Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
538 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 132 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 | #19632 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #19573 | C2 | 1 | 2 | yes | maximal |
| 3 | #19338 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #19188 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #18090 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10495 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10118 | C2 | 1 | 2 | yes | maximal |
| 8 | #9807 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #9633 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #9434 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #8245 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #3254 | C22 | 1 | 2 | yes | — |
| 13 | #3230 | C22 | 1 | 2 | yes | — |
| 14 | #3226 | C22 | 1 | 2 | yes | — |
| 15 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 16 | #3165 | C22 | 1 | 2 | yes | — |
| 17 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 18 | #2986 | C22 | 1 | 2 | yes | — |
| 19 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1997 | C22 | 1 | 2 | yes | — |
| 21 | #1723 | — | 2 | 2 | no | — |
| 22 | #1713 | — | 2 | 1 | no | — |
| 23 | #1707 | — | 2 | 1 | no | — |
| 24 | #1698 | — | 2 | 2 | no | — |
| 25 | #1696 | — | 2 | 2 | no | — |
| 26 | #1693 | — | 2 | 1 | no | — |
| 27 | #1689 | — | 2 | 1 | no | — |
| 28 | #1689 | — | 2 | 1 | no | — |
| 29 | #1689 | — | 2 | 1 | no | — |
| 30 | #1679 | — | 2 | 2 | no | — |
| 31 | #1678 | — | 2 | 2 | no | — |
| 32 | #1676 | — | 2 | 2 | no | — |
| 33 | #1669 | — | 2 | 2 | no | — |
| 34 | #1593 | — | 2 | 1 | no | — |
| 35 | #1584 | — | 2 | 2 | no | — |
| 36 | #1579 | — | 2 | 1 | no | — |
| 37 | #1574 | C22 | 1 | 2 | yes | — |
| 38 | #1566 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1561 | C22 | 1 | 2 | yes | — |
| 41 | #1548 | C22 | 1 | 2 | yes | — |
| 42 | #1546 | C22 | 1 | 2 | yes | — |
| 43 | #1543 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1525 | C22 | 1 | 1 | yes | characteristic |
| 45 | #1485 | C22 | 1 | 2 | yes | — |
| 46 | #1451 | C22 | 1 | 1 | yes | characteristic |
| 47 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 48 | #1299 | C22 | 1 | 1 | yes | characteristic |
| 49 | #1299 | C22 | 1 | 1 | yes | characteristic |
| 50 | #1299 | C22 | 1 | 2 | yes | — |
| 51 | C22×Q16 | — | 2 | 1 | no | — |
| 52 | C22×Q16 | — | 2 | 1 | no | — |
| 53 | C2×#131 | — | 2 | 1 | no | — |
| 54 | #533 | C23 | 1 | 2 | yes | — |
| 55 | #532 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 57 | #501 | — | 2 | 4 | no | — |
| 58 | #500 | — | 2 | 2 | no | — |
| 59 | #500 | — | 2 | 2 | no | — |
| 60 | #500 | — | 2 | 2 | no | — |
| 61 | #500 | D4 | 1 | 1 | yes | characteristic |
| 62 | #498 | — | 2 | 2 | no | — |
| 63 | #494 | — | 2 | 1 | no | — |
| 64 | #494 | — | 2 | 1 | no | — |
| 65 | #494 | — | 2 | 1 | no | — |
| 66 | #494 | — | 2 | 1 | no | — |
| 67 | #494 | D4 | 1 | 1 | yes | characteristic |
| 68 | #487 | — | 2 | 2 | no | — |
| 69 | #487 | D4 | 1 | 2 | yes | — |
| 70 | #486 | — | 2 | 2 | no | — |
| 71 | #484 | — | 4 | 2 | no | — |
| 72 | #480 | — | 4 | 2 | no | — |
| 73 | #477 | — | 2 | 2 | no | — |
| 74 | #477 | — | 2 | 2 | no | — |
| 75 | #477 | — | 2 | 2 | no | — |
| 76 | #474 | — | 2 | 2 | no | — |
| 77 | #474 | — | 2 | 2 | no | — |
| 78 | #474 | — | 2 | 2 | no | — |
| 79 | #468 | — | 4 | 2 | no | — |
| 80 | #467 | — | 2 | 1 | no | — |
| 81 | #467 | — | 2 | 1 | no | — |
| 82 | #461 | — | 2 | 2 | no | — |
| 83 | #460 | — | 2 | 2 | no | — |
| 84 | #446 | — | 2 | 2 | no | — |
| 85 | #446 | — | 2 | 2 | no | — |
| 86 | C8×Q8 | — | 2 | 2 | no | — |
| 87 | C8×Q8 | — | 2 | 2 | no | — |
| 88 | C4×Q16 | — | 2 | 1 | no | — |
| 89 | C4×Q16 | — | 2 | 1 | no | — |
| 90 | #436 | — | 2 | 1 | no | — |
| 91 | #436 | — | 2 | 1 | no | — |
| 92 | C8×D4 | — | 2 | 1 | no | — |
| 93 | C8×D4 | — | 2 | 1 | no | — |
| 94 | #433 | — | 4 | 2 | no | — |
| 95 | #431 | C23 | 1 | 1 | yes | characteristic |
| 96 | #431 | C23 | 1 | 1 | yes | characteristic |
| 97 | #431 | C23 | 1 | 2 | yes | — |
| 98 | #428 | — | 4 | 2 | no | — |
| 99 | #423 | C23 | 1 | 1 | yes | characteristic |
| 100 | #423 | C23 | 1 | 2 | yes | — |
| 101 | #408 | — | 2 | 1 | no | — |
| 102 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 103 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 105 | #343 | — | 4 | 2 | no | — |
| 106 | #335 | — | 2 | 2 | no | — |
| 107 | #335 | D4 | 1 | 2 | yes | — |
| 108 | #334 | — | 2 | 2 | no | — |
| 109 | #334 | — | 2 | 1 | no | — |
| 110 | #334 | — | 2 | 1 | no | — |
| 111 | #334 | D4 | 1 | 1 | yes | characteristic |
| 112 | #334 | D4 | 1 | 1 | yes | characteristic |
| 113 | #333 | — | 2 | 2 | no | — |
| 114 | #333 | — | 2 | 2 | no | — |
| 115 | #333 | D4 | 1 | 2 | yes | — |
| 116 | #332 | — | 2 | 2 | no | — |
| 117 | #332 | — | 2 | 2 | no | — |
| 118 | #332 | — | 2 | 2 | no | — |
| 119 | #332 | — | 2 | 2 | no | — |
| 120 | #331 | D4 | 1 | 2 | yes | — |
| 121 | #329 | — | 2 | 2 | no | — |
| 122 | Q8⋊C8 | — | 2 | 2 | no | — |
| 123 | Q8⋊C8 | — | 2 | 2 | no | — |
| 124 | #322 | — | 2 | 1 | no | — |
| 125 | #322 | — | 2 | 1 | no | — |
| 126 | #322 | — | 2 | 1 | no | — |
| 127 | C82 | — | 2 | 1 | no | — |
| 128 | C2×#41 | — | 2 | 1 | no | — |
| 129 | C22×Q8 | — | 2 | 2 | no | — |
| 130 | C2×Q16 | — | 2 | 1 | no | — |
| 131 | C2×Q16 | — | 2 | 1 | no | — |
| 132 | C2×Q16 | — | 4 | 1 | no | — |
| 133 | C2×Q16 | — | 4 | 1 | no | — |
| 134 | C2×Q16 | — | 4 | 1 | no | — |
| 135 | C2×Q16 | — | 4 | 1 | no | — |
| 136 | C2×Q16 | — | 8 | 2 | no | — |
| 137 | C2×Q16 | — | 8 | 2 | no | — |
| 138 | C2×SD16 | — | 4 | 2 | no | — |
| 139 | #131 | — | 8 | 2 | no | — |
| 140 | C2×M16 | — | 2 | 1 | no | — |
| 141 | C2×M16 | — | 2 | 1 | no | — |
| 142 | C2×M16 | — | 2 | 1 | no | — |
| 143 | C2×M16 | — | 2 | 2 | no | — |
| 144 | C2×M16 | — | 4 | 1 | no | — |
| 145 | C22×C8 | — | 4 | 1 | no | — |
| 146 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 147 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 148 | C4⋊Q8 | — | 2 | 2 | no | — |
| 149 | C4⋊Q8 | — | 2 | 2 | no | — |
| 150 | C4⋊Q8 | — | 2 | 2 | no | — |
| 151 | C4⋊Q8 | — | 2 | 2 | no | — |
| 152 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 153 | C4⋊Q8 | C2×D4 | 1 | 2 | yes | — |
| 154 | C4⋊Q8 | C2×D4 | 1 | 2 | yes | — |
| 155 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 156 | #125 | — | 2 | 2 | no | — |
| 157 | #124 | — | 2 | 1 | no | — |
| 158 | C22⋊Q8 | — | 2 | 2 | no | — |
| 159 | C22⋊Q8 | — | 2 | 2 | no | — |
| 160 | #121 | — | 2 | 1 | no | — |
| 161 | #121 | — | 2 | 1 | no | — |
| 162 | C4×Q8 | — | 2 | 2 | no | — |
| 163 | C4×Q8 | — | 2 | 2 | no | — |
| 164 | C4×D4 | — | 2 | 1 | no | — |
| 165 | C4×D4 | — | 2 | 1 | no | — |
| 166 | #117 | — | 2 | 2 | no | — |
| 167 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 168 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 169 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 170 | #107 | — | 2 | 1 | no | — |
| 171 | #107 | — | 2 | 1 | no | — |
| 172 | #107 | — | 4 | 2 | no | — |
| 173 | #107 | — | 4 | 4 | no | — |
| 174 | #107 | — | 4 | 2 | no | — |
| 175 | #107 | — | 4 | 2 | no | — |
| 176 | #107 | — | 4 | 2 | no | — |
| 177 | #107 | — | 4 | 2 | no | — |
| 178 | #106 | — | 4 | 4 | no | — |
| 179 | C4⋊C8 | — | 2 | 1 | no | — |
| 180 | C4⋊C8 | — | 2 | 1 | no | — |
| 181 | C4⋊C8 | — | 2 | 2 | no | — |
| 182 | C4⋊C8 | — | 2 | 2 | no | — |
| 183 | C4⋊C8 | — | 4 | 2 | no | — |
| 184 | C4⋊C8 | — | 4 | 2 | no | — |
| 185 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 186 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 187 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 188 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 189 | #103 | — | 2 | 2 | no | — |
| 190 | #103 | — | 2 | 2 | no | — |
| 191 | #103 | — | 4 | 2 | no | — |
| 192 | #103 | — | 4 | 2 | no | — |
| 193 | #103 | — | 4 | 2 | no | — |
| 194 | #102 | — | 4 | 2 | no | — |
| 195 | #102 | — | 4 | 2 | no | — |
| 196 | #102 | — | 4 | 2 | no | — |
| 197 | C22⋊C8 | — | 2 | 1 | no | — |
| 198 | C22⋊C8 | — | 2 | 2 | no | — |
| 199 | C22⋊C8 | — | 2 | 1 | no | — |
| 200 | C22⋊C8 | — | 4 | 1 | no | — |
| 201 | C22⋊C8 | — | 4 | 1 | no | — |
| 202 | #97 | — | 2 | 1 | no | — |
| 203 | #97 | — | 4 | 2 | no | — |
| 204 | #97 | #41 | 1 | 2 | yes | — |
| 205 | C4×C8 | — | 2 | 1 | no | — |
| 206 | C4×C8 | — | 2 | 1 | no | — |
| 207 | C4×C8 | — | 2 | 1 | no | — |
| 208 | C4×C8 | — | 2 | 2 | no | — |
| 209 | C4×C8 | — | 2 | 1 | no | — |
| 210 | C4×C8 | — | 4 | 2 | no | — |
| 211 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 212 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 213 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 214 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 215 | #41 | — | 8 | 1 | no | — |
| 216 | #41 | — | 8 | 1 | no | — |
| 217 | C2×Q8 | — | 2 | 2 | no | — |
| 218 | C2×Q8 | — | 2 | 1 | no | — |
| 219 | C2×Q8 | — | 2 | 2 | no | — |
| 220 | C2×Q8 | — | 4 | 2 | no | — |
| 221 | C2×Q8 | — | 4 | 2 | no | — |
| 222 | C2×Q8 | — | 8 | 2 | no | — |
| 223 | C2×Q8 | — | 8 | 2 | no | — |
| 224 | C2×D4 | — | 2 | 1 | no | — |
| 225 | C2×D4 | — | 2 | 1 | no | — |
| 226 | C2×D4 | — | 2 | 1 | no | — |
| 227 | C2×D4 | — | 4 | 1 | no | — |
| 228 | C22×C4 | — | 4 | 1 | no | — |
| 229 | C22×C4 | — | 4 | 2 | no | — |
| 230 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 231 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 232 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 233 | Q16 | — | 4 | 2 | no | — |
| 234 | Q16 | — | 8 | 1 | no | — |
| 235 | Q16 | — | 8 | 2 | no | — |
| 236 | Q16 | — | 8 | 2 | no | — |
| 237 | Q16 | — | 8 | 2 | no | — |
| 238 | Q16 | — | 8 | 2 | no | — |
| 239 | SD16 | — | 16 | 2 | no | — |
| 240 | M16 | — | 2 | 4 | no | — |
| 241 | M16 | — | 4 | 2 | no | — |
| 242 | M16 | — | 8 | 2 | no | — |
| 243 | M16 | — | 8 | 2 | no | — |
| 244 | M16 | — | 8 | 1 | no | — |
| 245 | C2×C8 | — | 2 | 2 | no | — |
| 246 | C2×C8 | — | 2 | 2 | no | — |
| 247 | C2×C8 | — | 2 | 2 | no | — |
| 248 | C2×C8 | — | 2 | 2 | no | — |
| 249 | C2×C8 | — | 2 | 2 | no | — |
| 250 | C2×C8 | — | 2 | 2 | no | — |
| 251 | C2×C8 | — | 4 | 1 | no | — |
| 252 | C2×C8 | — | 4 | 1 | no | — |
| 253 | C2×C8 | — | 4 | 2 | no | — |
| 254 | C2×C8 | — | 4 | 1 | no | — |
| 255 | C2×C8 | — | 8 | 2 | no | — |
| 256 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 260 | C4⋊C4 | — | 2 | 2 | no | — |
| 261 | C4⋊C4 | — | 2 | 2 | no | — |
| 262 | C4⋊C4 | — | 2 | 1 | no | — |
| 263 | C4⋊C4 | — | 2 | 1 | no | — |
| 264 | C4⋊C4 | — | 2 | 2 | no | — |
| 265 | C4⋊C4 | — | 2 | 2 | no | — |
| 266 | C4⋊C4 | — | 4 | 2 | no | — |
| 267 | C4⋊C4 | — | 4 | 2 | no | — |
| 268 | C4⋊C4 | — | 4 | 2 | no | — |
| 269 | C4⋊C4 | — | 4 | 2 | no | — |
| 270 | C4⋊C4 | — | 4 | 2 | no | — |
| 271 | C4⋊C4 | — | 4 | 2 | no | — |
| 272 | C22⋊C4 | — | 4 | 2 | no | — |
| 273 | C22⋊C4 | — | 4 | 1 | no | — |
| 274 | C22⋊C4 | — | 4 | 1 | no | — |
| 275 | C42 | — | 4 | 2 | no | — |
| 276 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 277 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 278 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 279 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 280 | C23 | — | 4 | 1 | no | — |
| 281 | C23 | #540 | 1 | 1 | yes | characteristic |
| 282 | Q8 | — | 4 | 2 | no | — |
| 283 | Q8 | — | 4 | 2 | no | — |
| 284 | Q8 | — | 4 | 2 | no | — |
| 285 | Q8 | — | 4 | 2 | no | — |
| 286 | Q8 | — | 8 | 1 | no | — |
| 287 | Q8 | — | 8 | 2 | no | — |
| 288 | Q8 | — | 8 | 2 | no | — |
| 289 | Q8 | — | 16 | 2 | no | — |
| 290 | D4 | — | 4 | 2 | no | — |
| 291 | D4 | — | 4 | 2 | no | — |
| 292 | D4 | — | 8 | 1 | no | — |
| 293 | D4 | — | 8 | 2 | no | — |
| 294 | C2×C4 | — | 2 | 2 | no | — |
| 295 | C2×C4 | — | 2 | 2 | no | — |
| 296 | C2×C4 | — | 2 | 1 | no | — |
| 297 | C2×C4 | — | 2 | 1 | no | — |
| 298 | C2×C4 | — | 4 | 2 | no | — |
| 299 | C2×C4 | — | 4 | 2 | no | — |
| 300 | C2×C4 | — | 4 | 1 | no | — |
| 301 | C2×C4 | — | 8 | 1 | no | — |
| 302 | C2×C4 | — | 8 | 1 | no | — |
| 303 | C2×C4 | — | 8 | 2 | no | — |
| 304 | C2×C4 | — | 16 | 2 | no | — |
| 305 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | #550 | 1 | 1 | yes | characteristic |
| 311 | C8 | — | 2 | 4 | no | — |
| 312 | C8 | — | 2 | 1 | no | — |
| 313 | C8 | — | 2 | 1 | no | — |
| 314 | C8 | — | 2 | 1 | no | — |
| 315 | C8 | — | 4 | 2 | no | — |
| 316 | C8 | — | 4 | 2 | no | — |
| 317 | C8 | — | 4 | 2 | no | — |
| 318 | C8 | — | 4 | 2 | no | — |
| 319 | C8 | — | 8 | 1 | no | — |
| 320 | C8 | — | 8 | 2 | no | — |
| 321 | C8 | — | 8 | 1 | no | — |
| 322 | C8 | #472 | 1 | 2 | yes | — |
| 323 | C22 | — | 2 | 1 | no | — |
| 324 | C22 | — | 2 | 1 | no | — |
| 325 | C22 | — | 2 | 1 | no | — |
| 326 | C22 | — | 8 | 1 | no | — |
| 327 | C22 | — | 8 | 1 | no | — |
| 328 | C22 | — | 8 | 1 | no | — |
| 329 | C22 | #2428 | 1 | 1 | yes | characteristic |
| 330 | C4 | — | 2 | 1 | no | — |
| 331 | C4 | — | 2 | 1 | no | — |
| 332 | C4 | — | 2 | 1 | no | — |
| 333 | C4 | — | 8 | 1 | no | — |
| 334 | C4 | — | 8 | 2 | no | — |
| 335 | C4 | — | 8 | 2 | no | — |
| 336 | C4 | — | 16 | 2 | no | — |
| 337 | C4 | C2×#472 | 1 | 1 | yes | characteristic |
| 338 | C4 | #3068 | 1 | 1 | yes | characteristic |
| 339 | C4 | #3289 | 1 | 2 | yes | — |
| 340 | C4 | #3293 | 1 | 2 | yes | — |
| 341 | C2 | — | 4 | 1 | no | — |
| 342 | C2 | — | 16 | 1 | no | — |
| 343 | C2 | #25404 | 1 | 1 | yes | characteristic |
| 344 | C2 | #25435 | 1 | 1 | yes | characteristic |
| 345 | C2 | #25445 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #532 | C23 |
| 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 |
|---|---|---|
| 1. #19632 | C2 | semidirect |
| 5. #18090 | C2 | semidirect |
| 6. #10495 | C2 | semidirect |
| 8. #9807 | C2 | semidirect |
| 9. #9633 | C2 | semidirect |
| 10. #9434 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19632 | 2 |
| #19573 | 2 |
| #19338 | 2 |
| #19188 | 2 |
| #18090 | 2 |
| #10495 | 2 |
| #10118 | 2 |
| #9807 | 2 |
| #9633 | 2 |
| #9434 | 2 |
| #8245 | 2 |