Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
389 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 46 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 | #30497 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13592 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13574 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #13051 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #12975 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #9435 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7496 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3152 | C22 | 1 | 1 | yes | characteristic |
| 9 | #3069 | — | 2 | 1 | no | — |
| 10 | #3022 | — | 2 | 1 | no | — |
| 11 | #3018 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2979 | C22 | 1 | 1 | yes | characteristic |
| 13 | #2132 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2132 | C22 | 1 | 1 | yes | characteristic |
| 15 | #2049 | — | 2 | 1 | no | — |
| 16 | #2009 | — | 2 | 1 | no | — |
| 17 | #2005 | — | 2 | 1 | no | — |
| 18 | #1932 | — | 2 | 1 | no | — |
| 19 | #1884 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1858 | — | 2 | 1 | no | — |
| 21 | #1791 | — | 2 | 1 | no | — |
| 22 | #1560 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1549 | — | 2 | 1 | no | — |
| 24 | #1411 | — | 2 | 2 | no | — |
| 25 | #1405 | — | 2 | 2 | no | — |
| 26 | #1395 | — | 2 | 1 | no | — |
| 27 | #1392 | — | 2 | 1 | no | — |
| 28 | #1299 | — | 2 | 1 | no | — |
| 29 | #585 | D4 | 1 | 1 | yes | characteristic |
| 30 | #575 | — | 2 | 1 | no | — |
| 31 | C2×#137 | — | 2 | 1 | no | — |
| 32 | C2×#137 | — | 2 | 1 | no | — |
| 33 | C2×#137 | — | 2 | 1 | no | — |
| 34 | #568 | D4 | 1 | 1 | yes | characteristic |
| 35 | #536 | — | 2 | 1 | no | — |
| 36 | #532 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 38 | #497 | — | 2 | 1 | no | — |
| 39 | #497 | — | 2 | 1 | no | — |
| 40 | #494 | — | 2 | 1 | no | — |
| 41 | #492 | — | 2 | 1 | no | — |
| 42 | #473 | — | 4 | 2 | no | — |
| 43 | #471 | — | 4 | 2 | no | — |
| 44 | #456 | — | 2 | 1 | no | — |
| 45 | #456 | — | 4 | 2 | no | — |
| 46 | #455 | — | 2 | 1 | no | — |
| 47 | #455 | — | 4 | 2 | no | — |
| 48 | #454 | — | 2 | 1 | no | — |
| 49 | #453 | — | 2 | 1 | no | — |
| 50 | #436 | — | 4 | 1 | no | — |
| 51 | C8×D4 | — | 4 | 1 | no | — |
| 52 | #431 | — | 2 | 1 | no | — |
| 53 | #430 | — | 2 | 1 | no | — |
| 54 | #430 | — | 4 | 1 | no | — |
| 55 | #423 | — | 2 | 1 | no | — |
| 56 | #421 | — | 2 | 1 | no | — |
| 57 | #421 | — | 2 | 1 | no | — |
| 58 | C2×#104 | — | 2 | 1 | no | — |
| 59 | #419 | — | 2 | 1 | no | — |
| 60 | #413 | — | 2 | 1 | no | — |
| 61 | #413 | — | 2 | 1 | no | — |
| 62 | #413 | — | 2 | 1 | no | — |
| 63 | #413 | — | 4 | 1 | no | — |
| 64 | #413 | D4 | 1 | 1 | yes | characteristic |
| 65 | C2×#101 | — | 2 | 1 | no | — |
| 66 | #410 | D4 | 1 | 1 | yes | characteristic |
| 67 | #408 | — | 2 | 1 | no | — |
| 68 | #408 | — | 2 | 1 | no | — |
| 69 | #408 | — | 4 | 1 | no | — |
| 70 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 71 | #356 | — | 2 | 1 | no | — |
| 72 | #356 | D4 | 1 | 2 | yes | — |
| 73 | #355 | — | 2 | 1 | no | — |
| 74 | #355 | — | 2 | 1 | no | — |
| 75 | #344 | — | 2 | 1 | no | — |
| 76 | #341 | — | 4 | 1 | no | — |
| 77 | #339 | — | 2 | 1 | no | — |
| 78 | #333 | — | 2 | 1 | no | — |
| 79 | #332 | — | 4 | 1 | no | — |
| 80 | #332 | — | 4 | 1 | no | — |
| 81 | #331 | — | 2 | 1 | no | — |
| 82 | 2−1+4 | — | 2 | 1 | no | — |
| 83 | 2−1+4 | — | 4 | 1 | no | — |
| 84 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 85 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 86 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 87 | C2×#41 | — | 2 | 1 | no | — |
| 88 | C2×#41 | — | 4 | 1 | no | — |
| 89 | C2×#41 | — | 8 | 1 | no | — |
| 90 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C22×Q8 | — | 2 | 1 | no | — |
| 93 | #137 | — | 4 | 2 | no | — |
| 94 | #137 | — | 4 | 2 | no | — |
| 95 | #137 | — | 4 | 2 | no | — |
| 96 | #137 | — | 4 | 1 | no | — |
| 97 | #137 | — | 4 | 1 | no | — |
| 98 | #137 | — | 8 | 1 | no | — |
| 99 | #137 | — | 8 | 1 | no | — |
| 100 | #137 | — | 8 | 1 | no | — |
| 101 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 102 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 103 | #135 | — | 8 | 1 | no | — |
| 104 | C2×Q16 | — | 4 | 1 | no | — |
| 105 | C2×Q16 | — | 4 | 1 | no | — |
| 106 | C2×Q16 | — | 4 | 1 | no | — |
| 107 | C2×SD16 | — | 4 | 1 | no | — |
| 108 | C2×SD16 | — | 4 | 1 | no | — |
| 109 | C2×SD16 | — | 4 | 1 | no | — |
| 110 | #131 | — | 4 | 2 | no | — |
| 111 | C2×M16 | — | 2 | 1 | no | — |
| 112 | C2×M16 | — | 2 | 1 | no | — |
| 113 | C2×M16 | — | 2 | 1 | no | — |
| 114 | C2×M16 | — | 2 | 1 | no | — |
| 115 | C2×M16 | — | 2 | 1 | no | — |
| 116 | C2×M16 | — | 4 | 1 | no | — |
| 117 | C2×M16 | — | 4 | 1 | no | — |
| 118 | C2×M16 | — | 4 | 1 | no | — |
| 119 | C2×M16 | — | 4 | 1 | no | — |
| 120 | C22×C8 | — | 2 | 1 | no | — |
| 121 | C22×C8 | — | 2 | 1 | no | — |
| 122 | C4⋊Q8 | — | 2 | 1 | no | — |
| 123 | C4⋊Q8 | — | 2 | 1 | no | — |
| 124 | C4⋊Q8 | — | 4 | 1 | no | — |
| 125 | C4⋊Q8 | — | 4 | 1 | no | — |
| 126 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 127 | #127 | #41 | 1 | 1 | yes | characteristic |
| 128 | #124 | — | 4 | 1 | no | — |
| 129 | #124 | #41 | 1 | 1 | yes | characteristic |
| 130 | #124 | #41 | 1 | 1 | yes | characteristic |
| 131 | #123 | — | 8 | 1 | no | — |
| 132 | C22⋊Q8 | — | 8 | 1 | no | — |
| 133 | #121 | — | 4 | 1 | no | — |
| 134 | C4×D4 | — | 4 | 1 | no | — |
| 135 | #117 | — | 2 | 1 | no | — |
| 136 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 137 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 138 | #108 | — | 4 | 2 | no | — |
| 139 | #108 | — | 8 | 2 | no | — |
| 140 | C4⋊C8 | — | 2 | 1 | no | — |
| 141 | C4⋊C8 | — | 2 | 1 | no | — |
| 142 | C4⋊C8 | — | 4 | 1 | no | — |
| 143 | C4⋊C8 | — | 4 | 1 | no | — |
| 144 | #104 | — | 4 | 2 | no | — |
| 145 | #104 | — | 4 | 2 | no | — |
| 146 | #104 | — | 4 | 1 | no | — |
| 147 | #104 | — | 4 | 1 | no | — |
| 148 | #103 | — | 8 | 1 | no | — |
| 149 | #101 | — | 2 | 2 | no | — |
| 150 | #101 | — | 2 | 2 | no | — |
| 151 | #101 | — | 2 | 1 | no | — |
| 152 | #101 | — | 4 | 2 | no | — |
| 153 | #101 | — | 4 | 1 | no | — |
| 154 | #101 | — | 8 | 1 | no | — |
| 155 | #100 | — | 2 | 2 | no | — |
| 156 | #100 | — | 2 | 1 | no | — |
| 157 | #100 | — | 4 | 1 | no | — |
| 158 | #100 | — | 4 | 1 | no | — |
| 159 | #100 | — | 8 | 1 | no | — |
| 160 | #99 | — | 4 | 1 | no | — |
| 161 | C22⋊C8 | — | 4 | 1 | no | — |
| 162 | C22⋊C8 | — | 4 | 1 | no | — |
| 163 | C22⋊C8 | — | 4 | 1 | no | — |
| 164 | C22⋊C8 | — | 8 | 2 | no | — |
| 165 | #97 | — | 2 | 1 | no | — |
| 166 | C4×C8 | — | 2 | 1 | no | — |
| 167 | C4×C8 | — | 4 | 1 | no | — |
| 168 | #41 | — | 4 | 1 | no | — |
| 169 | #41 | — | 4 | 1 | no | — |
| 170 | #41 | — | 4 | 1 | no | — |
| 171 | #41 | — | 4 | 1 | no | — |
| 172 | #41 | — | 4 | 1 | no | — |
| 173 | #41 | — | 4 | 1 | no | — |
| 174 | #41 | — | 4 | 1 | no | — |
| 175 | #41 | — | 4 | 1 | no | — |
| 176 | #41 | — | 8 | 1 | no | — |
| 177 | #41 | — | 8 | 1 | no | — |
| 178 | #41 | — | 8 | 2 | no | — |
| 179 | #41 | — | 8 | 1 | no | — |
| 180 | #41 | — | 8 | 1 | no | — |
| 181 | #41 | — | 8 | 1 | no | — |
| 182 | C2×Q8 | — | 2 | 1 | no | — |
| 183 | C2×Q8 | — | 4 | 1 | no | — |
| 184 | C2×Q8 | — | 4 | 1 | no | — |
| 185 | C2×Q8 | — | 4 | 1 | no | — |
| 186 | C2×Q8 | — | 4 | 1 | no | — |
| 187 | C2×Q8 | — | 4 | 1 | no | — |
| 188 | C2×Q8 | — | 4 | 1 | no | — |
| 189 | C2×Q8 | — | 4 | 1 | no | — |
| 190 | C2×Q8 | — | 8 | 1 | no | — |
| 191 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 192 | C2×Q8 | #121 | 1 | 1 | yes | characteristic |
| 193 | C2×D4 | — | 2 | 1 | no | — |
| 194 | C2×D4 | — | 2 | 1 | no | — |
| 195 | C2×D4 | — | 2 | 1 | no | — |
| 196 | C2×D4 | — | 4 | 1 | no | — |
| 197 | C2×D4 | — | 4 | 1 | no | — |
| 198 | C2×D4 | — | 4 | 2 | no | — |
| 199 | C2×D4 | — | 4 | 1 | no | — |
| 200 | C2×D4 | — | 8 | 1 | no | — |
| 201 | C2×D4 | — | 8 | 2 | no | — |
| 202 | C2×D4 | #123 | 1 | 1 | yes | characteristic |
| 203 | C2×D4 | #124 | 1 | 1 | yes | characteristic |
| 204 | C22×C4 | — | 2 | 1 | no | — |
| 205 | C22×C4 | — | 2 | 1 | no | — |
| 206 | C22×C4 | — | 2 | 1 | no | — |
| 207 | C22×C4 | — | 4 | 1 | no | — |
| 208 | C22×C4 | — | 4 | 2 | no | — |
| 209 | C22×C4 | — | 4 | 1 | no | — |
| 210 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 211 | Q16 | — | 8 | 2 | no | — |
| 212 | Q16 | — | 8 | 2 | no | — |
| 213 | Q16 | — | 8 | 2 | no | — |
| 214 | Q16 | — | 8 | 1 | no | — |
| 215 | SD16 | — | 8 | 2 | no | — |
| 216 | SD16 | — | 8 | 2 | no | — |
| 217 | SD16 | — | 8 | 2 | no | — |
| 218 | SD16 | — | 8 | 1 | no | — |
| 219 | SD16 | — | 8 | 1 | no | — |
| 220 | D8 | — | 8 | 1 | no | — |
| 221 | M16 | — | 2 | 2 | no | — |
| 222 | M16 | — | 2 | 2 | no | — |
| 223 | M16 | — | 4 | 1 | no | — |
| 224 | M16 | — | 4 | 1 | no | — |
| 225 | M16 | — | 4 | 2 | no | — |
| 226 | M16 | — | 4 | 2 | no | — |
| 227 | M16 | — | 4 | 1 | no | — |
| 228 | M16 | — | 4 | 1 | no | — |
| 229 | M16 | — | 8 | 1 | no | — |
| 230 | M16 | — | 8 | 1 | no | — |
| 231 | M16 | — | 8 | 1 | no | — |
| 232 | M16 | — | 8 | 1 | no | — |
| 233 | M16 | — | 8 | 1 | no | — |
| 234 | M16 | — | 16 | 1 | no | — |
| 235 | C2×C8 | — | 4 | 1 | no | — |
| 236 | C2×C8 | — | 4 | 1 | no | — |
| 237 | C2×C8 | — | 4 | 1 | no | — |
| 238 | C2×C8 | — | 4 | 1 | no | — |
| 239 | C2×C8 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 8 | 1 | no | — |
| 241 | C2×C8 | — | 8 | 1 | no | — |
| 242 | C2×C8 | — | 8 | 1 | no | — |
| 243 | C2×C8 | — | 8 | 1 | no | — |
| 244 | C2×C8 | — | 8 | 1 | no | — |
| 245 | C2×C8 | — | 8 | 1 | no | — |
| 246 | C4⋊C4 | — | 4 | 1 | no | — |
| 247 | C4⋊C4 | — | 4 | 1 | no | — |
| 248 | C4⋊C4 | — | 4 | 2 | no | — |
| 249 | C4⋊C4 | — | 8 | 1 | no | — |
| 250 | C4⋊C4 | — | 8 | 2 | no | — |
| 251 | C22⋊C4 | — | 4 | 1 | no | — |
| 252 | C22⋊C4 | — | 4 | 1 | no | — |
| 253 | C22⋊C4 | — | 4 | 1 | no | — |
| 254 | C22⋊C4 | — | 8 | 2 | no | — |
| 255 | C42 | — | 2 | 1 | no | — |
| 256 | C42 | — | 4 | 1 | no | — |
| 257 | C42 | #121 | 1 | 1 | yes | characteristic |
| 258 | C42 | #123 | 1 | 1 | yes | characteristic |
| 259 | C23 | — | 2 | 1 | no | — |
| 260 | C23 | — | 2 | 1 | no | — |
| 261 | C23 | — | 4 | 1 | no | — |
| 262 | C23 | — | 4 | 2 | no | — |
| 263 | C23 | #458 | 1 | 1 | yes | characteristic |
| 264 | Q8 | — | 4 | 1 | no | — |
| 265 | Q8 | — | 4 | 1 | no | — |
| 266 | Q8 | — | 4 | 1 | no | — |
| 267 | Q8 | — | 4 | 1 | no | — |
| 268 | Q8 | — | 8 | 1 | no | — |
| 269 | Q8 | — | 8 | 1 | no | — |
| 270 | Q8 | — | 8 | 1 | no | — |
| 271 | Q8 | — | 8 | 1 | no | — |
| 272 | Q8 | — | 8 | 1 | no | — |
| 273 | D4 | — | 4 | 1 | no | — |
| 274 | D4 | — | 4 | 1 | no | — |
| 275 | D4 | — | 4 | 1 | no | — |
| 276 | D4 | — | 4 | 1 | no | — |
| 277 | D4 | — | 8 | 1 | no | — |
| 278 | D4 | — | 8 | 1 | no | — |
| 279 | D4 | — | 8 | 1 | no | — |
| 280 | D4 | — | 8 | 1 | no | — |
| 281 | D4 | — | 8 | 1 | no | — |
| 282 | D4 | — | 8 | 1 | no | — |
| 283 | D4 | — | 8 | 2 | no | — |
| 284 | D4 | — | 8 | 1 | no | — |
| 285 | C2×C4 | — | 2 | 1 | no | — |
| 286 | C2×C4 | — | 2 | 1 | no | — |
| 287 | C2×C4 | — | 2 | 1 | no | — |
| 288 | C2×C4 | — | 2 | 1 | no | — |
| 289 | C2×C4 | — | 2 | 1 | no | — |
| 290 | C2×C4 | — | 2 | 1 | no | — |
| 291 | C2×C4 | — | 4 | 1 | no | — |
| 292 | C2×C4 | — | 4 | 1 | no | — |
| 293 | C2×C4 | — | 4 | 1 | no | — |
| 294 | C2×C4 | — | 4 | 1 | no | — |
| 295 | C2×C4 | — | 4 | 1 | no | — |
| 296 | C2×C4 | — | 4 | 1 | no | — |
| 297 | C2×C4 | — | 4 | 1 | no | — |
| 298 | C2×C4 | — | 4 | 2 | no | — |
| 299 | C2×C4 | — | 8 | 1 | no | — |
| 300 | C2×C4 | — | 8 | 2 | no | — |
| 301 | C2×C4 | — | 8 | 1 | no | — |
| 302 | C2×C4 | — | 8 | 1 | no | — |
| 303 | C2×C4 | — | 8 | 1 | no | — |
| 304 | C2×C4 | — | 8 | 1 | no | — |
| 305 | C2×C4 | — | 8 | 1 | no | — |
| 306 | C2×C4 | #394 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 308 | C8 | — | 4 | 2 | no | — |
| 309 | C8 | — | 4 | 2 | no | — |
| 310 | C8 | — | 8 | 1 | no | — |
| 311 | C8 | — | 8 | 2 | no | — |
| 312 | C8 | — | 8 | 1 | no | — |
| 313 | C8 | — | 8 | 1 | no | — |
| 314 | C8 | — | 16 | 1 | no | — |
| 315 | C8 | — | 16 | 1 | no | — |
| 316 | C22 | — | 2 | 1 | no | — |
| 317 | C22 | — | 4 | 1 | no | — |
| 318 | C22 | — | 4 | 1 | no | — |
| 319 | C22 | — | 4 | 1 | no | — |
| 320 | C22 | — | 8 | 1 | no | — |
| 321 | C22 | — | 8 | 1 | no | — |
| 322 | C22 | — | 8 | 1 | no | — |
| 323 | C22 | — | 16 | 1 | no | — |
| 324 | C22 | — | 16 | 2 | no | — |
| 325 | C22 | #2044 | 1 | 1 | yes | characteristic |
| 326 | C4 | — | 2 | 1 | no | — |
| 327 | C4 | — | 2 | 1 | no | — |
| 328 | C4 | — | 2 | 1 | no | — |
| 329 | C4 | — | 4 | 1 | no | — |
| 330 | C4 | — | 4 | 1 | no | — |
| 331 | C4 | — | 4 | 1 | no | — |
| 332 | C4 | — | 8 | 1 | no | — |
| 333 | C4 | — | 16 | 1 | no | — |
| 334 | C4 | — | 16 | 1 | no | — |
| 335 | C4 | #2196 | 1 | 1 | yes | characteristic |
| 336 | C4 | #2200 | 1 | 1 | yes | characteristic |
| 337 | C2 | — | 2 | 1 | no | — |
| 338 | C2 | — | 4 | 1 | no | — |
| 339 | C2 | — | 8 | 1 | no | — |
| 340 | C2 | — | 8 | 1 | no | — |
| 341 | C2 | — | 16 | 1 | no | — |
| 342 | C2 | #13422 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13422 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #3018 | C22 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13592 | C2 | semidirect |
| 5. #12975 | C2 | semidirect |
| 6. #9435 | C2 | semidirect |
| 7. #7496 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30497 | 2 |
| #13592 | 2 |
| #13574 | 2 |
| #13051 | 2 |
| #12975 | 2 |
| #9435 | 2 |
| #7496 | 2 |