Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1584 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1584 conjugacy classes of subgroups — too many to draw. Only its 194 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 | #23461 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #20615 | C2 | 1 | 3 | yes | maximal |
| 3 | #11082 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #11053 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#585 | C22 | 1 | 1 | yes | characteristic |
| 6 | #3458 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×#423 | C22 | 1 | 3 | yes | complemented |
| 8 | C22×#104 | C22 | 1 | 6 | yes | — |
| 9 | #2896 | C22 | 1 | 6 | yes | — |
| 10 | C2×#413 | C22 | 1 | 3 | yes | — |
| 11 | #2297 | C22 | 1 | 3 | yes | — |
| 12 | #2028 | — | 2 | 3 | no | — |
| 13 | #2028 | — | 2 | 3 | no | — |
| 14 | #2027 | — | 2 | 3 | no | — |
| 15 | #2027 | — | 2 | 3 | no | — |
| 16 | #1793 | — | 2 | 3 | no | — |
| 17 | #1793 | — | 2 | 3 | no | — |
| 18 | #1792 | — | 2 | 3 | no | — |
| 19 | #1792 | — | 2 | 3 | no | — |
| 20 | #1742 | C22 | 1 | 3 | yes | complemented |
| 21 | C2×#339 | C22 | 1 | 6 | yes | complemented |
| 22 | C2×#338 | C22 | 1 | 2 | yes | complemented |
| 23 | C4×#95 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #585 | — | 2 | 6 | no | — |
| 25 | #585 | — | 4 | 1 | no | — |
| 26 | #585 | — | 4 | 3 | no | — |
| 27 | #585 | Q8 | 1 | 4 | yes | complemented |
| 28 | C2×2−1+4 | — | 2 | 3 | no | — |
| 29 | C2×H(5,𝔽2) | — | 2 | 3 | no | — |
| 30 | C2×H(5,𝔽2) | — | 4 | 1 | no | — |
| 31 | C22×#41 | — | 4 | 3 | no | — |
| 32 | C22×#41 | C23 | 1 | 3 | yes | — |
| 33 | C22×M16 | C23 | 1 | 6 | yes | — |
| 34 | #544 | — | 2 | 3 | no | — |
| 35 | #544 | — | 2 | 3 | no | — |
| 36 | #543 | — | 4 | 1 | no | — |
| 37 | #533 | — | 4 | 3 | no | — |
| 38 | C2×#117 | C23 | 1 | 3 | yes | — |
| 39 | C22×C42 | C23 | 1 | 3 | yes | — |
| 40 | #423 | — | 4 | 6 | no | — |
| 41 | C2×C4⋊C8 | — | 4 | 3 | no | — |
| 42 | C2×#104 | — | 2 | 12 | no | — |
| 43 | C2×#104 | — | 2 | 6 | no | — |
| 44 | C2×#104 | — | 2 | 6 | no | — |
| 45 | C2×#104 | — | 4 | 6 | no | — |
| 46 | #419 | — | 2 | 6 | no | — |
| 47 | #418 | — | 2 | 6 | no | — |
| 48 | #417 | — | 2 | 6 | no | — |
| 49 | #417 | — | 2 | 6 | no | — |
| 50 | #416 | — | 4 | 6 | no | — |
| 51 | #413 | — | 4 | 3 | no | — |
| 52 | #413 | — | 4 | 3 | no | — |
| 53 | C2×#101 | — | 2 | 3 | no | — |
| 54 | C2×#100 | — | 2 | 3 | no | — |
| 55 | #391 | D4 | 1 | 3 | yes | — |
| 56 | #391 | D4 | 1 | 3 | yes | — |
| 57 | #390 | D4 | 1 | 3 | yes | — |
| 58 | #390 | D4 | 1 | 3 | yes | — |
| 59 | #386 | — | 4 | 3 | no | — |
| 60 | C4×C22⋊C4 | — | 4 | 3 | no | — |
| 61 | C2×#95 | — | 4 | 1 | no | — |
| 62 | #343 | — | 2 | 3 | no | — |
| 63 | #343 | — | 2 | 3 | no | — |
| 64 | #339 | — | 2 | 6 | no | — |
| 65 | #339 | — | 2 | 6 | no | — |
| 66 | #338 | — | 4 | 2 | no | — |
| 67 | 2−1+4 | — | 2 | 12 | no | — |
| 68 | 2−1+4 | — | 4 | 12 | no | — |
| 69 | 2−1+4 | — | 4 | 6 | no | — |
| 70 | H(5,𝔽2) | — | 2 | 12 | no | — |
| 71 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 72 | H(5,𝔽2) | — | 4 | 12 | no | — |
| 73 | H(5,𝔽2) | — | 4 | 6 | no | — |
| 74 | H(5,𝔽2) | — | 4 | 12 | no | — |
| 75 | C2×#41 | — | 2 | 3 | no | — |
| 76 | C2×#41 | — | 2 | 3 | no | — |
| 77 | C2×#41 | — | 2 | 6 | no | — |
| 78 | C2×#41 | — | 2 | 6 | no | — |
| 79 | C2×#41 | — | 2 | 6 | no | — |
| 80 | C2×#41 | — | 2 | 6 | no | — |
| 81 | C2×#41 | — | 4 | 12 | no | — |
| 82 | C2×#41 | — | 4 | 6 | no | — |
| 83 | C2×#41 | — | 4 | 12 | no | — |
| 84 | C2×#41 | — | 4 | 6 | no | — |
| 85 | C2×#41 | — | 4 | 12 | no | — |
| 86 | C2×#41 | — | 4 | 2 | no | — |
| 87 | C2×#41 | — | 4 | 12 | no | — |
| 88 | C2×#41 | C2×Q8 | 1 | 6 | yes | — |
| 89 | C2×#41 | #41 | 1 | 6 | yes | — |
| 90 | C22×Q8 | — | 4 | 3 | no | — |
| 91 | C22×Q8 | C2×D4 | 1 | 3 | yes | — |
| 92 | C22×D4 | — | 2 | 3 | no | — |
| 93 | C22×D4 | — | 4 | 3 | no | — |
| 94 | C22×D4 | — | 4 | 6 | no | — |
| 95 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 96 | C23×C4 | — | 2 | 3 | no | — |
| 97 | C23×C4 | — | 2 | 3 | no | — |
| 98 | C23×C4 | — | 4 | 2 | no | — |
| 99 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 100 | C2×M16 | — | 2 | 6 | no | — |
| 101 | C2×M16 | — | 2 | 6 | no | — |
| 102 | C2×M16 | — | 2 | 6 | no | — |
| 103 | C2×M16 | — | 2 | 6 | no | — |
| 104 | C2×M16 | — | 2 | 6 | no | — |
| 105 | C2×M16 | — | 2 | 6 | no | — |
| 106 | C22×C8 | — | 2 | 6 | no | — |
| 107 | C4⋊Q8 | — | 4 | 3 | no | — |
| 108 | C4⋊Q8 | — | 4 | 3 | no | — |
| 109 | #125 | — | 4 | 6 | no | — |
| 110 | C22⋊Q8 | — | 4 | 6 | no | — |
| 111 | C22⋊Q8 | — | 4 | 6 | no | — |
| 112 | C4×Q8 | — | 4 | 2 | no | — |
| 113 | C4×Q8 | — | 4 | 2 | no | — |
| 114 | #117 | — | 2 | 6 | no | — |
| 115 | #117 | — | 2 | 6 | no | — |
| 116 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 117 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 118 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 119 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 120 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 121 | C2×C42 | — | 2 | 6 | no | — |
| 122 | C2×C42 | — | 2 | 6 | no | — |
| 123 | C2×C42 | — | 2 | 3 | no | — |
| 124 | C2×C42 | — | 2 | 3 | no | — |
| 125 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 126 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 127 | C4⋊C8 | — | 4 | 12 | no | — |
| 128 | #104 | — | 4 | 12 | no | — |
| 129 | #104 | — | 4 | 12 | no | — |
| 130 | #103 | — | 4 | 6 | no | — |
| 131 | #103 | — | 4 | 6 | no | — |
| 132 | #102 | — | 4 | 6 | no | — |
| 133 | #102 | — | 4 | 6 | no | — |
| 134 | #101 | — | 4 | 6 | no | — |
| 135 | #100 | — | 4 | 6 | no | — |
| 136 | #95 | — | 4 | 3 | no | — |
| 137 | #95 | — | 4 | 3 | no | — |
| 138 | #95 | — | 4 | 4 | no | — |
| 139 | C24 | — | 2 | 3 | no | — |
| 140 | C24 | — | 4 | 2 | no | — |
| 141 | C24 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 142 | #41 | — | 2 | 12 | no | — |
| 143 | #41 | — | 2 | 12 | no | — |
| 144 | #41 | — | 2 | 12 | no | — |
| 145 | #41 | — | 2 | 12 | no | — |
| 146 | #41 | — | 4 | 24 | no | — |
| 147 | #41 | — | 4 | 24 | no | — |
| 148 | #41 | — | 4 | 8 | no | — |
| 149 | #41 | — | 4 | 8 | no | — |
| 150 | #41 | — | 4 | 24 | no | — |
| 151 | #41 | — | 4 | 24 | no | — |
| 152 | #41 | — | 4 | 6 | no | — |
| 153 | #41 | — | 4 | 6 | no | — |
| 154 | #41 | — | 4 | 6 | no | — |
| 155 | #41 | — | 4 | 6 | no | — |
| 156 | C2×Q8 | — | 2 | 3 | no | — |
| 157 | C2×Q8 | — | 2 | 6 | no | — |
| 158 | C2×Q8 | — | 2 | 6 | no | — |
| 159 | C2×Q8 | — | 4 | 12 | no | — |
| 160 | C2×Q8 | — | 4 | 2 | no | — |
| 161 | C2×Q8 | — | 4 | 12 | no | — |
| 162 | C2×Q8 | — | 4 | 2 | no | — |
| 163 | C2×Q8 | — | 4 | 2 | no | — |
| 164 | C2×Q8 | C22⋊Q8 | 1 | 6 | yes | — |
| 165 | C2×D4 | — | 2 | 6 | no | — |
| 166 | C2×D4 | — | 2 | 6 | no | — |
| 167 | C2×D4 | — | 2 | 3 | no | — |
| 168 | C2×D4 | — | 2 | 6 | no | — |
| 169 | C2×D4 | — | 2 | 6 | no | — |
| 170 | C2×D4 | — | 4 | 12 | no | — |
| 171 | C2×D4 | — | 4 | 24 | no | — |
| 172 | C2×D4 | — | 4 | 6 | no | — |
| 173 | C2×D4 | — | 4 | 24 | no | — |
| 174 | C2×D4 | — | 4 | 12 | no | — |
| 175 | C2×D4 | — | 4 | 6 | no | — |
| 176 | C2×D4 | — | 4 | 6 | no | — |
| 177 | C2×D4 | C22⋊Q8 | 1 | 6 | yes | — |
| 178 | C22×C4 | — | 2 | 3 | no | — |
| 179 | C22×C4 | — | 2 | 3 | no | — |
| 180 | C22×C4 | — | 2 | 6 | no | — |
| 181 | C22×C4 | — | 2 | 6 | no | — |
| 182 | C22×C4 | — | 2 | 3 | no | — |
| 183 | C22×C4 | — | 2 | 3 | no | — |
| 184 | C22×C4 | — | 2 | 3 | no | — |
| 185 | C22×C4 | — | 4 | 6 | no | — |
| 186 | C22×C4 | — | 4 | 6 | no | — |
| 187 | C22×C4 | — | 4 | 8 | no | — |
| 188 | C22×C4 | — | 4 | 8 | no | — |
| 189 | C22×C4 | — | 4 | 3 | no | — |
| 190 | C22×C4 | — | 4 | 3 | no | — |
| 191 | C22×C4 | — | 4 | 3 | no | — |
| 192 | C22×C4 | — | 4 | 3 | no | — |
| 193 | C22×C4 | — | 4 | 6 | no | — |
| 194 | C22×C4 | — | 8 | 3 | no | — |
| 195 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 196 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 197 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 198 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 199 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 200 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 201 | M16 | — | 4 | 6 | no | — |
| 202 | M16 | — | 4 | 6 | no | — |
| 203 | M16 | — | 4 | 6 | no | — |
| 204 | M16 | — | 4 | 6 | no | — |
| 205 | C2×C8 | — | 4 | 6 | no | — |
| 206 | C2×C8 | — | 4 | 6 | no | — |
| 207 | C2×C8 | — | 4 | 6 | no | — |
| 208 | C4⋊C4 | — | 2 | 6 | no | — |
| 209 | C4⋊C4 | — | 2 | 6 | no | — |
| 210 | C4⋊C4 | — | 4 | 6 | no | — |
| 211 | C4⋊C4 | — | 4 | 6 | no | — |
| 212 | C22⋊C4 | — | 4 | 6 | no | — |
| 213 | C22⋊C4 | — | 8 | 6 | no | — |
| 214 | C22⋊C4 | — | 8 | 6 | no | — |
| 215 | C42 | — | 2 | 6 | no | — |
| 216 | C42 | — | 2 | 6 | no | — |
| 217 | C42 | — | 4 | 3 | no | — |
| 218 | C42 | — | 4 | 3 | no | — |
| 219 | C42 | — | 4 | 3 | no | — |
| 220 | C42 | — | 4 | 3 | no | — |
| 221 | C23 | — | 2 | 6 | no | — |
| 222 | C23 | — | 4 | 1 | no | — |
| 223 | C23 | — | 4 | 1 | no | — |
| 224 | C23 | — | 4 | 8 | no | — |
| 225 | C23 | — | 4 | 3 | no | — |
| 226 | C23 | — | 4 | 3 | no | — |
| 227 | C23 | — | 8 | 3 | no | — |
| 228 | C23 | — | 8 | 3 | no | — |
| 229 | C23 | — | 16 | 2 | no | — |
| 230 | C23 | — | 16 | 2 | no | — |
| 231 | C23 | #534 | 1 | 3 | yes | — |
| 232 | C23 | #538 | 1 | 3 | yes | — |
| 233 | C23 | #543 | 1 | 1 | yes | characteristic |
| 234 | Q8 | — | 2 | 12 | no | — |
| 235 | Q8 | — | 4 | 8 | no | — |
| 236 | Q8 | — | 4 | 6 | no | — |
| 237 | Q8 | — | 16 | 2 | no | — |
| 238 | Q8 | — | 16 | 2 | no | — |
| 239 | D4 | — | 2 | 12 | no | — |
| 240 | D4 | — | 4 | 24 | no | — |
| 241 | D4 | — | 4 | 12 | no | — |
| 242 | D4 | — | 4 | 12 | no | — |
| 243 | D4 | — | 4 | 6 | no | — |
| 244 | C2×C4 | — | 2 | 6 | no | — |
| 245 | C2×C4 | — | 2 | 6 | no | — |
| 246 | C2×C4 | — | 4 | 3 | no | — |
| 247 | C2×C4 | — | 4 | 24 | no | — |
| 248 | C2×C4 | — | 4 | 3 | no | — |
| 249 | C2×C4 | — | 4 | 3 | no | — |
| 250 | C2×C4 | — | 4 | 3 | no | — |
| 251 | C2×C4 | — | 4 | 6 | no | — |
| 252 | C2×C4 | — | 4 | 6 | no | — |
| 253 | C2×C4 | — | 4 | 6 | no | — |
| 254 | C2×C4 | — | 4 | 6 | no | — |
| 255 | C2×C4 | — | 4 | 3 | no | — |
| 256 | C2×C4 | — | 4 | 3 | no | — |
| 257 | C2×C4 | — | 4 | 3 | no | — |
| 258 | C2×C4 | — | 4 | 3 | no | — |
| 259 | C2×C4 | — | 4 | 3 | no | — |
| 260 | C2×C4 | — | 8 | 12 | no | — |
| 261 | C2×C4 | — | 8 | 3 | no | — |
| 262 | C2×C4 | — | 8 | 3 | no | — |
| 263 | C2×C4 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 264 | C2×C4 | #535 | 1 | 3 | yes | — |
| 265 | C2×C4 | #538 | 1 | 3 | yes | — |
| 266 | C2×C4 | #543 | 1 | 3 | yes | — |
| 267 | C2×C4 | D42 | 1 | 3 | yes | — |
| 268 | C2×C4 | #546 | 1 | 6 | yes | — |
| 269 | C2×C4 | #551 | 1 | 3 | yes | — |
| 270 | C2×C4 | H(3,𝔽4) | 1 | 3 | yes | — |
| 271 | C2×C4 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 272 | C8 | — | 8 | 6 | no | — |
| 273 | C22 | — | 2 | 3 | no | — |
| 274 | C22 | — | 2 | 3 | no | — |
| 275 | C22 | — | 4 | 3 | no | — |
| 276 | C22 | — | 4 | 1 | no | — |
| 277 | C22 | — | 4 | 6 | no | — |
| 278 | C22 | — | 8 | 6 | no | — |
| 279 | C22 | — | 8 | 3 | no | — |
| 280 | C22 | — | 8 | 3 | no | — |
| 281 | C22 | — | 16 | 3 | no | — |
| 282 | C22 | — | 16 | 8 | no | — |
| 283 | C22 | #2626 | 1 | 3 | yes | — |
| 284 | C22 | #2679 | 1 | 1 | yes | characteristic |
| 285 | C22 | #2684 | 1 | 3 | yes | — |
| 286 | C4 | — | 4 | 6 | no | — |
| 287 | C4 | — | 4 | 6 | no | — |
| 288 | C4 | — | 8 | 3 | no | — |
| 289 | C4 | — | 8 | 3 | no | — |
| 290 | C4 | — | 8 | 3 | no | — |
| 291 | C4 | #2437 | 1 | 3 | yes | — |
| 292 | C4 | #2670 | 1 | 3 | yes | — |
| 293 | C4 | #2736 | 1 | 1 | yes | characteristic |
| 294 | C4 | #2840 | 1 | 1 | yes | characteristic |
| 295 | C2 | — | 2 | 3 | no | — |
| 296 | C2 | — | 2 | 3 | no | — |
| 297 | C2 | — | 8 | 6 | no | — |
| 298 | C2 | #16297 | 1 | 1 | yes | characteristic |
| 299 | C2 | #31076 | 1 | 1 | yes | characteristic |
| 300 | C2 | #31077 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2679 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23461 | C2 | semidirect |
| 3. #11082 | C2 | semidirect |
| 4. #11053 | C2 | semidirect |
| 6. #3458 | C22 | semidirect |
| 7. C2×#423 | C22 | semidirect |
| 20. #1742 | C22 | semidirect |
| 21. C2×#339 | C22 | semidirect |
| 22. C2×#338 | C22 | semidirect |
| 23. C4×#95 | C22 | semidirect |
| 27. #585 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23461 | 2 |
| #20615 | 2 |
| #11082 | 2 |
| #11053 | 2 |