Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
979 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 979 conjugacy classes of subgroups — too many to draw. Only its 152 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 | #62328 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #28498 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19639 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #19074 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #18715 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10108 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #9724 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #8241 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #3549 | C22 | 1 | 2 | yes | — |
| 10 | #3341 | — | 2 | 2 | no | — |
| 11 | #3325 | — | 2 | 4 | no | — |
| 12 | D4×D8 | — | 2 | 2 | no | — |
| 13 | #3253 | C22 | 1 | 2 | yes | — |
| 14 | #3227 | C22 | 1 | 4 | yes | complemented |
| 15 | #3199 | — | 2 | 2 | no | — |
| 16 | #3169 | C22 | 1 | 2 | yes | — |
| 17 | #3152 | C22 | 1 | 2 | yes | — |
| 18 | C2×#494 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | #3141 | C22 | 1 | 4 | yes | complemented |
| 20 | C2×#473 | — | 2 | 2 | no | — |
| 21 | #3058 | — | 2 | 2 | no | — |
| 22 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 23 | #2020 | — | 2 | 2 | no | — |
| 24 | #1966 | C22 | 1 | 2 | yes | complemented |
| 25 | C4×#108 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #1720 | — | 2 | 4 | no | — |
| 27 | #1709 | — | 2 | 2 | no | — |
| 28 | #1707 | — | 2 | 2 | no | — |
| 29 | #1659 | — | 2 | 8 | no | — |
| 30 | #1634 | — | 2 | 4 | no | — |
| 31 | #1629 | — | 2 | 4 | no | — |
| 32 | #1543 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1525 | C22 | 1 | 1 | yes | characteristic |
| 34 | #1525 | C22 | 1 | 4 | yes | complemented |
| 35 | #1484 | C22 | 1 | 4 | yes | complemented |
| 36 | C8×M16 | C22 | 1 | 2 | yes | complemented |
| 37 | #1299 | C22 | 1 | 2 | yes | complemented |
| 38 | C2×#137 | — | 2 | 4 | no | — |
| 39 | C22×Q16 | — | 2 | 1 | no | — |
| 40 | C22×Q16 | — | 2 | 1 | no | — |
| 41 | C22×D8 | — | 2 | 2 | no | — |
| 42 | C2×#131 | D4 | 1 | 2 | yes | — |
| 43 | H(3,𝔽4) | — | 2 | 4 | no | — |
| 44 | #555 | — | 2 | 4 | no | — |
| 45 | #555 | — | 2 | 2 | no | — |
| 46 | D42 | — | 2 | 4 | no | — |
| 47 | D42 | — | 2 | 2 | no | — |
| 48 | #535 | — | 2 | 4 | no | — |
| 49 | #535 | — | 4 | 4 | no | — |
| 50 | #532 | C23 | 1 | 4 | yes | — |
| 51 | #532 | C23 | 1 | 1 | yes | characteristic |
| 52 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 53 | #500 | D4 | 1 | 2 | yes | — |
| 54 | #498 | — | 2 | 4 | no | — |
| 55 | #497 | — | 2 | 4 | no | — |
| 56 | #494 | — | 2 | 2 | no | — |
| 57 | #494 | — | 2 | 2 | no | — |
| 58 | #494 | — | 2 | 2 | no | — |
| 59 | #493 | D4 | 1 | 2 | yes | — |
| 60 | #492 | — | 2 | 2 | no | — |
| 61 | #489 | — | 2 | 8 | no | — |
| 62 | #487 | — | 2 | 4 | no | — |
| 63 | #486 | — | 2 | 4 | no | — |
| 64 | #486 | — | 2 | 4 | no | — |
| 65 | #486 | D4 | 1 | 4 | yes | — |
| 66 | #482 | — | 2 | 8 | no | — |
| 67 | #481 | — | 4 | 4 | no | — |
| 68 | #476 | — | 2 | 4 | no | — |
| 69 | #476 | — | 2 | 8 | no | — |
| 70 | #473 | — | 4 | 8 | no | — |
| 71 | #469 | — | 2 | 4 | no | — |
| 72 | #469 | — | 2 | 4 | no | — |
| 73 | #466 | — | 2 | 4 | no | — |
| 74 | #466 | — | 2 | 2 | no | — |
| 75 | #466 | — | 2 | 2 | no | — |
| 76 | #459 | — | 2 | 4 | no | — |
| 77 | #447 | — | 2 | 8 | no | — |
| 78 | C4×D8 | — | 2 | 2 | no | — |
| 79 | C4×D8 | — | 2 | 2 | no | — |
| 80 | #436 | — | 2 | 2 | no | — |
| 81 | C8×D4 | — | 2 | 2 | no | — |
| 82 | #431 | C23 | 1 | 2 | yes | — |
| 83 | C2×#108 | — | 2 | 1 | no | — |
| 84 | #427 | — | 2 | 2 | no | — |
| 85 | #423 | C23 | 1 | 1 | yes | characteristic |
| 86 | #418 | — | 2 | 4 | no | — |
| 87 | #417 | — | 4 | 4 | no | — |
| 88 | C2×#101 | — | 2 | 4 | no | — |
| 89 | #408 | D4 | 1 | 2 | yes | — |
| 90 | C4×M16 | C23 | 1 | 2 | yes | — |
| 91 | C4×M16 | C23 | 1 | 2 | yes | — |
| 92 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 93 | #341 | — | 2 | 1 | no | — |
| 94 | #333 | D4 | 1 | 2 | yes | — |
| 95 | #331 | D4 | 1 | 2 | yes | — |
| 96 | #329 | — | 2 | 8 | no | — |
| 97 | #329 | D4 | 1 | 4 | yes | — |
| 98 | D4⋊C8 | — | 2 | 8 | no | — |
| 99 | #322 | — | 2 | 2 | no | — |
| 100 | C82 | — | 2 | 2 | no | — |
| 101 | C2×#41 | — | 2 | 4 | no | — |
| 102 | C2×#41 | C2×D4 | 1 | 2 | yes | — |
| 103 | C22×Q8 | — | 2 | 2 | no | — |
| 104 | C22×D4 | — | 2 | 4 | no | — |
| 105 | C22×D4 | — | 4 | 4 | no | — |
| 106 | #137 | — | 4 | 8 | no | — |
| 107 | #137 | — | 8 | 4 | no | — |
| 108 | C2×Q16 | — | 4 | 4 | no | — |
| 109 | C2×Q16 | — | 4 | 4 | no | — |
| 110 | C2×Q16 | — | 4 | 2 | no | — |
| 111 | C2×Q16 | — | 4 | 1 | no | — |
| 112 | C2×Q16 | — | 4 | 1 | no | — |
| 113 | C2×Q16 | — | 8 | 2 | no | — |
| 114 | C2×SD16 | — | 4 | 4 | no | — |
| 115 | C2×D8 | — | 2 | 4 | no | — |
| 116 | C2×D8 | — | 2 | 2 | no | — |
| 117 | C2×D8 | — | 2 | 2 | no | — |
| 118 | C2×D8 | — | 8 | 4 | no | — |
| 119 | #131 | — | 4 | 4 | no | — |
| 120 | C2×M16 | — | 2 | 2 | no | — |
| 121 | C2×M16 | — | 2 | 2 | no | — |
| 122 | C2×M16 | — | 2 | 2 | no | — |
| 123 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 124 | C22×C8 | — | 2 | 2 | no | — |
| 125 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 126 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | C4⋊Q8 | — | 2 | 2 | no | — |
| 128 | C4⋊Q8 | — | 2 | 4 | no | — |
| 129 | C4⋊Q8 | — | 2 | 2 | no | — |
| 130 | C4⋊Q8 | C2×D4 | 1 | 4 | yes | — |
| 131 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 132 | #127 | — | 4 | 4 | no | — |
| 133 | #127 | C2×D4 | 1 | 4 | yes | — |
| 134 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 135 | #124 | — | 2 | 4 | no | — |
| 136 | #124 | — | 4 | 8 | no | — |
| 137 | #124 | — | 4 | 4 | no | — |
| 138 | #124 | C2×D4 | 1 | 2 | yes | — |
| 139 | #121 | — | 2 | 2 | no | — |
| 140 | #121 | — | 2 | 8 | no | — |
| 141 | #121 | — | 2 | 4 | no | — |
| 142 | #121 | — | 2 | 4 | no | — |
| 143 | #121 | — | 4 | 4 | no | — |
| 144 | #121 | — | 4 | 4 | no | — |
| 145 | #120 | — | 4 | 8 | no | — |
| 146 | #120 | — | 4 | 4 | no | — |
| 147 | #120 | — | 4 | 4 | no | — |
| 148 | C4×D4 | — | 2 | 2 | no | — |
| 149 | C4×D4 | — | 2 | 4 | no | — |
| 150 | C4×D4 | — | 2 | 4 | no | — |
| 151 | #117 | — | 2 | 4 | no | — |
| 152 | #117 | — | 4 | 4 | no | — |
| 153 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 154 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 155 | #108 | — | 2 | 4 | no | — |
| 156 | #107 | — | 2 | 2 | no | — |
| 157 | #107 | — | 2 | 2 | no | — |
| 158 | #106 | — | 2 | 4 | no | — |
| 159 | #106 | — | 4 | 8 | no | — |
| 160 | C4⋊C8 | — | 2 | 2 | no | — |
| 161 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 162 | #103 | — | 4 | 8 | no | — |
| 163 | #102 | — | 2 | 8 | no | — |
| 164 | #102 | — | 2 | 4 | no | — |
| 165 | #102 | — | 2 | 4 | no | — |
| 166 | #102 | — | 4 | 8 | no | — |
| 167 | #102 | — | 4 | 8 | no | — |
| 168 | #101 | — | 4 | 8 | no | — |
| 169 | C22⋊C8 | — | 2 | 2 | no | — |
| 170 | C22⋊C8 | — | 2 | 4 | no | — |
| 171 | #97 | — | 2 | 2 | no | — |
| 172 | #97 | C2×D4 | 1 | 4 | yes | — |
| 173 | C4×C8 | — | 2 | 2 | no | — |
| 174 | C4×C8 | — | 2 | 2 | no | — |
| 175 | C4×C8 | — | 2 | 2 | no | — |
| 176 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 177 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 178 | C24 | — | 4 | 4 | no | — |
| 179 | #41 | — | 4 | 2 | no | — |
| 180 | #41 | — | 4 | 2 | no | — |
| 181 | #41 | — | 8 | 8 | no | — |
| 182 | C2×Q8 | — | 2 | 4 | no | — |
| 183 | C2×Q8 | — | 4 | 4 | no | — |
| 184 | C2×Q8 | — | 4 | 4 | no | — |
| 185 | C2×Q8 | — | 4 | 2 | no | — |
| 186 | C2×Q8 | — | 4 | 2 | no | — |
| 187 | C2×Q8 | #127 | 1 | 2 | yes | — |
| 188 | C2×D4 | — | 2 | 2 | no | — |
| 189 | C2×D4 | — | 2 | 4 | no | — |
| 190 | C2×D4 | — | 2 | 2 | no | — |
| 191 | C2×D4 | — | 2 | 4 | no | — |
| 192 | C2×D4 | — | 2 | 4 | no | — |
| 193 | C2×D4 | — | 4 | 4 | no | — |
| 194 | C2×D4 | — | 4 | 8 | no | — |
| 195 | C2×D4 | — | 4 | 4 | no | — |
| 196 | C2×D4 | — | 8 | 4 | no | — |
| 197 | C2×D4 | — | 8 | 4 | no | — |
| 198 | C2×D4 | — | 16 | 8 | no | — |
| 199 | C2×D4 | #127 | 1 | 2 | yes | — |
| 200 | C22×C4 | — | 2 | 2 | no | — |
| 201 | C22×C4 | — | 4 | 4 | no | — |
| 202 | C22×C4 | — | 4 | 2 | no | — |
| 203 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 204 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 205 | Q16 | — | 8 | 2 | no | — |
| 206 | Q16 | — | 8 | 4 | no | — |
| 207 | Q16 | — | 8 | 8 | no | — |
| 208 | Q16 | — | 8 | 2 | no | — |
| 209 | SD16 | — | 8 | 8 | no | — |
| 210 | D8 | — | 4 | 4 | no | — |
| 211 | D8 | — | 8 | 2 | no | — |
| 212 | D8 | — | 8 | 4 | no | — |
| 213 | M16 | — | 2 | 4 | no | — |
| 214 | M16 | — | 4 | 4 | no | — |
| 215 | M16 | — | 4 | 2 | no | — |
| 216 | M16 | — | 4 | 2 | no | — |
| 217 | M16 | — | 8 | 2 | no | — |
| 218 | C2×C8 | — | 2 | 2 | no | — |
| 219 | C2×C8 | — | 2 | 2 | no | — |
| 220 | C2×C8 | — | 2 | 4 | no | — |
| 221 | C2×C8 | — | 2 | 4 | no | — |
| 222 | C2×C8 | — | 4 | 1 | no | — |
| 223 | C2×C8 | — | 4 | 2 | no | — |
| 224 | C2×C8 | — | 4 | 4 | no | — |
| 225 | C2×C8 | #120 | 1 | 2 | yes | — |
| 226 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 227 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 228 | C4⋊C4 | — | 2 | 2 | no | — |
| 229 | C4⋊C4 | — | 2 | 4 | no | — |
| 230 | C4⋊C4 | — | 2 | 4 | no | — |
| 231 | C4⋊C4 | — | 4 | 4 | no | — |
| 232 | C4⋊C4 | — | 4 | 4 | no | — |
| 233 | C22⋊C4 | — | 2 | 4 | no | — |
| 234 | C22⋊C4 | — | 4 | 8 | no | — |
| 235 | C22⋊C4 | — | 4 | 4 | no | — |
| 236 | C22⋊C4 | — | 4 | 4 | no | — |
| 237 | C22⋊C4 | — | 4 | 4 | no | — |
| 238 | C42 | — | 4 | 4 | no | — |
| 239 | C42 | C22×D4 | 1 | 2 | yes | — |
| 240 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 241 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 242 | C23 | — | 2 | 2 | no | — |
| 243 | C23 | — | 4 | 4 | no | — |
| 244 | C23 | — | 16 | 4 | no | — |
| 245 | C23 | — | 16 | 8 | no | — |
| 246 | C23 | #535 | 1 | 1 | yes | characteristic |
| 247 | Q8 | — | 4 | 2 | no | — |
| 248 | Q8 | — | 8 | 4 | no | — |
| 249 | Q8 | — | 8 | 4 | no | — |
| 250 | Q8 | — | 8 | 2 | no | — |
| 251 | Q8 | — | 8 | 2 | no | — |
| 252 | D4 | — | 4 | 2 | no | — |
| 253 | D4 | — | 4 | 4 | no | — |
| 254 | D4 | — | 4 | 4 | no | — |
| 255 | D4 | — | 4 | 4 | no | — |
| 256 | D4 | — | 4 | 8 | no | — |
| 257 | D4 | — | 8 | 4 | no | — |
| 258 | D4 | — | 8 | 2 | no | — |
| 259 | D4 | — | 16 | 4 | no | — |
| 260 | D4 | — | 16 | 4 | no | — |
| 261 | D4 | — | 16 | 8 | no | — |
| 262 | C2×C4 | — | 2 | 1 | no | — |
| 263 | C2×C4 | — | 2 | 1 | no | — |
| 264 | C2×C4 | — | 2 | 4 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 4 | 2 | no | — |
| 267 | C2×C4 | — | 4 | 2 | no | — |
| 268 | C2×C4 | — | 4 | 4 | no | — |
| 269 | C2×C4 | — | 8 | 8 | no | — |
| 270 | C2×C4 | — | 8 | 4 | no | — |
| 271 | C2×C4 | — | 8 | 2 | no | — |
| 272 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | D42 | 1 | 2 | yes | — |
| 275 | C2×C4 | D42 | 1 | 2 | yes | — |
| 276 | C8 | — | 2 | 1 | no | — |
| 277 | C8 | — | 2 | 2 | no | — |
| 278 | C8 | — | 4 | 4 | no | — |
| 279 | C8 | — | 4 | 4 | no | — |
| 280 | C8 | — | 4 | 2 | no | — |
| 281 | C8 | — | 4 | 4 | no | — |
| 282 | C8 | #453 | 1 | 2 | yes | — |
| 283 | C22 | — | 2 | 1 | no | — |
| 284 | C22 | — | 2 | 2 | no | — |
| 285 | C22 | — | 4 | 2 | no | — |
| 286 | C22 | — | 4 | 4 | no | — |
| 287 | C22 | — | 8 | 4 | no | — |
| 288 | C22 | — | 8 | 4 | no | — |
| 289 | C22 | — | 8 | 4 | no | — |
| 290 | C22 | — | 32 | 8 | no | — |
| 291 | C22 | #2433 | 1 | 1 | yes | characteristic |
| 292 | C4 | — | 2 | 1 | no | — |
| 293 | C4 | — | 2 | 2 | no | — |
| 294 | C4 | — | 4 | 2 | no | — |
| 295 | C4 | — | 8 | 4 | no | — |
| 296 | C4 | — | 16 | 2 | no | — |
| 297 | C4 | C2×#453 | 1 | 1 | yes | characteristic |
| 298 | C4 | #3021 | 1 | 1 | yes | characteristic |
| 299 | C4 | #3290 | 1 | 4 | yes | — |
| 300 | C2 | — | 4 | 1 | no | — |
| 301 | C2 | — | 8 | 2 | no | — |
| 302 | C2 | — | 16 | 4 | no | — |
| 303 | C2 | #25739 | 1 | 1 | yes | characteristic |
| 304 | C2 | #25804 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62328 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #28498 | C2 | semidirect |
| 3. #19639 | C2 | semidirect |
| 4. #19074 | C2 | semidirect |
| 5. #18715 | C2 | semidirect |
| 6. #10108 | C2 | semidirect |
| 7. #9724 | C2 | semidirect |
| 8. #8241 | C2 | semidirect |
| 14. #3227 | C22 | semidirect |
| 18. C2×#494 | C22 | semidirect |
| 19. #3141 | C22 | semidirect |
| 24. #1966 | C22 | semidirect |
| 25. C4×#108 | C22 | semidirect |
| 34. #1525 | C22 | semidirect |
| 35. #1484 | C22 | semidirect |
| 36. C8×M16 | C22 | semidirect |
| 37. #1299 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62328 | 2 |
| #28498 | 2 |
| #19639 | 2 |
| #19074 | 2 |
| #18715 | 2 |
| #10108 | 2 |
| #9724 | 2 |
| #8241 | 2 |