Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3631 conjugacy classes of subgroups; only the 568 normal subgroups are shown.
This group has 3631 conjugacy classes of subgroups — too many to draw. Only its 568 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 | #63048 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #60389 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #33571 | C2 | 1 | 18 | yes | maximal, complemented |
| 4 | #33564 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #33371 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | H(7,𝔽2) | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | #3586 | — | 2 | 3 | no | — |
| 8 | #3561 | — | 2 | 9 | no | — |
| 9 | #3550 | C22 | 1 | 9 | yes | complemented |
| 10 | #3540 | — | 2 | 9 | no | — |
| 11 | #3529 | C22 | 1 | 3 | yes | complemented |
| 12 | #3518 | — | 2 | 3 | no | — |
| 13 | #3500 | C22 | 1 | 3 | yes | — |
| 14 | #3486 | — | 2 | 3 | no | — |
| 15 | #3022 | C22 | 1 | 18 | yes | complemented |
| 16 | #3019 | — | 2 | 9 | no | — |
| 17 | #3018 | C22 | 1 | 18 | yes | — |
| 18 | C2×#455 | — | 2 | 9 | no | — |
| 19 | #2901 | — | 2 | 3 | no | — |
| 20 | #2888 | C22 | 1 | 3 | yes | complemented |
| 21 | #2205 | C22 | 1 | 24 | yes | complemented |
| 22 | #2203 | C22 | 1 | 36 | yes | complemented |
| 23 | #2202 | C22 | 1 | 4 | yes | complemented |
| 24 | #2135 | C22 | 1 | 18 | yes | complemented |
| 25 | #2132 | C22 | 1 | 18 | yes | complemented |
| 26 | #585 | — | 2 | 3 | no | — |
| 27 | #585 | — | 4 | 4 | no | — |
| 28 | #585 | C23 | 1 | 6 | yes | — |
| 29 | C2×2−1+4 | D4 | 1 | 3 | yes | — |
| 30 | C2×H(5,𝔽2) | — | 2 | 9 | no | — |
| 31 | C2×H(5,𝔽2) | — | 4 | 4 | no | — |
| 32 | C2×H(5,𝔽2) | C23 | 1 | 1 | yes | characteristic |
| 33 | C23×Q8 | — | 2 | 1 | no | — |
| 34 | #578 | — | 4 | 12 | no | — |
| 35 | #577 | — | 4 | 12 | no | — |
| 36 | #575 | — | 2 | 18 | no | — |
| 37 | C2×#137 | — | 2 | 18 | no | — |
| 38 | #568 | — | 2 | 3 | no | — |
| 39 | #563 | — | 2 | 9 | no | — |
| 40 | #563 | C23 | 1 | 9 | yes | complemented |
| 41 | #562 | — | 2 | 9 | no | — |
| 42 | H(3,𝔽4) | — | 2 | 6 | no | — |
| 43 | H(3,𝔽4) | C23 | 1 | 1 | yes | characteristic, complemented |
| 44 | #560 | — | 2 | 9 | no | — |
| 45 | #559 | — | 2 | 9 | no | — |
| 46 | Q82 | — | 2 | 1 | no | — |
| 47 | Q82 | C23 | 1 | 6 | yes | complemented |
| 48 | #556 | — | 2 | 9 | no | — |
| 49 | #556 | — | 2 | 18 | no | — |
| 50 | #555 | — | 2 | 9 | no | — |
| 51 | #551 | — | 2 | 18 | no | — |
| 52 | D4×Q8 | — | 2 | 3 | no | — |
| 53 | D4×Q8 | — | 2 | 18 | no | — |
| 54 | #548 | — | 2 | 18 | no | — |
| 55 | #548 | — | 2 | 6 | no | — |
| 56 | #546 | — | 2 | 18 | no | — |
| 57 | #543 | — | 2 | 6 | no | — |
| 58 | #542 | — | 2 | 18 | no | — |
| 59 | #542 | — | 2 | 36 | no | — |
| 60 | #538 | — | 2 | 18 | no | — |
| 61 | #536 | — | 2 | 9 | no | — |
| 62 | #536 | — | 2 | 9 | no | — |
| 63 | #536 | C23 | 1 | 9 | yes | — |
| 64 | #535 | D4 | 1 | 9 | yes | — |
| 65 | #533 | — | 2 | 9 | no | — |
| 66 | #532 | C23 | 1 | 9 | yes | — |
| 67 | #529 | — | 2 | 9 | no | — |
| 68 | C2×#124 | D4 | 1 | 9 | yes | — |
| 69 | #525 | — | 2 | 9 | no | — |
| 70 | #525 | — | 2 | 9 | no | — |
| 71 | #519 | — | 2 | 3 | no | — |
| 72 | C4×#41 | D4 | 1 | 3 | yes | — |
| 73 | #456 | C23 | 1 | 36 | yes | — |
| 74 | #455 | — | 2 | 18 | no | — |
| 75 | #455 | — | 2 | 36 | no | — |
| 76 | #454 | C23 | 1 | 12 | yes | — |
| 77 | #453 | — | 2 | 18 | no | — |
| 78 | #421 | — | 2 | 18 | no | — |
| 79 | C2×#104 | — | 2 | 18 | no | — |
| 80 | #413 | C23 | 1 | 18 | yes | — |
| 81 | C2×#100 | — | 2 | 9 | no | — |
| 82 | #356 | C23 | 1 | 36 | yes | complemented |
| 83 | #355 | C23 | 1 | 12 | yes | complemented |
| 84 | 2−1+4 | — | 2 | 12 | no | — |
| 85 | 2−1+4 | — | 4 | 6 | no | — |
| 86 | 2−1+4 | — | 4 | 1 | no | — |
| 87 | 2−1+4 | — | 4 | 12 | no | — |
| 88 | 2−1+4 | C24 | 1 | 4 | yes | — |
| 89 | H(5,𝔽2) | — | 2 | 12 | no | — |
| 90 | H(5,𝔽2) | — | 2 | 18 | no | — |
| 91 | H(5,𝔽2) | — | 4 | 6 | no | — |
| 92 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 93 | H(5,𝔽2) | — | 4 | 36 | no | — |
| 94 | H(5,𝔽2) | — | 4 | 9 | no | — |
| 95 | C2×#41 | — | 2 | 9 | no | — |
| 96 | C2×#41 | — | 2 | 18 | no | — |
| 97 | C2×#41 | — | 4 | 18 | no | — |
| 98 | C2×#41 | — | 4 | 12 | no | — |
| 99 | C2×#41 | — | 4 | 12 | no | — |
| 100 | C2×#41 | C2×D4 | 1 | 9 | yes | — |
| 101 | C2×#41 | C2×D4 | 1 | 3 | yes | — |
| 102 | C2×#41 | C24 | 1 | 3 | yes | — |
| 103 | C22×Q8 | — | 2 | 9 | no | — |
| 104 | C22×Q8 | — | 2 | 3 | no | — |
| 105 | C22×Q8 | — | 8 | 4 | no | — |
| 106 | C22×Q8 | C2×D4 | 1 | 9 | yes | — |
| 107 | C22×D4 | — | 4 | 18 | no | — |
| 108 | C22×D4 | — | 4 | 6 | no | — |
| 109 | C22×D4 | C2×D4 | 1 | 9 | yes | — |
| 110 | C23×C4 | — | 2 | 3 | no | — |
| 111 | #137 | — | 2 | 36 | no | — |
| 112 | #137 | — | 4 | 18 | no | — |
| 113 | #137 | — | 4 | 36 | no | — |
| 114 | AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 115 | AGL(1,ℤ/8) | — | 4 | 18 | no | — |
| 116 | #135 | — | 4 | 36 | no | — |
| 117 | C2×Q16 | — | 4 | 18 | no | — |
| 118 | C2×SD16 | — | 4 | 18 | no | — |
| 119 | #131 | — | 4 | 12 | no | — |
| 120 | C2×M16 | — | 2 | 18 | no | — |
| 121 | C4⋊Q8 | — | 2 | 9 | no | — |
| 122 | C4⋊Q8 | — | 2 | 9 | no | — |
| 123 | C4⋊Q8 | — | 2 | 18 | no | — |
| 124 | C4⋊Q8 | C24 | 1 | 9 | yes | — |
| 125 | #127 | C2×D4 | 1 | 9 | yes | — |
| 126 | #126 | — | 2 | 36 | no | — |
| 127 | #126 | — | 2 | 18 | no | — |
| 128 | #125 | — | 2 | 9 | no | — |
| 129 | #124 | — | 2 | 18 | no | — |
| 130 | #124 | — | 2 | 36 | no | — |
| 131 | #124 | — | 2 | 18 | no | — |
| 132 | #124 | — | 2 | 3 | no | — |
| 133 | #124 | C2×D4 | 1 | 9 | yes | — |
| 134 | #124 | C2×D4 | 1 | 18 | yes | — |
| 135 | #124 | C24 | 1 | 3 | yes | — |
| 136 | #123 | — | 2 | 18 | no | — |
| 137 | #123 | — | 2 | 9 | no | — |
| 138 | #123 | — | 2 | 18 | no | — |
| 139 | #123 | — | 2 | 9 | no | — |
| 140 | #123 | — | 2 | 18 | no | — |
| 141 | C22⋊Q8 | — | 2 | 18 | no | — |
| 142 | C22⋊Q8 | — | 2 | 9 | no | — |
| 143 | C22⋊Q8 | — | 2 | 18 | no | — |
| 144 | C22⋊Q8 | — | 2 | 36 | no | — |
| 145 | C22⋊Q8 | — | 2 | 18 | no | — |
| 146 | C22⋊Q8 | — | 2 | 18 | no | — |
| 147 | #121 | — | 2 | 18 | no | — |
| 148 | #121 | — | 2 | 9 | no | — |
| 149 | #121 | — | 2 | 18 | no | — |
| 150 | #121 | — | 2 | 18 | no | — |
| 151 | #120 | — | 2 | 9 | no | — |
| 152 | #120 | — | 2 | 18 | no | — |
| 153 | #120 | — | 2 | 3 | no | — |
| 154 | C4×Q8 | — | 2 | 18 | no | — |
| 155 | C4×Q8 | — | 2 | 3 | no | — |
| 156 | C4×Q8 | — | 2 | 3 | no | — |
| 157 | C4×Q8 | — | 2 | 18 | no | — |
| 158 | C4×D4 | — | 2 | 9 | no | — |
| 159 | C4×D4 | — | 2 | 36 | no | — |
| 160 | C4×D4 | — | 2 | 9 | no | — |
| 161 | C4×D4 | — | 2 | 18 | no | — |
| 162 | #117 | — | 2 | 9 | no | — |
| 163 | #117 | — | 2 | 9 | no | — |
| 164 | #117 | — | 2 | 18 | no | — |
| 165 | #117 | C2×D4 | 1 | 9 | yes | — |
| 166 | C2×C22⋊C4 | — | 2 | 18 | no | — |
| 167 | C2×C42 | C2×D4 | 1 | 9 | yes | — |
| 168 | #104 | — | 2 | 12 | no | — |
| 169 | #104 | — | 2 | 36 | no | — |
| 170 | #101 | C24 | 1 | 12 | yes | — |
| 171 | #100 | — | 2 | 18 | no | — |
| 172 | C24 | — | 2 | 1 | no | — |
| 173 | C24 | — | 4 | 6 | no | — |
| 174 | C24 | #120 | 1 | 6 | yes | — |
| 175 | #41 | — | 2 | 12 | no | — |
| 176 | #41 | — | 2 | 36 | no | — |
| 177 | #41 | — | 2 | 12 | no | — |
| 178 | #41 | — | 4 | 6 | no | — |
| 179 | #41 | — | 4 | 18 | no | — |
| 180 | #41 | — | 4 | 18 | no | — |
| 181 | #41 | — | 4 | 4 | no | — |
| 182 | #41 | — | 4 | 36 | no | — |
| 183 | #41 | — | 4 | 36 | no | — |
| 184 | #41 | — | 4 | 4 | no | — |
| 185 | #41 | — | 4 | 6 | no | — |
| 186 | #41 | — | 8 | 18 | no | — |
| 187 | C2×Q8 | — | 2 | 1 | no | — |
| 188 | C2×Q8 | — | 2 | 6 | no | — |
| 189 | C2×Q8 | — | 2 | 6 | no | — |
| 190 | C2×Q8 | — | 2 | 9 | no | — |
| 191 | C2×Q8 | — | 2 | 18 | no | — |
| 192 | C2×Q8 | — | 4 | 18 | no | — |
| 193 | C2×Q8 | — | 4 | 4 | no | — |
| 194 | C2×Q8 | — | 4 | 12 | no | — |
| 195 | C2×Q8 | — | 4 | 4 | no | — |
| 196 | C2×Q8 | — | 8 | 18 | no | — |
| 197 | C2×Q8 | #120 | 1 | 1 | yes | characteristic |
| 198 | C2×Q8 | C22×D4 | 1 | 9 | yes | — |
| 199 | C2×Q8 | C25 | 1 | 1 | yes | characteristic |
| 200 | C2×D4 | — | 2 | 18 | no | — |
| 201 | C2×D4 | — | 2 | 3 | no | — |
| 202 | C2×D4 | — | 2 | 18 | no | — |
| 203 | C2×D4 | — | 2 | 9 | no | — |
| 204 | C2×D4 | — | 4 | 18 | no | — |
| 205 | C2×D4 | — | 4 | 36 | no | — |
| 206 | C2×D4 | — | 4 | 36 | no | — |
| 207 | C2×D4 | — | 4 | 36 | no | — |
| 208 | C2×D4 | — | 4 | 18 | no | — |
| 209 | C2×D4 | #120 | 1 | 9 | yes | — |
| 210 | C2×D4 | C22×D4 | 1 | 3 | yes | — |
| 211 | C2×D4 | C22×D4 | 1 | 9 | yes | — |
| 212 | C22×C4 | — | 2 | 9 | no | — |
| 213 | C22×C4 | — | 2 | 9 | no | — |
| 214 | C22×C4 | — | 2 | 9 | no | — |
| 215 | C22×C4 | — | 2 | 9 | no | — |
| 216 | C22×C4 | — | 4 | 18 | no | — |
| 217 | C22×C4 | — | 4 | 6 | no | — |
| 218 | C22×C4 | — | 8 | 6 | no | — |
| 219 | C22×C4 | C22×D4 | 1 | 9 | yes | — |
| 220 | Q16 | — | 4 | 36 | no | — |
| 221 | SD16 | — | 4 | 36 | no | — |
| 222 | SD16 | — | 4 | 12 | no | — |
| 223 | D8 | — | 4 | 12 | no | — |
| 224 | M16 | — | 2 | 12 | no | — |
| 225 | M16 | — | 4 | 18 | no | — |
| 226 | C2×C8 | — | 4 | 18 | no | — |
| 227 | C4⋊C4 | — | 2 | 9 | no | — |
| 228 | C4⋊C4 | — | 2 | 9 | no | — |
| 229 | C4⋊C4 | — | 2 | 18 | no | — |
| 230 | C4⋊C4 | — | 2 | 18 | no | — |
| 231 | C4⋊C4 | — | 2 | 9 | no | — |
| 232 | C4⋊C4 | — | 2 | 18 | no | — |
| 233 | C22⋊C4 | — | 2 | 9 | no | — |
| 234 | C22⋊C4 | — | 2 | 3 | no | — |
| 235 | C22⋊C4 | — | 2 | 18 | no | — |
| 236 | C22⋊C4 | — | 2 | 18 | no | — |
| 237 | C22⋊C4 | — | 2 | 36 | no | — |
| 238 | C22⋊C4 | — | 2 | 9 | no | — |
| 239 | C22⋊C4 | — | 2 | 6 | no | — |
| 240 | C42 | — | 2 | 18 | no | — |
| 241 | C42 | — | 2 | 9 | no | — |
| 242 | C42 | C22×D4 | 1 | 3 | yes | — |
| 243 | C42 | C22×D4 | 1 | 9 | yes | — |
| 244 | C23 | — | 2 | 3 | no | — |
| 245 | C23 | — | 4 | 6 | no | — |
| 246 | C23 | — | 4 | 18 | no | — |
| 247 | C23 | — | 8 | 1 | no | — |
| 248 | C23 | — | 8 | 2 | no | — |
| 249 | C23 | — | 8 | 6 | no | — |
| 250 | C23 | — | 32 | 6 | no | — |
| 251 | C23 | C2×#120 | 1 | 9 | yes | — |
| 252 | Q8 | — | 2 | 12 | no | — |
| 253 | Q8 | — | 4 | 1 | no | — |
| 254 | Q8 | — | 4 | 9 | no | — |
| 255 | Q8 | — | 4 | 6 | no | — |
| 256 | Q8 | — | 4 | 18 | no | — |
| 257 | Q8 | — | 8 | 1 | no | — |
| 258 | Q8 | — | 8 | 6 | no | — |
| 259 | Q8 | — | 8 | 6 | no | — |
| 260 | Q8 | B(4,𝔽2) | 1 | 4 | yes | — |
| 261 | D4 | — | 2 | 12 | no | — |
| 262 | D4 | — | 2 | 18 | no | — |
| 263 | D4 | — | 4 | 9 | no | — |
| 264 | D4 | — | 4 | 3 | no | — |
| 265 | D4 | — | 4 | 36 | no | — |
| 266 | D4 | — | 4 | 4 | no | — |
| 267 | D4 | — | 4 | 6 | no | — |
| 268 | D4 | — | 8 | 9 | no | — |
| 269 | D4 | — | 8 | 3 | no | — |
| 270 | D4 | — | 8 | 18 | no | — |
| 271 | C2×C4 | — | 2 | 18 | no | — |
| 272 | C2×C4 | — | 2 | 9 | no | — |
| 273 | C2×C4 | — | 2 | 3 | no | — |
| 274 | C2×C4 | — | 2 | 9 | no | — |
| 275 | C2×C4 | — | 4 | 12 | no | — |
| 276 | C2×C4 | — | 4 | 18 | no | — |
| 277 | C2×C4 | — | 4 | 18 | no | — |
| 278 | C2×C4 | — | 4 | 12 | no | — |
| 279 | C2×C4 | — | 8 | 9 | no | — |
| 280 | C2×C4 | — | 8 | 9 | no | — |
| 281 | C2×C4 | — | 8 | 9 | no | — |
| 282 | C2×C4 | C2×#120 | 1 | 3 | yes | — |
| 283 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 284 | C8 | — | 4 | 12 | no | — |
| 285 | C22 | — | 2 | 9 | no | — |
| 286 | C22 | — | 4 | 9 | no | — |
| 287 | C22 | — | 4 | 6 | no | — |
| 288 | C22 | — | 4 | 3 | no | — |
| 289 | C22 | — | 4 | 4 | no | — |
| 290 | C22 | — | 8 | 4 | no | — |
| 291 | C22 | — | 16 | 6 | no | — |
| 292 | C22 | — | 16 | 18 | no | — |
| 293 | C22 | C22×#120 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 3 | no | — |
| 295 | C4 | — | 4 | 3 | no | — |
| 296 | C4 | — | 4 | 9 | no | — |
| 297 | C4 | — | 4 | 9 | no | — |
| 298 | C4 | — | 4 | 4 | no | — |
| 299 | C4 | C2×B(4,𝔽2) | 1 | 6 | yes | — |
| 300 | C2 | — | 2 | 1 | no | — |
| 301 | C2 | — | 4 | 9 | no | — |
| 302 | C2 | — | 8 | 3 | no | — |
| 303 | C2 | — | 8 | 4 | no | — |
| 304 | C2 | C22×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C25 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #63048 | C2 | semidirect |
| 2. #60389 | C2 | semidirect |
| 3. #33571 | C2 | semidirect |
| 4. #33564 | C2 | semidirect |
| 5. #33371 | C2 | semidirect |
| 6. H(7,𝔽2) | C22 | semidirect |
| 9. #3550 | C22 | semidirect |
| 11. #3529 | C22 | semidirect |
| 15. #3022 | C22 | semidirect |
| 20. #2888 | C22 | semidirect |
| 21. #2205 | C22 | semidirect |
| 22. #2203 | C22 | semidirect |
| 23. #2202 | C22 | semidirect |
| 24. #2135 | C22 | semidirect |
| 25. #2132 | C22 | semidirect |
| 40. #563 | C23 | semidirect |
| 43. H(3,𝔽4) | C23 | semidirect |
| 47. Q82 | C23 | semidirect |
| 82. #356 | C23 | semidirect |
| 83. #355 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63048 | 2 |
| #60389 | 2 |
| #33571 | 2 |
| #33564 | 2 |
| #33371 | 2 |