Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7181 conjugacy classes of subgroups.
The subgroup lattice has 291 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #63040 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #62415 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #62328 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | #62001 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #61661 | C2 | 1 | 18 | yes | maximal, complemented |
| 6 | #61436 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #61089 | C2 | 1 | 9 | yes | maximal, complemented |
| 8 | #60308 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#576 | C22 | 1 | 6 | yes | complemented |
| 10 | #3549 | — | 2 | 18 | no | — |
| 11 | #3549 | C22 | 1 | 18 | yes | complemented |
| 12 | #3498 | — | 2 | 12 | no | — |
| 13 | #3498 | C22 | 1 | 18 | yes | complemented |
| 14 | #3498 | C22 | 1 | 12 | yes | complemented |
| 15 | #3483 | — | 2 | 2 | no | — |
| 16 | #3483 | C22 | 1 | 1 | yes | characteristic |
| 17 | #3483 | C22 | 1 | 12 | yes | complemented |
| 18 | #3391 | C22 | 1 | 12 | yes | complemented |
| 19 | #3391 | C22 | 1 | 6 | yes | complemented |
| 20 | Q8×Q16 | C22 | 1 | 2 | yes | complemented |
| 21 | #3362 | C22 | 1 | 12 | yes | complemented |
| 22 | #3341 | C22 | 1 | 18 | yes | complemented |
| 23 | #3325 | C22 | 1 | 72 | yes | complemented |
| 24 | #3305 | C22 | 1 | 36 | yes | complemented |
| 25 | #3287 | C22 | 1 | 18 | yes | complemented |
| 26 | D4×D8 | C22 | 1 | 36 | yes | complemented |
| 27 | D4×D8 | C22 | 1 | 36 | yes | complemented |
| 28 | #3274 | C22 | 1 | 72 | yes | complemented |
| 29 | #3253 | C22 | 1 | 18 | yes | complemented |
| 30 | #3205 | C22 | 1 | 36 | yes | complemented |
| 31 | #3199 | C22 | 1 | 36 | yes | complemented |
| 32 | #3169 | C22 | 1 | 18 | yes | complemented |
| 33 | C2×#493 | C22 | 1 | 9 | yes | complemented |
| 34 | #3106 | C22 | 1 | 36 | yes | complemented |
| 35 | #3058 | C22 | 1 | 18 | yes | complemented |
| 36 | #3038 | C22 | 1 | 18 | yes | complemented |
| 37 | #3008 | C22 | 1 | 36 | yes | complemented |
| 38 | #2991 | C22 | 1 | 6 | yes | complemented |
| 39 | #2979 | C22 | 1 | 9 | yes | complemented |
| 40 | C8×#41 | C22 | 1 | 6 | yes | complemented |
| 41 | C4×#135 | C22 | 1 | 9 | yes | complemented |
| 42 | #2931 | C22 | 1 | 9 | yes | complemented |
| 43 | C2×H(5,𝔽2) | — | 2 | 12 | no | — |
| 44 | C2×H(5,𝔽2) | C23 | 1 | 12 | yes | — |
| 45 | #576 | — | 4 | 24 | no | — |
| 46 | C2×AGL(1,ℤ/8) | C23 | 1 | 36 | yes | — |
| 47 | C2×#135 | C23 | 1 | 9 | yes | — |
| 48 | C22×D8 | C23 | 1 | 18 | yes | — |
| 49 | C22×D8 | C23 | 1 | 18 | yes | — |
| 50 | C2×#131 | C23 | 1 | 6 | yes | — |
| 51 | H(3,𝔽4) | — | 2 | 36 | no | — |
| 52 | Q82 | D4 | 1 | 1 | yes | characteristic |
| 53 | Q82 | C23 | 1 | 4 | yes | — |
| 54 | #555 | — | 2 | 18 | no | — |
| 55 | #555 | — | 2 | 36 | no | — |
| 56 | #555 | D4 | 1 | 9 | yes | — |
| 57 | #555 | C23 | 1 | 36 | yes | — |
| 58 | #550 | — | 2 | 12 | no | — |
| 59 | #550 | — | 2 | 12 | no | — |
| 60 | #550 | D4 | 1 | 6 | yes | — |
| 61 | #550 | C23 | 1 | 36 | yes | — |
| 62 | #550 | C23 | 1 | 12 | yes | — |
| 63 | D42 | — | 2 | 18 | no | — |
| 64 | D42 | — | 2 | 72 | no | — |
| 65 | D42 | — | 2 | 36 | no | — |
| 66 | D42 | C23 | 1 | 72 | yes | — |
| 67 | #535 | — | 2 | 36 | no | — |
| 68 | #535 | — | 2 | 72 | no | — |
| 69 | #535 | — | 2 | 36 | no | — |
| 70 | #535 | C23 | 1 | 36 | yes | — |
| 71 | #532 | — | 2 | 18 | no | — |
| 72 | #532 | — | 2 | 36 | no | — |
| 73 | #532 | C23 | 1 | 36 | yes | — |
| 74 | #532 | C23 | 1 | 9 | yes | — |
| 75 | C2×#127 | — | 2 | 18 | no | — |
| 76 | C2×#127 | C23 | 1 | 18 | yes | — |
| 77 | C4×#41 | — | 2 | 12 | no | — |
| 78 | C4×#41 | C23 | 1 | 18 | yes | — |
| 79 | C4×#41 | C23 | 1 | 6 | yes | — |
| 80 | #500 | C23 | 1 | 6 | yes | — |
| 81 | #493 | C23 | 1 | 18 | yes | — |
| 82 | #493 | C23 | 1 | 36 | yes | — |
| 83 | #486 | C23 | 1 | 36 | yes | — |
| 84 | #482 | C23 | 1 | 72 | yes | — |
| 85 | #477 | C23 | 1 | 12 | yes | — |
| 86 | #476 | C23 | 1 | 36 | yes | — |
| 87 | #469 | C23 | 1 | 72 | yes | — |
| 88 | #466 | C23 | 1 | 36 | yes | — |
| 89 | #466 | C23 | 1 | 36 | yes | — |
| 90 | #460 | C23 | 1 | 36 | yes | — |
| 91 | #459 | C23 | 1 | 36 | yes | — |
| 92 | #459 | C23 | 1 | 72 | yes | — |
| 93 | #449 | C23 | 1 | 72 | yes | — |
| 94 | #447 | C23 | 1 | 144 | yes | — |
| 95 | C8×Q8 | C23 | 1 | 6 | yes | — |
| 96 | C8×Q8 | C23 | 1 | 2 | yes | — |
| 97 | C4×Q16 | C23 | 1 | 6 | yes | — |
| 98 | C4×SD16 | C23 | 1 | 36 | yes | — |
| 99 | C4×D8 | C23 | 1 | 18 | yes | — |
| 100 | C4×D8 | C23 | 1 | 36 | yes | — |
| 101 | #436 | C23 | 1 | 36 | yes | — |
| 102 | C8×D4 | C23 | 1 | 18 | yes | — |
| 103 | C8×D4 | C23 | 1 | 18 | yes | — |
| 104 | #431 | C23 | 1 | 18 | yes | — |
| 105 | #427 | C23 | 1 | 9 | yes | — |
| 106 | #423 | C23 | 1 | 9 | yes | — |
| 107 | #418 | C23 | 1 | 36 | yes | — |
| 108 | #416 | C23 | 1 | 18 | yes | — |
| 109 | #408 | C23 | 1 | 18 | yes | — |
| 110 | C2×C4×C8 | C23 | 1 | 9 | yes | — |
| 111 | H(5,𝔽2) | — | 4 | 24 | no | — |
| 112 | H(5,𝔽2) | — | 8 | 12 | no | — |
| 113 | H(5,𝔽2) | — | 8 | 48 | no | — |
| 114 | C2×#41 | — | 2 | 12 | no | — |
| 115 | C2×#41 | — | 2 | 12 | no | — |
| 116 | C2×#41 | — | 2 | 36 | no | — |
| 117 | C2×#41 | C2×D4 | 1 | 6 | yes | — |
| 118 | C2×#41 | C24 | 1 | 18 | yes | — |
| 119 | C2×#41 | C24 | 1 | 6 | yes | — |
| 120 | C22×D4 | — | 2 | 36 | no | — |
| 121 | C22×D4 | — | 2 | 72 | no | — |
| 122 | C22×D4 | — | 2 | 36 | no | — |
| 123 | C22×D4 | C24 | 1 | 36 | yes | — |
| 124 | AGL(1,ℤ/8) | — | 4 | 72 | no | — |
| 125 | #135 | — | 2 | 36 | no | — |
| 126 | C2×Q16 | C24 | 1 | 1 | yes | characteristic |
| 127 | C2×SD16 | C24 | 1 | 18 | yes | — |
| 128 | C2×D8 | — | 4 | 36 | no | — |
| 129 | C2×D8 | — | 8 | 18 | no | — |
| 130 | C2×D8 | C24 | 1 | 9 | yes | — |
| 131 | C2×D8 | C24 | 1 | 36 | yes | — |
| 132 | C2×D8 | C24 | 1 | 36 | yes | — |
| 133 | #131 | — | 4 | 12 | no | — |
| 134 | C2×M16 | C24 | 1 | 18 | yes | — |
| 135 | C22×C8 | C24 | 1 | 9 | yes | — |
| 136 | C22×C8 | C24 | 1 | 9 | yes | — |
| 137 | C4⋊Q8 | — | 2 | 12 | no | — |
| 138 | C4⋊Q8 | C2×D4 | 1 | 9 | yes | — |
| 139 | C4⋊Q8 | C24 | 1 | 12 | yes | — |
| 140 | #127 | — | 2 | 36 | no | — |
| 141 | #127 | — | 2 | 36 | no | — |
| 142 | #127 | — | 2 | 18 | no | — |
| 143 | #127 | — | 2 | 18 | no | — |
| 144 | #127 | C2×D4 | 1 | 9 | yes | — |
| 145 | #127 | C24 | 1 | 36 | yes | — |
| 146 | #124 | — | 2 | 36 | no | — |
| 147 | #124 | — | 2 | 72 | no | — |
| 148 | #124 | — | 2 | 36 | no | — |
| 149 | #124 | C2×D4 | 1 | 18 | yes | — |
| 150 | #121 | — | 2 | 36 | no | — |
| 151 | #121 | — | 2 | 36 | no | — |
| 152 | #121 | — | 2 | 36 | no | — |
| 153 | #121 | — | 2 | 72 | no | — |
| 154 | #121 | — | 2 | 72 | no | — |
| 155 | #121 | C2×D4 | 1 | 36 | yes | — |
| 156 | #121 | C24 | 1 | 72 | yes | — |
| 157 | #120 | — | 2 | 72 | no | — |
| 158 | #120 | — | 2 | 144 | no | — |
| 159 | C4×Q8 | — | 2 | 4 | no | — |
| 160 | C4×Q8 | C2×D4 | 1 | 6 | yes | — |
| 161 | C4×Q8 | C24 | 1 | 12 | yes | — |
| 162 | C4×Q8 | C24 | 1 | 2 | yes | — |
| 163 | C4×D4 | — | 2 | 36 | no | — |
| 164 | C4×D4 | — | 2 | 36 | no | — |
| 165 | C4×D4 | — | 2 | 36 | no | — |
| 166 | C4×D4 | C2×D4 | 1 | 18 | yes | — |
| 167 | C4×D4 | C24 | 1 | 36 | yes | — |
| 168 | C4×D4 | C24 | 1 | 18 | yes | — |
| 169 | #117 | — | 2 | 36 | no | — |
| 170 | #117 | — | 2 | 18 | no | — |
| 171 | #117 | C2×D4 | 1 | 18 | yes | — |
| 172 | #117 | C24 | 1 | 18 | yes | — |
| 173 | C2×C42 | — | 2 | 18 | no | — |
| 174 | C2×C42 | C24 | 1 | 9 | yes | — |
| 175 | #107 | C24 | 1 | 6 | yes | — |
| 176 | #107 | C24 | 1 | 12 | yes | — |
| 177 | #106 | C24 | 1 | 18 | yes | — |
| 178 | C4⋊C8 | C24 | 1 | 18 | yes | — |
| 179 | C4⋊C8 | C24 | 1 | 6 | yes | — |
| 180 | #103 | C24 | 1 | 12 | yes | — |
| 181 | #102 | C24 | 1 | 36 | yes | — |
| 182 | #102 | C24 | 1 | 72 | yes | — |
| 183 | C22⋊C8 | C24 | 1 | 36 | yes | — |
| 184 | C4×C8 | C24 | 1 | 6 | yes | — |
| 185 | C4×C8 | C24 | 1 | 18 | yes | — |
| 186 | C24 | — | 2 | 36 | no | — |
| 187 | #41 | — | 2 | 24 | no | — |
| 188 | #41 | — | 2 | 36 | no | — |
| 189 | #41 | — | 4 | 6 | no | — |
| 190 | #41 | — | 4 | 6 | no | — |
| 191 | #41 | — | 4 | 18 | no | — |
| 192 | #41 | — | 4 | 48 | no | — |
| 193 | #41 | — | 8 | 72 | no | — |
| 194 | #41 | — | 8 | 24 | no | — |
| 195 | C2×Q8 | — | 2 | 12 | no | — |
| 196 | C2×Q8 | C22×D4 | 1 | 6 | yes | — |
| 197 | C2×Q8 | C22×D4 | 1 | 2 | yes | — |
| 198 | C2×Q8 | C25 | 1 | 2 | yes | — |
| 199 | C2×D4 | — | 2 | 18 | no | — |
| 200 | C2×D4 | — | 2 | 72 | no | — |
| 201 | C2×D4 | — | 2 | 36 | no | — |
| 202 | C2×D4 | — | 2 | 36 | no | — |
| 203 | C2×D4 | — | 2 | 36 | no | — |
| 204 | C2×D4 | — | 4 | 36 | no | — |
| 205 | C2×D4 | — | 4 | 36 | no | — |
| 206 | C2×D4 | — | 8 | 72 | no | — |
| 207 | C2×D4 | — | 8 | 144 | no | — |
| 208 | C2×D4 | — | 8 | 72 | no | — |
| 209 | C2×D4 | C22×D4 | 1 | 18 | yes | — |
| 210 | C2×D4 | C22×D4 | 1 | 18 | yes | — |
| 211 | C2×D4 | C25 | 1 | 18 | yes | — |
| 212 | C22×C4 | — | 2 | 18 | no | — |
| 213 | C22×C4 | — | 2 | 36 | no | — |
| 214 | C22×C4 | C22×D4 | 1 | 18 | yes | — |
| 215 | C22×C4 | C25 | 1 | 9 | yes | — |
| 216 | Q16 | H(5,𝔽2) | 1 | 4 | yes | complemented |
| 217 | SD16 | — | 2 | 36 | no | — |
| 218 | D8 | — | 2 | 18 | no | — |
| 219 | D8 | — | 4 | 36 | no | — |
| 220 | D8 | — | 4 | 36 | no | — |
| 221 | M16 | — | 4 | 18 | no | — |
| 222 | C2×C8 | — | 2 | 18 | no | — |
| 223 | C2×C8 | — | 4 | 9 | no | — |
| 224 | C2×C8 | C25 | 1 | 6 | yes | — |
| 225 | C2×C8 | C25 | 1 | 9 | yes | — |
| 226 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 227 | C4⋊C4 | — | 2 | 12 | no | — |
| 228 | C4⋊C4 | — | 2 | 12 | no | — |
| 229 | C4⋊C4 | C22×D4 | 1 | 18 | yes | — |
| 230 | C4⋊C4 | C22×D4 | 1 | 6 | yes | — |
| 231 | C4⋊C4 | C25 | 1 | 12 | yes | — |
| 232 | C22⋊C4 | — | 2 | 18 | no | — |
| 233 | C22⋊C4 | — | 2 | 72 | no | — |
| 234 | C22⋊C4 | — | 2 | 36 | no | — |
| 235 | C22⋊C4 | C22×D4 | 1 | 36 | yes | — |
| 236 | C42 | — | 2 | 12 | no | — |
| 237 | C42 | — | 2 | 18 | no | — |
| 238 | C42 | C22×D4 | 1 | 9 | yes | — |
| 239 | C42 | C22×D4 | 1 | 9 | yes | — |
| 240 | C42 | C25 | 1 | 6 | yes | — |
| 241 | C23 | — | 2 | 18 | no | — |
| 242 | C23 | — | 8 | 36 | no | — |
| 243 | C23 | — | 8 | 72 | no | — |
| 244 | C23 | C23×D4 | 1 | 9 | yes | — |
| 245 | Q8 | — | 2 | 2 | no | — |
| 246 | Q8 | — | 4 | 6 | no | — |
| 247 | Q8 | — | 8 | 12 | no | — |
| 248 | Q8 | #576 | 1 | 8 | yes | complemented |
| 249 | Q8 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 250 | D4 | — | 2 | 18 | no | — |
| 251 | D4 | — | 2 | 18 | no | — |
| 252 | D4 | — | 2 | 36 | no | — |
| 253 | D4 | — | 4 | 18 | no | — |
| 254 | D4 | — | 4 | 36 | no | — |
| 255 | D4 | — | 4 | 72 | no | — |
| 256 | D4 | — | 8 | 36 | no | — |
| 257 | D4 | — | 8 | 36 | no | — |
| 258 | D4 | — | 8 | 72 | no | — |
| 259 | C2×C4 | — | 2 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 9 | no | — |
| 261 | C2×C4 | — | 2 | 9 | no | — |
| 262 | C2×C4 | — | 2 | 36 | no | — |
| 263 | C2×C4 | — | 2 | 12 | no | — |
| 264 | C2×C4 | — | 4 | 36 | no | — |
| 265 | C2×C4 | — | 4 | 72 | no | — |
| 266 | C2×C4 | C23×D4 | 1 | 9 | yes | — |
| 267 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 268 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 269 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 270 | C8 | — | 2 | 9 | no | — |
| 271 | C8 | — | 2 | 6 | no | — |
| 272 | C8 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 273 | C22 | — | 2 | 9 | no | — |
| 274 | C22 | — | 2 | 18 | no | — |
| 275 | C22 | — | 4 | 18 | no | — |
| 276 | C22 | — | 4 | 36 | no | — |
| 277 | C22 | — | 32 | 36 | no | — |
| 278 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 2 | 9 | no | — |
| 280 | C4 | — | 2 | 4 | no | — |
| 281 | C4 | — | 2 | 6 | no | — |
| 282 | C4 | — | 4 | 12 | no | — |
| 283 | C4 | C2×#576 | 1 | 12 | yes | — |
| 284 | C4 | C22×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 285 | C4 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 286 | C2 | — | 4 | 9 | no | — |
| 287 | C2 | — | 8 | 18 | no | — |
| 288 | C2 | D4×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 289 | C2 | C22×#576 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3483 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #63040 | C2 | semidirect |
| 2. #62415 | C2 | semidirect |
| 3. #62328 | C2 | semidirect |
| 4. #62001 | C2 | semidirect |
| 5. #61661 | C2 | semidirect |
| 6. #61436 | C2 | semidirect |
| 7. #61089 | C2 | semidirect |
| 8. #60308 | C2 | semidirect |
| 9. C2×#576 | C22 | semidirect |
| 11. #3549 | C22 | semidirect |
| 13. #3498 | C22 | semidirect |
| 14. #3498 | C22 | semidirect |
| 17. #3483 | C22 | semidirect |
| 18. #3391 | C22 | semidirect |
| 19. #3391 | C22 | semidirect |
| 20. Q8×Q16 | C22 | semidirect |
| 21. #3362 | C22 | semidirect |
| 22. #3341 | C22 | semidirect |
| 23. #3325 | C22 | semidirect |
| 24. #3305 | C22 | semidirect |
| 25. #3287 | C22 | semidirect |
| 26. D4×D8 | C22 | semidirect |
| 27. D4×D8 | C22 | semidirect |
| 28. #3274 | C22 | semidirect |
| 29. #3253 | C22 | semidirect |
| 30. #3205 | C22 | semidirect |
| 31. #3199 | C22 | semidirect |
| 32. #3169 | C22 | semidirect |
| 33. C2×#493 | C22 | semidirect |
| 34. #3106 | C22 | semidirect |
| 35. #3058 | C22 | semidirect |
| 36. #3038 | C22 | semidirect |
| 37. #3008 | C22 | semidirect |
| 38. #2991 | C22 | semidirect |
| 39. #2979 | C22 | semidirect |
| 40. C8×#41 | C22 | semidirect |
| 41. C4×#135 | C22 | semidirect |
| 42. #2931 | C22 | semidirect |
| 216. Q16 | H(5,𝔽2) | semidirect |
| 248. Q8 | #576 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63040 | 2 |
| #62415 | 2 |
| #62328 | 2 |
| #62001 | 2 |
| #61661 | 2 |
| #61436 | 2 |
| #61089 | 2 |
| #60308 | 2 |