Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
859 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 859 conjugacy classes of subgroups — too many to draw. Only its 154 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 | #61409 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2248 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33583 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #28762 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13338 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #13266 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #13257 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13019 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | C4×#359 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3508 | — | 2 | 1 | no | — |
| 11 | C2×#510 | C22 | 1 | 2 | yes | — |
| 12 | C22×SD32 | C22 | 1 | 4 | yes | — |
| 13 | #3399 | — | 2 | 4 | no | — |
| 14 | #3396 | — | 2 | 2 | no | — |
| 15 | #3391 | — | 2 | 2 | no | — |
| 16 | #3155 | C22 | 1 | 2 | yes | — |
| 17 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 19 | #3139 | — | 2 | 1 | no | — |
| 20 | #3041 | — | 2 | 2 | no | — |
| 21 | #3038 | — | 2 | 1 | no | — |
| 22 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 23 | #2948 | C22 | 1 | 2 | yes | — |
| 24 | C4×#135 | C22 | 1 | 1 | yes | characteristic |
| 25 | #2248 | — | 2 | 4 | no | — |
| 26 | #2209 | — | 2 | 2 | no | — |
| 27 | #2208 | — | 2 | 2 | no | — |
| 28 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#361 | C22 | 1 | 4 | yes | — |
| 30 | C2×#359 | C22 | 1 | 2 | yes | — |
| 31 | #2141 | C22 | 1 | 2 | yes | — |
| 32 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 33 | #2047 | — | 2 | 2 | no | — |
| 34 | #1979 | C22 | 1 | 4 | yes | — |
| 35 | #1966 | — | 2 | 1 | no | — |
| 36 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1382 | C22 | 1 | 2 | yes | — |
| 38 | #1379 | C22 | 1 | 4 | yes | — |
| 39 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 40 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 41 | C22×Q16 | — | 2 | 1 | no | — |
| 42 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 43 | C22×D8 | — | 2 | 1 | no | — |
| 44 | C22×D8 | C23 | 1 | 2 | yes | — |
| 45 | #559 | — | 2 | 4 | no | — |
| 46 | #550 | — | 2 | 4 | no | — |
| 47 | #546 | — | 4 | 4 | no | — |
| 48 | #540 | — | 4 | 2 | no | — |
| 49 | #535 | — | 2 | 2 | no | — |
| 50 | #533 | D4 | 1 | 1 | yes | characteristic |
| 51 | #532 | — | 4 | 2 | no | — |
| 52 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 53 | C2×#127 | — | 2 | 1 | no | — |
| 54 | #518 | — | 2 | 2 | no | — |
| 55 | C4×#41 | — | 2 | 1 | no | — |
| 56 | #510 | — | 4 | 4 | no | — |
| 57 | C2×Q32 | — | 2 | 2 | no | — |
| 58 | C2×SD32 | — | 2 | 8 | no | — |
| 59 | C2×SD32 | — | 2 | 2 | no | — |
| 60 | C2×SD32 | — | 2 | 8 | no | — |
| 61 | C2×SD32 | — | 4 | 4 | no | — |
| 62 | C2×M32 | C23 | 1 | 2 | yes | — |
| 63 | C22×C16 | C23 | 1 | 2 | yes | — |
| 64 | #496 | — | 2 | 4 | no | — |
| 65 | #495 | — | 2 | 2 | no | — |
| 66 | #494 | — | 2 | 2 | no | — |
| 67 | #494 | — | 2 | 2 | no | — |
| 68 | #493 | — | 2 | 2 | no | — |
| 69 | #493 | — | 2 | 2 | no | — |
| 70 | #493 | D4 | 1 | 4 | yes | — |
| 71 | #491 | — | 4 | 2 | no | — |
| 72 | #487 | — | 4 | 2 | no | — |
| 73 | #463 | — | 2 | 8 | no | — |
| 74 | #460 | — | 2 | 4 | no | — |
| 75 | #459 | — | 2 | 4 | no | — |
| 76 | #459 | — | 2 | 8 | no | — |
| 77 | #446 | — | 2 | 2 | no | — |
| 78 | C8×Q8 | — | 2 | 2 | no | — |
| 79 | #442 | — | 2 | 4 | no | — |
| 80 | C4×Q16 | D4 | 1 | 2 | yes | — |
| 81 | C4×SD16 | — | 2 | 2 | no | — |
| 82 | C4×D8 | — | 2 | 4 | no | — |
| 83 | C4×D8 | D4 | 1 | 2 | yes | — |
| 84 | C2×#108 | C23 | 1 | 2 | yes | — |
| 85 | #427 | C23 | 1 | 1 | yes | characteristic |
| 86 | #424 | D4 | 1 | 2 | yes | — |
| 87 | #423 | D4 | 1 | 1 | yes | characteristic |
| 88 | #419 | — | 2 | 4 | no | — |
| 89 | #418 | — | 2 | 2 | no | — |
| 90 | #416 | — | 2 | 1 | no | — |
| 91 | C2×#101 | — | 2 | 4 | no | — |
| 92 | #405 | C23 | 1 | 2 | yes | — |
| 93 | C4×M16 | — | 2 | 1 | no | — |
| 94 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 95 | C4⋊C16 | — | 2 | 2 | no | — |
| 96 | #361 | — | 4 | 4 | no | — |
| 97 | #359 | — | 2 | 4 | no | — |
| 98 | #358 | — | 2 | 4 | no | — |
| 99 | C4×C16 | — | 2 | 2 | no | — |
| 100 | #341 | — | 2 | 1 | no | — |
| 101 | C2×#41 | — | 2 | 1 | no | — |
| 102 | C2×#41 | — | 4 | 4 | no | — |
| 103 | C22×Q8 | — | 2 | 2 | no | — |
| 104 | C22×D4 | — | 2 | 2 | no | — |
| 105 | C22×D4 | — | 4 | 2 | no | — |
| 106 | AGL(1,ℤ/8) | — | 8 | 4 | no | — |
| 107 | #135 | — | 2 | 2 | no | — |
| 108 | #135 | — | 2 | 2 | no | — |
| 109 | C2×Q16 | — | 2 | 2 | no | — |
| 110 | C2×Q16 | — | 2 | 2 | no | — |
| 111 | C2×Q16 | — | 2 | 4 | no | — |
| 112 | C2×Q16 | — | 2 | 4 | no | — |
| 113 | C2×Q16 | — | 4 | 2 | no | — |
| 114 | C2×Q16 | — | 8 | 2 | no | — |
| 115 | C2×Q16 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×SD16 | — | 2 | 4 | no | — |
| 117 | C2×SD16 | — | 2 | 1 | no | — |
| 118 | C2×D8 | — | 2 | 4 | no | — |
| 119 | C2×D8 | — | 2 | 4 | no | — |
| 120 | C2×D8 | — | 2 | 2 | no | — |
| 121 | C2×D8 | — | 2 | 4 | no | — |
| 122 | C2×D8 | — | 2 | 4 | no | — |
| 123 | C2×D8 | — | 8 | 2 | no | — |
| 124 | C2×D8 | C2×D4 | 1 | 4 | yes | — |
| 125 | C2×D8 | C2×D4 | 1 | 1 | yes | characteristic |
| 126 | C2×M16 | — | 2 | 2 | no | — |
| 127 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 128 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 129 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 130 | C4⋊Q8 | — | 4 | 2 | no | — |
| 131 | C4⋊Q8 | — | 4 | 1 | no | — |
| 132 | C4⋊Q8 | — | 4 | 1 | no | — |
| 133 | C4⋊Q8 | D8 | 1 | 2 | yes | — |
| 134 | #127 | — | 2 | 4 | no | — |
| 135 | #127 | — | 2 | 4 | no | — |
| 136 | #127 | — | 4 | 1 | no | — |
| 137 | #127 | — | 4 | 1 | no | — |
| 138 | #125 | D8 | 1 | 2 | yes | — |
| 139 | #124 | — | 2 | 4 | no | — |
| 140 | #124 | — | 4 | 4 | no | — |
| 141 | #123 | — | 4 | 8 | no | — |
| 142 | C22⋊Q8 | — | 2 | 2 | no | — |
| 143 | #121 | — | 4 | 4 | no | — |
| 144 | #121 | — | 4 | 8 | no | — |
| 145 | #121 | — | 4 | 4 | no | — |
| 146 | #120 | — | 4 | 4 | no | — |
| 147 | C4×Q8 | — | 2 | 2 | no | — |
| 148 | C4×Q8 | — | 2 | 2 | no | — |
| 149 | C4×D4 | — | 2 | 8 | no | — |
| 150 | C4×D4 | — | 2 | 2 | no | — |
| 151 | C4×D4 | — | 4 | 4 | no | — |
| 152 | C4×D4 | — | 4 | 2 | no | — |
| 153 | C4×D4 | — | 4 | 4 | no | — |
| 154 | #117 | — | 2 | 1 | no | — |
| 155 | #117 | — | 4 | 1 | no | — |
| 156 | #117 | C2×D4 | 1 | 2 | yes | — |
| 157 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 158 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 159 | C2×C42 | — | 4 | 1 | no | — |
| 160 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 161 | Q32 | — | 4 | 4 | no | — |
| 162 | SD32 | — | 4 | 16 | no | — |
| 163 | SD32 | — | 4 | 4 | no | — |
| 164 | M32 | — | 2 | 4 | no | — |
| 165 | C2×C16 | — | 2 | 2 | no | — |
| 166 | C2×C16 | — | 2 | 2 | no | — |
| 167 | C2×C16 | — | 2 | 2 | no | — |
| 168 | C2×C16 | #41 | 1 | 4 | yes | complemented |
| 169 | #108 | — | 2 | 2 | no | — |
| 170 | #108 | — | 2 | 2 | no | — |
| 171 | #107 | C2×D4 | 1 | 2 | yes | — |
| 172 | #106 | — | 2 | 1 | no | — |
| 173 | C4⋊C8 | — | 2 | 4 | no | — |
| 174 | C4⋊C8 | — | 2 | 2 | no | — |
| 175 | #103 | — | 2 | 2 | no | — |
| 176 | #103 | — | 4 | 8 | no | — |
| 177 | #102 | — | 2 | 8 | no | — |
| 178 | #102 | — | 2 | 2 | no | — |
| 179 | #101 | — | 8 | 4 | no | — |
| 180 | #97 | — | 2 | 2 | no | — |
| 181 | #97 | — | 4 | 1 | no | — |
| 182 | C4×C8 | — | 2 | 2 | no | — |
| 183 | C4×C8 | — | 4 | 1 | no | — |
| 184 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 185 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 186 | C24 | — | 4 | 2 | no | — |
| 187 | #41 | — | 4 | 2 | no | — |
| 188 | #41 | — | 4 | 2 | no | — |
| 189 | #41 | — | 16 | 8 | no | — |
| 190 | C2×Q8 | — | 2 | 2 | no | — |
| 191 | C2×Q8 | — | 2 | 1 | no | — |
| 192 | C2×Q8 | — | 4 | 2 | no | — |
| 193 | C2×Q8 | — | 4 | 4 | no | — |
| 194 | C2×Q8 | — | 4 | 2 | no | — |
| 195 | C2×Q8 | — | 4 | 4 | no | — |
| 196 | C2×D4 | — | 2 | 4 | no | — |
| 197 | C2×D4 | — | 2 | 1 | no | — |
| 198 | C2×D4 | — | 4 | 8 | no | — |
| 199 | C2×D4 | — | 4 | 1 | no | — |
| 200 | C2×D4 | — | 4 | 4 | no | — |
| 201 | C2×D4 | — | 4 | 2 | no | — |
| 202 | C2×D4 | — | 4 | 4 | no | — |
| 203 | C2×D4 | — | 4 | 4 | no | — |
| 204 | C2×D4 | — | 4 | 4 | no | — |
| 205 | C2×D4 | — | 16 | 4 | no | — |
| 206 | C22×C4 | — | 4 | 8 | no | — |
| 207 | C22×C4 | — | 4 | 1 | no | — |
| 208 | C22×C4 | — | 4 | 2 | no | — |
| 209 | C22×C4 | — | 4 | 2 | no | — |
| 210 | C22×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 211 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 212 | Q16 | — | 2 | 1 | no | — |
| 213 | Q16 | — | 2 | 1 | no | — |
| 214 | Q16 | — | 4 | 2 | no | — |
| 215 | Q16 | — | 4 | 2 | no | — |
| 216 | Q16 | — | 4 | 4 | no | — |
| 217 | Q16 | — | 8 | 4 | no | — |
| 218 | SD16 | — | 4 | 2 | no | — |
| 219 | SD16 | — | 8 | 4 | no | — |
| 220 | D8 | — | 2 | 1 | no | — |
| 221 | D8 | — | 2 | 1 | no | — |
| 222 | D8 | — | 2 | 8 | no | — |
| 223 | D8 | — | 4 | 2 | no | — |
| 224 | D8 | — | 4 | 2 | no | — |
| 225 | D8 | — | 8 | 4 | no | — |
| 226 | D8 | — | 8 | 4 | no | — |
| 227 | M16 | — | 2 | 4 | no | — |
| 228 | M16 | — | 4 | 2 | no | — |
| 229 | M16 | — | 4 | 2 | no | — |
| 230 | C2×C8 | — | 2 | 2 | no | — |
| 231 | C2×C8 | — | 2 | 2 | no | — |
| 232 | C2×C8 | — | 4 | 2 | no | — |
| 233 | C2×C8 | #120 | 1 | 2 | yes | — |
| 234 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 237 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 238 | C4⋊C4 | — | 2 | 2 | no | — |
| 239 | C4⋊C4 | — | 2 | 4 | no | — |
| 240 | C4⋊C4 | — | 2 | 2 | no | — |
| 241 | C4⋊C4 | — | 4 | 2 | no | — |
| 242 | C4⋊C4 | — | 8 | 2 | no | — |
| 243 | C4⋊C4 | — | 8 | 2 | no | — |
| 244 | C4⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 245 | C22⋊C4 | — | 2 | 2 | no | — |
| 246 | C22⋊C4 | — | 4 | 8 | no | — |
| 247 | C22⋊C4 | — | 4 | 8 | no | — |
| 248 | C22⋊C4 | — | 4 | 1 | no | — |
| 249 | C22⋊C4 | — | 4 | 2 | no | — |
| 250 | C42 | — | 2 | 2 | no | — |
| 251 | C42 | — | 4 | 1 | no | — |
| 252 | C42 | — | 4 | 2 | no | — |
| 253 | C42 | #120 | 1 | 2 | yes | — |
| 254 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 255 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 256 | C16 | — | 2 | 4 | no | — |
| 257 | C16 | — | 4 | 2 | no | — |
| 258 | C23 | — | 4 | 4 | no | — |
| 259 | C23 | — | 4 | 1 | no | — |
| 260 | C23 | — | 8 | 4 | no | — |
| 261 | C23 | — | 16 | 4 | no | — |
| 262 | C23 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 263 | Q8 | — | 4 | 4 | no | — |
| 264 | Q8 | — | 4 | 1 | no | — |
| 265 | Q8 | — | 4 | 1 | no | — |
| 266 | Q8 | — | 8 | 4 | no | — |
| 267 | Q8 | — | 8 | 2 | no | — |
| 268 | Q8 | — | 8 | 2 | no | — |
| 269 | D4 | — | 4 | 4 | no | — |
| 270 | D4 | — | 4 | 1 | no | — |
| 271 | D4 | — | 4 | 1 | no | — |
| 272 | D4 | — | 4 | 4 | no | — |
| 273 | D4 | — | 8 | 2 | no | — |
| 274 | D4 | — | 8 | 4 | no | — |
| 275 | D4 | — | 8 | 8 | no | — |
| 276 | D4 | — | 16 | 4 | no | — |
| 277 | D4 | — | 16 | 8 | no | — |
| 278 | C2×C4 | — | 2 | 4 | no | — |
| 279 | C2×C4 | — | 2 | 4 | no | — |
| 280 | C2×C4 | — | 4 | 1 | no | — |
| 281 | C2×C4 | — | 4 | 2 | no | — |
| 282 | C2×C4 | — | 8 | 2 | no | — |
| 283 | C2×C4 | — | 8 | 4 | no | — |
| 284 | C2×C4 | — | 8 | 4 | no | — |
| 285 | C2×C4 | — | 8 | 2 | no | — |
| 286 | C2×C4 | — | 16 | 8 | no | — |
| 287 | C2×C4 | #447 | 1 | 2 | yes | — |
| 288 | C2×C4 | #447 | 1 | 2 | yes | — |
| 289 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #550 | 1 | 2 | yes | — |
| 291 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #576 | 1 | 2 | yes | — |
| 293 | C8 | — | 2 | 2 | no | — |
| 294 | C8 | — | 4 | 2 | no | — |
| 295 | C8 | — | 8 | 2 | no | — |
| 296 | C8 | #532 | 1 | 1 | yes | characteristic |
| 297 | C8 | #535 | 1 | 1 | yes | characteristic |
| 298 | C8 | #546 | 1 | 2 | yes | — |
| 299 | C22 | — | 2 | 2 | no | — |
| 300 | C22 | — | 8 | 1 | no | — |
| 301 | C22 | — | 8 | 4 | no | — |
| 302 | C22 | — | 8 | 2 | no | — |
| 303 | C22 | — | 8 | 8 | no | — |
| 304 | C22 | — | 32 | 4 | no | — |
| 305 | C22 | C2×#447 | 1 | 1 | yes | characteristic |
| 306 | C22 | #3399 | 1 | 2 | yes | — |
| 307 | C4 | — | 2 | 2 | no | — |
| 308 | C4 | — | 4 | 4 | no | — |
| 309 | C4 | — | 8 | 1 | no | — |
| 310 | C4 | — | 8 | 2 | no | — |
| 311 | C4 | — | 16 | 2 | no | — |
| 312 | C4 | #2194 | 1 | 4 | yes | — |
| 313 | C4 | #2431 | 1 | 1 | yes | characteristic |
| 314 | C4 | #3008 | 1 | 1 | yes | characteristic |
| 315 | C4 | #3391 | 1 | 2 | yes | — |
| 316 | C2 | — | 2 | 2 | no | — |
| 317 | C2 | — | 16 | 1 | no | — |
| 318 | C2 | — | 16 | 4 | no | — |
| 319 | C2 | #25631 | 1 | 1 | yes | characteristic |
| 320 | C2 | C2×#2194 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61409 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2248 | C2 | semidirect |
| 3. #33583 | C2 | semidirect |
| 4. #28762 | C2 | semidirect |
| 5. #13338 | C2 | semidirect |
| 6. #13266 | C2 | semidirect |
| 7. #13257 | C2 | semidirect |
| 8. #13019 | C2 | semidirect |
| 9. C4×#359 | C2 | semidirect |
| 168. C2×C16 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #61409 | 2 |
| C2×#2248 | 2 |
| #33583 | 2 |
| #28762 | 2 |
| #13338 | 2 |
| #13266 | 2 |
| #13257 | 2 |
| #13019 | 2 |
| C4×#359 | 2 |