← #8097764
Subgroups — #8097764
Order 512.
6120 conjugacy classes of subgroups.
The subgroup lattice has 244 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #52990 | C2 | 1 | 27 | yes | maximal, complemented |
| 2 | #52958 | C2 | 1 | 9 | yes | maximal, complemented |
| 3 | #45479 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | #36675 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #36575 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #36082 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #35948 | C2 | 1 | 9 | yes | maximal, complemented |
| 8 | C22×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C24×Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×H(3,𝔽4) | C22 | 1 | 6 | yes | complemented |
| 11 | C2×Q82 | C22 | 1 | 3 | yes | complemented |
| 12 | C2×#548 | C22 | 1 | 9 | yes | — |
| 13 | #2847 | C22 | 1 | 9 | yes | complemented |
| 14 | #2840 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #2840 | C22 | 1 | 18 | yes | complemented |
| 16 | #2835 | C22 | 1 | 9 | yes | complemented |
| 17 | #2832 | C22 | 1 | 54 | yes | complemented |
| 18 | #2824 | C22 | 1 | 27 | yes | complemented |
| 19 | #2812 | C22 | 1 | 9 | yes | complemented |
| 20 | #2811 | C22 | 1 | 6 | yes | complemented |
| 21 | #2736 | C22 | 1 | 18 | yes | complemented |
| 22 | #2712 | C22 | 1 | 54 | yes | complemented |
| 23 | #2684 | C22 | 1 | 54 | yes | complemented |
| 24 | #2640 | C22 | 1 | 27 | yes | complemented |
| 25 | #2616 | C22 | 1 | 9 | yes | complemented |
| 26 | #2592 | C22 | 1 | 27 | yes | complemented |
| 27 | #2547 | C22 | 1 | 27 | yes | complemented |
| 28 | #2492 | C22 | 1 | 54 | yes | complemented |
| 29 | #2491 | C22 | 1 | 54 | yes | complemented |
| 30 | #2436 | C22 | 1 | 27 | yes | complemented |
| 31 | #2435 | C22 | 1 | 27 | yes | complemented |
| 32 | #2409 | C22 | 1 | 27 | yes | complemented |
| 33 | #2408 | C22 | 1 | 27 | yes | — |
| 34 | #2398 | C22 | 1 | 27 | yes | complemented |
| 35 | #2388 | C22 | 1 | 9 | yes | — |
| 36 | #2341 | C22 | 1 | 9 | yes | complemented |
| 37 | Q8×C22⋊C4 | C22 | 1 | 9 | yes | complemented |
| 38 | C4×#120 | C22 | 1 | 9 | yes | complemented |
| 39 | #2276 | C22 | 1 | 3 | yes | complemented |
| 40 | C2×H(5,𝔽2) | — | 4 | 12 | no | — |
| 41 | C22×#41 | C23 | 1 | 3 | yes | — |
| 42 | C22×#41 | C23 | 1 | 3 | yes | — |
| 43 | C23×Q8 | — | 4 | 12 | no | — |
| 44 | C23×Q8 | C23 | 1 | 3 | yes | — |
| 45 | C23×Q8 | C23 | 1 | 9 | yes | — |
| 46 | C23×D4 | C23 | 1 | 9 | yes | — |
| 47 | C24×C4 | C23 | 1 | 3 | yes | — |
| 48 | H(3,𝔽4) | — | 4 | 24 | no | — |
| 49 | Q82 | — | 4 | 12 | no | — |
| 50 | #548 | — | 4 | 36 | no | — |
| 51 | C2×C4⋊Q8 | C23 | 1 | 27 | yes | — |
| 52 | C2×#124 | C23 | 1 | 27 | yes | — |
| 53 | C2×#124 | C23 | 1 | 54 | yes | — |
| 54 | C2×C22⋊Q8 | C23 | 1 | 27 | yes | — |
| 55 | C2×#121 | C23 | 1 | 27 | yes | — |
| 56 | C2×#120 | C23 | 1 | 9 | yes | — |
| 57 | C2×#120 | C23 | 1 | 18 | yes | — |
| 58 | C2×C4×Q8 | C23 | 1 | 9 | yes | — |
| 59 | C2×C4×Q8 | C23 | 1 | 9 | yes | — |
| 60 | C2×C4×D4 | C23 | 1 | 27 | yes | — |
| 61 | #400 | C23 | 1 | 54 | yes | — |
| 62 | #397 | C23 | 1 | 27 | yes | — |
| 63 | #397 | C23 | 1 | 54 | yes | — |
| 64 | #397 | C23 | 1 | 27 | yes | — |
| 65 | #395 | C23 | 1 | 18 | yes | — |
| 66 | #395 | C23 | 1 | 9 | yes | — |
| 67 | #394 | C23 | 1 | 27 | yes | — |
| 68 | #394 | C23 | 1 | 54 | yes | — |
| 69 | #394 | C23 | 1 | 54 | yes | — |
| 70 | #393 | C23 | 1 | 9 | yes | — |
| 71 | #393 | C23 | 1 | 27 | yes | — |
| 72 | #393 | C23 | 1 | 54 | yes | — |
| 73 | #393 | C23 | 1 | 108 | yes | — |
| 74 | #393 | C23 | 1 | 3 | yes | — |
| 75 | #392 | C23 | 1 | 27 | yes | — |
| 76 | #391 | C23 | 1 | 54 | yes | — |
| 77 | #391 | C23 | 1 | 54 | yes | — |
| 78 | #390 | C23 | 1 | 54 | yes | — |
| 79 | #388 | C23 | 1 | 108 | yes | — |
| 80 | #386 | C23 | 1 | 27 | yes | — |
| 81 | #386 | C23 | 1 | 54 | yes | — |
| 82 | #385 | C23 | 1 | 27 | yes | — |
| 83 | #380 | C23 | 1 | 27 | yes | — |
| 84 | #380 | C23 | 1 | 27 | yes | — |
| 85 | #379 | C23 | 1 | 54 | yes | — |
| 86 | #379 | C23 | 1 | 3 | yes | — |
| 87 | C4×C4⋊C4 | C23 | 1 | 27 | yes | — |
| 88 | C4×C22⋊C4 | C23 | 1 | 27 | yes | — |
| 89 | C4×C22⋊C4 | C23 | 1 | 27 | yes | — |
| 90 | #376 | C23 | 1 | 9 | yes | — |
| 91 | C25 | C24 | 1 | 6 | yes | — |
| 92 | C25 | C24 | 1 | 1 | yes | characteristic |
| 93 | H(5,𝔽2) | — | 4 | 64 | no | — |
| 94 | C2×#41 | — | 2 | 36 | no | — |
| 95 | C2×#41 | — | 4 | 18 | no | — |
| 96 | C22×Q8 | — | 2 | 36 | no | — |
| 97 | C22×Q8 | — | 4 | 54 | no | — |
| 98 | C22×Q8 | — | 4 | 64 | no | — |
| 99 | C22×Q8 | C24 | 1 | 9 | yes | — |
| 100 | C22×Q8 | C24 | 1 | 3 | yes | — |
| 101 | C22×Q8 | C24 | 1 | 18 | yes | — |
| 102 | C22×Q8 | C24 | 1 | 6 | yes | — |
| 103 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 104 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 105 | C22×D4 | — | 4 | 54 | no | — |
| 106 | C22×D4 | C24 | 1 | 9 | yes | — |
| 107 | C22×D4 | C24 | 1 | 27 | yes | — |
| 108 | C22×D4 | C24 | 1 | 9 | yes | — |
| 109 | C23×C4 | — | 4 | 18 | no | — |
| 110 | C23×C4 | C24 | 1 | 9 | yes | — |
| 111 | C23×C4 | C24 | 1 | 9 | yes | — |
| 112 | C23×C4 | C24 | 1 | 9 | yes | — |
| 113 | C4⋊Q8 | — | 4 | 54 | no | — |
| 114 | #124 | — | 4 | 54 | no | — |
| 115 | #124 | — | 4 | 108 | no | — |
| 116 | C22⋊Q8 | — | 4 | 54 | no | — |
| 117 | #121 | — | 4 | 54 | no | — |
| 118 | #120 | — | 2 | 36 | no | — |
| 119 | #120 | — | 4 | 36 | no | — |
| 120 | C4×Q8 | — | 2 | 36 | no | — |
| 121 | C4×Q8 | — | 4 | 18 | no | — |
| 122 | C4×D4 | — | 4 | 54 | no | — |
| 123 | C2×C4⋊C4 | C24 | 1 | 27 | yes | — |
| 124 | C2×C4⋊C4 | C24 | 1 | 27 | yes | — |
| 125 | C2×C4⋊C4 | C24 | 1 | 27 | yes | — |
| 126 | C2×C22⋊C4 | C24 | 1 | 27 | yes | — |
| 127 | C2×C22⋊C4 | C24 | 1 | 9 | yes | — |
| 128 | C2×C22⋊C4 | C24 | 1 | 27 | yes | — |
| 129 | C2×C22⋊C4 | C24 | 1 | 54 | yes | — |
| 130 | C2×C22⋊C4 | C24 | 1 | 54 | yes | — |
| 131 | C2×C22⋊C4 | C24 | 1 | 9 | yes | — |
| 132 | C2×C22⋊C4 | C24 | 1 | 54 | yes | — |
| 133 | C2×C42 | C24 | 1 | 27 | yes | — |
| 134 | C2×C42 | C24 | 1 | 18 | yes | — |
| 135 | #95 | C24 | 1 | 3 | yes | — |
| 136 | #95 | C24 | 1 | 27 | yes | — |
| 137 | #95 | C24 | 1 | 9 | yes | — |
| 138 | #95 | C24 | 1 | 27 | yes | — |
| 139 | #95 | C24 | 1 | 54 | yes | — |
| 140 | #95 | C24 | 1 | 54 | yes | — |
| 141 | C24 | — | 4 | 3 | no | — |
| 142 | C24 | — | 4 | 3 | no | — |
| 143 | C24 | — | 4 | 18 | no | — |
| 144 | C24 | — | 4 | 6 | no | — |
| 145 | C24 | — | 4 | 18 | no | — |
| 146 | C24 | H(5,𝔽2) | 1 | 4 | yes | complemented |
| 147 | C24 | C25 | 1 | 9 | yes | — |
| 148 | C24 | C25 | 1 | 3 | yes | — |
| 149 | #41 | — | 4 | 48 | no | — |
| 150 | #41 | — | 4 | 48 | no | — |
| 151 | C2×Q8 | — | 2 | 54 | no | — |
| 152 | C2×Q8 | — | 4 | 3 | no | — |
| 153 | C2×Q8 | — | 4 | 18 | no | — |
| 154 | C2×Q8 | — | 4 | 6 | no | — |
| 155 | C2×Q8 | — | 4 | 3 | no | — |
| 156 | C2×Q8 | — | 4 | 18 | no | — |
| 157 | C2×Q8 | — | 4 | 18 | no | — |
| 158 | C2×Q8 | — | 4 | 18 | no | — |
| 159 | C2×Q8 | — | 4 | 144 | no | — |
| 160 | C2×Q8 | — | 4 | 48 | no | — |
| 161 | C2×Q8 | H(5,𝔽2) | 1 | 12 | yes | — |
| 162 | C2×D4 | — | 2 | 54 | no | — |
| 163 | C2×D4 | — | 2 | 54 | no | — |
| 164 | C2×D4 | — | 4 | 27 | no | — |
| 165 | C2×D4 | — | 4 | 27 | no | — |
| 166 | C2×D4 | — | 4 | 27 | no | — |
| 167 | C2×D4 | — | 4 | 144 | no | — |
| 168 | C22×C4 | — | 2 | 54 | no | — |
| 169 | C22×C4 | — | 2 | 54 | no | — |
| 170 | C22×C4 | — | 4 | 27 | no | — |
| 171 | C22×C4 | — | 4 | 48 | no | — |
| 172 | C22×C4 | C25 | 1 | 9 | yes | — |
| 173 | C22×C4 | C25 | 1 | 9 | yes | — |
| 174 | C22×C4 | C25 | 1 | 3 | yes | — |
| 175 | C22×C4 | C25 | 1 | 27 | yes | — |
| 176 | C22×C4 | C25 | 1 | 3 | yes | — |
| 177 | C4⋊C4 | — | 2 | 54 | no | — |
| 178 | C4⋊C4 | — | 4 | 27 | no | — |
| 179 | C4⋊C4 | — | 4 | 27 | no | — |
| 180 | C22⋊C4 | — | 2 | 54 | no | — |
| 181 | C22⋊C4 | — | 4 | 9 | no | — |
| 182 | C22⋊C4 | — | 4 | 54 | no | — |
| 183 | C22⋊C4 | — | 4 | 9 | no | — |
| 184 | C22⋊C4 | — | 4 | 27 | no | — |
| 185 | C22⋊C4 | — | 4 | 54 | no | — |
| 186 | C22⋊C4 | — | 4 | 54 | no | — |
| 187 | C42 | — | 2 | 54 | no | — |
| 188 | C42 | — | 4 | 18 | no | — |
| 189 | C23 | — | 2 | 9 | no | — |
| 190 | C23 | — | 2 | 9 | no | — |
| 191 | C23 | — | 2 | 9 | no | — |
| 192 | C23 | — | 2 | 27 | no | — |
| 193 | C23 | — | 2 | 27 | no | — |
| 194 | C23 | — | 4 | 12 | no | — |
| 195 | C23 | — | 4 | 4 | no | — |
| 196 | C23 | — | 4 | 12 | no | — |
| 197 | C23 | — | 4 | 24 | no | — |
| 198 | C23 | — | 16 | 18 | no | — |
| 199 | C23 | — | 16 | 18 | no | — |
| 200 | C23 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 201 | C23 | C26 | 1 | 1 | yes | characteristic |
| 202 | Q8 | — | 4 | 12 | no | — |
| 203 | Q8 | — | 4 | 24 | no | — |
| 204 | Q8 | — | 4 | 4 | no | — |
| 205 | Q8 | — | 4 | 36 | no | — |
| 206 | Q8 | — | 16 | 18 | no | — |
| 207 | Q8 | H(3,𝔽4) | 1 | 16 | yes | complemented |
| 208 | D4 | — | 4 | 36 | no | — |
| 209 | D4 | — | 4 | 36 | no | — |
| 210 | D4 | — | 8 | 54 | no | — |
| 211 | C2×C4 | — | 2 | 9 | no | — |
| 212 | C2×C4 | — | 2 | 27 | no | — |
| 213 | C2×C4 | — | 2 | 27 | no | — |
| 214 | C2×C4 | — | 2 | 27 | no | — |
| 215 | C2×C4 | — | 2 | 9 | no | — |
| 216 | C2×C4 | — | 2 | 18 | no | — |
| 217 | C2×C4 | — | 2 | 27 | no | — |
| 218 | C2×C4 | — | 4 | 36 | no | — |
| 219 | C2×C4 | — | 4 | 36 | no | — |
| 220 | C2×C4 | — | 4 | 36 | no | — |
| 221 | C2×C4 | C2×H(5,𝔽2) | 1 | 18 | yes | — |
| 222 | C22 | — | 2 | 18 | no | — |
| 223 | C22 | — | 4 | 16 | no | — |
| 224 | C22 | — | 4 | 9 | no | — |
| 225 | C22 | — | 4 | 3 | no | — |
| 226 | C22 | — | 4 | 27 | no | — |
| 227 | C22 | — | 4 | 27 | no | — |
| 228 | C22 | — | 4 | 9 | no | — |
| 229 | C22 | — | 16 | 24 | no | — |
| 230 | C22 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 231 | C22 | C22×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 232 | C22 | 2−1+6 | 1 | 3 | yes | — |
| 233 | C4 | — | 4 | 9 | no | — |
| 234 | C4 | — | 4 | 27 | no | — |
| 235 | C4 | — | 4 | 3 | no | — |
| 236 | C4 | — | 4 | 9 | no | — |
| 237 | C4 | C2×H(3,𝔽4) | 1 | 12 | yes | — |
| 238 | C2 | — | 4 | 6 | no | — |
| 239 | C2 | — | 8 | 9 | no | — |
| 240 | C2 | C22×H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 241 | C2 | #62895 | 1 | 3 | yes | — |
| 242 | C2 | #63048 | 1 | 3 | yes | — |
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