 | no | — |
| 96 | C4×M16 | — | 3 | 1 | no | — |
| 97 | #333 | — | 6 | 1 | no | — |
| 98 | #332 | — | 2 | 1 | no | — |
| 99 | #332 | — | 2 | 1 | no | — |
| 100 | #332 | — | 6 | 1 | no | — |
| 101 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 102 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 103 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 104 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 105 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 106 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 107 | C22×C3×C4 | — | 2 | 2 | no | — |
| 108 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 111 | C2×Dic6 | — | 2 | 1 | no | — |
| 112 | C2×Dic6 | C22⋊C4 | 1 | 2 | yes | — |
| 113 | C2×C3×C8 | — | 2 | 1 | no | — |
| 114 | C2×C3×C8 | — | 2 | 1 | no | — |
| 115 | C2×C3×C8 | — | 4 | 1 | no | — |
| 116 | C2×C3×C8 | — | 4 | 1 | no | — |
| 117 | C2×C3×C8 | — | 4 | 1 | no | — |
| 118 | C2×C3×C8 | — | 8 | 1 | no | — |
| 119 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 120 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 121 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 122 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 123 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 125 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 126 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 127 | #217 | — | 2 | 2 | no | — |
| 128 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 129 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 130 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 131 | C4⋊Dic3 | C22⋊C4 | 1 | 2 | yes | — |
| 132 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 133 | Dic3⋊C4 | — | 2 | 2 | no | — |
| 134 | C4×Dic3 | — | 2 | 2 | no | — |
| 135 | #208 | — | 2 | 1 | no | — |
| 136 | #208 | — | 2 | 1 | no | — |
| 137 | #208 | — | 2 | 1 | no | — |
| 138 | #208 | — | 4 | 1 | no | — |
| 139 | #208 | — | 4 | 1 | no | — |
| 140 | #208 | SD16 | 1 | 2 | yes | — |
| 141 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 142 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 143 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 144 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 145 | C22×Q8 | — | 2 | 1 | no | — |
| 146 | C2×M16 | — | 3 | 1 | no | — |
| 147 | C2×M16 | — | 3 | 1 | no | — |
| 148 | C2×M16 | — | 6 | 1 | no | — |
| 149 | C22×C8 | — | 2 | 1 | no | — |
| 150 | C22×C8 | — | 2 | 1 | no | — |
| 151 | C4⋊Q8 | — | 2 | 1 | no | — |
| 152 | C4⋊Q8 | — | 2 | 1 | no | — |
| 153 | C4⋊Q8 | — | 2 | 1 | no | — |
| 154 | C4⋊Q8 | — | 2 | 1 | no | — |
| 155 | C4⋊Q8 | — | 3 | 1 | no | — |
| 156 | C4⋊Q8 | — | 3 | 1 | no | — |
| 157 | #125 | — | 6 | 1 | no | — |
| 158 | C22⋊Q8 | — | 3 | 2 | no | — |
| 159 | C22⋊Q8 | — | 6 | 1 | no | — |
| 160 | C4×Q8 | — | 3 | 2 | no | — |
| 161 | C4×Q8 | — | 6 | 1 | no | — |
| 162 | #117 | — | 6 | 1 | no | — |
| 163 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 164 | C2×C4⋊C4 | C4×D3 | 1 | 2 | yes | — |
| 165 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 166 | #106 | — | 6 | 4 | no | — |
| 167 | C4⋊C8 | — | 2 | 1 | no | — |
| 168 | C4⋊C8 | — | 2 | 1 | no | — |
| 169 | C4⋊C8 | — | 2 | 1 | no | — |
| 170 | C4⋊C8 | — | 6 | 2 | no | — |
| 171 | C4⋊C8 | — | 6 | 2 | no | — |
| 172 | C4⋊C8 | — | 6 | 1 | no | — |
| 173 | C4⋊C8 | — | 6 | 1 | no | — |
| 174 | C4⋊C8 | — | 6 | 1 | no | — |
| 175 | C4⋊C8 | #67 | 1 | 1 | yes | characteristic |
| 176 | C4⋊C8 | #67 | 1 | 1 | yes | characteristic |
| 177 | #103 | — | 6 | 4 | no | — |
| 178 | #101 | — | 12 | 2 | no | — |
| 179 | #101 | — | 12 | 2 | no | — |
| 180 | C22⋊C8 | — | 3 | 2 | no | — |
| 181 | #97 | — | 6 | 2 | no | — |
| 182 | #97 | — | 6 | 1 | no | — |
| 183 | C4×C8 | — | 6 | 2 | no | — |
| 184 | C4×C8 | — | 6 | 1 | no | — |
| 185 | C4×C8 | — | 6 | 1 | no | — |
| 186 | C23×C3 | C4×D4 | 1 | 1 | yes | characteristic |
| 187 | C3×Q8 | — | 8 | 1 | no | — |
| 188 | C3×Q8 | — | 8 | 1 | no | — |
| 189 | C3×Q8 | — | 8 | 1 | no | — |
| 190 | C3×Q8 | — | 8 | 1 | no | — |
| 191 | C2×C3×C4 | — | 2 | 1 | no | — |
| 192 | C2×C3×C4 | — | 2 | 1 | no | — |
| 193 | C2×C3×C4 | — | 4 | 1 | no | — |
| 194 | C2×C3×C4 | — | 4 | 2 | 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 | C4×D4 | 1 | 1 | yes | characteristic |
| 199 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 200 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 201 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 202 | C2×C3×C4 | #123 | 1 | 1 | yes | characteristic |
| 203 | C2×C3×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 204 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 205 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 206 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 207 | C2×Dic3 | — | 2 | 2 | no | — |
| 208 | C2×Dic3 | — | 2 | 2 | no | — |
| 209 | Dic6 | — | 4 | 1 | no | — |
| 210 | Dic6 | — | 4 | 1 | no | — |
| 211 | Dic6 | — | 4 | 2 | no | — |
| 212 | C3×C8 | — | 4 | 1 | no | — |
| 213 | C3×C8 | — | 4 | 1 | no | — |
| 214 | C3×C8 | — | 8 | 1 | no | — |
| 215 | C3⋊C8 | — | 2 | 2 | no | — |
| 216 | C3⋊C8 | — | 4 | 2 | no | — |
| 217 | C3⋊C8 | — | 4 | 1 | no | — |
| 218 | C3⋊C8 | — | 8 | 1 | no | — |
| 219 | C2×Q8 | — | 3 | 2 | no | — |
| 220 | C2×Q8 | — | 4 | 1 | no | — |
| 221 | C2×Q8 | — | 4 | 1 | no | — |
| 222 | C2×Q8 | — | 4 | 1 | no | — |
| 223 | C2×Q8 | — | 4 | 1 | no | — |
| 224 | C2×Q8 | — | 4 | 1 | no | — |
| 225 | C2×Q8 | — | 4 | 1 | no | — |
| 226 | C2×Q8 | — | 6 | 1 | no | — |
| 227 | C22×C4 | — | 2 | 2 | no | — |
| 228 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 229 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 230 | C22×C4 | #237 | 1 | 1 | yes | characteristic |
| 231 | M16 | — | 3 | 2 | no | — |
| 232 | M16 | — | 6 | 1 | no | — |
| 233 | M16 | — | 6 | 1 | no | — |
| 234 | M16 | — | 6 | 1 | 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 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 4 | 1 | no | — |
| 241 | C2×C8 | — | 4 | 1 | no | — |
| 242 | C2×C8 | — | 6 | 1 | no | — |
| 243 | C2×C8 | — | 6 | 2 | no | — |
| 244 | C2×C8 | — | 6 | 1 | no | — |
| 245 | C2×C8 | — | 8 | 1 | no | — |
| 246 | C2×C8 | — | 12 | 1 | no | — |
| 247 | C4⋊C4 | — | 2 | 2 | no | — |
| 248 | C4⋊C4 | — | 2 | 2 | no | — |
| 249 | C4⋊C4 | — | 3 | 2 | no | — |
| 250 | C4⋊C4 | — | 4 | 1 | no | — |
| 251 | C4⋊C4 | — | 4 | 1 | no | — |
| 252 | C4⋊C4 | — | 6 | 2 | no | — |
| 253 | C4⋊C4 | — | 6 | 1 | no | — |
| 254 | C4⋊C4 | — | 6 | 1 | no | — |
| 255 | C4⋊C4 | — | 6 | 2 | no | — |
| 256 | C4⋊C4 | — | 6 | 1 | no | — |
| 257 | C22⋊C4 | — | 6 | 2 | no | — |
| 258 | C42 | — | 6 | 2 | no | — |
| 259 | C42 | #235 | 1 | 1 | yes | characteristic |
| 260 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 261 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 262 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 263 | C22×C3 | — | 2 | 1 | no | — |
| 264 | C22×C3 | — | 2 | 1 | no | — |
| 265 | C22×C3 | #386 | 1 | 1 | yes | characteristic |
| 266 | C22×C3 | C4×SD16 | 1 | 1 | yes | characteristic |
| 267 | C22×C3 | #440 | 1 | 1 | yes | characteristic |
| 268 | C3×C4 | — | 2 | 1 | no | — |
| 269 | C3×C4 | — | 2 | 1 | no | — |
| 270 | C3×C4 | — | 2 | 1 | no | — |
| 271 | C3×C4 | — | 8 | 2 | no | — |
| 272 | C3×C4 | #448 | 1 | 1 | yes | characteristic |
| 273 | C3×C4 | #449 | 1 | 1 | yes | characteristic |
| 274 | C3×C4 | #460 | 1 | 1 | yes | characteristic |
| 275 | C3×C4 | #464 | 1 | 1 | yes | characteristic |
| 276 | C3×C4 | #471 | 1 | 1 | yes | characteristic |
| 277 | C3×C4 | #473 | 1 | 1 | yes | characteristic |
| 278 | Dic3 | — | 4 | 2 | no | — |
| 279 | Dic3 | — | 4 | 2 | no | — |
| 280 | C23 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 8 | 1 | no | — |
| 282 | Q8 | — | 8 | 1 | no | — |
| 283 | Q8 | — | 8 | 1 | no | — |
| 284 | Q8 | — | 8 | 1 | no | — |
| 285 | Q8 | — | 12 | 2 | no | — |
| 286 | Q8 | — | 12 | 1 | no | — |
| 287 | Q8 | — | 12 | 1 | no | — |
| 288 | C2×C4 | — | 2 | 1 | no | — |
| 289 | C2×C4 | — | 2 | 1 | no | — |
| 290 | C2×C4 | — | 4 | 1 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 2 | no | — |
| 293 | C2×C4 | — | 4 | 2 | no | — |
| 294 | C2×C4 | — | 6 | 2 | no | — |
| 295 | C2×C4 | — | 6 | 2 | no | — |
| 296 | C2×C4 | D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #883 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #884 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | #911 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #912 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | D3×SD16 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | #915 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C4×#67 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | #937 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #938 | 1 | 1 | yes | characteristic |
| 306 | C8 | — | 4 | 1 | no | — |
| 307 | C8 | — | 4 | 1 | no | — |
| 308 | C8 | — | 6 | 2 | no | — |
| 309 | C8 | — | 8 | 1 | no | — |
| 310 | C8 | — | 12 | 2 | no | — |
| 311 | C8 | — | 12 | 1 | no | — |
| 312 | C8 | — | 24 | 1 | no | — |
| 313 | C2×C3 | — | 2 | 1 | no | — |
| 314 | C2×C3 | — | 2 | 1 | no | — |
| 315 | C2×C3 | #1880 | 1 | 1 | yes | characteristic |
| 316 | C2×C3 | #1908 | 1 | 1 | yes | characteristic |
| 317 | C2×C3 | #1915 | 1 | 1 | yes | characteristic |
| 318 | C22 | — | 2 | 1 | no | — |
| 319 | C22 | — | 2 | 1 | no | — |
| 320 | C22 | #4757 | 1 | 1 | yes | characteristic |
| 321 | C22 | #4758 | 1 | 1 | yes | characteristic |
| 322 | C22 | #4956 | 1 | 1 | yes | characteristic |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 2 | 1 | no | — |
| 326 | C4 | — | 8 | 2 | no | — |
| 327 | C4 | — | 12 | 2 | no | — |
| 328 | C4 | — | 12 | 2 | no | — |
| 329 | C4 | #4762 | 1 | 1 | yes | characteristic |
| 330 | C4 | #4767 | 1 | 1 | yes | characteristic |
| 331 | C4 | #5158 | 1 | 1 | yes | characteristic |
| 332 | C4 | #5170 | 1 | 1 | yes | characteristic |
| 333 | C4 | #5177 | 1 | 1 | yes | characteristic |
| 334 | C4 | #5192 | 1 | 1 | yes | characteristic |
| 335 | C3 | #9272 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 336 | C2 | — | 2 | 1 | no | — |
| 337 | C2 | — | 2 | 1 | no | — |
| 338 | C2 | #70124 | 1 | 1 | yes | characteristic |
| 339 | C2 | #70213 | 1 | 1 | yes | characteristic |
| 340 | C2 | #70286 | 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