Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1247 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1247 conjugacy classes of subgroups — too many to draw. Only its 222 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 | Q8×#127 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q8×#99 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #13298 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #12797 | C2 | 1 | 3 | yes | maximal |
| 5 | C4×#353 | C2 | 1 | 3 | yes | maximal |
| 6 | C2×D4×Q8 | — | 2 | 2 | no | — |
| 7 | C2×D4×Q8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×D4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2868 | C4 | 1 | 3 | yes | complemented |
| 10 | #2542 | — | 2 | 6 | no | — |
| 11 | #2542 | C4 | 1 | 3 | yes | complemented |
| 12 | #2542 | C22 | 1 | 3 | yes | — |
| 13 | Q8×C22⋊C4 | — | 2 | 2 | no | — |
| 14 | C4×#127 | C22 | 1 | 3 | yes | — |
| 15 | C42×Q8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#353 | C22 | 1 | 3 | yes | — |
| 17 | C2×#353 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1996 | — | 2 | 6 | no | — |
| 19 | #1799 | C22 | 1 | 6 | yes | — |
| 20 | C4×#99 | C22 | 1 | 6 | yes | — |
| 21 | #1393 | C22 | 1 | 6 | yes | — |
| 22 | #1393 | C22 | 1 | 6 | yes | — |
| 23 | C23×Q8 | — | 2 | 1 | no | — |
| 24 | C23×Q8 | D4 | 1 | 2 | yes | — |
| 25 | D4×Q8 | — | 4 | 1 | no | — |
| 26 | D4×Q8 | — | 4 | 3 | no | — |
| 27 | D4×Q8 | — | 4 | 3 | no | — |
| 28 | D4×Q8 | — | 4 | 1 | no | — |
| 29 | D4×Q8 | — | 4 | 16 | no | — |
| 30 | C2×C4⋊Q8 | — | 2 | 6 | no | — |
| 31 | C2×C4⋊Q8 | C2×C4 | 1 | 3 | yes | — |
| 32 | C2×C4⋊Q8 | C2×C4 | 1 | 3 | yes | — |
| 33 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C22⋊Q8 | — | 2 | 6 | no | — |
| 35 | C2×C22⋊Q8 | — | 2 | 6 | no | — |
| 36 | C2×C22⋊Q8 | — | 2 | 3 | no | — |
| 37 | C2×C22⋊Q8 | D4 | 1 | 6 | yes | — |
| 38 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 39 | C2×C4×Q8 | — | 4 | 2 | no | — |
| 40 | C2×C4×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×C4×D4 | — | 2 | 6 | no | — |
| 42 | C2×C4×D4 | C2×C4 | 1 | 3 | yes | — |
| 43 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 44 | C2×#99 | C23 | 1 | 6 | yes | — |
| 45 | C2×#99 | C23 | 1 | 2 | yes | — |
| 46 | #399 | — | 2 | 6 | no | — |
| 47 | #399 | — | 2 | 6 | no | — |
| 48 | #391 | — | 4 | 6 | no | — |
| 49 | #390 | — | 2 | 6 | no | — |
| 50 | #390 | C2×C4 | 1 | 3 | yes | — |
| 51 | #390 | C23 | 1 | 3 | yes | — |
| 52 | #384 | — | 2 | 6 | no | — |
| 53 | #384 | C2×C4 | 1 | 3 | yes | — |
| 54 | #384 | C2×C4 | 1 | 6 | yes | — |
| 55 | #380 | — | 2 | 6 | no | — |
| 56 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 57 | C4×C4⋊C4 | C2×C4 | 1 | 3 | yes | — |
| 58 | C4×C22⋊C4 | — | 2 | 6 | no | — |
| 59 | C43 | C2×C4 | 1 | 3 | yes | — |
| 60 | #353 | — | 2 | 6 | no | — |
| 61 | #353 | Q8 | 1 | 4 | yes | direct product factor |
| 62 | #342 | — | 2 | 6 | no | — |
| 63 | #342 | — | 2 | 6 | no | — |
| 64 | C22×Q8 | — | 2 | 2 | no | — |
| 65 | C22×Q8 | — | 2 | 6 | no | — |
| 66 | C22×Q8 | — | 2 | 3 | no | — |
| 67 | C22×Q8 | — | 4 | 8 | no | — |
| 68 | C22×Q8 | — | 4 | 8 | no | — |
| 69 | C22×Q8 | — | 4 | 3 | no | — |
| 70 | C22×Q8 | — | 4 | 1 | no | — |
| 71 | C22×Q8 | C22⋊C4 | 1 | 6 | yes | — |
| 72 | C22×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C22×D4 | — | 2 | 2 | no | — |
| 74 | C22×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C22×D4 | C24 | 1 | 1 | yes | characteristic |
| 76 | C23×C4 | — | 2 | 3 | no | — |
| 77 | C23×C4 | C2×D4 | 1 | 6 | yes | — |
| 78 | C4⋊Q8 | — | 4 | 3 | no | — |
| 79 | C4⋊Q8 | — | 4 | 3 | no | — |
| 80 | C4⋊Q8 | — | 4 | 3 | no | — |
| 81 | C4⋊Q8 | — | 4 | 3 | no | — |
| 82 | C4⋊Q8 | — | 4 | 24 | no | — |
| 83 | #127 | — | 2 | 2 | no | — |
| 84 | #127 | C2×Q8 | 1 | 2 | yes | — |
| 85 | #127 | #41 | 1 | 2 | yes | — |
| 86 | C22⋊Q8 | — | 4 | 6 | no | — |
| 87 | C22⋊Q8 | — | 4 | 6 | no | — |
| 88 | C22⋊Q8 | — | 4 | 24 | no | — |
| 89 | C22⋊Q8 | — | 4 | 24 | no | — |
| 90 | C22⋊Q8 | — | 4 | 6 | no | — |
| 91 | C22⋊Q8 | — | 4 | 6 | no | — |
| 92 | C4×Q8 | — | 2 | 8 | no | — |
| 93 | C4×Q8 | — | 2 | 3 | no | — |
| 94 | C4×Q8 | — | 2 | 1 | no | — |
| 95 | C4×Q8 | — | 8 | 8 | no | — |
| 96 | C4×D4 | — | 4 | 3 | no | — |
| 97 | C4×D4 | — | 4 | 3 | no | — |
| 98 | C4×D4 | — | 4 | 24 | no | — |
| 99 | C4×D4 | — | 4 | 3 | no | — |
| 100 | C4×D4 | — | 4 | 3 | no | — |
| 101 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 102 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 103 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 104 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 105 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 106 | C2×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 107 | C2×C4⋊C4 | C22⋊C4 | 1 | 6 | yes | — |
| 108 | C2×C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 109 | C2×C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 110 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 111 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 112 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 113 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 114 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 115 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 116 | C2×C42 | — | 2 | 3 | no | — |
| 117 | C2×C42 | — | 2 | 3 | no | — |
| 118 | C2×C42 | — | 4 | 6 | no | — |
| 119 | C2×C42 | C22×C4 | 1 | 3 | yes | — |
| 120 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 121 | C2×C42 | C22×C4 | 1 | 3 | yes | — |
| 122 | #99 | — | 2 | 2 | no | — |
| 123 | #99 | — | 2 | 6 | no | — |
| 124 | #99 | — | 2 | 6 | no | — |
| 125 | #99 | C2×Q8 | 1 | 4 | yes | complemented |
| 126 | #95 | — | 2 | 3 | no | — |
| 127 | #95 | — | 4 | 6 | no | — |
| 128 | #95 | — | 4 | 6 | no | — |
| 129 | C24 | — | 2 | 1 | no | — |
| 130 | C24 | C22×D4 | 1 | 2 | yes | — |
| 131 | C2×Q8 | — | 2 | 4 | no | — |
| 132 | C2×Q8 | — | 2 | 6 | no | — |
| 133 | C2×Q8 | — | 2 | 6 | no | — |
| 134 | C2×Q8 | — | 4 | 12 | no | — |
| 135 | C2×Q8 | — | 4 | 12 | no | — |
| 136 | C2×Q8 | — | 4 | 8 | no | — |
| 137 | C2×Q8 | — | 4 | 3 | no | — |
| 138 | C2×Q8 | — | 4 | 3 | no | — |
| 139 | C2×Q8 | — | 8 | 12 | no | — |
| 140 | C2×Q8 | — | 8 | 12 | no | — |
| 141 | C2×Q8 | — | 8 | 4 | no | — |
| 142 | C2×Q8 | #99 | 1 | 3 | yes | — |
| 143 | C2×Q8 | #99 | 1 | 1 | yes | characteristic |
| 144 | C2×D4 | — | 2 | 4 | no | — |
| 145 | C2×D4 | — | 2 | 4 | no | — |
| 146 | C2×D4 | — | 2 | 1 | no | — |
| 147 | C2×D4 | — | 4 | 1 | no | — |
| 148 | C2×D4 | — | 4 | 1 | no | — |
| 149 | C2×D4 | — | 4 | 1 | no | — |
| 150 | C2×D4 | — | 4 | 1 | no | — |
| 151 | C2×D4 | — | 4 | 4 | no | — |
| 152 | C2×D4 | — | 4 | 4 | no | — |
| 153 | C2×D4 | C4×Q8 | 1 | 2 | yes | complemented |
| 154 | C2×D4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 155 | C2×D4 | C2×#41 | 1 | 1 | yes | characteristic |
| 156 | C2×D4 | 2−1+4 | 1 | 2 | yes | — |
| 157 | C22×C4 | — | 2 | 1 | no | — |
| 158 | C22×C4 | — | 2 | 3 | no | — |
| 159 | C22×C4 | — | 2 | 6 | no | — |
| 160 | C22×C4 | — | 2 | 6 | no | — |
| 161 | C22×C4 | — | 2 | 3 | no | — |
| 162 | C22×C4 | — | 4 | 12 | no | — |
| 163 | C22×C4 | — | 4 | 12 | no | — |
| 164 | C22×C4 | — | 4 | 2 | no | — |
| 165 | C22×C4 | — | 4 | 3 | no | — |
| 166 | C22×C4 | — | 4 | 3 | no | — |
| 167 | C22×C4 | — | 4 | 6 | no | — |
| 168 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 169 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 170 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 171 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 172 | C4⋊C4 | — | 2 | 12 | no | — |
| 173 | C4⋊C4 | — | 2 | 3 | no | — |
| 174 | C4⋊C4 | — | 2 | 3 | no | — |
| 175 | C4⋊C4 | — | 4 | 6 | no | — |
| 176 | C4⋊C4 | — | 4 | 3 | no | — |
| 177 | C4⋊C4 | — | 4 | 3 | no | — |
| 178 | C4⋊C4 | — | 4 | 12 | no | — |
| 179 | C4⋊C4 | — | 4 | 24 | no | — |
| 180 | C4⋊C4 | — | 4 | 6 | no | — |
| 181 | C4⋊C4 | — | 8 | 12 | no | — |
| 182 | C22⋊C4 | — | 2 | 4 | no | — |
| 183 | C22⋊C4 | — | 4 | 6 | no | — |
| 184 | C22⋊C4 | — | 4 | 12 | no | — |
| 185 | C22⋊C4 | — | 4 | 12 | no | — |
| 186 | C22⋊C4 | — | 4 | 6 | no | — |
| 187 | C22⋊C4 | — | 4 | 2 | no | — |
| 188 | C22⋊C4 | — | 4 | 6 | no | — |
| 189 | C22⋊C4 | — | 4 | 6 | no | — |
| 190 | C42 | — | 2 | 12 | no | — |
| 191 | C42 | — | 2 | 1 | no | — |
| 192 | C42 | — | 2 | 3 | no | — |
| 193 | C42 | — | 2 | 3 | no | — |
| 194 | C42 | — | 4 | 3 | no | — |
| 195 | C42 | — | 4 | 6 | no | — |
| 196 | C42 | — | 8 | 12 | no | — |
| 197 | C42 | C4×Q8 | 1 | 2 | yes | complemented |
| 198 | C23 | — | 2 | 2 | no | — |
| 199 | C23 | — | 2 | 2 | no | — |
| 200 | C23 | — | 2 | 2 | no | — |
| 201 | C23 | — | 4 | 2 | no | — |
| 202 | C23 | — | 4 | 2 | no | — |
| 203 | C23 | — | 4 | 2 | no | — |
| 204 | C23 | — | 4 | 2 | no | — |
| 205 | C23 | — | 4 | 1 | no | — |
| 206 | C23 | — | 4 | 1 | no | — |
| 207 | C23 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 208 | C23 | #548 | 1 | 2 | yes | — |
| 209 | C23 | D4×Q8 | 1 | 2 | yes | — |
| 210 | Q8 | — | 2 | 6 | no | — |
| 211 | Q8 | — | 4 | 12 | no | — |
| 212 | Q8 | — | 8 | 6 | no | — |
| 213 | Q8 | — | 8 | 6 | no | — |
| 214 | Q8 | — | 8 | 6 | no | — |
| 215 | Q8 | #353 | 1 | 4 | yes | direct product factor |
| 216 | D4 | — | 4 | 1 | no | — |
| 217 | D4 | — | 4 | 1 | no | — |
| 218 | D4 | — | 4 | 2 | no | — |
| 219 | D4 | — | 4 | 2 | no | — |
| 220 | D4 | — | 4 | 8 | no | — |
| 221 | D4 | — | 4 | 8 | no | — |
| 222 | D4 | — | 4 | 2 | no | — |
| 223 | C2×C4 | — | 2 | 2 | no | — |
| 224 | C2×C4 | — | 2 | 2 | no | — |
| 225 | C2×C4 | — | 2 | 2 | no | — |
| 226 | C2×C4 | — | 2 | 1 | no | — |
| 227 | C2×C4 | — | 2 | 1 | no | — |
| 228 | C2×C4 | — | 2 | 6 | no | — |
| 229 | C2×C4 | — | 2 | 6 | no | — |
| 230 | C2×C4 | — | 2 | 3 | no | — |
| 231 | C2×C4 | — | 2 | 6 | no | — |
| 232 | C2×C4 | — | 4 | 3 | no | — |
| 233 | C2×C4 | — | 4 | 3 | no | — |
| 234 | C2×C4 | — | 4 | 12 | no | — |
| 235 | C2×C4 | — | 4 | 3 | no | — |
| 236 | C2×C4 | — | 4 | 6 | no | — |
| 237 | C2×C4 | — | 4 | 4 | no | — |
| 238 | C2×C4 | — | 4 | 6 | no | — |
| 239 | C2×C4 | — | 4 | 3 | no | — |
| 240 | C2×C4 | — | 8 | 6 | no | — |
| 241 | C2×C4 | — | 8 | 6 | no | — |
| 242 | C2×C4 | — | 8 | 2 | no | — |
| 243 | C2×C4 | — | 8 | 2 | no | — |
| 244 | C2×C4 | — | 8 | 6 | no | — |
| 245 | C2×C4 | — | 8 | 6 | no | — |
| 246 | C2×C4 | — | 8 | 6 | no | — |
| 247 | C2×C4 | — | 8 | 6 | no | — |
| 248 | C2×C4 | C2×#99 | 1 | 3 | yes | — |
| 249 | C2×C4 | C2×#99 | 1 | 3 | yes | — |
| 250 | C2×C4 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 251 | C2×C4 | #519 | 1 | 1 | yes | characteristic |
| 252 | C22 | — | 2 | 1 | no | — |
| 253 | C22 | — | 2 | 1 | no | — |
| 254 | C22 | — | 2 | 2 | no | — |
| 255 | C22 | — | 2 | 2 | no | — |
| 256 | C22 | — | 4 | 2 | no | — |
| 257 | C22 | — | 4 | 2 | no | — |
| 258 | C22 | — | 4 | 4 | no | — |
| 259 | C22 | — | 4 | 2 | no | — |
| 260 | C22 | — | 4 | 2 | no | — |
| 261 | C22 | — | 4 | 2 | no | — |
| 262 | C22 | — | 8 | 2 | no | — |
| 263 | C22 | — | 8 | 2 | no | — |
| 264 | C22 | — | 8 | 1 | no | — |
| 265 | C22 | — | 8 | 1 | no | — |
| 266 | C22 | Q8×C22⋊C4 | 1 | 1 | yes | characteristic |
| 267 | C22 | C22×#99 | 1 | 1 | yes | characteristic |
| 268 | C22 | #2884 | 1 | 1 | yes | characteristic |
| 269 | C4 | — | 2 | 4 | no | — |
| 270 | C4 | — | 2 | 3 | no | — |
| 271 | C4 | — | 2 | 1 | no | — |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 4 | 6 | no | — |
| 274 | C4 | — | 4 | 6 | no | — |
| 275 | C4 | — | 4 | 3 | no | — |
| 276 | C4 | — | 8 | 4 | no | — |
| 277 | C4 | — | 8 | 3 | no | — |
| 278 | C4 | — | 16 | 6 | no | — |
| 279 | C4 | C2×#353 | 1 | 6 | yes | — |
| 280 | C2 | — | 2 | 1 | no | — |
| 281 | C2 | — | 2 | 1 | no | — |
| 282 | C2 | — | 4 | 2 | no | — |
| 283 | C2 | — | 4 | 2 | no | — |
| 284 | C2 | — | 8 | 2 | no | — |
| 285 | C2 | Q8×#99 | 1 | 1 | yes | characteristic |
| 286 | C2 | C22×#353 | 1 | 1 | yes | characteristic |
| 287 | C2 | #33360 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C2×D4×Q8 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×#99 | C2 | semidirect |
| 3. #13298 | C2 | semidirect |
| 7. C2×D4×Q8 | C4 | semidirect |
| 9. #2868 | C4 | semidirect |
| 11. #2542 | C4 | semidirect |
| 15. C42×Q8 | C4 | semidirect |
| 61. #353 | Q8 | direct (N × H) |
| 125. #99 | C2×Q8 | semidirect |
| 153. C2×D4 | C4×Q8 | semidirect |
| 197. C42 | C4×Q8 | semidirect |
| 215. Q8 | #353 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#127 | 2 |
| Q8×#99 | 2 |
| #13298 | 2 |
| #12797 | 2 |
| C4×#353 | 2 |