← #498570
Subgroups — #498570
Order 512.
643 conjugacy classes of subgroups.
The subgroup lattice has 251 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #33716 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33465 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23196 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13259 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13243 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13016 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #12671 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #12665 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#508 | C22 | 1 | 2 | yes | — |
| 10 | C2×#508 | C22 | 1 | 2 | yes | — |
| 11 | #3147 | C22 | 1 | 2 | yes | — |
| 12 | #2957 | — | 2 | 2 | no | — |
| 13 | #2957 | C22 | 1 | 4 | yes | — |
| 14 | #2251 | C4 | 1 | 4 | yes | complemented |
| 15 | #2250 | C4 | 1 | 4 | yes | complemented |
| 16 | #2179 | — | 2 | 4 | no | — |
| 17 | #2178 | — | 2 | 4 | no | — |
| 18 | C2×#368 | C22 | 1 | 1 | yes | characteristic |
| 19 | #2158 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2145 | C22 | 1 | 4 | yes | — |
| 22 | #2143 | C22 | 1 | 4 | yes | — |
| 23 | #2108 | C22 | 1 | 2 | yes | — |
| 24 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1976 | — | 2 | 2 | no | — |
| 26 | #1976 | C22 | 1 | 4 | yes | — |
| 27 | #1884 | — | 2 | 4 | no | — |
| 28 | #1772 | C22 | 1 | 2 | yes | — |
| 29 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1382 | C22 | 1 | 2 | yes | — |
| 31 | #1379 | C22 | 1 | 2 | yes | — |
| 32 | C2×#135 | — | 2 | 2 | no | — |
| 33 | C2×#135 | C23 | 1 | 4 | yes | — |
| 34 | #568 | — | 2 | 4 | no | — |
| 35 | #532 | — | 2 | 2 | no | — |
| 36 | #508 | — | 2 | 2 | no | — |
| 37 | #508 | — | 2 | 2 | no | — |
| 38 | #508 | — | 2 | 2 | no | — |
| 39 | #508 | — | 2 | 2 | no | — |
| 40 | #508 | — | 4 | 2 | no | — |
| 41 | #508 | — | 4 | 2 | no | — |
| 42 | C2×Q32 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×Q32 | C2×C4 | 1 | 2 | yes | — |
| 44 | C2×SD32 | C2×C4 | 1 | 4 | yes | — |
| 45 | C2×SD32 | C2×C4 | 1 | 4 | yes | — |
| 46 | C2×D16 | C2×C4 | 1 | 2 | yes | — |
| 47 | C2×D16 | C2×C4 | 1 | 2 | yes | — |
| 48 | C2×M32 | C23 | 1 | 2 | yes | — |
| 49 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 50 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 51 | #495 | — | 2 | 2 | no | — |
| 52 | #495 | C2×C4 | 1 | 4 | yes | — |
| 53 | #494 | C2×C4 | 1 | 2 | yes | — |
| 54 | #493 | C2×C4 | 1 | 2 | yes | — |
| 55 | #492 | — | 2 | 2 | no | — |
| 56 | #444 | — | 4 | 4 | no | — |
| 57 | #444 | — | 4 | 4 | no | — |
| 58 | #443 | — | 2 | 8 | no | — |
| 59 | #443 | — | 4 | 4 | no | — |
| 60 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×#108 | C23 | 1 | 2 | yes | — |
| 62 | #427 | D4 | 1 | 1 | yes | characteristic |
| 63 | #427 | C23 | 1 | 2 | yes | — |
| 64 | #421 | — | 2 | 4 | no | — |
| 65 | #421 | — | 2 | 4 | no | — |
| 66 | #410 | — | 2 | 4 | no | — |
| 67 | #405 | — | 2 | 2 | no | — |
| 68 | #405 | C23 | 1 | 2 | yes | — |
| 69 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 70 | #368 | D4 | 1 | 4 | yes | complemented |
| 71 | #367 | — | 2 | 1 | no | — |
| 72 | #366 | — | 2 | 1 | no | — |
| 73 | #364 | — | 2 | 2 | no | — |
| 74 | #360 | — | 4 | 4 | no | — |
| 75 | #359 | — | 2 | 8 | no | — |
| 76 | #346 | — | 2 | 2 | no | — |
| 77 | #346 | C2×C4 | 1 | 4 | yes | — |
| 78 | C4×C16 | — | 2 | 2 | no | — |
| 79 | #339 | — | 2 | 2 | no | — |
| 80 | #339 | D4 | 1 | 2 | yes | — |
| 81 | C2×#41 | — | 2 | 4 | no | — |
| 82 | #135 | — | 2 | 4 | no | — |
| 83 | #135 | — | 2 | 4 | no | — |
| 84 | #135 | — | 2 | 4 | no | — |
| 85 | #135 | — | 2 | 4 | no | — |
| 86 | #135 | — | 4 | 2 | no | — |
| 87 | #135 | — | 4 | 2 | no | — |
| 88 | #135 | — | 4 | 2 | no | — |
| 89 | #135 | — | 4 | 2 | no | — |
| 90 | C2×Q16 | — | 2 | 2 | no | — |
| 91 | C2×Q16 | C22×C4 | 1 | 4 | yes | — |
| 92 | C2×SD16 | — | 2 | 4 | no | — |
| 93 | C2×SD16 | — | 2 | 4 | no | — |
| 94 | C2×D8 | — | 2 | 2 | no | — |
| 95 | C2×D8 | C22×C4 | 1 | 4 | yes | — |
| 96 | #131 | — | 4 | 8 | no | — |
| 97 | #131 | — | 4 | 8 | no | — |
| 98 | C2×M16 | — | 2 | 2 | no | — |
| 99 | C2×M16 | — | 8 | 4 | no | — |
| 100 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 101 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 103 | C4⋊Q8 | — | 2 | 2 | no | — |
| 104 | #127 | — | 2 | 2 | no | — |
| 105 | #124 | — | 2 | 4 | no | — |
| 106 | #121 | — | 8 | 2 | no | — |
| 107 | C4×D4 | — | 8 | 2 | no | — |
| 108 | #117 | — | 2 | 2 | no | — |
| 109 | #117 | C2×D4 | 1 | 2 | yes | — |
| 110 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | Q32 | — | 2 | 2 | no | — |
| 112 | Q32 | — | 2 | 2 | no | — |
| 113 | Q32 | — | 4 | 2 | no | — |
| 114 | SD32 | — | 2 | 4 | no | — |
| 115 | SD32 | — | 2 | 4 | no | — |
| 116 | SD32 | — | 4 | 4 | no | — |
| 117 | D16 | — | 2 | 2 | no | — |
| 118 | D16 | — | 2 | 2 | no | — |
| 119 | D16 | — | 4 | 2 | no | — |
| 120 | M32 | — | 2 | 2 | no | — |
| 121 | M32 | — | 2 | 2 | no | — |
| 122 | C2×C16 | — | 2 | 1 | no | — |
| 123 | C2×C16 | — | 2 | 1 | no | — |
| 124 | C2×C16 | — | 2 | 2 | no | — |
| 125 | C2×C16 | C22×C4 | 1 | 2 | yes | — |
| 126 | C2×C16 | C22×C4 | 1 | 2 | yes | — |
| 127 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 128 | C2×C16 | #41 | 1 | 1 | yes | characteristic |
| 129 | C2×C16 | #41 | 1 | 1 | yes | characteristic |
| 130 | #108 | — | 2 | 2 | no | — |
| 131 | #108 | D8 | 1 | 4 | yes | — |
| 132 | #108 | C2×D4 | 1 | 4 | yes | — |
| 133 | #107 | — | 2 | 1 | no | — |
| 134 | #107 | — | 2 | 2 | no | — |
| 135 | #106 | — | 2 | 1 | no | — |
| 136 | #106 | — | 2 | 2 | no | — |
| 137 | #104 | — | 4 | 8 | no | — |
| 138 | #104 | — | 4 | 8 | no | — |
| 139 | #99 | — | 8 | 4 | no | — |
| 140 | #97 | — | 2 | 2 | no | — |
| 141 | #97 | — | 4 | 2 | no | — |
| 142 | C4×C8 | — | 2 | 2 | no | — |
| 143 | C4×C8 | — | 4 | 2 | no | — |
| 144 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 145 | C4×C8 | #41 | 1 | 2 | yes | — |
| 146 | #41 | — | 4 | 4 | no | — |
| 147 | #41 | — | 4 | 4 | no | — |
| 148 | #41 | — | 4 | 4 | no | — |
| 149 | #41 | — | 4 | 4 | no | — |
| 150 | C2×Q8 | — | 2 | 4 | no | — |
| 151 | C2×D4 | — | 2 | 4 | no | — |
| 152 | C2×D4 | — | 4 | 4 | no | — |
| 153 | C2×D4 | — | 8 | 2 | no | — |
| 154 | C22×C4 | — | 2 | 1 | no | — |
| 155 | C22×C4 | — | 4 | 4 | no | — |
| 156 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 157 | Q16 | — | 2 | 4 | no | — |
| 158 | Q16 | — | 2 | 4 | no | — |
| 159 | Q16 | — | 4 | 2 | no | — |
| 160 | Q16 | — | 4 | 2 | no | — |
| 161 | SD16 | — | 4 | 4 | no | — |
| 162 | SD16 | — | 4 | 4 | no | — |
| 163 | SD16 | — | 4 | 4 | no | — |
| 164 | SD16 | — | 4 | 4 | no | — |
| 165 | D8 | — | 2 | 4 | no | — |
| 166 | D8 | — | 2 | 4 | no | — |
| 167 | D8 | — | 4 | 2 | no | — |
| 168 | D8 | — | 4 | 2 | no | — |
| 169 | M16 | — | 2 | 2 | no | — |
| 170 | M16 | — | 2 | 4 | no | — |
| 171 | M16 | — | 4 | 2 | no | — |
| 172 | M16 | — | 8 | 8 | no | — |
| 173 | M16 | C2×D8 | 1 | 4 | yes | — |
| 174 | C2×C8 | — | 2 | 2 | no | — |
| 175 | C2×C8 | — | 4 | 2 | no | — |
| 176 | C2×C8 | — | 8 | 8 | no | — |
| 177 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 178 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 179 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 180 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 181 | C2×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 182 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 183 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 187 | C4⋊C4 | — | 2 | 2 | no | — |
| 188 | C4⋊C4 | — | 4 | 2 | no | — |
| 189 | C4⋊C4 | — | 8 | 2 | no | — |
| 190 | C22⋊C4 | — | 2 | 2 | no | — |
| 191 | C22⋊C4 | — | 4 | 2 | no | — |
| 192 | C22⋊C4 | — | 8 | 4 | no | — |
| 193 | C42 | — | 2 | 1 | no | — |
| 194 | C42 | — | 2 | 2 | no | — |
| 195 | C42 | — | 4 | 2 | no | — |
| 196 | C42 | C4×D4 | 1 | 2 | yes | — |
| 197 | C16 | — | 2 | 2 | no | — |
| 198 | C16 | — | 4 | 2 | no | — |
| 199 | C16 | C4×D4 | 1 | 2 | yes | — |
| 200 | C16 | C4×D4 | 1 | 2 | yes | — |
| 201 | C23 | — | 4 | 4 | no | — |
| 202 | C23 | #576 | 1 | 1 | yes | characteristic |
| 203 | Q8 | — | 4 | 4 | no | — |
| 204 | Q8 | — | 4 | 4 | no | — |
| 205 | D4 | — | 4 | 4 | no | — |
| 206 | D4 | — | 4 | 4 | no | — |
| 207 | D4 | — | 8 | 4 | no | — |
| 208 | D4 | — | 8 | 4 | no | — |
| 209 | D4 | — | 16 | 4 | no | — |
| 210 | C2×C4 | — | 2 | 2 | no | — |
| 211 | C2×C4 | — | 4 | 4 | no | — |
| 212 | C2×C4 | — | 4 | 2 | no | — |
| 213 | C2×C4 | — | 8 | 1 | no | — |
| 214 | C2×C4 | — | 8 | 4 | no | — |
| 215 | C2×C4 | — | 8 | 4 | no | — |
| 216 | C2×C4 | — | 8 | 2 | no | — |
| 217 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | #550 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 223 | C8 | — | 2 | 4 | no | — |
| 224 | C8 | — | 4 | 4 | no | — |
| 225 | C8 | C4×D8 | 1 | 2 | yes | — |
| 226 | C8 | C4×D8 | 1 | 2 | yes | — |
| 227 | C8 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 228 | C8 | #518 | 1 | 1 | yes | characteristic |
| 229 | C8 | #550 | 1 | 1 | yes | characteristic |
| 230 | C8 | #559 | 1 | 1 | yes | characteristic |
| 231 | C22 | — | 4 | 1 | no | — |
| 232 | C22 | — | 8 | 4 | no | — |
| 233 | C22 | — | 16 | 4 | no | — |
| 234 | C22 | #2948 | 1 | 1 | yes | characteristic |
| 235 | C22 | #3287 | 1 | 1 | yes | characteristic |
| 236 | C22 | #3391 | 1 | 1 | yes | characteristic |
| 237 | C4 | — | 4 | 2 | no | — |
| 238 | C4 | — | 8 | 2 | no | — |
| 239 | C4 | — | 8 | 4 | no | — |
| 240 | C4 | — | 16 | 2 | no | — |
| 241 | C4 | #2366 | 1 | 1 | yes | characteristic |
| 242 | C4 | C2×C4×D8 | 1 | 1 | yes | characteristic |
| 243 | C4 | D4×D8 | 1 | 1 | yes | characteristic |
| 244 | C4 | #3396 | 1 | 1 | yes | characteristic |
| 245 | C2 | — | 2 | 1 | no | — |
| 246 | C2 | — | 2 | 1 | no | — |
| 247 | C2 | — | 2 | 1 | no | — |
| 248 | C2 | — | 16 | 4 | no | — |
| 249 | C2 | #23185 | 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