Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
422 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 80 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 | #22202 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13010 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #12931 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#1339 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11336 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7254 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7194 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#469 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#430 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#424 | — | 2 | 1 | no | — |
| 11 | #2324 | — | 2 | 1 | no | — |
| 12 | C2×#365 | — | 2 | 1 | no | — |
| 13 | C2×#361 | — | 2 | 1 | no | — |
| 14 | C2×#349 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#349 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1926 | — | 2 | 1 | no | — |
| 17 | #1920 | — | 2 | 1 | no | — |
| 18 | #1893 | — | 2 | 1 | no | — |
| 19 | #1879 | — | 2 | 1 | no | — |
| 20 | #1862 | — | 2 | 1 | no | — |
| 21 | #1809 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1733 | — | 2 | 1 | no | — |
| 23 | C2×H(3,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 24 | #1729 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1387 | — | 2 | 1 | no | — |
| 26 | #1383 | — | 2 | 1 | no | — |
| 27 | #1339 | C4 | 1 | 4 | yes | complemented |
| 28 | C22×D8 | D4 | 1 | 1 | yes | characteristic |
| 29 | C22×M16 | — | 2 | 1 | no | — |
| 30 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 31 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×#121 | — | 2 | 1 | no | — |
| 33 | C2×C4×D4 | — | 2 | 1 | no | — |
| 34 | C2×#117 | — | 2 | 1 | no | — |
| 35 | C2×#117 | D4 | 1 | 1 | yes | characteristic |
| 36 | C2×M32 | — | 2 | 1 | no | — |
| 37 | C2×M32 | — | 2 | 1 | no | — |
| 38 | #469 | — | 2 | 1 | no | — |
| 39 | #469 | — | 2 | 1 | no | — |
| 40 | #469 | C2×C4 | 1 | 2 | yes | — |
| 41 | #469 | Q8 | 1 | 2 | yes | — |
| 42 | #430 | — | 4 | 2 | no | — |
| 43 | C2×#108 | — | 2 | 1 | no | — |
| 44 | C2×#108 | — | 2 | 1 | no | — |
| 45 | C2×#107 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×#106 | D4 | 1 | 1 | yes | characteristic |
| 47 | C2×#106 | D4 | 1 | 1 | yes | characteristic |
| 48 | #424 | — | 4 | 2 | no | — |
| 49 | #424 | — | 4 | 2 | no | — |
| 50 | C2×C4⋊C8 | — | 4 | 2 | no | — |
| 51 | C2×#102 | — | 2 | 1 | no | — |
| 52 | C2×#102 | — | 2 | 1 | no | — |
| 53 | C2×#102 | — | 2 | 1 | no | — |
| 54 | #390 | — | 2 | 1 | no | — |
| 55 | #390 | — | 4 | 1 | no | — |
| 56 | #389 | — | 2 | 1 | no | — |
| 57 | #388 | — | 4 | 1 | no | — |
| 58 | #386 | — | 4 | 1 | no | — |
| 59 | #385 | — | 4 | 1 | no | — |
| 60 | #380 | — | 2 | 1 | no | — |
| 61 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 62 | #365 | — | 4 | 2 | no | — |
| 63 | #361 | — | 4 | 2 | no | — |
| 64 | #349 | — | 2 | 1 | no | — |
| 65 | #349 | — | 2 | 1 | no | — |
| 66 | #349 | — | 2 | 1 | no | — |
| 67 | #349 | C2×C4 | 1 | 2 | yes | complemented |
| 68 | #340 | — | 2 | 1 | no | — |
| 69 | #340 | — | 2 | 1 | no | — |
| 70 | #340 | — | 2 | 1 | no | — |
| 71 | #339 | — | 4 | 2 | no | — |
| 72 | H(3,ℤ/4) | — | 2 | 1 | no | — |
| 73 | H(3,ℤ/4) | C2×C4 | 1 | 2 | yes | complemented |
| 74 | #336 | — | 2 | 1 | no | — |
| 75 | #336 | — | 2 | 1 | no | — |
| 76 | C22×D4 | — | 2 | 1 | no | — |
| 77 | C22×D4 | — | 4 | 1 | no | — |
| 78 | C23×C4 | — | 4 | 1 | no | — |
| 79 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×D8 | — | 2 | 1 | no | — |
| 81 | C2×D8 | — | 4 | 1 | no | — |
| 82 | C2×D8 | — | 4 | 1 | no | — |
| 83 | C2×D8 | C4⋊C4 | 1 | 2 | yes | — |
| 84 | C2×M16 | — | 2 | 2 | no | — |
| 85 | C2×M16 | — | 4 | 1 | no | — |
| 86 | C2×M16 | — | 4 | 1 | no | — |
| 87 | C2×M16 | — | 4 | 1 | no | — |
| 88 | C2×M16 | — | 4 | 1 | no | — |
| 89 | C2×M16 | — | 4 | 1 | no | — |
| 90 | C2×M16 | — | 4 | 1 | no | — |
| 91 | C2×M16 | — | 4 | 1 | no | — |
| 92 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C2×M16 | C2×Q8 | 1 | 1 | yes | characteristic |
| 94 | C2×M16 | #41 | 1 | 1 | yes | characteristic |
| 95 | C2×M16 | #41 | 1 | 1 | yes | characteristic |
| 96 | C22×C8 | — | 2 | 1 | no | — |
| 97 | C22×C8 | — | 2 | 1 | no | — |
| 98 | C22×C8 | — | 2 | 1 | no | — |
| 99 | C22×C8 | — | 4 | 1 | no | — |
| 100 | C22×C8 | — | 4 | 1 | no | — |
| 101 | C22×C8 | — | 4 | 1 | no | — |
| 102 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | #121 | — | 2 | 2 | no | — |
| 105 | #121 | — | 2 | 2 | no | — |
| 106 | #121 | — | 4 | 1 | no | — |
| 107 | #121 | — | 4 | 1 | no | — |
| 108 | C4×D4 | — | 8 | 1 | no | — |
| 109 | C4×D4 | — | 8 | 1 | no | — |
| 110 | #117 | — | 2 | 2 | no | — |
| 111 | #117 | — | 2 | 2 | no | — |
| 112 | #117 | — | 4 | 1 | no | — |
| 113 | #117 | — | 4 | 1 | no | — |
| 114 | #117 | — | 4 | 1 | no | — |
| 115 | #117 | — | 4 | 1 | no | — |
| 116 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 117 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 118 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 119 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 120 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 121 | C2×C4⋊C4 | SD16 | 1 | 1 | yes | characteristic |
| 122 | C2×C4⋊C4 | SD16 | 1 | 1 | yes | characteristic |
| 123 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 124 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 125 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 126 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 127 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 128 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 129 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 130 | C2×C42 | — | 2 | 2 | no | — |
| 131 | C2×C42 | — | 2 | 1 | no | — |
| 132 | C2×C42 | — | 4 | 1 | no | — |
| 133 | M32 | — | 2 | 2 | no | — |
| 134 | M32 | — | 4 | 1 | no | — |
| 135 | M32 | — | 4 | 1 | no | — |
| 136 | M32 | — | 4 | 1 | no | — |
| 137 | C2×C16 | — | 4 | 1 | no | — |
| 138 | C2×C16 | — | 4 | 1 | no | — |
| 139 | #108 | — | 2 | 2 | no | — |
| 140 | #108 | — | 2 | 2 | no | — |
| 141 | #108 | — | 4 | 1 | no | — |
| 142 | #108 | — | 4 | 1 | no | — |
| 143 | #107 | — | 4 | 1 | no | — |
| 144 | #106 | — | 2 | 1 | no | — |
| 145 | #106 | — | 4 | 1 | no | — |
| 146 | #106 | C4⋊C4 | 1 | 2 | yes | — |
| 147 | C4⋊C8 | — | 4 | 8 | no | — |
| 148 | #102 | — | 2 | 2 | no | — |
| 149 | #102 | — | 2 | 2 | no | — |
| 150 | #102 | — | 8 | 1 | no | — |
| 151 | #102 | — | 8 | 1 | no | — |
| 152 | #95 | — | 4 | 1 | no | — |
| 153 | #95 | — | 4 | 1 | no | — |
| 154 | #95 | — | 4 | 1 | no | — |
| 155 | C24 | — | 4 | 1 | no | — |
| 156 | C24 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 157 | C2×D4 | — | 2 | 2 | no | — |
| 158 | C2×D4 | — | 4 | 2 | no | — |
| 159 | C2×D4 | — | 4 | 2 | no | — |
| 160 | C2×D4 | — | 4 | 1 | no | — |
| 161 | C2×D4 | — | 4 | 1 | no | — |
| 162 | C2×D4 | — | 4 | 1 | no | — |
| 163 | C2×D4 | — | 4 | 1 | no | — |
| 164 | C2×D4 | — | 4 | 1 | no | — |
| 165 | C2×D4 | — | 16 | 1 | no | — |
| 166 | C2×D4 | — | 16 | 1 | no | — |
| 167 | C22×C4 | — | 2 | 1 | no | — |
| 168 | C22×C4 | — | 2 | 1 | no | — |
| 169 | C22×C4 | — | 2 | 1 | no | — |
| 170 | C22×C4 | — | 2 | 1 | no | — |
| 171 | C22×C4 | — | 2 | 1 | no | — |
| 172 | C22×C4 | — | 2 | 1 | no | — |
| 173 | C22×C4 | — | 4 | 2 | no | — |
| 174 | C22×C4 | — | 4 | 1 | no | — |
| 175 | C22×C4 | — | 4 | 1 | no | — |
| 176 | C22×C4 | — | 4 | 1 | no | — |
| 177 | C22×C4 | — | 8 | 2 | no | — |
| 178 | C22×C4 | — | 16 | 1 | no | — |
| 179 | C22×C4 | — | 16 | 1 | no | — |
| 180 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 181 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 182 | C22×C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 183 | C22×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 184 | C22×C4 | #135 | 1 | 1 | yes | characteristic |
| 185 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 186 | D8 | — | 4 | 2 | no | — |
| 187 | D8 | — | 8 | 1 | no | — |
| 188 | M16 | — | 4 | 1 | no | — |
| 189 | M16 | — | 4 | 1 | no | — |
| 190 | M16 | — | 4 | 1 | no | — |
| 191 | M16 | — | 4 | 1 | no | — |
| 192 | M16 | — | 4 | 2 | no | — |
| 193 | M16 | — | 4 | 2 | no | — |
| 194 | M16 | — | 8 | 1 | no | — |
| 195 | M16 | — | 8 | 1 | no | — |
| 196 | C2×C8 | — | 2 | 1 | no | — |
| 197 | C2×C8 | — | 2 | 1 | no | — |
| 198 | C2×C8 | — | 2 | 1 | no | — |
| 199 | C2×C8 | — | 2 | 1 | no | — |
| 200 | C2×C8 | — | 4 | 2 | no | — |
| 201 | C2×C8 | — | 4 | 2 | no | — |
| 202 | C2×C8 | — | 4 | 1 | no | — |
| 203 | C2×C8 | — | 4 | 1 | no | — |
| 204 | C2×C8 | — | 4 | 1 | no | — |
| 205 | C2×C8 | — | 8 | 1 | no | — |
| 206 | C2×C8 | — | 8 | 1 | no | — |
| 207 | C2×C8 | — | 8 | 1 | no | — |
| 208 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 209 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 210 | C2×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 211 | C2×C8 | #123 | 1 | 1 | yes | characteristic |
| 212 | C4⋊C4 | — | 2 | 2 | no | — |
| 213 | C4⋊C4 | — | 4 | 1 | no | — |
| 214 | C4⋊C4 | — | 4 | 1 | no | — |
| 215 | C4⋊C4 | — | 4 | 2 | no | — |
| 216 | C4⋊C4 | — | 4 | 2 | no | — |
| 217 | C4⋊C4 | — | 4 | 1 | no | — |
| 218 | C4⋊C4 | — | 4 | 1 | no | — |
| 219 | C4⋊C4 | — | 4 | 1 | no | — |
| 220 | C4⋊C4 | — | 4 | 1 | no | — |
| 221 | C4⋊C4 | — | 4 | 1 | no | — |
| 222 | C4⋊C4 | — | 16 | 1 | no | — |
| 223 | C22⋊C4 | — | 4 | 2 | no | — |
| 224 | C22⋊C4 | — | 4 | 2 | no | — |
| 225 | C22⋊C4 | — | 4 | 2 | no | — |
| 226 | C22⋊C4 | — | 4 | 2 | no | — |
| 227 | C22⋊C4 | — | 8 | 1 | no | — |
| 228 | C22⋊C4 | — | 8 | 2 | no | — |
| 229 | C22⋊C4 | — | 16 | 1 | no | — |
| 230 | C22⋊C4 | — | 16 | 1 | no | — |
| 231 | C22⋊C4 | — | 16 | 1 | no | — |
| 232 | C42 | — | 2 | 4 | no | — |
| 233 | C42 | — | 4 | 2 | no | — |
| 234 | C42 | — | 4 | 1 | no | — |
| 235 | C42 | — | 4 | 1 | no | — |
| 236 | C42 | — | 16 | 1 | no | — |
| 237 | C16 | — | 4 | 2 | no | — |
| 238 | C16 | — | 8 | 1 | no | — |
| 239 | C23 | — | 2 | 1 | no | — |
| 240 | C23 | — | 2 | 1 | no | — |
| 241 | C23 | — | 4 | 1 | no | — |
| 242 | C23 | — | 4 | 1 | no | — |
| 243 | C23 | — | 4 | 2 | no | — |
| 244 | C23 | — | 8 | 1 | no | — |
| 245 | C23 | — | 8 | 1 | no | — |
| 246 | C23 | — | 8 | 1 | no | — |
| 247 | C23 | — | 8 | 1 | no | — |
| 248 | C23 | — | 8 | 1 | no | — |
| 249 | C23 | — | 8 | 1 | no | — |
| 250 | C23 | #442 | 1 | 1 | yes | characteristic |
| 251 | C23 | #450 | 1 | 1 | yes | characteristic |
| 252 | C23 | #478 | 1 | 1 | yes | characteristic |
| 253 | D4 | — | 4 | 2 | no | — |
| 254 | D4 | — | 4 | 2 | no | — |
| 255 | D4 | — | 8 | 1 | no | — |
| 256 | D4 | — | 8 | 1 | no | — |
| 257 | D4 | — | 16 | 2 | no | — |
| 258 | D4 | — | 16 | 2 | no | — |
| 259 | C2×C4 | — | 2 | 1 | no | — |
| 260 | C2×C4 | — | 2 | 1 | no | — |
| 261 | C2×C4 | — | 2 | 1 | no | — |
| 262 | C2×C4 | — | 2 | 1 | no | — |
| 263 | C2×C4 | — | 2 | 1 | no | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 2 | 1 | no | — |
| 267 | C2×C4 | — | 2 | 1 | no | — |
| 268 | C2×C4 | — | 2 | 1 | no | — |
| 269 | C2×C4 | — | 4 | 4 | no | — |
| 270 | C2×C4 | — | 4 | 1 | no | — |
| 271 | C2×C4 | — | 4 | 1 | no | — |
| 272 | C2×C4 | — | 4 | 1 | no | — |
| 273 | C2×C4 | — | 4 | 1 | no | — |
| 274 | C2×C4 | — | 4 | 2 | no | — |
| 275 | C2×C4 | — | 4 | 1 | no | — |
| 276 | C2×C4 | — | 4 | 1 | no | — |
| 277 | C2×C4 | — | 8 | 1 | no | — |
| 278 | C2×C4 | — | 8 | 1 | no | — |
| 279 | C2×C4 | — | 8 | 2 | no | — |
| 280 | C2×C4 | — | 8 | 2 | no | — |
| 281 | C2×C4 | — | 8 | 1 | no | — |
| 282 | C2×C4 | — | 8 | 1 | no | — |
| 283 | C2×C4 | — | 8 | 1 | no | — |
| 284 | C2×C4 | — | 8 | 1 | no | — |
| 285 | C2×C4 | — | 8 | 1 | no | — |
| 286 | C2×C4 | — | 8 | 1 | no | — |
| 287 | C2×C4 | — | 16 | 4 | no | — |
| 288 | C2×C4 | #385 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | C4×SD16 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #449 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | #453 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #456 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #476 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 2 | 1 | no | — |
| 297 | C8 | — | 2 | 1 | no | — |
| 298 | C8 | — | 2 | 1 | no | — |
| 299 | C8 | — | 2 | 1 | no | — |
| 300 | C8 | — | 8 | 1 | no | — |
| 301 | C8 | — | 8 | 2 | no | — |
| 302 | C22 | — | 2 | 1 | no | — |
| 303 | C22 | — | 2 | 1 | no | — |
| 304 | C22 | — | 2 | 1 | no | — |
| 305 | C22 | — | 2 | 1 | no | — |
| 306 | C22 | — | 4 | 1 | no | — |
| 307 | C22 | — | 4 | 1 | no | — |
| 308 | C22 | — | 4 | 1 | no | — |
| 309 | C22 | — | 4 | 1 | no | — |
| 310 | C22 | — | 8 | 1 | no | — |
| 311 | C22 | — | 8 | 2 | no | — |
| 312 | C22 | — | 8 | 2 | no | — |
| 313 | C22 | — | 8 | 2 | no | — |
| 314 | C22 | — | 16 | 1 | no | — |
| 315 | C22 | — | 16 | 1 | no | — |
| 316 | C22 | — | 16 | 1 | no | — |
| 317 | C22 | — | 16 | 1 | no | — |
| 318 | C22 | #1604 | 1 | 1 | yes | characteristic |
| 319 | C22 | #1613 | 1 | 1 | yes | characteristic |
| 320 | C22 | #1864 | 1 | 1 | yes | characteristic |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 8 | 2 | no | — |
| 324 | C4 | — | 8 | 4 | no | — |
| 325 | C4 | — | 8 | 1 | no | — |
| 326 | C4 | — | 8 | 1 | no | — |
| 327 | C4 | — | 16 | 1 | no | — |
| 328 | C4 | — | 16 | 1 | no | — |
| 329 | C4 | #1600 | 1 | 1 | yes | characteristic |
| 330 | C4 | #1609 | 1 | 1 | yes | characteristic |
| 331 | C4 | #1859 | 1 | 1 | yes | characteristic |
| 332 | C4 | #1867 | 1 | 1 | yes | characteristic |
| 333 | C2 | — | 2 | 1 | no | — |
| 334 | C2 | — | 2 | 1 | no | — |
| 335 | C2 | — | 4 | 1 | no | — |
| 336 | C2 | — | 4 | 1 | no | — |
| 337 | C2 | — | 16 | 2 | no | — |
| 338 | C2 | #8873 | 1 | 1 | yes | characteristic |
| 339 | C2 | #12301 | 1 | 1 | yes | characteristic |
| 340 | C2 | #12303 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12931 | C2 | semidirect |
| 5. #11336 | C2 | semidirect |
| 6. #7254 | C2 | semidirect |
| 7. #7194 | C2 | semidirect |
| 27. #1339 | C4 | semidirect |
| 67. #349 | C2×C4 | semidirect |
| 73. H(3,ℤ/4) | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22202 | 2 |
| #13010 | 2 |
| #12931 | 2 |
| C2×#1339 | 2 |
| #11336 | 2 |
| #7254 | 2 |
| #7194 | 2 |