← #135821
Subgroups — #135821
Order 512.
312 conjugacy classes of subgroups.
The subgroup lattice has 266 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #17789 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×#350 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #12009 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#1372 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #8605 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #7159 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7155 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #2988 | — | 2 | 1 | no | — |
| 9 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#350 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#350 | C4 | 1 | 1 | yes | characteristic |
| 12 | C4×M32 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 14 | #1565 | — | 2 | 1 | no | — |
| 15 | #1565 | — | 2 | 2 | no | — |
| 16 | #1564 | — | 2 | 1 | no | — |
| 17 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#335 | — | 2 | 1 | no | — |
| 20 | C2×#335 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#335 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1477 | — | 2 | 1 | no | — |
| 23 | #1379 | C4 | 1 | 1 | yes | characteristic |
| 24 | #1379 | C4 | 1 | 1 | yes | characteristic |
| 25 | #1372 | — | 2 | 2 | no | — |
| 26 | #1372 | — | 2 | 1 | no | — |
| 27 | #1371 | — | 2 | 1 | no | — |
| 28 | #1283 | — | 2 | 1 | no | — |
| 29 | C2×#131 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C2×#131 | Q8 | 1 | 1 | yes | characteristic |
| 31 | C4×#41 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×M32 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×M32 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C8×Q8 | — | 4 | 1 | no | — |
| 37 | C8×D4 | — | 4 | 2 | no | — |
| 38 | C8×D4 | — | 4 | 1 | no | — |
| 39 | #431 | — | 2 | 2 | no | — |
| 40 | #431 | — | 2 | 1 | no | — |
| 41 | #431 | — | 2 | 1 | no | — |
| 42 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 43 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 44 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 45 | #405 | — | 2 | 1 | no | — |
| 46 | #405 | — | 2 | 1 | no | — |
| 47 | #405 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | #405 | Q8 | 1 | 1 | yes | characteristic |
| 49 | C4×M16 | — | 2 | 1 | no | — |
| 50 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 51 | C2×C4×C8 | — | 2 | 1 | no | — |
| 52 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 53 | #350 | — | 2 | 2 | no | — |
| 54 | #350 | C8 | 1 | 4 | yes | complemented |
| 55 | #346 | — | 2 | 1 | no | — |
| 56 | C4×C16 | — | 2 | 1 | no | — |
| 57 | C4×C16 | — | 2 | 1 | no | — |
| 58 | C4×C16 | — | 2 | 1 | no | — |
| 59 | #335 | — | 2 | 2 | no | — |
| 60 | #335 | — | 2 | 1 | no | — |
| 61 | #335 | — | 2 | 1 | no | — |
| 62 | #335 | — | 2 | 1 | no | — |
| 63 | #335 | — | 2 | 1 | no | — |
| 64 | #335 | D4 | 1 | 1 | yes | characteristic |
| 65 | #335 | D4 | 1 | 2 | yes | — |
| 66 | #335 | D4 | 1 | 1 | yes | characteristic |
| 67 | #334 | — | 2 | 2 | no | — |
| 68 | #334 | — | 2 | 1 | no | — |
| 69 | #334 | — | 2 | 1 | no | — |
| 70 | Q8⋊C8 | — | 4 | 1 | no | — |
| 71 | D4⋊C8 | — | 4 | 1 | no | — |
| 72 | C2×#41 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 73 | #131 | — | 2 | 1 | no | — |
| 74 | #131 | — | 2 | 1 | no | — |
| 75 | #131 | — | 2 | 2 | no | — |
| 76 | #131 | — | 4 | 1 | no | — |
| 77 | #131 | C2×C8 | 1 | 2 | yes | — |
| 78 | #131 | M16 | 1 | 2 | yes | — |
| 79 | C2×M16 | — | 2 | 1 | no | — |
| 80 | C2×M16 | — | 2 | 1 | no | — |
| 81 | C2×M16 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C2×M16 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C22×C8 | — | 2 | 1 | no | — |
| 84 | C22×C8 | — | 2 | 1 | no | — |
| 85 | C22×C8 | — | 4 | 2 | no | — |
| 86 | C22×C8 | — | 4 | 1 | no | — |
| 87 | C22×C8 | — | 4 | 1 | no | — |
| 88 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C22×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 90 | C4×Q8 | D8 | 1 | 1 | yes | characteristic |
| 91 | C4×Q8 | Q16 | 1 | 1 | yes | characteristic |
| 92 | C4×D4 | — | 2 | 1 | no | — |
| 93 | C4×D4 | — | 2 | 1 | no | — |
| 94 | C4×D4 | D8 | 1 | 1 | yes | characteristic |
| 95 | C4×D4 | Q16 | 1 | 1 | yes | characteristic |
| 96 | #117 | — | 2 | 1 | no | — |
| 97 | #117 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C2×C42 | — | 2 | 1 | no | — |
| 99 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | M32 | — | 2 | 1 | no | — |
| 101 | M32 | — | 2 | 1 | no | — |
| 102 | M32 | — | 2 | 1 | no | — |
| 103 | M32 | C2×C8 | 1 | 2 | yes | complemented |
| 104 | C2×C16 | — | 2 | 1 | no | — |
| 105 | C2×C16 | — | 2 | 1 | no | — |
| 106 | C2×C16 | — | 2 | 1 | no | — |
| 107 | C2×C16 | — | 2 | 1 | no | — |
| 108 | C2×C16 | — | 2 | 1 | no | — |
| 109 | C2×C16 | — | 2 | 1 | no | — |
| 110 | C2×C16 | — | 2 | 1 | no | — |
| 111 | C2×C16 | C2×C8 | 1 | 2 | yes | complemented |
| 112 | C4⋊C8 | — | 2 | 1 | no | — |
| 113 | C4⋊C8 | — | 2 | 2 | no | — |
| 114 | C4⋊C8 | — | 2 | 1 | no | — |
| 115 | C4⋊C8 | — | 4 | 2 | no | — |
| 116 | C4⋊C8 | — | 4 | 1 | no | — |
| 117 | C4⋊C8 | — | 4 | 1 | no | — |
| 118 | C4⋊C8 | — | 4 | 1 | no | — |
| 119 | C4⋊C8 | — | 4 | 1 | no | — |
| 120 | C22⋊C8 | — | 4 | 2 | no | — |
| 121 | C22⋊C8 | — | 4 | 1 | no | — |
| 122 | C22⋊C8 | — | 4 | 1 | no | — |
| 123 | #97 | — | 2 | 1 | no | — |
| 124 | #97 | — | 2 | 1 | no | — |
| 125 | #97 | — | 2 | 1 | no | — |
| 126 | C4×C8 | — | 2 | 1 | no | — |
| 127 | C4×C8 | — | 2 | 1 | no | — |
| 128 | C4×C8 | — | 2 | 1 | no | — |
| 129 | C4×C8 | — | 4 | 2 | no | — |
| 130 | C4×C8 | — | 4 | 1 | no | — |
| 131 | C4×C8 | — | 4 | 1 | no | — |
| 132 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 133 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 134 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 135 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 136 | #41 | — | 2 | 1 | no | — |
| 137 | #41 | — | 2 | 1 | no | — |
| 138 | #41 | C4⋊C8 | 1 | 2 | yes | — |
| 139 | #41 | #108 | 1 | 2 | yes | — |
| 140 | C2×Q8 | #107 | 1 | 1 | yes | characteristic |
| 141 | C2×D4 | — | 2 | 1 | no | — |
| 142 | C2×D4 | #107 | 1 | 1 | yes | characteristic |
| 143 | C22×C4 | — | 2 | 1 | no | — |
| 144 | C22×C4 | — | 2 | 1 | no | — |
| 145 | C22×C4 | — | 2 | 1 | no | — |
| 146 | C22×C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 147 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 148 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 149 | M16 | — | 2 | 1 | no | — |
| 150 | M16 | — | 2 | 2 | no | — |
| 151 | M16 | — | 2 | 2 | no | — |
| 152 | M16 | — | 2 | 1 | no | — |
| 153 | M16 | — | 2 | 1 | no | — |
| 154 | M16 | — | 4 | 2 | no | — |
| 155 | M16 | C4⋊C8 | 1 | 2 | yes | — |
| 156 | C2×C8 | — | 2 | 1 | no | — |
| 157 | C2×C8 | — | 2 | 1 | no | — |
| 158 | C2×C8 | — | 2 | 1 | no | — |
| 159 | C2×C8 | — | 2 | 2 | no | — |
| 160 | C2×C8 | — | 2 | 2 | no | — |
| 161 | C2×C8 | — | 2 | 1 | no | — |
| 162 | C2×C8 | — | 4 | 2 | no | — |
| 163 | C2×C8 | — | 4 | 1 | no | — |
| 164 | C2×C8 | — | 4 | 2 | no | — |
| 165 | C2×C8 | — | 4 | 1 | no | — |
| 166 | C2×C8 | — | 8 | 1 | no | — |
| 167 | C2×C8 | — | 8 | 1 | no | — |
| 168 | C2×C8 | — | 16 | 1 | no | — |
| 169 | C2×C8 | — | 16 | 2 | no | — |
| 170 | C2×C8 | — | 16 | 1 | no | — |
| 171 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 175 | C2×C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 176 | C2×C8 | C2×M16 | 1 | 1 | yes | characteristic |
| 177 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 178 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 179 | C4⋊C4 | — | 2 | 1 | no | — |
| 180 | C4⋊C4 | — | 2 | 1 | no | — |
| 181 | C4⋊C4 | #107 | 1 | 1 | yes | characteristic |
| 182 | C4⋊C4 | #107 | 1 | 1 | yes | characteristic |
| 183 | C22⋊C4 | — | 2 | 1 | no | — |
| 184 | C22⋊C4 | — | 2 | 1 | no | — |
| 185 | C22⋊C4 | — | 2 | 1 | no | — |
| 186 | C42 | — | 2 | 1 | no | — |
| 187 | C42 | — | 2 | 1 | no | — |
| 188 | C42 | — | 2 | 1 | no | — |
| 189 | C42 | #120 | 1 | 1 | yes | characteristic |
| 190 | C42 | #123 | 1 | 1 | yes | characteristic |
| 191 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 192 | C42 | C2×Q16 | 1 | 1 | yes | characteristic |
| 193 | C16 | — | 2 | 2 | no | — |
| 194 | C16 | — | 2 | 2 | no | — |
| 195 | C16 | — | 4 | 1 | no | — |
| 196 | C16 | — | 4 | 1 | no | — |
| 197 | C23 | — | 2 | 1 | no | — |
| 198 | C23 | #428 | 1 | 1 | yes | characteristic |
| 199 | Q8 | — | 2 | 1 | no | — |
| 200 | Q8 | #335 | 1 | 2 | yes | — |
| 201 | D4 | — | 2 | 2 | no | — |
| 202 | D4 | — | 2 | 2 | no | — |
| 203 | D4 | — | 2 | 1 | no | — |
| 204 | D4 | #335 | 1 | 2 | yes | — |
| 205 | C2×C4 | — | 2 | 1 | no | — |
| 206 | C2×C4 | — | 2 | 2 | no | — |
| 207 | C2×C4 | — | 2 | 2 | no | — |
| 208 | C2×C4 | — | 2 | 1 | no | — |
| 209 | C2×C4 | — | 2 | 1 | no | — |
| 210 | C2×C4 | — | 2 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 4 | 1 | no | — |
| 214 | C2×C4 | — | 4 | 1 | no | — |
| 215 | C2×C4 | — | 8 | 1 | no | — |
| 216 | C2×C4 | #385 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | C2×C4⋊C8 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | #424 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | C2×#107 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | C2×#108 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #430 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #451 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #480 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | #484 | 1 | 1 | yes | characteristic |
| 226 | C8 | — | 2 | 2 | no | — |
| 227 | C8 | — | 2 | 1 | no | — |
| 228 | C8 | — | 2 | 1 | no | — |
| 229 | C8 | — | 4 | 1 | no | — |
| 230 | C8 | — | 8 | 1 | no | — |
| 231 | C8 | — | 8 | 2 | no | — |
| 232 | C8 | — | 8 | 1 | no | — |
| 233 | C8 | — | 16 | 1 | no | — |
| 234 | C8 | — | 16 | 1 | no | — |
| 235 | C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 236 | C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 237 | C8 | #435 | 1 | 1 | yes | characteristic |
| 238 | C8 | #435 | 1 | 1 | yes | characteristic |
| 239 | C22 | — | 2 | 1 | no | — |
| 240 | C22 | — | 2 | 1 | no | — |
| 241 | C22 | — | 2 | 2 | no | — |
| 242 | C22 | — | 4 | 1 | no | — |
| 243 | C22 | — | 4 | 1 | no | — |
| 244 | C22 | #1565 | 1 | 1 | yes | characteristic |
| 245 | C22 | #1573 | 1 | 1 | yes | characteristic |
| 246 | C22 | #1852 | 1 | 1 | yes | characteristic |
| 247 | C4 | — | 2 | 2 | no | — |
| 248 | C4 | — | 2 | 1 | no | — |
| 249 | C4 | — | 2 | 1 | no | — |
| 250 | C4 | — | 2 | 1 | no | — |
| 251 | C4 | — | 4 | 1 | no | — |
| 252 | C4 | — | 4 | 1 | no | — |
| 253 | C4 | C2×#335 | 1 | 1 | yes | characteristic |
| 254 | C4 | #1571 | 1 | 1 | yes | characteristic |
| 255 | C4 | #1850 | 1 | 1 | yes | characteristic |
| 256 | C4 | #1855 | 1 | 1 | yes | characteristic |
| 257 | C4 | #2193 | 1 | 1 | yes | characteristic |
| 258 | C4 | #2194 | 1 | 1 | yes | characteristic |
| 259 | C2 | — | 2 | 1 | no | — |
| 260 | C2 | — | 2 | 1 | no | — |
| 261 | C2 | — | 4 | 2 | no | — |
| 262 | C2 | #8829 | 1 | 1 | yes | characteristic |
| 263 | C2 | #12926 | 1 | 1 | yes | characteristic |
| 264 | C2 | #12927 | 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