Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
427 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #33779 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19639 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11965 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11678 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #9787 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7165 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×Q32 | — | 2 | 1 | no | — |
| 9 | C22×SD32 | — | 2 | 1 | no | — |
| 10 | #3227 | — | 2 | 1 | no | — |
| 11 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#495 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3141 | — | 2 | 1 | no | — |
| 15 | #2251 | — | 2 | 2 | no | — |
| 16 | #2251 | — | 2 | 2 | no | — |
| 17 | #2250 | — | 2 | 1 | no | — |
| 18 | #2243 | — | 2 | 1 | no | — |
| 19 | #2210 | — | 2 | 2 | no | — |
| 20 | #2185 | — | 2 | 1 | no | — |
| 21 | #2184 | — | 2 | 1 | no | — |
| 22 | #2163 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1720 | — | 2 | 2 | no | — |
| 25 | #1709 | — | 2 | 1 | no | — |
| 26 | #1707 | — | 2 | 1 | no | — |
| 27 | #1568 | C22 | 1 | 1 | yes | characteristic |
| 28 | #1527 | — | 2 | 1 | no | — |
| 29 | #1469 | — | 2 | 1 | no | — |
| 30 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1362 | — | 2 | 2 | no | — |
| 32 | #1333 | — | 2 | 2 | no | — |
| 33 | C22×Q16 | — | 2 | 1 | no | — |
| 34 | C22×Q16 | — | 2 | 1 | no | — |
| 35 | C22×Q16 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×SD16 | — | 2 | 1 | no | — |
| 37 | C22×D8 | D4 | 1 | 1 | yes | characteristic |
| 38 | #532 | — | 2 | 1 | no | — |
| 39 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 40 | C2×#124 | — | 2 | 1 | no | — |
| 41 | C2×Q32 | — | 2 | 4 | no | — |
| 42 | C2×Q32 | — | 4 | 4 | no | — |
| 43 | C2×SD32 | — | 2 | 4 | no | — |
| 44 | C2×SD32 | — | 2 | 2 | no | — |
| 45 | C2×SD32 | — | 4 | 4 | no | — |
| 46 | C2×D16 | — | 2 | 2 | no | — |
| 47 | C22×C16 | — | 2 | 1 | no | — |
| 48 | #500 | D4 | 1 | 1 | yes | characteristic |
| 49 | #498 | — | 2 | 2 | no | — |
| 50 | #495 | — | 2 | 1 | no | — |
| 51 | #495 | — | 2 | 1 | no | — |
| 52 | #495 | — | 2 | 1 | no | — |
| 53 | #495 | — | 2 | 1 | no | — |
| 54 | #494 | — | 2 | 1 | no | — |
| 55 | #494 | — | 2 | 1 | no | — |
| 56 | #494 | — | 2 | 1 | no | — |
| 57 | #494 | D4 | 1 | 2 | yes | — |
| 58 | #493 | D4 | 1 | 1 | yes | characteristic |
| 59 | #489 | — | 4 | 2 | no | — |
| 60 | #487 | — | 2 | 2 | no | — |
| 61 | #486 | — | 2 | 2 | no | — |
| 62 | #486 | — | 2 | 2 | no | — |
| 63 | #486 | — | 2 | 1 | no | — |
| 64 | #481 | — | 4 | 2 | no | — |
| 65 | #476 | — | 4 | 2 | no | — |
| 66 | #466 | — | 2 | 1 | no | — |
| 67 | #466 | — | 2 | 1 | no | — |
| 68 | C4×D8 | — | 2 | 1 | no | — |
| 69 | C4×D8 | — | 2 | 1 | no | — |
| 70 | #432 | — | 2 | 1 | no | — |
| 71 | #431 | D4 | 1 | 1 | yes | characteristic |
| 72 | #417 | — | 4 | 2 | no | — |
| 73 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 74 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 75 | C4⋊C16 | D4 | 1 | 2 | yes | — |
| 76 | #358 | — | 2 | 2 | no | — |
| 77 | #357 | — | 2 | 2 | no | — |
| 78 | C22⋊C16 | — | 2 | 1 | no | — |
| 79 | #346 | — | 2 | 1 | no | — |
| 80 | #346 | — | 2 | 1 | no | — |
| 81 | #346 | D4 | 1 | 2 | yes | — |
| 82 | #335 | — | 2 | 1 | no | — |
| 83 | #334 | — | 2 | 1 | no | — |
| 84 | #329 | — | 4 | 2 | no | — |
| 85 | #323 | — | 4 | 2 | no | — |
| 86 | #322 | — | 2 | 1 | no | — |
| 87 | C82 | — | 2 | 1 | no | — |
| 88 | C2×#41 | — | 4 | 1 | no | — |
| 89 | C22×Q8 | — | 2 | 1 | no | — |
| 90 | C22×Q8 | — | 4 | 1 | no | — |
| 91 | C22×D4 | — | 2 | 1 | no | — |
| 92 | C2×Q16 | — | 2 | 2 | no | — |
| 93 | C2×Q16 | — | 2 | 1 | no | — |
| 94 | C2×Q16 | — | 2 | 1 | no | — |
| 95 | C2×Q16 | — | 2 | 2 | no | — |
| 96 | C2×Q16 | — | 4 | 2 | no | — |
| 97 | C2×Q16 | — | 4 | 2 | no | — |
| 98 | C2×Q16 | — | 4 | 2 | no | — |
| 99 | C2×Q16 | — | 4 | 1 | no | — |
| 100 | C2×Q16 | — | 4 | 1 | no | — |
| 101 | C2×Q16 | — | 8 | 2 | no | — |
| 102 | C2×SD16 | — | 4 | 1 | no | — |
| 103 | C2×SD16 | — | 4 | 1 | no | — |
| 104 | C2×SD16 | — | 8 | 2 | no | — |
| 105 | C2×D8 | — | 2 | 1 | no | — |
| 106 | C2×D8 | — | 2 | 1 | no | — |
| 107 | C2×D8 | — | 2 | 1 | no | — |
| 108 | C2×D8 | — | 2 | 1 | no | — |
| 109 | C2×D8 | — | 4 | 2 | no | — |
| 110 | C2×M16 | — | 2 | 1 | no | — |
| 111 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 113 | C4⋊Q8 | — | 2 | 2 | no | — |
| 114 | C4⋊Q8 | — | 2 | 1 | no | — |
| 115 | C4⋊Q8 | — | 4 | 1 | no | — |
| 116 | C4⋊Q8 | — | 4 | 1 | no | — |
| 117 | C4⋊Q8 | — | 4 | 1 | no | — |
| 118 | #127 | — | 2 | 1 | no | — |
| 119 | #124 | — | 4 | 1 | no | — |
| 120 | #124 | — | 4 | 1 | no | — |
| 121 | #124 | — | 4 | 1 | no | — |
| 122 | #124 | — | 4 | 1 | no | — |
| 123 | #124 | — | 4 | 1 | no | — |
| 124 | #121 | — | 4 | 1 | no | — |
| 125 | #121 | — | 4 | 1 | no | — |
| 126 | C4×D4 | — | 4 | 1 | no | — |
| 127 | C4×D4 | — | 4 | 1 | no | — |
| 128 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 129 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 130 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 131 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 132 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 133 | Q32 | — | 4 | 8 | no | — |
| 134 | SD32 | — | 4 | 8 | no | — |
| 135 | SD32 | — | 8 | 2 | no | — |
| 136 | D16 | — | 8 | 2 | no | — |
| 137 | C2×C16 | — | 2 | 4 | no | — |
| 138 | C2×C16 | — | 2 | 2 | no | — |
| 139 | C2×C16 | — | 2 | 2 | no | — |
| 140 | #107 | — | 2 | 1 | no | — |
| 141 | #107 | — | 2 | 1 | no | — |
| 142 | #106 | — | 4 | 4 | no | — |
| 143 | C4⋊C8 | — | 2 | 1 | no | — |
| 144 | C4⋊C8 | — | 4 | 1 | no | — |
| 145 | #103 | — | 4 | 4 | no | — |
| 146 | #102 | — | 4 | 4 | no | — |
| 147 | #102 | — | 4 | 4 | no | — |
| 148 | #102 | — | 4 | 1 | no | — |
| 149 | #102 | — | 4 | 1 | no | — |
| 150 | C22⋊C8 | — | 4 | 1 | no | — |
| 151 | C22⋊C8 | D8 | 1 | 2 | yes | — |
| 152 | #97 | — | 4 | 1 | no | — |
| 153 | #97 | D8 | 1 | 2 | yes | — |
| 154 | C4×C8 | — | 2 | 1 | no | — |
| 155 | C4×C8 | — | 2 | 1 | no | — |
| 156 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 157 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 158 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 159 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 160 | C24 | — | 4 | 1 | no | — |
| 161 | #41 | — | 8 | 4 | no | — |
| 162 | C2×Q8 | — | 4 | 1 | no | — |
| 163 | C2×Q8 | — | 4 | 2 | no | — |
| 164 | C2×Q8 | — | 4 | 1 | no | — |
| 165 | C2×Q8 | — | 4 | 1 | no | — |
| 166 | C2×Q8 | — | 8 | 2 | no | — |
| 167 | C2×Q8 | — | 8 | 2 | no | — |
| 168 | C2×Q8 | — | 8 | 1 | no | — |
| 169 | C2×Q8 | — | 8 | 1 | no | — |
| 170 | C2×Q8 | — | 8 | 1 | no | — |
| 171 | C2×D4 | — | 4 | 1 | no | — |
| 172 | C2×D4 | — | 4 | 1 | no | — |
| 173 | C2×D4 | — | 4 | 1 | no | — |
| 174 | C2×D4 | — | 4 | 1 | no | — |
| 175 | C2×D4 | — | 4 | 1 | no | — |
| 176 | C2×D4 | — | 8 | 1 | no | — |
| 177 | C2×D4 | — | 8 | 1 | no | — |
| 178 | C2×D4 | — | 8 | 1 | no | — |
| 179 | C22×C4 | — | 4 | 1 | no | — |
| 180 | C22×C4 | — | 8 | 1 | no | — |
| 181 | C22×C4 | — | 8 | 1 | no | — |
| 182 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 183 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 184 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 185 | Q16 | — | 4 | 2 | no | — |
| 186 | Q16 | — | 4 | 2 | no | — |
| 187 | Q16 | — | 4 | 2 | no | — |
| 188 | Q16 | — | 4 | 2 | no | — |
| 189 | Q16 | — | 8 | 2 | no | — |
| 190 | Q16 | — | 8 | 2 | no | — |
| 191 | Q16 | — | 8 | 2 | no | — |
| 192 | SD16 | — | 8 | 2 | no | — |
| 193 | SD16 | — | 8 | 2 | no | — |
| 194 | D8 | — | 4 | 2 | no | — |
| 195 | D8 | — | 4 | 2 | no | — |
| 196 | D8 | — | 4 | 2 | no | — |
| 197 | D8 | — | 4 | 2 | no | — |
| 198 | M16 | — | 4 | 2 | no | — |
| 199 | C2×C8 | — | 2 | 2 | no | — |
| 200 | C2×C8 | — | 2 | 2 | no | — |
| 201 | C2×C8 | — | 2 | 2 | no | — |
| 202 | C2×C8 | — | 2 | 2 | no | — |
| 203 | C2×C8 | — | 8 | 1 | no | — |
| 204 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 205 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 206 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 207 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 208 | C4⋊C4 | — | 4 | 1 | no | — |
| 209 | C4⋊C4 | — | 4 | 2 | no | — |
| 210 | C4⋊C4 | — | 4 | 2 | no | — |
| 211 | C4⋊C4 | — | 4 | 1 | no | — |
| 212 | C4⋊C4 | — | 8 | 1 | no | — |
| 213 | C4⋊C4 | — | 8 | 1 | no | — |
| 214 | C22⋊C4 | — | 8 | 1 | no | — |
| 215 | C22⋊C4 | — | 8 | 1 | no | — |
| 216 | C22⋊C4 | — | 8 | 1 | no | — |
| 217 | C22⋊C4 | — | 8 | 1 | no | — |
| 218 | C22⋊C4 | — | 8 | 1 | no | — |
| 219 | C22⋊C4 | — | 8 | 1 | no | — |
| 220 | C42 | #120 | 1 | 1 | yes | characteristic |
| 221 | C42 | #127 | 1 | 1 | yes | characteristic |
| 222 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 223 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 224 | C16 | — | 2 | 4 | no | — |
| 225 | C16 | — | 4 | 2 | no | — |
| 226 | C23 | — | 8 | 1 | no | — |
| 227 | C23 | — | 8 | 1 | no | — |
| 228 | C23 | — | 8 | 2 | no | — |
| 229 | C23 | — | 16 | 1 | no | — |
| 230 | C23 | — | 16 | 1 | no | — |
| 231 | C23 | #459 | 1 | 1 | yes | characteristic |
| 232 | Q8 | — | 8 | 2 | no | — |
| 233 | Q8 | — | 8 | 1 | no | — |
| 234 | Q8 | — | 8 | 2 | no | — |
| 235 | Q8 | — | 8 | 1 | no | — |
| 236 | Q8 | — | 8 | 2 | no | — |
| 237 | Q8 | — | 16 | 1 | no | — |
| 238 | Q8 | — | 16 | 2 | no | — |
| 239 | D4 | — | 8 | 2 | no | — |
| 240 | D4 | — | 8 | 2 | no | — |
| 241 | D4 | — | 8 | 1 | no | — |
| 242 | D4 | — | 8 | 1 | no | — |
| 243 | D4 | — | 8 | 1 | no | — |
| 244 | D4 | — | 8 | 1 | no | — |
| 245 | D4 | — | 16 | 1 | no | — |
| 246 | D4 | — | 16 | 2 | no | — |
| 247 | C2×C4 | — | 2 | 2 | no | — |
| 248 | C2×C4 | — | 2 | 1 | no | — |
| 249 | C2×C4 | — | 2 | 1 | no | — |
| 250 | C2×C4 | — | 4 | 1 | no | — |
| 251 | C2×C4 | — | 8 | 1 | no | — |
| 252 | C2×C4 | — | 8 | 1 | no | — |
| 253 | C2×C4 | — | 8 | 2 | no | — |
| 254 | C2×C4 | — | 16 | 2 | no | — |
| 255 | C2×C4 | — | 16 | 2 | no | — |
| 256 | C2×C4 | — | 16 | 1 | no | — |
| 257 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 260 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 261 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | #496 | 1 | 1 | yes | characteristic |
| 263 | C8 | — | 2 | 1 | no | — |
| 264 | C8 | — | 2 | 1 | no | — |
| 265 | C8 | — | 4 | 2 | no | — |
| 266 | C8 | — | 4 | 2 | no | — |
| 267 | C8 | — | 16 | 1 | no | — |
| 268 | C8 | #453 | 1 | 2 | yes | — |
| 269 | C8 | #472 | 1 | 2 | yes | — |
| 270 | C22 | — | 2 | 1 | no | — |
| 271 | C22 | — | 2 | 1 | no | — |
| 272 | C22 | — | 16 | 1 | no | — |
| 273 | C22 | — | 16 | 1 | no | — |
| 274 | C22 | — | 16 | 1 | no | — |
| 275 | C22 | — | 16 | 1 | no | — |
| 276 | C22 | — | 16 | 1 | no | — |
| 277 | C22 | — | 16 | 1 | no | — |
| 278 | C22 | — | 32 | 2 | no | — |
| 279 | C22 | #2011 | 1 | 1 | yes | characteristic |
| 280 | C22 | #2212 | 1 | 2 | yes | — |
| 281 | C4 | — | 2 | 1 | no | — |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 2 | 1 | no | — |
| 284 | C4 | — | 2 | 1 | no | — |
| 285 | C4 | — | 2 | 1 | no | — |
| 286 | C4 | — | 16 | 1 | no | — |
| 287 | C4 | — | 16 | 2 | no | — |
| 288 | C4 | — | 16 | 1 | no | — |
| 289 | C4 | #2002 | 1 | 1 | yes | characteristic |
| 290 | C4 | #2018 | 1 | 1 | yes | characteristic |
| 291 | C2 | — | 2 | 2 | no | — |
| 292 | C2 | — | 32 | 1 | no | — |
| 293 | C2 | — | 32 | 1 | no | — |
| 294 | C2 | #10144 | 1 | 1 | yes | characteristic |
| 295 | C2 | #13332 | 1 | 1 | yes | characteristic |
| 296 | C2 | #13335 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33779 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2251 | C2 | semidirect |
| 3. #19639 | C2 | semidirect |
| 4. #11965 | C2 | semidirect |
| 5. #11678 | C2 | semidirect |
| 6. #9787 | C2 | semidirect |
| 7. #7165 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33779 | 2 |
| C2×#2251 | 2 |
| #19639 | 2 |
| #11965 | 2 |
| #11678 | 2 |
| #9787 | 2 |
| #7165 | 2 |