Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1110 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1110 conjugacy classes of subgroups — too many to draw. Only its 202 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 | C23×M32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2218 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#2215 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #32930 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13061 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #13059 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12836 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12835 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C24×C8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×D16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×M32 | — | 2 | 8 | no | — |
| 12 | C22×M32 | — | 2 | 4 | no | — |
| 13 | C22×M32 | C22 | 1 | 2 | yes | — |
| 14 | C22×M32 | C22 | 1 | 2 | yes | — |
| 15 | C23×C16 | C22 | 1 | 1 | yes | characteristic |
| 16 | C23×C16 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#466 | C22 | 1 | 2 | yes | — |
| 18 | C2×#466 | C22 | 1 | 4 | yes | — |
| 19 | #2847 | — | 2 | 2 | no | — |
| 20 | #2218 | — | 2 | 8 | no | — |
| 21 | #2215 | — | 2 | 4 | no | — |
| 22 | C2×#367 | C22 | 1 | 2 | yes | complemented |
| 23 | C2×#366 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×#357 | C22 | 1 | 4 | yes | complemented |
| 25 | C2×#357 | C22 | 1 | 2 | yes | complemented |
| 26 | #1893 | — | 2 | 2 | no | — |
| 27 | #1893 | — | 2 | 4 | no | — |
| 28 | #1808 | C22 | 1 | 2 | yes | — |
| 29 | #1808 | C22 | 1 | 4 | yes | — |
| 30 | #1380 | C22 | 1 | 4 | yes | complemented |
| 31 | #1380 | C22 | 1 | 2 | yes | complemented |
| 32 | C24×C4 | D4 | 1 | 1 | yes | characteristic |
| 33 | C22×D8 | C23 | 1 | 2 | yes | — |
| 34 | C23×C8 | — | 2 | 4 | no | — |
| 35 | C23×C8 | — | 2 | 2 | no | — |
| 36 | C23×C8 | — | 2 | 4 | no | — |
| 37 | C23×C8 | — | 2 | 2 | no | — |
| 38 | C23×C8 | D4 | 1 | 2 | yes | — |
| 39 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 40 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 41 | C23×C8 | C23 | 1 | 2 | yes | — |
| 42 | C2×#121 | — | 2 | 2 | no | — |
| 43 | C2×#121 | — | 2 | 4 | no | — |
| 44 | C2×#120 | — | 4 | 2 | no | — |
| 45 | C2×D16 | — | 2 | 2 | no | — |
| 46 | C2×D16 | — | 2 | 4 | no | — |
| 47 | C2×M32 | — | 2 | 32 | no | — |
| 48 | C2×M32 | — | 2 | 4 | no | — |
| 49 | C2×M32 | — | 2 | 4 | no | — |
| 50 | C2×M32 | — | 2 | 4 | no | — |
| 51 | C2×M32 | — | 2 | 4 | no | — |
| 52 | C2×M32 | — | 2 | 4 | no | — |
| 53 | C2×M32 | D4 | 1 | 8 | yes | complemented |
| 54 | C22×C16 | — | 2 | 2 | no | — |
| 55 | C22×C16 | — | 2 | 4 | no | — |
| 56 | C22×C16 | — | 2 | 4 | no | — |
| 57 | C22×C16 | D4 | 1 | 4 | yes | complemented |
| 58 | C22×C16 | C23 | 1 | 2 | yes | — |
| 59 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 60 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 61 | #466 | — | 2 | 4 | no | — |
| 62 | #466 | — | 2 | 8 | no | — |
| 63 | #466 | — | 2 | 4 | no | — |
| 64 | #466 | — | 2 | 8 | no | — |
| 65 | C2×#107 | C23 | 1 | 2 | yes | — |
| 66 | C2×#107 | C23 | 1 | 4 | yes | — |
| 67 | C2×#102 | — | 2 | 2 | no | — |
| 68 | C2×#102 | — | 2 | 4 | no | — |
| 69 | #386 | — | 4 | 4 | no | — |
| 70 | #386 | — | 4 | 2 | no | — |
| 71 | #380 | — | 2 | 2 | no | — |
| 72 | #380 | — | 2 | 4 | no | — |
| 73 | #367 | — | 2 | 4 | no | — |
| 74 | #366 | — | 2 | 2 | no | — |
| 75 | #357 | — | 2 | 4 | no | — |
| 76 | #357 | — | 2 | 8 | no | — |
| 77 | #340 | — | 2 | 2 | no | — |
| 78 | #340 | — | 2 | 4 | no | — |
| 79 | C25 | D8 | 1 | 1 | yes | characteristic |
| 80 | C22×D4 | — | 2 | 2 | no | — |
| 81 | C22×D4 | — | 4 | 4 | no | — |
| 82 | C22×D4 | — | 4 | 2 | no | — |
| 83 | C23×C4 | — | 2 | 4 | no | — |
| 84 | C23×C4 | — | 2 | 2 | no | — |
| 85 | C23×C4 | — | 2 | 4 | no | — |
| 86 | C23×C4 | — | 2 | 2 | no | — |
| 87 | C23×C4 | D8 | 1 | 2 | yes | — |
| 88 | C23×C4 | D8 | 1 | 1 | yes | characteristic |
| 89 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 90 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×D8 | — | 2 | 2 | no | — |
| 92 | C2×D8 | — | 2 | 4 | no | — |
| 93 | C2×D8 | — | 2 | 2 | no | — |
| 94 | C2×D8 | — | 2 | 4 | no | — |
| 95 | C22×C8 | — | 2 | 8 | no | — |
| 96 | C22×C8 | — | 2 | 16 | no | — |
| 97 | C22×C8 | — | 2 | 8 | no | — |
| 98 | C22×C8 | — | 2 | 4 | no | — |
| 99 | C22×C8 | — | 2 | 2 | no | — |
| 100 | C22×C8 | — | 2 | 4 | no | — |
| 101 | C22×C8 | — | 2 | 4 | no | — |
| 102 | C22×C8 | — | 2 | 4 | no | — |
| 103 | C22×C8 | — | 2 | 2 | no | — |
| 104 | C22×C8 | — | 2 | 2 | no | — |
| 105 | C22×C8 | — | 2 | 2 | no | — |
| 106 | C22×C8 | — | 2 | 2 | no | — |
| 107 | C22×C8 | — | 2 | 2 | no | — |
| 108 | C22×C8 | — | 2 | 2 | no | — |
| 109 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 111 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 112 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 113 | C22×C8 | #41 | 1 | 2 | yes | — |
| 114 | C22×C8 | #41 | 1 | 4 | yes | — |
| 115 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 116 | #121 | — | 4 | 4 | no | — |
| 117 | #121 | — | 4 | 8 | no | — |
| 118 | #121 | — | 4 | 8 | no | — |
| 119 | #121 | — | 4 | 4 | no | — |
| 120 | #120 | — | 8 | 8 | no | — |
| 121 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 122 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 123 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 124 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 125 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 126 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 127 | D16 | — | 4 | 4 | no | — |
| 128 | M32 | — | 2 | 16 | no | — |
| 129 | M32 | — | 2 | 16 | no | — |
| 130 | C2×C16 | — | 2 | 8 | no | — |
| 131 | C2×C16 | — | 2 | 2 | no | — |
| 132 | C2×C16 | — | 2 | 2 | no | — |
| 133 | C2×C16 | — | 2 | 2 | no | — |
| 134 | C2×C16 | — | 2 | 8 | no | — |
| 135 | C2×C16 | C2×D4 | 1 | 4 | yes | — |
| 136 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 137 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 138 | C2×C16 | C2×D4 | 1 | 4 | yes | — |
| 139 | #107 | — | 2 | 2 | no | — |
| 140 | #107 | — | 2 | 4 | no | — |
| 141 | #107 | — | 2 | 2 | no | — |
| 142 | #107 | — | 2 | 4 | no | — |
| 143 | #102 | — | 4 | 4 | no | — |
| 144 | #102 | — | 4 | 8 | no | — |
| 145 | #95 | — | 4 | 4 | no | — |
| 146 | #95 | — | 4 | 2 | no | — |
| 147 | C24 | — | 2 | 4 | no | — |
| 148 | C24 | — | 2 | 2 | no | — |
| 149 | C24 | — | 2 | 4 | no | — |
| 150 | C24 | — | 4 | 2 | no | — |
| 151 | C24 | — | 4 | 2 | no | — |
| 152 | C24 | C2×D8 | 1 | 2 | yes | — |
| 153 | C24 | C2×D8 | 1 | 1 | yes | characteristic |
| 154 | C2×D4 | — | 4 | 2 | no | — |
| 155 | C2×D4 | — | 4 | 4 | no | — |
| 156 | C2×D4 | — | 4 | 2 | no | — |
| 157 | C2×D4 | — | 4 | 4 | no | — |
| 158 | C2×D4 | — | 8 | 8 | no | — |
| 159 | C2×D4 | — | 8 | 8 | no | — |
| 160 | C2×D4 | — | 8 | 8 | no | — |
| 161 | C2×D4 | — | 8 | 4 | no | — |
| 162 | C2×D4 | — | 8 | 4 | no | — |
| 163 | C2×D4 | — | 8 | 4 | no | — |
| 164 | C22×C4 | — | 2 | 8 | no | — |
| 165 | C22×C4 | — | 2 | 8 | no | — |
| 166 | C22×C4 | — | 2 | 16 | no | — |
| 167 | C22×C4 | — | 2 | 4 | no | — |
| 168 | C22×C4 | — | 2 | 2 | no | — |
| 169 | C22×C4 | — | 2 | 2 | no | — |
| 170 | C22×C4 | — | 2 | 2 | no | — |
| 171 | C22×C4 | — | 2 | 2 | no | — |
| 172 | C22×C4 | — | 2 | 2 | no | — |
| 173 | C22×C4 | — | 2 | 2 | no | — |
| 174 | C22×C4 | — | 2 | 2 | no | — |
| 175 | C22×C4 | — | 2 | 4 | no | — |
| 176 | C22×C4 | — | 2 | 4 | no | — |
| 177 | C22×C4 | — | 2 | 4 | no | — |
| 178 | C22×C4 | — | 4 | 2 | no | — |
| 179 | C22×C4 | — | 4 | 4 | no | — |
| 180 | C22×C4 | #121 | 1 | 4 | yes | — |
| 181 | C22×C4 | #121 | 1 | 2 | yes | — |
| 182 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 183 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 184 | C22×C4 | #135 | 1 | 4 | yes | — |
| 185 | C22×C4 | #135 | 1 | 2 | yes | — |
| 186 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 187 | D8 | — | 4 | 2 | no | — |
| 188 | D8 | — | 4 | 2 | no | — |
| 189 | D8 | — | 4 | 4 | no | — |
| 190 | C2×C8 | — | 2 | 16 | no | — |
| 191 | C2×C8 | — | 2 | 8 | no | — |
| 192 | C2×C8 | — | 2 | 2 | no | — |
| 193 | C2×C8 | — | 2 | 2 | no | — |
| 194 | C2×C8 | — | 2 | 2 | no | — |
| 195 | C2×C8 | — | 2 | 2 | no | — |
| 196 | C2×C8 | — | 2 | 4 | no | — |
| 197 | C2×C8 | — | 2 | 2 | no | — |
| 198 | C2×C8 | — | 2 | 4 | no | — |
| 199 | C2×C8 | — | 2 | 4 | no | — |
| 200 | C2×C8 | — | 2 | 4 | no | — |
| 201 | C2×C8 | — | 2 | 4 | no | — |
| 202 | C2×C8 | #121 | 1 | 2 | yes | — |
| 203 | C2×C8 | #121 | 1 | 4 | yes | — |
| 204 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 205 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 206 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 207 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 208 | C4⋊C4 | — | 4 | 2 | no | — |
| 209 | C4⋊C4 | — | 4 | 4 | no | — |
| 210 | C4⋊C4 | — | 4 | 2 | no | — |
| 211 | C4⋊C4 | — | 4 | 4 | no | — |
| 212 | C22⋊C4 | — | 8 | 8 | no | — |
| 213 | C22⋊C4 | — | 8 | 4 | no | — |
| 214 | C22⋊C4 | — | 8 | 4 | no | — |
| 215 | C22⋊C4 | — | 8 | 8 | no | — |
| 216 | C16 | — | 2 | 4 | no | — |
| 217 | C16 | — | 2 | 2 | no | — |
| 218 | C16 | #120 | 1 | 4 | yes | complemented |
| 219 | C23 | — | 2 | 2 | no | — |
| 220 | C23 | — | 2 | 2 | no | — |
| 221 | C23 | — | 2 | 8 | no | — |
| 222 | C23 | — | 2 | 4 | no | — |
| 223 | C23 | — | 2 | 4 | no | — |
| 224 | C23 | — | 2 | 2 | no | — |
| 225 | C23 | — | 2 | 2 | no | — |
| 226 | C23 | — | 4 | 16 | no | — |
| 227 | C23 | — | 4 | 4 | no | — |
| 228 | C23 | — | 4 | 4 | no | — |
| 229 | C23 | — | 8 | 2 | no | — |
| 230 | C23 | — | 8 | 2 | no | — |
| 231 | C23 | — | 8 | 4 | no | — |
| 232 | C23 | — | 8 | 2 | no | — |
| 233 | C23 | — | 8 | 4 | no | — |
| 234 | C23 | #466 | 1 | 4 | yes | — |
| 235 | C23 | #466 | 1 | 2 | yes | — |
| 236 | C23 | #509 | 1 | 4 | yes | — |
| 237 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 238 | D4 | — | 8 | 2 | no | — |
| 239 | D4 | — | 8 | 2 | no | — |
| 240 | D4 | — | 8 | 4 | no | — |
| 241 | D4 | — | 16 | 16 | no | — |
| 242 | D4 | — | 16 | 8 | no | — |
| 243 | C2×C4 | — | 2 | 8 | no | — |
| 244 | C2×C4 | — | 2 | 16 | no | — |
| 245 | C2×C4 | — | 2 | 2 | no | — |
| 246 | C2×C4 | — | 2 | 2 | no | — |
| 247 | C2×C4 | — | 2 | 2 | no | — |
| 248 | C2×C4 | — | 2 | 2 | no | — |
| 249 | C2×C4 | — | 2 | 4 | no | — |
| 250 | C2×C4 | — | 2 | 2 | no | — |
| 251 | C2×C4 | — | 2 | 4 | no | — |
| 252 | C2×C4 | — | 2 | 4 | no | — |
| 253 | C2×C4 | — | 2 | 4 | no | — |
| 254 | C2×C4 | — | 2 | 4 | no | — |
| 255 | C2×C4 | — | 8 | 2 | no | — |
| 256 | C2×C4 | — | 8 | 4 | no | — |
| 257 | C2×C4 | — | 8 | 2 | no | — |
| 258 | C2×C4 | — | 8 | 4 | no | — |
| 259 | C2×C4 | — | 8 | 2 | no | — |
| 260 | C2×C4 | — | 8 | 4 | no | — |
| 261 | C2×C4 | #466 | 1 | 2 | yes | — |
| 262 | C2×C4 | #466 | 1 | 4 | yes | — |
| 263 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 265 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 267 | C8 | — | 2 | 2 | no | — |
| 268 | C8 | — | 2 | 4 | no | — |
| 269 | C8 | C2×#120 | 1 | 1 | yes | characteristic |
| 270 | C8 | #525 | 1 | 1 | yes | characteristic |
| 271 | C8 | #525 | 1 | 2 | yes | — |
| 272 | C22 | — | 2 | 2 | no | — |
| 273 | C22 | — | 2 | 8 | no | — |
| 274 | C22 | — | 2 | 2 | no | — |
| 275 | C22 | — | 2 | 8 | no | — |
| 276 | C22 | — | 2 | 4 | no | — |
| 277 | C22 | — | 4 | 16 | no | — |
| 278 | C22 | — | 4 | 2 | no | — |
| 279 | C22 | — | 4 | 2 | no | — |
| 280 | C22 | — | 4 | 4 | no | — |
| 281 | C22 | — | 16 | 2 | no | — |
| 282 | C22 | — | 16 | 2 | no | — |
| 283 | C22 | — | 16 | 4 | no | — |
| 284 | C22 | — | 16 | 4 | no | — |
| 285 | C22 | — | 16 | 2 | no | — |
| 286 | C22 | #2218 | 1 | 4 | yes | complemented |
| 287 | C22 | C2×#466 | 1 | 1 | yes | characteristic |
| 288 | C22 | C2×#466 | 1 | 2 | yes | — |
| 289 | C22 | C2×#509 | 1 | 2 | yes | — |
| 290 | C22 | #3414 | 1 | 2 | yes | — |
| 291 | C4 | — | 2 | 4 | no | — |
| 292 | C4 | — | 2 | 2 | no | — |
| 293 | C4 | — | 16 | 2 | no | — |
| 294 | C4 | — | 16 | 4 | no | — |
| 295 | C4 | #2847 | 1 | 1 | yes | characteristic |
| 296 | C4 | #3050 | 1 | 1 | yes | characteristic |
| 297 | C4 | #3050 | 1 | 2 | yes | — |
| 298 | C2 | — | 2 | 4 | no | — |
| 299 | C2 | — | 4 | 4 | no | — |
| 300 | C2 | — | 32 | 2 | no | — |
| 301 | C2 | #32930 | 1 | 1 | yes | characteristic |
| 302 | C2 | C2×#2218 | 1 | 1 | yes | characteristic |
| 303 | C2 | #33604 | 1 | 1 | yes | characteristic |
| 304 | C2 | #33605 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×M32 | C2 | semidirect |
| 2. C2×#2218 | C2 | semidirect |
| 3. C2×#2215 | C2 | semidirect |
| 4. #32930 | C2 | semidirect |
| 5. #13061 | C2 | semidirect |
| 6. #13059 | C2 | semidirect |
| 7. #12836 | C2 | semidirect |
| 8. #12835 | C2 | semidirect |
| 10. C22×D16 | C22 | semidirect |
| 22. C2×#367 | C22 | semidirect |
| 23. C2×#366 | C22 | semidirect |
| 24. C2×#357 | C22 | semidirect |
| 25. C2×#357 | C22 | semidirect |
| 30. #1380 | C22 | semidirect |
| 31. #1380 | C22 | semidirect |
| 53. C2×M32 | D4 | semidirect |
| 57. C22×C16 | D4 | semidirect |
| 218. C16 | #120 | semidirect |
| 286. C22 | #2218 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M32 | 2 |
| C2×#2218 | 2 |
| C2×#2215 | 2 |
| #32930 | 2 |
| #13061 | 2 |
| #13059 | 2 |
| #12836 | 2 |
| #12835 | 2 |