← #10584237
Subgroups — #10584237
Order 512.
6114 conjugacy classes of subgroups.
The subgroup lattice has 274 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #62989 | C2 | 1 | 2 | yes | maximal |
| 2 | #62472 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #62246 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | #62100 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #61698 | C2 | 1 | 18 | yes | maximal, complemented |
| 6 | Q16×#41 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C2×#3338 | C2 | 1 | 9 | yes | maximal, complemented |
| 8 | C8×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #3504 | — | 2 | 12 | no | — |
| 10 | #3503 | C22 | 1 | 12 | yes | — |
| 11 | #3501 | — | 2 | 18 | no | — |
| 12 | #3501 | C22 | 1 | 18 | yes | — |
| 13 | Q8×#41 | C22 | 1 | 12 | yes | — |
| 14 | C2×#554 | C22 | 1 | 18 | yes | — |
| 15 | C4×H(5,𝔽2) | — | 2 | 2 | no | — |
| 16 | C4×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 17 | #3395 | C22 | 1 | 2 | yes | complemented |
| 18 | Q8×Q16 | C22 | 1 | 6 | yes | complemented |
| 19 | Q8×Q16 | C22 | 1 | 12 | yes | complemented |
| 20 | #3363 | C22 | 1 | 12 | yes | complemented |
| 21 | #3338 | C22 | 1 | 36 | yes | complemented |
| 22 | #3338 | C22 | 1 | 72 | yes | complemented |
| 23 | #3338 | C22 | 1 | 36 | yes | complemented |
| 24 | #3338 | C22 | 1 | 18 | yes | complemented |
| 25 | #3299 | C22 | 1 | 36 | yes | complemented |
| 26 | #3299 | C22 | 1 | 72 | yes | complemented |
| 27 | D4×Q16 | C22 | 1 | 18 | yes | complemented |
| 28 | #3249 | C22 | 1 | 18 | yes | complemented |
| 29 | #3230 | C22 | 1 | 36 | yes | complemented |
| 30 | #3189 | C22 | 1 | 36 | yes | complemented |
| 31 | #3170 | C22 | 1 | 18 | yes | complemented |
| 32 | C2×#500 | C22 | 1 | 9 | yes | complemented |
| 33 | #3104 | C22 | 1 | 36 | yes | complemented |
| 34 | C2×#484 | C22 | 1 | 36 | yes | complemented |
| 35 | C2×#477 | C22 | 1 | 18 | yes | complemented |
| 36 | C2×#467 | C22 | 1 | 18 | yes | complemented |
| 37 | #2988 | C22 | 1 | 6 | yes | — |
| 38 | #2972 | C22 | 1 | 9 | yes | — |
| 39 | C8×#41 | C22 | 1 | 6 | yes | — |
| 40 | C2×C4×Q16 | C22 | 1 | 9 | yes | complemented |
| 41 | C2×C8×D4 | C22 | 1 | 9 | yes | — |
| 42 | C22×#107 | C22 | 1 | 6 | yes | complemented |
| 43 | C2×H(5,𝔽2) | D4 | 1 | 1 | yes | characteristic |
| 44 | C22×Q16 | C23 | 1 | 9 | yes | — |
| 45 | C23×C8 | C23 | 1 | 6 | yes | — |
| 46 | Q82 | C23 | 1 | 12 | yes | — |
| 47 | #557 | — | 2 | 12 | no | — |
| 48 | #557 | — | 2 | 12 | no | — |
| 49 | #557 | C23 | 1 | 4 | yes | — |
| 50 | #554 | — | 2 | 72 | no | — |
| 51 | #554 | — | 2 | 36 | no | — |
| 52 | #554 | — | 2 | 72 | no | — |
| 53 | #554 | C23 | 1 | 36 | yes | — |
| 54 | #554 | C23 | 1 | 72 | yes | — |
| 55 | D4×Q8 | — | 2 | 36 | no | — |
| 56 | D4×Q8 | C23 | 1 | 36 | yes | — |
| 57 | #544 | — | 2 | 36 | no | — |
| 58 | #544 | — | 2 | 18 | no | — |
| 59 | #543 | — | 2 | 36 | no | — |
| 60 | #533 | — | 2 | 36 | no | — |
| 61 | #533 | C23 | 1 | 36 | yes | — |
| 62 | C2×C4⋊Q8 | C23 | 1 | 18 | yes | — |
| 63 | C2×#125 | — | 2 | 18 | no | — |
| 64 | C2×C22⋊Q8 | — | 2 | 36 | no | — |
| 65 | C2×C22⋊Q8 | C23 | 1 | 36 | yes | — |
| 66 | #520 | — | 2 | 12 | no | — |
| 67 | #520 | D4 | 1 | 6 | yes | — |
| 68 | #518 | — | 2 | 18 | no | — |
| 69 | #518 | D4 | 1 | 9 | yes | — |
| 70 | C4×#41 | — | 2 | 12 | no | — |
| 71 | C4×#41 | C23 | 1 | 6 | yes | — |
| 72 | C2×C4×Q8 | C23 | 1 | 18 | yes | — |
| 73 | C2×C4×D4 | — | 2 | 18 | no | — |
| 74 | C2×C4×D4 | C23 | 1 | 9 | yes | — |
| 75 | C22×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 76 | #500 | C23 | 1 | 18 | yes | — |
| 77 | #500 | C23 | 1 | 36 | yes | — |
| 78 | #494 | C23 | 1 | 6 | yes | — |
| 79 | #487 | C23 | 1 | 36 | yes | — |
| 80 | #484 | C23 | 1 | 72 | yes | — |
| 81 | #484 | C23 | 1 | 144 | yes | — |
| 82 | #477 | C23 | 1 | 36 | yes | — |
| 83 | #477 | C23 | 1 | 72 | yes | — |
| 84 | #477 | C23 | 1 | 36 | yes | — |
| 85 | #467 | C23 | 1 | 36 | yes | — |
| 86 | #467 | C23 | 1 | 36 | yes | — |
| 87 | #467 | C23 | 1 | 72 | yes | — |
| 88 | #462 | C23 | 1 | 12 | yes | — |
| 89 | #462 | C23 | 1 | 36 | yes | — |
| 90 | #451 | C23 | 1 | 72 | yes | — |
| 91 | C8×Q8 | C23 | 1 | 6 | yes | — |
| 92 | C8×Q8 | C23 | 1 | 2 | yes | — |
| 93 | C4×Q16 | C23 | 1 | 6 | yes | — |
| 94 | C4×Q16 | C23 | 1 | 36 | yes | — |
| 95 | C4×Q16 | C23 | 1 | 18 | yes | — |
| 96 | C4×Q16 | C23 | 1 | 36 | yes | — |
| 97 | C8×D4 | C23 | 1 | 18 | yes | — |
| 98 | C8×D4 | C23 | 1 | 36 | yes | — |
| 99 | C8×D4 | C23 | 1 | 18 | yes | — |
| 100 | #431 | C23 | 1 | 18 | yes | — |
| 101 | C2×#107 | C23 | 1 | 36 | yes | — |
| 102 | C2×#107 | C23 | 1 | 18 | yes | — |
| 103 | C2×#107 | C23 | 1 | 18 | yes | — |
| 104 | C2×#107 | C23 | 1 | 9 | yes | — |
| 105 | C2×C4⋊C8 | C23 | 1 | 9 | yes | — |
| 106 | C2×#103 | C23 | 1 | 36 | yes | — |
| 107 | C2×#103 | C23 | 1 | 18 | yes | — |
| 108 | C2×C22⋊C8 | C23 | 1 | 18 | yes | — |
| 109 | C2×C4×C8 | C23 | 1 | 9 | yes | — |
| 110 | H(5,𝔽2) | Q16 | 1 | 16 | yes | complemented |
| 111 | C2×#41 | C2×D4 | 1 | 6 | yes | — |
| 112 | C22×Q8 | — | 2 | 18 | no | — |
| 113 | C22×Q8 | C24 | 1 | 18 | yes | — |
| 114 | C22×D4 | C2×D4 | 1 | 9 | yes | — |
| 115 | C23×C4 | — | 2 | 12 | no | — |
| 116 | C23×C4 | C24 | 1 | 6 | yes | — |
| 117 | C2×Q16 | — | 4 | 18 | no | — |
| 118 | C2×Q16 | C24 | 1 | 6 | yes | — |
| 119 | C2×Q16 | C24 | 1 | 18 | yes | — |
| 120 | C2×Q16 | C24 | 1 | 12 | yes | — |
| 121 | C22×C8 | — | 4 | 12 | no | — |
| 122 | C22×C8 | C24 | 1 | 9 | yes | — |
| 123 | C22×C8 | C24 | 1 | 18 | yes | — |
| 124 | C22×C8 | C24 | 1 | 9 | yes | — |
| 125 | C4⋊Q8 | — | 2 | 36 | no | — |
| 126 | C4⋊Q8 | — | 2 | 36 | no | — |
| 127 | C4⋊Q8 | — | 2 | 36 | no | — |
| 128 | C4⋊Q8 | C24 | 1 | 12 | yes | — |
| 129 | C4⋊Q8 | C24 | 1 | 36 | yes | — |
| 130 | #125 | — | 2 | 36 | no | — |
| 131 | #125 | — | 2 | 72 | no | — |
| 132 | #125 | — | 2 | 12 | no | — |
| 133 | #125 | — | 2 | 36 | no | — |
| 134 | C22⋊Q8 | — | 2 | 72 | no | — |
| 135 | C22⋊Q8 | — | 2 | 36 | no | — |
| 136 | C22⋊Q8 | — | 2 | 72 | no | — |
| 137 | C22⋊Q8 | — | 2 | 144 | no | — |
| 138 | C22⋊Q8 | — | 2 | 72 | no | — |
| 139 | C22⋊Q8 | C24 | 1 | 72 | yes | — |
| 140 | C4×Q8 | — | 2 | 4 | no | — |
| 141 | C4×Q8 | — | 2 | 12 | no | — |
| 142 | C4×Q8 | — | 2 | 36 | no | — |
| 143 | C4×Q8 | — | 2 | 36 | no | — |
| 144 | C4×Q8 | C2×D4 | 1 | 6 | yes | — |
| 145 | C4×Q8 | C24 | 1 | 12 | yes | — |
| 146 | C4×Q8 | C24 | 1 | 36 | yes | — |
| 147 | C4×Q8 | C24 | 1 | 2 | yes | — |
| 148 | C4×D4 | — | 2 | 36 | no | — |
| 149 | C4×D4 | — | 2 | 72 | no | — |
| 150 | C4×D4 | — | 2 | 36 | no | — |
| 151 | C4×D4 | C2×D4 | 1 | 36 | yes | — |
| 152 | C4×D4 | C2×D4 | 1 | 18 | yes | — |
| 153 | C4×D4 | C24 | 1 | 18 | yes | — |
| 154 | #117 | — | 2 | 36 | no | — |
| 155 | #117 | C2×D4 | 1 | 18 | yes | — |
| 156 | C2×C4⋊C4 | — | 2 | 36 | no | — |
| 157 | C2×C4⋊C4 | — | 2 | 18 | no | — |
| 158 | C2×C4⋊C4 | — | 2 | 18 | no | — |
| 159 | C2×C4⋊C4 | — | 2 | 36 | no | — |
| 160 | C2×C4⋊C4 | C2×D4 | 1 | 9 | yes | — |
| 161 | C2×C4⋊C4 | C24 | 1 | 18 | yes | — |
| 162 | C2×C4⋊C4 | C24 | 1 | 36 | yes | — |
| 163 | C2×C22⋊C4 | — | 2 | 36 | no | — |
| 164 | C2×C22⋊C4 | C2×D4 | 1 | 18 | yes | — |
| 165 | C2×C42 | — | 2 | 18 | no | — |
| 166 | C2×C42 | C24 | 1 | 9 | yes | — |
| 167 | #107 | C24 | 1 | 1 | yes | characteristic |
| 168 | #107 | C24 | 1 | 9 | yes | — |
| 169 | #107 | C24 | 1 | 18 | yes | — |
| 170 | #107 | C24 | 1 | 36 | yes | — |
| 171 | #107 | C24 | 1 | 36 | yes | — |
| 172 | C4⋊C8 | C24 | 1 | 18 | yes | — |
| 173 | C4⋊C8 | C24 | 1 | 6 | yes | — |
| 174 | #103 | C24 | 1 | 12 | yes | — |
| 175 | #103 | C24 | 1 | 72 | yes | — |
| 176 | #103 | C24 | 1 | 36 | yes | — |
| 177 | C22⋊C8 | C24 | 1 | 36 | yes | — |
| 178 | C4×C8 | C24 | 1 | 6 | yes | — |
| 179 | C4×C8 | C24 | 1 | 18 | yes | — |
| 180 | C24 | C22×D4 | 1 | 6 | yes | — |
| 181 | #41 | C2×Q16 | 1 | 48 | yes | complemented |
| 182 | C2×Q8 | — | 2 | 18 | no | — |
| 183 | C2×Q8 | — | 2 | 36 | no | — |
| 184 | C2×Q8 | — | 4 | 18 | no | — |
| 185 | C2×Q8 | — | 4 | 18 | no | — |
| 186 | C2×Q8 | — | 16 | 18 | no | — |
| 187 | C2×Q8 | C22×D4 | 1 | 2 | yes | — |
| 188 | C2×Q8 | C25 | 1 | 12 | yes | — |
| 189 | C2×D4 | C2×Q16 | 1 | 72 | yes | complemented |
| 190 | C2×D4 | C22×D4 | 1 | 18 | yes | — |
| 191 | C22×C4 | — | 2 | 18 | no | — |
| 192 | C22×C4 | — | 2 | 18 | no | — |
| 193 | C22×C4 | — | 2 | 36 | no | — |
| 194 | C22×C4 | — | 4 | 6 | no | — |
| 195 | C22×C4 | — | 4 | 6 | no | — |
| 196 | C22×C4 | — | 16 | 12 | no | — |
| 197 | C22×C4 | C22×D4 | 1 | 9 | yes | — |
| 198 | C22×C4 | C22×D4 | 1 | 18 | yes | — |
| 199 | C22×C4 | C22×D4 | 1 | 9 | yes | — |
| 200 | C22×C4 | C25 | 1 | 9 | yes | — |
| 201 | Q16 | — | 2 | 12 | no | — |
| 202 | Q16 | — | 4 | 18 | no | — |
| 203 | Q16 | — | 4 | 12 | no | — |
| 204 | C2×C8 | — | 2 | 18 | no | — |
| 205 | C2×C8 | — | 4 | 18 | no | — |
| 206 | C2×C8 | — | 4 | 9 | no | — |
| 207 | C2×C8 | C25 | 1 | 6 | yes | — |
| 208 | C2×C8 | C25 | 1 | 9 | yes | — |
| 209 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 210 | C4⋊C4 | — | 2 | 18 | no | — |
| 211 | C4⋊C4 | — | 2 | 12 | no | — |
| 212 | C4⋊C4 | — | 2 | 36 | no | — |
| 213 | C4⋊C4 | — | 2 | 36 | no | — |
| 214 | C4⋊C4 | — | 2 | 72 | no | — |
| 215 | C4⋊C4 | — | 2 | 36 | no | — |
| 216 | C4⋊C4 | — | 2 | 12 | no | — |
| 217 | C4⋊C4 | — | 2 | 12 | no | — |
| 218 | C4⋊C4 | C22×D4 | 1 | 18 | yes | — |
| 219 | C4⋊C4 | C22×D4 | 1 | 6 | yes | — |
| 220 | C4⋊C4 | C25 | 1 | 2 | yes | — |
| 221 | C4⋊C4 | C25 | 1 | 18 | yes | — |
| 222 | C22⋊C4 | — | 2 | 72 | no | — |
| 223 | C22⋊C4 | C22×D4 | 1 | 36 | yes | — |
| 224 | C42 | — | 2 | 12 | no | — |
| 225 | C42 | — | 2 | 36 | no | — |
| 226 | C42 | C22×D4 | 1 | 18 | yes | — |
| 227 | C42 | C25 | 1 | 6 | yes | — |
| 228 | C23 | — | 4 | 6 | no | — |
| 229 | C23 | — | 4 | 6 | no | — |
| 230 | C23 | C22×Q16 | 1 | 24 | yes | — |
| 231 | C23 | C23×D4 | 1 | 9 | yes | — |
| 232 | Q8 | — | 2 | 12 | no | — |
| 233 | Q8 | — | 2 | 12 | no | — |
| 234 | Q8 | — | 8 | 18 | no | — |
| 235 | Q8 | — | 8 | 36 | no | — |
| 236 | Q8 | C22×Q16 | 1 | 8 | yes | — |
| 237 | D4 | C22×Q16 | 1 | 72 | yes | — |
| 238 | C2×C4 | — | 2 | 9 | no | — |
| 239 | C2×C4 | — | 2 | 9 | no | — |
| 240 | C2×C4 | — | 2 | 2 | no | — |
| 241 | C2×C4 | — | 2 | 12 | no | — |
| 242 | C2×C4 | — | 2 | 18 | no | — |
| 243 | C2×C4 | — | 4 | 18 | no | — |
| 244 | C2×C4 | — | 4 | 9 | no | — |
| 245 | C2×C4 | — | 8 | 18 | no | — |
| 246 | C2×C4 | — | 8 | 36 | no | — |
| 247 | C2×C4 | — | 8 | 18 | no | — |
| 248 | C2×C4 | C22×Q16 | 1 | 36 | yes | — |
| 249 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 250 | C2×C4 | C23×D4 | 1 | 9 | yes | — |
| 251 | C2×C4 | C23×D4 | 1 | 6 | yes | — |
| 252 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 253 | C8 | — | 2 | 9 | no | — |
| 254 | C8 | — | 2 | 6 | no | — |
| 255 | C8 | #585 | 1 | 2 | yes | — |
| 256 | C22 | — | 2 | 9 | no | — |
| 257 | C22 | — | 2 | 9 | no | — |
| 258 | C22 | — | 4 | 24 | no | — |
| 259 | C22 | C23×Q16 | 1 | 18 | yes | — |
| 260 | C22 | C24×D4 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 2 | 9 | no | — |
| 262 | C4 | — | 2 | 6 | no | — |
| 263 | C4 | — | 4 | 12 | no | — |
| 264 | C4 | — | 4 | 2 | no | — |
| 265 | C4 | — | 4 | 18 | no | — |
| 266 | C4 | C23×Q16 | 1 | 12 | yes | — |
| 267 | C4 | C2×#585 | 1 | 1 | yes | characteristic |
| 268 | C4 | 2−1+6 | 1 | 1 | yes | characteristic |
| 269 | C2 | — | 2 | 18 | no | — |
| 270 | C2 | #62988 | 1 | 1 | yes | characteristic |
| 271 | C2 | C24×Q16 | 1 | 1 | yes | characteristic |
| 272 | C2 | #63091 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta