Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
8208 conjugacy classes of subgroups.
The subgroup lattice has 298 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62988 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #62471 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #62245 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | #62095 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #61694 | C2 | 1 | 18 | yes | maximal, complemented |
| 6 | D8×#41 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C2×#3334 | C2 | 1 | 9 | yes | maximal, complemented |
| 8 | C8×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#585 | — | 2 | 2 | no | — |
| 10 | #3499 | C22 | 1 | 12 | yes | complemented |
| 11 | #3496 | C22 | 1 | 18 | yes | complemented |
| 12 | #3489 | — | 2 | 12 | no | — |
| 13 | #3485 | — | 2 | 18 | no | — |
| 14 | D4×#41 | C22 | 1 | 12 | yes | complemented |
| 15 | C2×#547 | C22 | 1 | 18 | yes | complemented |
| 16 | C4×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 17 | #3391 | C22 | 1 | 2 | yes | complemented |
| 18 | Q8×D8 | C22 | 1 | 6 | yes | complemented |
| 19 | Q8×D8 | C22 | 1 | 12 | yes | complemented |
| 20 | #3358 | C22 | 1 | 12 | yes | complemented |
| 21 | #3334 | C22 | 1 | 36 | yes | complemented |
| 22 | #3334 | C22 | 1 | 36 | yes | complemented |
| 23 | #3334 | C22 | 1 | 18 | yes | complemented |
| 24 | #3334 | C22 | 1 | 72 | yes | complemented |
| 25 | #3294 | C22 | 1 | 72 | yes | complemented |
| 26 | #3294 | C22 | 1 | 36 | yes | complemented |
| 27 | D4×D8 | C22 | 1 | 18 | yes | complemented |
| 28 | #3245 | C22 | 1 | 18 | yes | complemented |
| 29 | #3226 | C22 | 1 | 36 | yes | complemented |
| 30 | #3186 | C22 | 1 | 36 | yes | complemented |
| 31 | #3169 | C22 | 1 | 18 | yes | complemented |
| 32 | C2×#500 | C22 | 1 | 9 | yes | complemented |
| 33 | #3100 | C22 | 1 | 36 | yes | complemented |
| 34 | C2×#480 | C22 | 1 | 36 | yes | complemented |
| 35 | C2×#474 | C22 | 1 | 18 | yes | complemented |
| 36 | C2×#466 | C22 | 1 | 18 | yes | complemented |
| 37 | #2988 | C22 | 1 | 6 | yes | complemented |
| 38 | #2972 | C22 | 1 | 9 | yes | complemented |
| 39 | C8×#41 | C22 | 1 | 6 | yes | complemented |
| 40 | C2×C4×D8 | C22 | 1 | 9 | yes | complemented |
| 41 | C2×C8×D4 | C22 | 1 | 9 | yes | complemented |
| 42 | C22×#107 | C22 | 1 | 6 | yes | complemented |
| 43 | #585 | — | 4 | 32 | no | — |
| 44 | C2×2−1+4 | — | 2 | 12 | no | — |
| 45 | C2×H(5,𝔽2) | — | 2 | 18 | no | — |
| 46 | C2×H(5,𝔽2) | D4 | 1 | 1 | yes | characteristic |
| 47 | C22×#41 | — | 2 | 12 | no | — |
| 48 | C22×#41 | — | 2 | 18 | no | — |
| 49 | C22×D8 | C23 | 1 | 9 | yes | — |
| 50 | C23×C8 | C23 | 1 | 6 | yes | — |
| 51 | #550 | C23 | 1 | 4 | yes | — |
| 52 | D4×Q8 | — | 2 | 12 | no | — |
| 53 | D4×Q8 | C23 | 1 | 12 | yes | — |
| 54 | #548 | — | 2 | 12 | no | — |
| 55 | #547 | — | 2 | 72 | no | — |
| 56 | #547 | — | 2 | 36 | no | — |
| 57 | #547 | C23 | 1 | 36 | yes | — |
| 58 | #547 | C23 | 1 | 72 | yes | — |
| 59 | #546 | — | 2 | 36 | no | — |
| 60 | #546 | — | 2 | 72 | no | — |
| 61 | D42 | C23 | 1 | 36 | yes | — |
| 62 | #537 | — | 2 | 18 | no | — |
| 63 | #536 | — | 2 | 36 | no | — |
| 64 | #534 | — | 2 | 36 | no | — |
| 65 | #532 | C23 | 1 | 36 | yes | — |
| 66 | C2×C4⋊Q8 | C23 | 1 | 18 | yes | — |
| 67 | #525 | — | 2 | 36 | no | — |
| 68 | C2×#123 | — | 2 | 36 | no | — |
| 69 | C2×C22⋊Q8 | — | 2 | 18 | no | — |
| 70 | C2×#121 | C23 | 1 | 36 | yes | — |
| 71 | #520 | D4 | 1 | 6 | yes | — |
| 72 | #518 | D4 | 1 | 9 | yes | — |
| 73 | C4×#41 | C23 | 1 | 6 | yes | — |
| 74 | C2×C4×D4 | C23 | 1 | 18 | yes | — |
| 75 | C2×C4×D4 | C23 | 1 | 9 | yes | — |
| 76 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 77 | #500 | C23 | 1 | 18 | yes | — |
| 78 | #500 | C23 | 1 | 36 | yes | — |
| 79 | #493 | C23 | 1 | 6 | yes | — |
| 80 | #486 | C23 | 1 | 36 | yes | — |
| 81 | #480 | C23 | 1 | 72 | yes | — |
| 82 | #480 | C23 | 1 | 144 | yes | — |
| 83 | #474 | C23 | 1 | 36 | yes | — |
| 84 | #474 | C23 | 1 | 36 | yes | — |
| 85 | #474 | C23 | 1 | 72 | yes | — |
| 86 | #466 | C23 | 1 | 36 | yes | — |
| 87 | #466 | C23 | 1 | 36 | yes | — |
| 88 | #466 | C23 | 1 | 72 | yes | — |
| 89 | #459 | C23 | 1 | 12 | yes | — |
| 90 | #459 | C23 | 1 | 36 | yes | — |
| 91 | #447 | C23 | 1 | 72 | yes | — |
| 92 | C8×Q8 | C23 | 1 | 6 | yes | — |
| 93 | C8×Q8 | C23 | 1 | 2 | yes | — |
| 94 | C4×D8 | C23 | 1 | 6 | yes | — |
| 95 | C4×D8 | C23 | 1 | 36 | yes | — |
| 96 | C4×D8 | C23 | 1 | 18 | yes | — |
| 97 | C4×D8 | C23 | 1 | 36 | yes | — |
| 98 | C8×D4 | C23 | 1 | 18 | yes | — |
| 99 | C8×D4 | C23 | 1 | 36 | yes | — |
| 100 | C8×D4 | C23 | 1 | 18 | yes | — |
| 101 | #431 | C23 | 1 | 18 | yes | — |
| 102 | C2×#107 | C23 | 1 | 18 | yes | — |
| 103 | C2×#107 | C23 | 1 | 36 | yes | — |
| 104 | C2×#107 | C23 | 1 | 18 | yes | — |
| 105 | C2×#107 | C23 | 1 | 9 | yes | — |
| 106 | C2×C4⋊C8 | C23 | 1 | 9 | yes | — |
| 107 | C2×#102 | C23 | 1 | 18 | yes | — |
| 108 | C2×#102 | C23 | 1 | 36 | yes | — |
| 109 | C2×C22⋊C8 | C23 | 1 | 18 | yes | — |
| 110 | C2×C4×C8 | C23 | 1 | 9 | yes | — |
| 111 | 2−1+4 | — | 4 | 96 | no | — |
| 112 | H(5,𝔽2) | — | 4 | 144 | no | — |
| 113 | H(5,𝔽2) | D8 | 1 | 16 | yes | complemented |
| 114 | C2×#41 | — | 2 | 4 | no | — |
| 115 | C2×#41 | — | 2 | 36 | no | — |
| 116 | C2×#41 | — | 2 | 72 | no | — |
| 117 | C2×#41 | — | 2 | 36 | no | — |
| 118 | C2×#41 | — | 4 | 96 | no | — |
| 119 | C2×#41 | — | 4 | 144 | no | — |
| 120 | C2×#41 | C2×D4 | 1 | 6 | yes | — |
| 121 | C22×Q8 | — | 2 | 12 | no | — |
| 122 | C22×Q8 | — | 2 | 18 | no | — |
| 123 | C22×D4 | — | 2 | 36 | no | — |
| 124 | C22×D4 | — | 2 | 36 | no | — |
| 125 | C22×D4 | C2×D4 | 1 | 9 | yes | — |
| 126 | C22×D4 | C24 | 1 | 18 | yes | — |
| 127 | C23×C4 | — | 2 | 12 | no | — |
| 128 | C23×C4 | — | 2 | 18 | no | — |
| 129 | C23×C4 | C24 | 1 | 6 | yes | — |
| 130 | C2×D8 | — | 4 | 18 | no | — |
| 131 | C2×D8 | C24 | 1 | 6 | yes | — |
| 132 | C2×D8 | C24 | 1 | 18 | yes | — |
| 133 | C2×D8 | C24 | 1 | 12 | yes | — |
| 134 | C22×C8 | — | 4 | 12 | no | — |
| 135 | C22×C8 | C24 | 1 | 9 | yes | — |
| 136 | C22×C8 | C24 | 1 | 18 | yes | — |
| 137 | C22×C8 | C24 | 1 | 9 | yes | — |
| 138 | C4⋊Q8 | — | 2 | 36 | no | — |
| 139 | C4⋊Q8 | C24 | 1 | 36 | yes | — |
| 140 | #127 | C24 | 1 | 12 | yes | — |
| 141 | #124 | — | 2 | 36 | no | — |
| 142 | #123 | — | 2 | 72 | no | — |
| 143 | #123 | — | 2 | 144 | no | — |
| 144 | C22⋊Q8 | — | 2 | 36 | no | — |
| 145 | C22⋊Q8 | — | 2 | 72 | no | — |
| 146 | C22⋊Q8 | — | 2 | 36 | no | — |
| 147 | #121 | — | 2 | 72 | no | — |
| 148 | #121 | — | 2 | 36 | no | — |
| 149 | #121 | — | 2 | 12 | no | — |
| 150 | #121 | — | 2 | 36 | no | — |
| 151 | #121 | C24 | 1 | 72 | yes | — |
| 152 | #120 | — | 2 | 72 | no | — |
| 153 | C4×Q8 | C2×D4 | 1 | 6 | yes | — |
| 154 | C4×Q8 | C24 | 1 | 2 | yes | — |
| 155 | C4×D4 | — | 2 | 36 | no | — |
| 156 | C4×D4 | — | 2 | 36 | no | — |
| 157 | C4×D4 | C2×D4 | 1 | 36 | yes | — |
| 158 | C4×D4 | C2×D4 | 1 | 18 | yes | — |
| 159 | C4×D4 | C24 | 1 | 12 | yes | — |
| 160 | C4×D4 | C24 | 1 | 36 | yes | — |
| 161 | C4×D4 | C24 | 1 | 18 | yes | — |
| 162 | #117 | C2×D4 | 1 | 18 | yes | — |
| 163 | C2×C4⋊C4 | — | 2 | 36 | no | — |
| 164 | C2×C4⋊C4 | — | 2 | 18 | no | — |
| 165 | C2×C4⋊C4 | C2×D4 | 1 | 9 | yes | — |
| 166 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 167 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 168 | C2×C22⋊C4 | — | 2 | 18 | no | — |
| 169 | C2×C22⋊C4 | — | 2 | 36 | no | — |
| 170 | C2×C22⋊C4 | C2×D4 | 1 | 18 | yes | — |
| 171 | C2×C42 | C24 | 1 | 9 | yes | — |
| 172 | #107 | C24 | 1 | 1 | yes | characteristic |
| 173 | #107 | C24 | 1 | 9 | yes | — |
| 174 | #107 | C24 | 1 | 18 | yes | — |
| 175 | #107 | C24 | 1 | 36 | yes | — |
| 176 | #107 | C24 | 1 | 36 | yes | — |
| 177 | C4⋊C8 | C24 | 1 | 18 | yes | — |
| 178 | C4⋊C8 | C24 | 1 | 6 | yes | — |
| 179 | #102 | C24 | 1 | 12 | yes | — |
| 180 | #102 | C24 | 1 | 72 | yes | — |
| 181 | #102 | C24 | 1 | 36 | yes | — |
| 182 | C22⋊C8 | C24 | 1 | 36 | yes | — |
| 183 | C4×C8 | C24 | 1 | 6 | yes | — |
| 184 | C4×C8 | C24 | 1 | 18 | yes | — |
| 185 | C24 | — | 2 | 18 | no | — |
| 186 | C24 | C22×D4 | 1 | 6 | yes | — |
| 187 | #41 | — | 4 | 16 | no | — |
| 188 | #41 | — | 4 | 144 | no | — |
| 189 | #41 | — | 4 | 288 | no | — |
| 190 | #41 | — | 4 | 144 | no | — |
| 191 | #41 | C2×D8 | 1 | 48 | yes | complemented |
| 192 | C2×Q8 | — | 2 | 36 | no | — |
| 193 | C2×Q8 | — | 4 | 48 | no | — |
| 194 | C2×Q8 | — | 4 | 72 | no | — |
| 195 | C2×Q8 | C22×D4 | 1 | 2 | yes | — |
| 196 | C2×D4 | — | 2 | 72 | no | — |
| 197 | C2×D4 | — | 2 | 12 | no | — |
| 198 | C2×D4 | — | 2 | 12 | no | — |
| 199 | C2×D4 | — | 2 | 18 | no | — |
| 200 | C2×D4 | — | 4 | 18 | no | — |
| 201 | C2×D4 | — | 4 | 18 | no | — |
| 202 | C2×D4 | — | 4 | 144 | no | — |
| 203 | C2×D4 | — | 4 | 144 | no | — |
| 204 | C2×D4 | C2×D8 | 1 | 72 | yes | complemented |
| 205 | C2×D4 | C22×D4 | 1 | 18 | yes | — |
| 206 | C2×D4 | C25 | 1 | 12 | yes | — |
| 207 | C22×C4 | — | 2 | 18 | no | — |
| 208 | C22×C4 | — | 2 | 36 | no | — |
| 209 | C22×C4 | — | 2 | 12 | no | — |
| 210 | C22×C4 | — | 2 | 36 | no | — |
| 211 | C22×C4 | — | 4 | 6 | no | — |
| 212 | C22×C4 | — | 4 | 6 | no | — |
| 213 | C22×C4 | — | 4 | 48 | no | — |
| 214 | C22×C4 | — | 4 | 72 | no | — |
| 215 | C22×C4 | C22×D4 | 1 | 9 | yes | — |
| 216 | C22×C4 | C22×D4 | 1 | 18 | yes | — |
| 217 | C22×C4 | C22×D4 | 1 | 9 | yes | — |
| 218 | C22×C4 | C25 | 1 | 9 | yes | — |
| 219 | D8 | — | 2 | 12 | no | — |
| 220 | D8 | — | 4 | 18 | no | — |
| 221 | D8 | — | 4 | 12 | no | — |
| 222 | C2×C8 | — | 2 | 18 | no | — |
| 223 | C2×C8 | — | 4 | 18 | no | — |
| 224 | C2×C8 | — | 4 | 9 | no | — |
| 225 | C2×C8 | C25 | 1 | 6 | yes | — |
| 226 | C2×C8 | C25 | 1 | 9 | yes | — |
| 227 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 228 | C4⋊C4 | — | 2 | 18 | no | — |
| 229 | C4⋊C4 | — | 2 | 36 | no | — |
| 230 | C4⋊C4 | — | 2 | 36 | no | — |
| 231 | C4⋊C4 | C22×D4 | 1 | 18 | yes | — |
| 232 | C4⋊C4 | C22×D4 | 1 | 6 | yes | — |
| 233 | C4⋊C4 | C25 | 1 | 2 | yes | — |
| 234 | C4⋊C4 | C25 | 1 | 18 | yes | — |
| 235 | C22⋊C4 | — | 2 | 12 | no | — |
| 236 | C22⋊C4 | — | 2 | 72 | no | — |
| 237 | C22⋊C4 | — | 2 | 36 | no | — |
| 238 | C22⋊C4 | C22×D4 | 1 | 36 | yes | — |
| 239 | C42 | C22×D4 | 1 | 18 | yes | — |
| 240 | C42 | C25 | 1 | 6 | yes | — |
| 241 | C23 | — | 2 | 12 | no | — |
| 242 | C23 | — | 4 | 6 | no | — |
| 243 | C23 | — | 4 | 6 | no | — |
| 244 | C23 | — | 4 | 36 | no | — |
| 245 | C23 | — | 8 | 18 | no | — |
| 246 | C23 | — | 8 | 18 | no | — |
| 247 | C23 | C22×D8 | 1 | 24 | yes | — |
| 248 | C23 | C23×D4 | 1 | 9 | yes | — |
| 249 | Q8 | — | 4 | 72 | no | — |
| 250 | Q8 | C22×D8 | 1 | 8 | yes | — |
| 251 | D4 | — | 2 | 12 | no | — |
| 252 | D4 | — | 2 | 12 | no | — |
| 253 | D4 | — | 4 | 144 | no | — |
| 254 | D4 | — | 4 | 24 | no | — |
| 255 | D4 | — | 8 | 12 | no | — |
| 256 | D4 | — | 8 | 18 | no | — |
| 257 | D4 | C22×D8 | 1 | 72 | yes | — |
| 258 | C2×C4 | — | 2 | 9 | no | — |
| 259 | C2×C4 | — | 2 | 9 | no | — |
| 260 | C2×C4 | — | 2 | 2 | no | — |
| 261 | C2×C4 | — | 2 | 18 | no | — |
| 262 | C2×C4 | — | 4 | 18 | no | — |
| 263 | C2×C4 | — | 4 | 9 | no | — |
| 264 | C2×C4 | — | 4 | 36 | no | — |
| 265 | C2×C4 | — | 4 | 72 | no | — |
| 266 | C2×C4 | — | 4 | 24 | no | — |
| 267 | C2×C4 | — | 4 | 72 | no | — |
| 268 | C2×C4 | C22×D8 | 1 | 36 | yes | — |
| 269 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 270 | C2×C4 | C23×D4 | 1 | 9 | yes | — |
| 271 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 272 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 273 | C8 | — | 2 | 9 | no | — |
| 274 | C8 | — | 2 | 6 | no | — |
| 275 | C8 | #585 | 1 | 2 | yes | complemented |
| 276 | C22 | — | 2 | 9 | no | — |
| 277 | C22 | — | 2 | 9 | no | — |
| 278 | C22 | — | 4 | 24 | no | — |
| 279 | C22 | — | 4 | 12 | no | — |
| 280 | C22 | — | 4 | 12 | no | — |
| 281 | C22 | — | 4 | 12 | no | — |
| 282 | C22 | — | 16 | 36 | no | — |
| 283 | C22 | C23×D8 | 1 | 18 | yes | — |
| 284 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 9 | no | — |
| 286 | C4 | — | 2 | 6 | no | — |
| 287 | C4 | — | 4 | 2 | no | — |
| 288 | C4 | — | 4 | 18 | no | — |
| 289 | C4 | C23×D8 | 1 | 12 | yes | — |
| 290 | C4 | C2×#585 | 1 | 1 | yes | characteristic |
| 291 | C4 | 2−1+6 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 2 | 18 | no | — |
| 293 | C2 | — | 8 | 12 | no | — |
| 294 | C2 | #62988 | 1 | 1 | yes | characteristic |
| 295 | C2 | C24×D8 | 1 | 1 | yes | characteristic |
| 296 | C2 | #63093 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #62988 | C2 | semidirect |
| 2. #62471 | C2 | semidirect |
| 3. #62245 | C2 | semidirect |
| 4. #62095 | C2 | semidirect |
| 5. #61694 | C2 | semidirect |
| 6. D8×#41 | C2 | semidirect |
| 7. C2×#3334 | C2 | semidirect |
| 8. C8×H(5,𝔽2) | C2 | semidirect |
| 10. #3499 | C22 | semidirect |
| 11. #3496 | C22 | semidirect |
| 14. D4×#41 | C22 | semidirect |
| 15. C2×#547 | C22 | semidirect |
| 17. #3391 | C22 | semidirect |
| 18. Q8×D8 | C22 | semidirect |
| 19. Q8×D8 | C22 | semidirect |
| 20. #3358 | C22 | semidirect |
| 21. #3334 | C22 | semidirect |
| 22. #3334 | C22 | semidirect |
| 23. #3334 | C22 | semidirect |
| 24. #3334 | C22 | semidirect |
| 25. #3294 | C22 | semidirect |
| 26. #3294 | C22 | semidirect |
| 27. D4×D8 | C22 | semidirect |
| 28. #3245 | C22 | semidirect |
| 29. #3226 | C22 | semidirect |
| 30. #3186 | C22 | semidirect |
| 31. #3169 | C22 | semidirect |
| 32. C2×#500 | C22 | semidirect |
| 33. #3100 | C22 | semidirect |
| 34. C2×#480 | C22 | semidirect |
| 35. C2×#474 | C22 | semidirect |
| 36. C2×#466 | C22 | semidirect |
| 37. #2988 | C22 | semidirect |
| 38. #2972 | C22 | semidirect |
| 39. C8×#41 | C22 | semidirect |
| 40. C2×C4×D8 | C22 | semidirect |
| 41. C2×C8×D4 | C22 | semidirect |
| 42. C22×#107 | C22 | semidirect |
| 113. H(5,𝔽2) | D8 | semidirect |
| 191. #41 | C2×D8 | semidirect |
| 204. C2×D4 | C2×D8 | semidirect |
| 275. C8 | #585 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62988 | 2 |
| #62471 | 2 |
| #62245 | 2 |
| #62095 | 2 |
| #61694 | 2 |
| D8×#41 | 2 |
| C2×#3334 | 2 |
| C8×H(5,𝔽2) | 2 |