Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1700 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60664 | C2 | 1 | 4 | yes | maximal |
| 2 | #28826 | C2 | 1 | 8 | yes | maximal, complemented |
| 3 | #28823 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #23560 | C2 | 1 | 2 | yes | maximal |
| 5 | C2×#2061 | C2 | 1 | 2 | yes | maximal |
| 6 | #21182 | C2 | 1 | 4 | yes | maximal |
| 7 | #20801 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #20445 | C2 | 1 | 2 | yes | maximal |
| 9 | #3458 | C22 | 1 | 2 | yes | — |
| 10 | C2×#544 | C22 | 1 | 2 | yes | — |
| 11 | #3166 | — | 2 | 8 | no | — |
| 12 | #3165 | — | 2 | 8 | no | — |
| 13 | #3164 | — | 2 | 8 | no | — |
| 14 | #3164 | C22 | 1 | 16 | yes | — |
| 15 | #3163 | — | 2 | 8 | no | — |
| 16 | #3163 | C22 | 1 | 16 | yes | — |
| 17 | C2×#430 | C22 | 1 | 2 | yes | — |
| 18 | #2911 | C22 | 1 | 4 | yes | — |
| 19 | #2911 | C22 | 1 | 2 | yes | — |
| 20 | C2×#428 | C22 | 1 | 4 | yes | — |
| 21 | C2×#428 | C22 | 1 | 4 | yes | — |
| 22 | #2873 | C22 | 1 | 4 | yes | — |
| 23 | C2×C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 24 | #2285 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#382 | C22 | 1 | 1 | yes | characteristic |
| 26 | #2061 | C22 | 1 | 8 | yes | — |
| 27 | #2061 | C22 | 1 | 8 | yes | — |
| 28 | #2061 | C22 | 1 | 8 | yes | — |
| 29 | #2060 | C22 | 1 | 8 | yes | — |
| 30 | #1952 | C22 | 1 | 16 | yes | — |
| 31 | #1951 | C22 | 1 | 16 | yes | — |
| 32 | #1847 | C22 | 1 | 4 | yes | — |
| 33 | #1846 | C22 | 1 | 4 | yes | — |
| 34 | #1773 | C22 | 1 | 8 | yes | — |
| 35 | #1772 | C22 | 1 | 8 | yes | — |
| 36 | #1742 | C22 | 1 | 2 | yes | — |
| 37 | C2×#343 | C22 | 1 | 2 | yes | — |
| 38 | C2×#339 | C22 | 1 | 4 | yes | — |
| 39 | C22×M16 | C23 | 1 | 4 | yes | — |
| 40 | C22×M16 | C23 | 1 | 2 | yes | — |
| 41 | #544 | — | 2 | 8 | no | — |
| 42 | #544 | — | 2 | 4 | no | — |
| 43 | #544 | — | 2 | 4 | no | — |
| 44 | #544 | — | 4 | 2 | no | — |
| 45 | #544 | C23 | 1 | 8 | yes | — |
| 46 | #544 | C23 | 1 | 8 | yes | — |
| 47 | #543 | — | 4 | 2 | no | — |
| 48 | #533 | — | 2 | 8 | no | — |
| 49 | #533 | — | 4 | 2 | no | — |
| 50 | C2×C4⋊Q8 | — | 2 | 4 | no | — |
| 51 | C2×#125 | — | 2 | 4 | no | — |
| 52 | C2×C22⋊Q8 | — | 4 | 2 | no | — |
| 53 | C2×#117 | C23 | 1 | 4 | yes | — |
| 54 | C2×#117 | C23 | 1 | 2 | yes | — |
| 55 | C22×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 56 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 57 | #501 | — | 2 | 16 | no | — |
| 58 | #501 | — | 2 | 16 | no | — |
| 59 | #501 | — | 2 | 16 | no | — |
| 60 | #501 | — | 2 | 16 | no | — |
| 61 | #500 | — | 2 | 16 | no | — |
| 62 | #499 | — | 2 | 16 | no | — |
| 63 | #499 | — | 2 | 16 | no | — |
| 64 | #498 | — | 2 | 16 | no | — |
| 65 | #430 | Q8 | 1 | 16 | yes | complemented |
| 66 | C2×#108 | C23 | 1 | 8 | yes | — |
| 67 | #428 | — | 2 | 8 | no | — |
| 68 | #428 | — | 2 | 8 | no | — |
| 69 | #428 | — | 2 | 8 | no | — |
| 70 | #428 | — | 2 | 4 | no | — |
| 71 | #428 | — | 2 | 4 | no | — |
| 72 | #428 | D4 | 1 | 8 | yes | — |
| 73 | #428 | D4 | 1 | 8 | yes | — |
| 74 | #428 | C23 | 1 | 8 | yes | — |
| 75 | #428 | C23 | 1 | 8 | yes | — |
| 76 | #428 | C23 | 1 | 16 | yes | — |
| 77 | #427 | — | 2 | 8 | no | — |
| 78 | #427 | C23 | 1 | 8 | yes | — |
| 79 | C2×#107 | — | 2 | 4 | no | — |
| 80 | C2×#107 | C23 | 1 | 8 | yes | — |
| 81 | C2×#106 | — | 2 | 4 | no | — |
| 82 | C2×#106 | C23 | 1 | 8 | yes | — |
| 83 | #405 | — | 2 | 8 | no | — |
| 84 | #405 | C23 | 1 | 16 | yes | — |
| 85 | C4×M16 | — | 2 | 8 | no | — |
| 86 | C4×M16 | Q8 | 1 | 8 | yes | complemented |
| 87 | C2×#97 | C23 | 1 | 2 | yes | — |
| 88 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 89 | #384 | C23 | 1 | 2 | yes | — |
| 90 | #382 | — | 4 | 1 | no | — |
| 91 | #382 | C23 | 1 | 2 | yes | — |
| 92 | #382 | C23 | 1 | 2 | yes | — |
| 93 | #382 | C23 | 1 | 2 | yes | — |
| 94 | #380 | — | 4 | 1 | no | — |
| 95 | C4×C4⋊C4 | — | 4 | 1 | no | — |
| 96 | C2×#95 | — | 4 | 1 | no | — |
| 97 | #343 | C23 | 1 | 8 | yes | — |
| 98 | #343 | C23 | 1 | 8 | yes | — |
| 99 | #339 | C23 | 1 | 8 | yes | — |
| 100 | #339 | C23 | 1 | 8 | yes | — |
| 101 | C22×Q8 | — | 4 | 2 | no | — |
| 102 | C23×C4 | — | 2 | 1 | no | — |
| 103 | C23×C4 | — | 2 | 2 | no | — |
| 104 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 105 | C2×M16 | — | 2 | 4 | no | — |
| 106 | C2×M16 | — | 2 | 8 | no | — |
| 107 | C2×M16 | C2×D4 | 1 | 8 | yes | — |
| 108 | C2×M16 | C2×Q8 | 1 | 16 | yes | — |
| 109 | C2×M16 | C2×Q8 | 1 | 8 | yes | — |
| 110 | C2×M16 | C2×Q8 | 1 | 8 | yes | — |
| 111 | C2×M16 | C24 | 1 | 8 | yes | — |
| 112 | C22×C8 | — | 2 | 4 | no | — |
| 113 | C22×C8 | C24 | 1 | 4 | yes | — |
| 114 | C4⋊Q8 | — | 2 | 16 | no | — |
| 115 | C4⋊Q8 | — | 2 | 8 | no | — |
| 116 | C4⋊Q8 | — | 2 | 8 | no | — |
| 117 | C4⋊Q8 | — | 4 | 8 | no | — |
| 118 | C4⋊Q8 | — | 4 | 4 | no | — |
| 119 | #125 | — | 2 | 16 | no | — |
| 120 | #125 | — | 2 | 8 | no | — |
| 121 | #125 | — | 2 | 8 | no | — |
| 122 | #125 | — | 4 | 8 | no | — |
| 123 | #125 | — | 4 | 2 | no | — |
| 124 | #125 | — | 4 | 2 | no | — |
| 125 | C22⋊Q8 | — | 4 | 8 | no | — |
| 126 | C22⋊Q8 | — | 4 | 8 | no | — |
| 127 | C22⋊Q8 | — | 4 | 8 | no | — |
| 128 | C22⋊Q8 | — | 4 | 8 | no | — |
| 129 | C22⋊Q8 | — | 4 | 8 | no | — |
| 130 | C4×Q8 | — | 4 | 8 | no | — |
| 131 | #117 | — | 2 | 4 | no | — |
| 132 | #117 | — | 2 | 4 | no | — |
| 133 | #117 | — | 2 | 4 | no | — |
| 134 | #117 | — | 2 | 4 | no | — |
| 135 | #117 | C2×D4 | 1 | 8 | yes | — |
| 136 | #117 | C24 | 1 | 8 | yes | — |
| 137 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 138 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 139 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 140 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 141 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 142 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 143 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 144 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 145 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 146 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 147 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 148 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 149 | C2×C42 | — | 2 | 4 | no | — |
| 150 | C2×C42 | — | 2 | 1 | no | — |
| 151 | C2×C42 | — | 2 | 2 | no | — |
| 152 | C2×C42 | — | 2 | 2 | no | — |
| 153 | C2×C42 | C2×Q8 | 1 | 4 | yes | — |
| 154 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 155 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 156 | #108 | C2×Q8 | 1 | 32 | yes | complemented |
| 157 | #107 | — | 2 | 16 | no | — |
| 158 | #107 | — | 2 | 16 | no | — |
| 159 | #107 | — | 2 | 8 | no | — |
| 160 | #107 | — | 2 | 8 | no | — |
| 161 | #107 | C2×D4 | 1 | 16 | yes | — |
| 162 | #106 | — | 2 | 16 | no | — |
| 163 | #106 | — | 2 | 16 | no | — |
| 164 | #106 | — | 2 | 8 | no | — |
| 165 | #106 | — | 2 | 8 | no | — |
| 166 | #106 | C2×D4 | 1 | 8 | yes | — |
| 167 | #106 | C2×D4 | 1 | 8 | yes | — |
| 168 | #97 | — | 2 | 16 | no | — |
| 169 | #97 | C2×Q8 | 1 | 8 | yes | complemented |
| 170 | C4×C8 | — | 2 | 16 | no | — |
| 171 | C4×C8 | C2×Q8 | 1 | 8 | yes | complemented |
| 172 | #95 | — | 4 | 2 | no | — |
| 173 | #95 | — | 4 | 2 | no | — |
| 174 | #95 | — | 4 | 2 | no | — |
| 175 | C24 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 176 | C2×Q8 | — | 4 | 8 | no | — |
| 177 | C2×Q8 | — | 4 | 8 | no | — |
| 178 | C2×Q8 | — | 16 | 4 | no | — |
| 179 | C22×C4 | — | 2 | 1 | no | — |
| 180 | C22×C4 | — | 2 | 1 | no | — |
| 181 | C22×C4 | — | 2 | 4 | no | — |
| 182 | C22×C4 | — | 2 | 4 | no | — |
| 183 | C22×C4 | — | 2 | 4 | no | — |
| 184 | C22×C4 | — | 2 | 4 | no | — |
| 185 | C22×C4 | — | 2 | 4 | no | — |
| 186 | C22×C4 | — | 4 | 2 | no | — |
| 187 | C22×C4 | — | 4 | 2 | no | — |
| 188 | C22×C4 | — | 8 | 2 | no | — |
| 189 | C22×C4 | — | 8 | 4 | no | — |
| 190 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 191 | C22×C4 | C22×Q8 | 1 | 2 | yes | — |
| 192 | C22×C4 | C22×Q8 | 1 | 4 | yes | — |
| 193 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 194 | C22×C4 | 2−1+4 | 1 | 2 | yes | — |
| 195 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 196 | M16 | — | 2 | 16 | no | — |
| 197 | M16 | — | 2 | 8 | no | — |
| 198 | M16 | C4⋊Q8 | 1 | 32 | yes | — |
| 199 | M16 | C22×Q8 | 1 | 16 | yes | — |
| 200 | C2×C8 | — | 2 | 16 | no | — |
| 201 | C2×C8 | — | 2 | 8 | no | — |
| 202 | C2×C8 | C22×D4 | 1 | 8 | yes | — |
| 203 | C2×C8 | C22×Q8 | 1 | 8 | yes | — |
| 204 | C2×C8 | C22×Q8 | 1 | 8 | yes | — |
| 205 | C4⋊C4 | — | 2 | 4 | no | — |
| 206 | C4⋊C4 | — | 2 | 4 | no | — |
| 207 | C4⋊C4 | — | 2 | 4 | no | — |
| 208 | C4⋊C4 | — | 2 | 8 | no | — |
| 209 | C4⋊C4 | — | 2 | 4 | no | — |
| 210 | C4⋊C4 | — | 4 | 8 | no | — |
| 211 | C4⋊C4 | — | 4 | 16 | no | — |
| 212 | C4⋊C4 | — | 4 | 8 | no | — |
| 213 | C4⋊C4 | — | 4 | 8 | no | — |
| 214 | C4⋊C4 | — | 4 | 8 | no | — |
| 215 | C4⋊C4 | — | 4 | 4 | no | — |
| 216 | C4⋊C4 | — | 8 | 4 | no | — |
| 217 | C4⋊C4 | C22×D4 | 1 | 8 | yes | — |
| 218 | C4⋊C4 | C22×D4 | 1 | 8 | yes | — |
| 219 | C22⋊C4 | — | 2 | 8 | no | — |
| 220 | C22⋊C4 | — | 2 | 8 | no | — |
| 221 | C22⋊C4 | — | 4 | 4 | no | — |
| 222 | C42 | — | 2 | 4 | no | — |
| 223 | C42 | — | 2 | 8 | no | — |
| 224 | C42 | — | 2 | 8 | no | — |
| 225 | C42 | — | 2 | 1 | no | — |
| 226 | C42 | — | 2 | 1 | no | — |
| 227 | C42 | — | 4 | 2 | no | — |
| 228 | C42 | — | 4 | 2 | no | — |
| 229 | C42 | C22×Q8 | 1 | 2 | yes | — |
| 230 | C42 | C22×Q8 | 1 | 2 | yes | — |
| 231 | C23 | — | 4 | 2 | no | — |
| 232 | C23 | #544 | 1 | 2 | yes | — |
| 233 | C23 | D42 | 1 | 2 | yes | — |
| 234 | C23 | Q82 | 1 | 2 | yes | — |
| 235 | C23 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 236 | Q8 | — | 16 | 8 | no | — |
| 237 | Q8 | — | 16 | 8 | no | — |
| 238 | C2×C4 | — | 2 | 2 | no | — |
| 239 | C2×C4 | — | 2 | 2 | no | — |
| 240 | C2×C4 | — | 2 | 8 | no | — |
| 241 | C2×C4 | — | 2 | 8 | no | — |
| 242 | C2×C4 | — | 4 | 2 | no | — |
| 243 | C2×C4 | — | 4 | 2 | no | — |
| 244 | C2×C4 | — | 4 | 4 | no | — |
| 245 | C2×C4 | — | 4 | 4 | no | — |
| 246 | C2×C4 | — | 8 | 4 | no | — |
| 247 | C2×C4 | — | 8 | 4 | no | — |
| 248 | C2×C4 | — | 8 | 4 | no | — |
| 249 | C2×C4 | — | 8 | 4 | no | — |
| 250 | C2×C4 | — | 8 | 4 | no | — |
| 251 | C2×C4 | — | 8 | 4 | no | — |
| 252 | C2×C4 | — | 16 | 2 | no | — |
| 253 | C2×C4 | C2×C4⋊Q8 | 1 | 8 | yes | — |
| 254 | C2×C4 | #544 | 1 | 2 | yes | — |
| 255 | C2×C4 | D42 | 1 | 2 | yes | — |
| 256 | C2×C4 | D4×Q8 | 1 | 8 | yes | — |
| 257 | C2×C4 | Q82 | 1 | 2 | yes | — |
| 258 | C2×C4 | C23×D4 | 1 | 2 | yes | — |
| 259 | C2×C4 | C23×Q8 | 1 | 1 | yes | characteristic |
| 260 | C2×C4 | C23×Q8 | 1 | 2 | yes | — |
| 261 | C2×C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 262 | C8 | — | 2 | 16 | no | — |
| 263 | C8 | C2×C4⋊Q8 | 1 | 16 | yes | — |
| 264 | C22 | — | 2 | 2 | no | — |
| 265 | C22 | — | 2 | 2 | no | — |
| 266 | C22 | — | 2 | 2 | no | — |
| 267 | C22 | — | 4 | 2 | no | — |
| 268 | C22 | — | 4 | 2 | no | — |
| 269 | C22 | #2551 | 1 | 4 | yes | — |
| 270 | C22 | C2×#544 | 1 | 1 | yes | characteristic |
| 271 | C22 | C2×D42 | 1 | 1 | yes | characteristic |
| 272 | C22 | C2×Q82 | 1 | 1 | yes | characteristic |
| 273 | C4 | — | 4 | 2 | no | — |
| 274 | C4 | — | 4 | 2 | no | — |
| 275 | C4 | — | 4 | 8 | no | — |
| 276 | C4 | — | 4 | 8 | no | — |
| 277 | C4 | — | 8 | 4 | no | — |
| 278 | C4 | — | 8 | 4 | no | — |
| 279 | C4 | #2551 | 1 | 4 | yes | — |
| 280 | C4 | C22×C4⋊Q8 | 1 | 2 | yes | — |
| 281 | C4 | C2×D4×Q8 | 1 | 2 | yes | — |
| 282 | C2 | — | 2 | 1 | no | — |
| 283 | C2 | — | 2 | 1 | no | — |
| 284 | C2 | — | 2 | 4 | no | — |
| 285 | C2 | C2×#2551 | 1 | 1 | yes | characteristic |
| 286 | C2 | #61109 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #28826 | C2 | semidirect |
| 3. #28823 | C2 | semidirect |
| 65. #430 | Q8 | semidirect |
| 86. C4×M16 | Q8 | semidirect |
| 156. #108 | C2×Q8 | semidirect |
| 169. #97 | C2×Q8 | semidirect |
| 171. C4×C8 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60664 | 2 |
| #28826 | 2 |
| #28823 | 2 |
| #23560 | 2 |
| C2×#2061 | 2 |
| #21182 | 2 |
| #20801 | 2 |
| #20445 | 2 |