← #10586098
Subgroups — #10586098
Order 512.
33208 conjugacy classes of subgroups.
The subgroup lattice has 278 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #63004 | C2 | 1 | 12 | yes | maximal, complemented |
| 2 | Q8×2−1+4 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | #62989 | C2 | 1 | 20 | yes | maximal, complemented |
| 4 | Q8×H(5,𝔽2) | C2 | 1 | 20 | yes | maximal, complemented |
| 5 | #62854 | C2 | 1 | 30 | yes | maximal, complemented |
| 6 | #62813 | C2 | 1 | 15 | yes | maximal, complemented |
| 7 | C2×#3503 | C2 | 1 | 15 | yes | maximal, complemented |
| 8 | #62654 | C2 | 1 | 2 | yes | maximal |
| 9 | C4×#585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C2×#585 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3504 | C22 | 1 | 240 | yes | complemented |
| 12 | #3504 | C22 | 1 | 120 | yes | complemented |
| 13 | #3503 | C22 | 1 | 240 | yes | complemented |
| 14 | #3503 | C22 | 1 | 120 | yes | complemented |
| 15 | #3503 | C22 | 1 | 120 | yes | complemented |
| 16 | #3503 | C22 | 1 | 60 | yes | complemented |
| 17 | #3503 | C22 | 1 | 120 | yes | complemented |
| 18 | #3503 | C22 | 1 | 20 | yes | complemented |
| 19 | #3502 | C22 | 1 | 30 | yes | complemented |
| 20 | #3502 | C22 | 1 | 120 | yes | complemented |
| 21 | #3502 | C22 | 1 | 30 | yes | complemented |
| 22 | #3501 | C22 | 1 | 90 | yes | complemented |
| 23 | #3501 | C22 | 1 | 360 | yes | complemented |
| 24 | #3501 | C22 | 1 | 90 | yes | complemented |
| 25 | Q8×#41 | C22 | 1 | 120 | yes | complemented |
| 26 | Q8×#41 | C22 | 1 | 120 | yes | complemented |
| 27 | C2×Q82 | C22 | 1 | 30 | yes | complemented |
| 28 | C2×#557 | C22 | 1 | 30 | yes | complemented |
| 29 | C2×#554 | C22 | 1 | 90 | yes | complemented |
| 30 | C2×D4×Q8 | C22 | 1 | 90 | yes | complemented |
| 31 | #3458 | C22 | 1 | 90 | yes | complemented |
| 32 | C2×#544 | C22 | 1 | 90 | yes | complemented |
| 33 | C2×#533 | C22 | 1 | 45 | yes | complemented |
| 34 | C22×C4⋊Q8 | C22 | 1 | 15 | yes | complemented |
| 35 | C4×2−1+4 | C22 | 1 | 6 | yes | — |
| 36 | C4×2−1+4 | C22 | 1 | 24 | yes | — |
| 37 | C4×2−1+4 | C22 | 1 | 6 | yes | — |
| 38 | C4×H(5,𝔽2) | C22 | 1 | 10 | yes | — |
| 39 | C4×H(5,𝔽2) | C22 | 1 | 40 | yes | — |
| 40 | C4×H(5,𝔽2) | C22 | 1 | 10 | yes | — |
| 41 | #3428 | C22 | 1 | 30 | yes | — |
| 42 | #3428 | C22 | 1 | 15 | yes | — |
| 43 | C2×#520 | C22 | 1 | 30 | yes | — |
| 44 | C2×C4×#41 | C22 | 1 | 15 | yes | — |
| 45 | #585 | Q8 | 1 | 32 | yes | complemented |
| 46 | C2×2−1+4 | C23 | 1 | 6 | yes | — |
| 47 | C2×H(5,𝔽2) | C23 | 1 | 10 | yes | — |
| 48 | C22×#41 | C23 | 1 | 15 | yes | — |
| 49 | C23×Q8 | C23 | 1 | 30 | yes | — |
| 50 | Q82 | C23 | 1 | 120 | yes | — |
| 51 | Q82 | C23 | 1 | 240 | yes | — |
| 52 | Q82 | C23 | 1 | 40 | yes | — |
| 53 | #557 | C23 | 1 | 120 | yes | — |
| 54 | #557 | C23 | 1 | 240 | yes | — |
| 55 | #557 | C23 | 1 | 40 | yes | — |
| 56 | #557 | C23 | 1 | 240 | yes | — |
| 57 | #557 | C23 | 1 | 240 | yes | — |
| 58 | #554 | C23 | 1 | 360 | yes | — |
| 59 | #554 | C23 | 1 | 720 | yes | — |
| 60 | #554 | C23 | 1 | 120 | yes | — |
| 61 | #554 | C23 | 1 | 720 | yes | — |
| 62 | #554 | C23 | 1 | 720 | yes | — |
| 63 | D4×Q8 | C23 | 1 | 360 | yes | — |
| 64 | D4×Q8 | C23 | 1 | 720 | yes | — |
| 65 | D4×Q8 | C23 | 1 | 120 | yes | — |
| 66 | #544 | C23 | 1 | 360 | yes | — |
| 67 | #544 | C23 | 1 | 720 | yes | — |
| 68 | #544 | C23 | 1 | 120 | yes | — |
| 69 | #544 | C23 | 1 | 180 | yes | — |
| 70 | #544 | C23 | 1 | 180 | yes | — |
| 71 | #543 | C23 | 1 | 360 | yes | — |
| 72 | #533 | C23 | 1 | 60 | yes | — |
| 73 | #533 | C23 | 1 | 180 | yes | — |
| 74 | #533 | C23 | 1 | 180 | yes | — |
| 75 | #533 | C23 | 1 | 180 | yes | — |
| 76 | #533 | C23 | 1 | 720 | yes | — |
| 77 | C2×C4⋊Q8 | C23 | 1 | 45 | yes | — |
| 78 | C2×C4⋊Q8 | C23 | 1 | 90 | yes | — |
| 79 | C2×C4⋊Q8 | C23 | 1 | 90 | yes | — |
| 80 | C2×C4⋊Q8 | C23 | 1 | 90 | yes | — |
| 81 | C2×C4⋊Q8 | C23 | 1 | 15 | yes | — |
| 82 | C2×C4⋊Q8 | C23 | 1 | 180 | yes | — |
| 83 | C2×#125 | C23 | 1 | 90 | yes | — |
| 84 | C2×#125 | C23 | 1 | 180 | yes | — |
| 85 | C2×C22⋊Q8 | C23 | 1 | 90 | yes | — |
| 86 | C2×C22⋊Q8 | C23 | 1 | 90 | yes | — |
| 87 | C2×C22⋊Q8 | C23 | 1 | 360 | yes | — |
| 88 | #520 | C23 | 1 | 120 | yes | — |
| 89 | #520 | C23 | 1 | 240 | yes | — |
| 90 | #520 | C23 | 1 | 40 | yes | — |
| 91 | #520 | C23 | 1 | 60 | yes | — |
| 92 | #520 | C23 | 1 | 60 | yes | — |
| 93 | #519 | C23 | 1 | 60 | yes | — |
| 94 | #519 | C23 | 1 | 30 | yes | — |
| 95 | #518 | C23 | 1 | 180 | yes | — |
| 96 | #518 | C23 | 1 | 90 | yes | — |
| 97 | C4×#41 | C23 | 1 | 20 | yes | — |
| 98 | C4×#41 | C23 | 1 | 60 | yes | — |
| 99 | C4×#41 | C23 | 1 | 60 | yes | — |
| 100 | C4×#41 | C23 | 1 | 60 | yes | — |
| 101 | C4×#41 | C23 | 1 | 240 | yes | — |
| 102 | C2×C4×Q8 | C23 | 1 | 15 | yes | — |
| 103 | C2×C4×Q8 | C23 | 1 | 15 | yes | — |
| 104 | C2×C4×Q8 | C23 | 1 | 60 | yes | — |
| 105 | C2×C4×Q8 | C23 | 1 | 90 | yes | — |
| 106 | C2×C4×Q8 | C23 | 1 | 90 | yes | — |
| 107 | C2×C4×D4 | C23 | 1 | 45 | yes | — |
| 108 | C2×C4×D4 | C23 | 1 | 45 | yes | — |
| 109 | C2×C4×D4 | C23 | 1 | 180 | yes | — |
| 110 | C2×#117 | C23 | 1 | 45 | yes | — |
| 111 | C2×#117 | C23 | 1 | 90 | yes | — |
| 112 | C22×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 113 | C22×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 114 | C22×C42 | C23 | 1 | 15 | yes | — |
| 115 | 2−1+4 | C2×Q8 | 1 | 96 | yes | complemented |
| 116 | H(5,𝔽2) | C2×Q8 | 1 | 160 | yes | complemented |
| 117 | C2×#41 | C2×Q8 | 1 | 240 | yes | complemented |
| 118 | C2×#41 | C24 | 1 | 60 | yes | — |
| 119 | C2×#41 | C24 | 1 | 20 | yes | — |
| 120 | C22×Q8 | — | 8 | 60 | no | — |
| 121 | C22×Q8 | C24 | 1 | 30 | yes | — |
| 122 | C22×Q8 | C24 | 1 | 90 | yes | — |
| 123 | C22×Q8 | C24 | 1 | 180 | yes | — |
| 124 | C22×Q8 | C24 | 1 | 15 | yes | — |
| 125 | C22×D4 | C24 | 1 | 45 | yes | — |
| 126 | C23×C4 | C24 | 1 | 15 | yes | — |
| 127 | C23×C4 | C24 | 1 | 15 | yes | — |
| 128 | C23×C4 | C24 | 1 | 60 | yes | — |
| 129 | C23×C4 | C24 | 1 | 15 | yes | — |
| 130 | C4⋊Q8 | C24 | 1 | 10 | yes | — |
| 131 | C4⋊Q8 | C24 | 1 | 6 | yes | — |
| 132 | C4⋊Q8 | C24 | 1 | 90 | yes | — |
| 133 | C4⋊Q8 | C24 | 1 | 90 | yes | — |
| 134 | C4⋊Q8 | C24 | 1 | 90 | yes | — |
| 135 | C4⋊Q8 | C24 | 1 | 360 | yes | — |
| 136 | C4⋊Q8 | C24 | 1 | 720 | yes | — |
| 137 | C4⋊Q8 | C24 | 1 | 30 | yes | — |
| 138 | C4⋊Q8 | C24 | 1 | 30 | yes | — |
| 139 | C4⋊Q8 | C24 | 1 | 30 | yes | — |
| 140 | C4⋊Q8 | C24 | 1 | 240 | yes | — |
| 141 | #125 | C24 | 1 | 120 | yes | — |
| 142 | #125 | C24 | 1 | 720 | yes | — |
| 143 | #125 | C24 | 1 | 360 | yes | — |
| 144 | #125 | C24 | 1 | 240 | yes | — |
| 145 | C22⋊Q8 | C24 | 1 | 120 | yes | — |
| 146 | C22⋊Q8 | C24 | 1 | 120 | yes | — |
| 147 | C22⋊Q8 | C24 | 1 | 1440 | yes | — |
| 148 | C22⋊Q8 | C24 | 1 | 360 | yes | — |
| 149 | C22⋊Q8 | C24 | 1 | 360 | yes | — |
| 150 | C22⋊Q8 | C24 | 1 | 480 | yes | — |
| 151 | C4×Q8 | C24 | 1 | 20 | yes | — |
| 152 | C4×Q8 | C24 | 1 | 20 | yes | — |
| 153 | C4×Q8 | C24 | 1 | 120 | yes | — |
| 154 | C4×Q8 | C24 | 1 | 120 | yes | — |
| 155 | C4×Q8 | C24 | 1 | 240 | yes | — |
| 156 | C4×Q8 | C24 | 1 | 60 | yes | — |
| 157 | C4×Q8 | C24 | 1 | 60 | yes | — |
| 158 | C4×Q8 | C24 | 1 | 360 | yes | — |
| 159 | C4×Q8 | C24 | 1 | 360 | yes | — |
| 160 | C4×Q8 | C24 | 1 | 80 | yes | — |
| 161 | C4×D4 | C24 | 1 | 60 | yes | — |
| 162 | C4×D4 | C24 | 1 | 60 | yes | — |
| 163 | C4×D4 | C24 | 1 | 720 | yes | — |
| 164 | C4×D4 | C24 | 1 | 180 | yes | — |
| 165 | C4×D4 | C24 | 1 | 180 | yes | — |
| 166 | C4×D4 | C24 | 1 | 240 | yes | — |
| 167 | #117 | C24 | 1 | 60 | yes | — |
| 168 | #117 | C24 | 1 | 120 | yes | — |
| 169 | #117 | C24 | 1 | 360 | yes | — |
| 170 | #117 | C24 | 1 | 180 | yes | — |
| 171 | C2×C4⋊C4 | C24 | 1 | 90 | yes | — |
| 172 | C2×C4⋊C4 | C24 | 1 | 90 | yes | — |
| 173 | C2×C4⋊C4 | C24 | 1 | 180 | yes | — |
| 174 | C2×C4⋊C4 | C24 | 1 | 180 | yes | — |
| 175 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 176 | C2×C4⋊C4 | C24 | 1 | 45 | yes | — |
| 177 | C2×C4⋊C4 | C24 | 1 | 45 | yes | — |
| 178 | C2×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 179 | C2×C4⋊C4 | C24 | 1 | 180 | yes | — |
| 180 | C2×C4⋊C4 | C24 | 1 | 180 | yes | — |
| 181 | C2×C22⋊C4 | C24 | 1 | 180 | yes | — |
| 182 | C2×C22⋊C4 | C24 | 1 | 90 | yes | — |
| 183 | C2×C42 | C24 | 1 | 45 | yes | — |
| 184 | C2×C42 | C24 | 1 | 180 | yes | — |
| 185 | C2×C42 | C24 | 1 | 45 | yes | — |
| 186 | C2×C42 | C24 | 1 | 45 | yes | — |
| 187 | C2×C42 | C24 | 1 | 15 | yes | — |
| 188 | C24 | C25 | 1 | 15 | yes | — |
| 189 | #41 | C22×Q8 | 1 | 160 | yes | complemented |
| 190 | #41 | C22×Q8 | 1 | 480 | yes | complemented |
| 191 | C2×Q8 | — | 4 | 180 | no | — |
| 192 | C2×Q8 | — | 4 | 60 | no | — |
| 193 | C2×Q8 | — | 8 | 180 | no | — |
| 194 | C2×Q8 | C22×Q8 | 1 | 120 | yes | complemented |
| 195 | C2×Q8 | C25 | 1 | 12 | yes | — |
| 196 | C2×Q8 | C25 | 1 | 20 | yes | — |
| 197 | C2×Q8 | C25 | 1 | 60 | yes | — |
| 198 | C2×Q8 | C25 | 1 | 180 | yes | — |
| 199 | C2×Q8 | C25 | 1 | 20 | yes | — |
| 200 | C2×D4 | C22×Q8 | 1 | 360 | yes | complemented |
| 201 | C2×D4 | C25 | 1 | 60 | yes | — |
| 202 | C22×C4 | — | 4 | 30 | no | — |
| 203 | C22×C4 | — | 4 | 30 | no | — |
| 204 | C22×C4 | — | 8 | 60 | no | — |
| 205 | C22×C4 | C22×Q8 | 1 | 120 | yes | complemented |
| 206 | C22×C4 | C25 | 1 | 15 | yes | — |
| 207 | C22×C4 | C25 | 1 | 15 | yes | — |
| 208 | C22×C4 | C25 | 1 | 60 | yes | — |
| 209 | C22×C4 | C25 | 1 | 45 | yes | — |
| 210 | C22×C4 | C25 | 1 | 45 | yes | — |
| 211 | C22×C4 | C25 | 1 | 180 | yes | — |
| 212 | C22×C4 | C25 | 1 | 45 | yes | — |
| 213 | C22×C4 | C25 | 1 | 15 | yes | — |
| 214 | C4⋊C4 | C25 | 1 | 20 | yes | — |
| 215 | C4⋊C4 | C25 | 1 | 20 | yes | — |
| 216 | C4⋊C4 | C25 | 1 | 12 | yes | — |
| 217 | C4⋊C4 | C25 | 1 | 180 | yes | — |
| 218 | C4⋊C4 | C25 | 1 | 12 | yes | — |
| 219 | C4⋊C4 | C25 | 1 | 180 | yes | — |
| 220 | C4⋊C4 | C25 | 1 | 60 | yes | — |
| 221 | C4⋊C4 | C25 | 1 | 60 | yes | — |
| 222 | C4⋊C4 | C25 | 1 | 240 | yes | — |
| 223 | C4⋊C4 | C25 | 1 | 240 | yes | — |
| 224 | C4⋊C4 | C25 | 1 | 60 | yes | — |
| 225 | C4⋊C4 | C25 | 1 | 60 | yes | — |
| 226 | C22⋊C4 | C25 | 1 | 240 | yes | — |
| 227 | C22⋊C4 | C25 | 1 | 120 | yes | — |
| 228 | C42 | C25 | 1 | 1 | yes | characteristic |
| 229 | C42 | C25 | 1 | 15 | yes | — |
| 230 | C42 | C25 | 1 | 15 | yes | — |
| 231 | C42 | C25 | 1 | 15 | yes | — |
| 232 | C42 | C25 | 1 | 240 | yes | — |
| 233 | C42 | C25 | 1 | 90 | yes | — |
| 234 | C23 | — | 4 | 15 | no | — |
| 235 | C23 | — | 4 | 15 | no | — |
| 236 | C23 | C23×Q8 | 1 | 60 | yes | — |
| 237 | C23 | C26 | 1 | 15 | yes | — |
| 238 | Q8 | — | 2 | 40 | no | — |
| 239 | Q8 | — | 2 | 24 | no | — |
| 240 | Q8 | — | 4 | 180 | no | — |
| 241 | Q8 | — | 4 | 60 | no | — |
| 242 | Q8 | C23×Q8 | 1 | 80 | yes | — |
| 243 | D4 | C23×Q8 | 1 | 240 | yes | — |
| 244 | C2×C4 | — | 2 | 30 | no | — |
| 245 | C2×C4 | — | 2 | 30 | no | — |
| 246 | C2×C4 | — | 4 | 45 | no | — |
| 247 | C2×C4 | — | 4 | 45 | no | — |
| 248 | C2×C4 | — | 4 | 180 | no | — |
| 249 | C2×C4 | — | 4 | 60 | no | — |
| 250 | C2×C4 | C23×Q8 | 1 | 60 | yes | — |
| 251 | C2×C4 | C23×Q8 | 1 | 180 | yes | — |
| 252 | C2×C4 | C26 | 1 | 24 | yes | — |
| 253 | C2×C4 | C26 | 1 | 15 | yes | — |
| 254 | C2×C4 | C26 | 1 | 15 | yes | — |
| 255 | C2×C4 | C26 | 1 | 40 | yes | — |
| 256 | C2×C4 | C26 | 1 | 15 | yes | — |
| 257 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | C26 | 1 | 1 | yes | characteristic |
| 260 | C22 | — | 2 | 15 | no | — |
| 261 | C22 | — | 2 | 15 | no | — |
| 262 | C22 | — | 4 | 60 | no | — |
| 263 | C22 | C24×Q8 | 1 | 30 | yes | — |
| 264 | C22 | C27 | 1 | 1 | yes | characteristic |
| 265 | C4 | — | 2 | 15 | no | — |
| 266 | C4 | — | 2 | 40 | no | — |
| 267 | C4 | — | 2 | 15 | no | — |
| 268 | C4 | — | 2 | 24 | no | — |
| 269 | C4 | C24×Q8 | 1 | 2 | yes | — |
| 270 | C4 | C24×Q8 | 1 | 30 | yes | — |
| 271 | C4 | H(7,𝔽2) | 1 | 2 | yes | — |
| 272 | C4 | 2−1+6 | 1 | 2 | yes | — |
| 273 | C2 | — | 2 | 30 | no | — |
| 274 | C2 | C25×Q8 | 1 | 1 | yes | characteristic |
| 275 | C2 | C2×H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 276 | C2 | C2×2−1+6 | 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