← #30652354
Subgroups — #30652354
Order 16256.
1202 conjugacy classes of subgroups.
The subgroup lattice has 286 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22550606 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22550592 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22550590 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #22550588 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #22550546 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22550544 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #22550423 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #22550422 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | C4×#14523248 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C4×#14523247 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #532×C127 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#14523248 | C22 | 1 | 2 | yes | — |
| 13 | C2×#14523248 | C22 | 1 | 4 | yes | — |
| 14 | C2×#14523247 | C22 | 1 | 4 | yes | — |
| 15 | #16861208 | C22 | 1 | 4 | yes | — |
| 16 | #16861207 | C22 | 1 | 4 | yes | — |
| 17 | #16861205 | C22 | 1 | 4 | yes | — |
| 18 | #16861205 | C22 | 1 | 4 | yes | — |
| 19 | #16861198 | C22 | 1 | 4 | yes | — |
| 20 | #16861196 | C22 | 1 | 4 | yes | — |
| 21 | #16861195 | C22 | 1 | 8 | yes | — |
| 22 | #16861184 | C22 | 1 | 4 | yes | — |
| 23 | #16861182 | C22 | 1 | 8 | yes | — |
| 24 | #16861181 | C22 | 1 | 8 | yes | — |
| 25 | #16861181 | C22 | 1 | 8 | yes | — |
| 26 | #16861180 | C22 | 1 | 4 | yes | — |
| 27 | C4×Dic254 | C22 | 1 | 2 | yes | — |
| 28 | C4×Dic254 | C22 | 1 | 2 | yes | — |
| 29 | #16861174 | C22 | 1 | 4 | yes | — |
| 30 | C4×D508 | C22 | 1 | 2 | yes | — |
| 31 | C4×D508 | C22 | 1 | 2 | yes | — |
| 32 | D4×Dic127 | C22 | 1 | 4 | yes | — |
| 33 | C4×C22⋊D127 | C22 | 1 | 4 | yes | — |
| 34 | C4×C22⋊D127 | C22 | 1 | 8 | yes | — |
| 35 | C4×C22⋊D127 | C22 | 1 | 4 | yes | — |
| 36 | (C22⋊D127)⋊C4 | C22 | 1 | 8 | yes | — |
| 37 | #16861168 | C22 | 1 | 2 | yes | — |
| 38 | #16861167 | C22 | 1 | 4 | yes | — |
| 39 | #16861166 | C22 | 1 | 4 | yes | — |
| 40 | #16861165 | C22 | 1 | 8 | yes | — |
| 41 | C2×C4×Dic127 | C22 | 1 | 4 | yes | — |
| 42 | C42×D127 | C22 | 1 | 4 | yes | — |
| 43 | C2×#41×C127 | C22 | 1 | 2 | yes | — |
| 44 | C4⋊Q8×C127 | C22 | 1 | 1 | yes | characteristic |
| 45 | #127×C127 | C22 | 1 | 1 | yes | characteristic |
| 46 | #124×C127 | C22 | 1 | 2 | yes | — |
| 47 | #121×C127 | C22 | 1 | 4 | yes | — |
| 48 | C4×D4×C127 | C22 | 1 | 4 | yes | — |
| 49 | C2×C42×C127 | C22 | 1 | 1 | yes | characteristic |
| 50 | #14523248 | — | 2 | 8 | no | — |
| 51 | #14523248 | — | 4 | 8 | no | — |
| 52 | #14523247 | — | 2 | 16 | no | — |
| 53 | C2×Dic254 | C23 | 1 | 4 | yes | — |
| 54 | C2×Dic254 | C23 | 1 | 2 | yes | — |
| 55 | C2×D508 | C23 | 1 | 2 | yes | — |
| 56 | C2×D508 | C23 | 1 | 4 | yes | — |
| 57 | C2×C22⋊D127 | C23 | 1 | 8 | yes | — |
| 58 | C2×C22⋊D127 | C23 | 1 | 4 | yes | — |
| 59 | C2×C22⋊D127 | C23 | 1 | 8 | yes | — |
| 60 | C22×Dic127 | C23 | 1 | 8 | yes | — |
| 61 | C2×C4×D127 | C23 | 1 | 4 | yes | — |
| 62 | C2×C4×D127 | C23 | 1 | 4 | yes | — |
| 63 | C2×C4×D127 | C23 | 1 | 4 | yes | — |
| 64 | C4⋊Dic127 | C23 | 1 | 2 | yes | — |
| 65 | C4⋊Dic127 | C23 | 1 | 2 | yes | — |
| 66 | Dic127⋊C4 | C23 | 1 | 4 | yes | — |
| 67 | Dic127⋊C4 | C23 | 1 | 8 | yes | — |
| 68 | Dic127⋊C4 | C23 | 1 | 4 | yes | — |
| 69 | C22⋊Dic127 | C23 | 1 | 8 | yes | — |
| 70 | C22⋊Dic127 | C23 | 1 | 4 | yes | — |
| 71 | (C2×D127)⋊C4 | C23 | 1 | 8 | yes | — |
| 72 | (C2×D127)⋊C4 | C23 | 1 | 4 | yes | — |
| 73 | (C2×D127)⋊C4 | C23 | 1 | 4 | yes | — |
| 74 | C4×Dic127 | C23 | 1 | 4 | yes | — |
| 75 | C4×Dic127 | C23 | 1 | 4 | yes | — |
| 76 | C4×Dic127 | C23 | 1 | 4 | yes | — |
| 77 | C4×Dic127 | C23 | 1 | 4 | yes | — |
| 78 | C4×Dic127 | C23 | 1 | 4 | yes | — |
| 79 | #41×C127 | — | 4 | 4 | no | — |
| 80 | C2×Q8×C127 | C23 | 1 | 2 | yes | — |
| 81 | C2×D4×C127 | C23 | 1 | 4 | yes | — |
| 82 | C2×D4×C127 | C23 | 1 | 4 | yes | — |
| 83 | C2×D4×C127 | C23 | 1 | 2 | yes | — |
| 84 | C22×C4×C127 | C23 | 1 | 2 | yes | — |
| 85 | C22×C4×C127 | C23 | 1 | 4 | yes | — |
| 86 | C22×C4×C127 | C23 | 1 | 1 | yes | characteristic |
| 87 | C4⋊C4×C127 | C23 | 1 | 4 | yes | — |
| 88 | C22⋊C4×C127 | C23 | 1 | 8 | yes | — |
| 89 | C42×C127 | C23 | 1 | 2 | yes | — |
| 90 | C42×C127 | C23 | 1 | 1 | yes | characteristic |
| 91 | C42×C127 | C23 | 1 | 1 | yes | characteristic |
| 92 | C22×D127 | C24 | 1 | 4 | yes | — |
| 93 | Dic254 | — | 4 | 4 | no | — |
| 94 | Dic254 | #41 | 1 | 8 | yes | complemented |
| 95 | D508 | — | 2 | 4 | no | — |
| 96 | D508 | — | 4 | 4 | no | — |
| 97 | C22⋊D127 | — | 2 | 16 | no | — |
| 98 | C22⋊D127 | — | 2 | 8 | no | — |
| 99 | C22⋊D127 | — | 4 | 8 | no | — |
| 100 | C2×Dic127 | — | 2 | 16 | no | — |
| 101 | C2×Dic127 | C24 | 1 | 4 | yes | — |
| 102 | C2×Dic127 | C24 | 1 | 4 | yes | — |
| 103 | C2×Dic127 | C24 | 1 | 4 | yes | — |
| 104 | C4×D127 | — | 2 | 8 | no | — |
| 105 | C4×D127 | — | 2 | 8 | no | — |
| 106 | C4×D127 | — | 2 | 8 | no | — |
| 107 | C23×C127 | C24 | 1 | 4 | yes | — |
| 108 | C23×C127 | C24 | 1 | 1 | yes | characteristic |
| 109 | Q8×C127 | — | 4 | 2 | no | — |
| 110 | D4×C127 | — | 2 | 8 | no | — |
| 111 | D4×C127 | — | 4 | 2 | no | — |
| 112 | D4×C127 | — | 4 | 4 | no | — |
| 113 | C2×C4×C127 | — | 2 | 4 | no | — |
| 114 | C2×C4×C127 | — | 2 | 2 | no | — |
| 115 | C2×C4×C127 | — | 2 | 8 | no | — |
| 116 | C2×C4×C127 | C24 | 1 | 2 | yes | — |
| 117 | C2×C4×C127 | C24 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C127 | C24 | 1 | 4 | yes | — |
| 119 | C2×C4×C127 | C24 | 1 | 2 | yes | — |
| 120 | C2×C4×C127 | C24 | 1 | 1 | yes | characteristic |
| 121 | C22×C127 | — | 2 | 2 | no | — |
| 122 | C22×C127 | — | 2 | 8 | no | — |
| 123 | C22×C127 | — | 2 | 1 | no | — |
| 124 | C22×C127 | — | 2 | 4 | no | — |
| 125 | C22×C127 | C25 | 1 | 1 | yes | characteristic |
| 126 | C2×D127 | — | 2 | 4 | no | — |
| 127 | C2×D127 | — | 2 | 4 | no | — |
| 128 | C2×D127 | — | 2 | 4 | no | — |
| 129 | C4×C127 | — | 2 | 2 | no | — |
| 130 | C4×C127 | — | 2 | 1 | no | — |
| 131 | C4×C127 | — | 2 | 4 | no | — |
| 132 | C4×C127 | C2×#41 | 1 | 4 | yes | — |
| 133 | C4×C127 | H(5,𝔽2) | 1 | 2 | yes | — |
| 134 | Dic127 | — | 2 | 4 | no | — |
| 135 | Dic127 | — | 2 | 4 | no | — |
| 136 | Dic127 | C2×#41 | 1 | 8 | yes | — |
| 137 | C2×C127 | — | 4 | 1 | no | — |
| 138 | C2×C127 | — | 4 | 4 | no | — |
| 139 | C2×C127 | C22×#41 | 1 | 2 | yes | — |
| 140 | C2×C127 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 141 | D127 | — | 4 | 4 | no | — |
| 142 | #3483 | — | 127 | 1 | no | maximal, Sylow |
| 143 | C127 | #3483 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 144 | Q82 | — | 127 | 1 | no | — |
| 145 | #555 | — | 127 | 1 | no | — |
| 146 | #555 | — | 127 | 4 | no | — |
| 147 | #555 | — | 127 | 4 | no | — |
| 148 | #550 | — | 127 | 2 | no | — |
| 149 | #550 | — | 127 | 4 | no | — |
| 150 | #532 | — | 127 | 4 | no | — |
| 151 | #532 | — | 127 | 4 | no | — |
| 152 | #532 | D127 | 1 | 1 | yes | characteristic, complemented |
| 153 | C4×#41 | — | 127 | 4 | no | — |
| 154 | C4×#41 | — | 127 | 2 | no | — |
| 155 | C2×#41 | — | 127 | 4 | no | — |
| 156 | C2×#41 | — | 127 | 4 | no | — |
| 157 | C2×#41 | — | 127 | 2 | no | — |
| 158 | C2×#41 | C2×D127 | 1 | 2 | yes | — |
| 159 | C4⋊Q8 | — | 127 | 4 | no | — |
| 160 | C4⋊Q8 | — | 127 | 4 | no | — |
| 161 | C4⋊Q8 | C2×D127 | 1 | 1 | yes | characteristic |
| 162 | #127 | — | 127 | 4 | no | — |
| 163 | #127 | — | 127 | 4 | no | — |
| 164 | #127 | C2×D127 | 1 | 1 | yes | characteristic |
| 165 | #124 | — | 127 | 4 | no | — |
| 166 | #124 | — | 127 | 4 | no | — |
| 167 | #124 | — | 127 | 8 | no | — |
| 168 | #124 | C2×D127 | 1 | 2 | yes | — |
| 169 | #121 | — | 127 | 4 | no | — |
| 170 | #121 | — | 127 | 8 | no | — |
| 171 | #121 | — | 127 | 8 | no | — |
| 172 | #121 | — | 127 | 4 | no | — |
| 173 | #121 | — | 127 | 8 | no | — |
| 174 | #121 | C2×D127 | 1 | 4 | yes | — |
| 175 | C4×Q8 | — | 127 | 4 | no | — |
| 176 | C4×Q8 | — | 127 | 2 | no | — |
| 177 | C4×Q8 | — | 127 | 2 | no | — |
| 178 | C4×D4 | — | 127 | 4 | no | — |
| 179 | C4×D4 | — | 127 | 8 | no | — |
| 180 | C4×D4 | — | 127 | 4 | no | — |
| 181 | C4×D4 | — | 127 | 4 | no | — |
| 182 | C4×D4 | — | 127 | 8 | no | — |
| 183 | C4×D4 | — | 127 | 2 | no | — |
| 184 | C4×D4 | — | 127 | 2 | no | — |
| 185 | C4×D4 | C2×D127 | 1 | 4 | yes | — |
| 186 | #117 | — | 127 | 2 | no | — |
| 187 | #117 | — | 127 | 4 | no | — |
| 188 | #117 | — | 127 | 8 | no | — |
| 189 | #117 | — | 127 | 4 | no | — |
| 190 | C2×C42 | — | 127 | 4 | no | — |
| 191 | C2×C42 | — | 127 | 4 | no | — |
| 192 | C2×C42 | C2×D127 | 1 | 1 | yes | characteristic |
| 193 | #41 | — | 4 | 4 | no | — |
| 194 | #41 | — | 508 | 8 | no | — |
| 195 | #41 | — | 254 | 8 | no | — |
| 196 | #41 | — | 254 | 16 | no | — |
| 197 | C2×Q8 | — | 127 | 2 | no | — |
| 198 | C2×Q8 | — | 127 | 4 | no | — |
| 199 | C2×Q8 | C22×D127 | 1 | 2 | yes | — |
| 200 | C2×D4 | — | 127 | 8 | no | — |
| 201 | C2×D4 | — | 127 | 4 | no | — |
| 202 | C2×D4 | — | 127 | 8 | no | — |
| 203 | C2×D4 | — | 127 | 4 | no | — |
| 204 | C2×D4 | — | 127 | 2 | no | — |
| 205 | C2×D4 | C22×D127 | 1 | 4 | yes | — |
| 206 | C2×D4 | C22×D127 | 1 | 2 | yes | — |
| 207 | C2×D4 | C22×D127 | 1 | 4 | yes | — |
| 208 | C22×C4 | — | 127 | 8 | no | — |
| 209 | C22×C4 | — | 127 | 4 | no | — |
| 210 | C22×C4 | — | 127 | 4 | no | — |
| 211 | C22×C4 | — | 127 | 4 | no | — |
| 212 | C22×C4 | C22×D127 | 1 | 2 | yes | — |
| 213 | C22×C4 | C22×D127 | 1 | 4 | yes | — |
| 214 | C22×C4 | C22×D127 | 1 | 1 | yes | characteristic |
| 215 | C4⋊C4 | — | 127 | 4 | no | — |
| 216 | C4⋊C4 | — | 127 | 8 | no | — |
| 217 | C4⋊C4 | — | 127 | 4 | no | — |
| 218 | C4⋊C4 | — | 127 | 2 | no | — |
| 219 | C4⋊C4 | — | 127 | 2 | no | — |
| 220 | C4⋊C4 | C22×D127 | 1 | 4 | yes | — |
| 221 | C22⋊C4 | — | 127 | 4 | no | — |
| 222 | C22⋊C4 | — | 127 | 8 | no | — |
| 223 | C22⋊C4 | — | 127 | 8 | no | — |
| 224 | C22⋊C4 | — | 127 | 4 | no | — |
| 225 | C22⋊C4 | — | 127 | 4 | no | — |
| 226 | C22⋊C4 | C22×D127 | 1 | 8 | yes | — |
| 227 | C42 | — | 127 | 4 | no | — |
| 228 | C42 | — | 127 | 4 | no | — |
| 229 | C42 | — | 127 | 4 | no | — |
| 230 | C42 | — | 127 | 4 | no | — |
| 231 | C42 | — | 127 | 4 | no | — |
| 232 | C42 | C22×D127 | 1 | 1 | yes | characteristic |
| 233 | C42 | C22×D127 | 1 | 1 | yes | characteristic |
| 234 | C42 | C22×D127 | 1 | 2 | yes | — |
| 235 | C23 | — | 127 | 4 | no | — |
| 236 | C23 | C23×D127 | 1 | 1 | yes | characteristic |
| 237 | C23 | C23×D127 | 1 | 4 | yes | — |
| 238 | Q8 | — | 4 | 2 | no | — |
| 239 | Q8 | — | 127 | 8 | no | — |
| 240 | Q8 | — | 508 | 4 | no | — |
| 241 | D4 | — | 2 | 8 | no | — |
| 242 | D4 | — | 4 | 2 | no | — |
| 243 | D4 | — | 4 | 4 | no | — |
| 244 | D4 | — | 508 | 4 | no | — |
| 245 | D4 | — | 508 | 8 | no | — |
| 246 | D4 | — | 254 | 8 | no | — |
| 247 | D4 | — | 254 | 16 | no | — |
| 248 | D4 | — | 254 | 4 | no | — |
| 249 | C2×C4 | — | 2 | 4 | no | — |
| 250 | C2×C4 | — | 2 | 2 | no | — |
| 251 | C2×C4 | — | 2 | 8 | no | — |
| 252 | C2×C4 | — | 127 | 4 | no | — |
| 253 | C2×C4 | — | 127 | 4 | no | — |
| 254 | C2×C4 | — | 127 | 4 | no | — |
| 255 | C2×C4 | — | 254 | 8 | no | — |
| 256 | C2×C4 | — | 254 | 16 | no | — |
| 257 | C2×C4 | — | 254 | 8 | no | — |
| 258 | C2×C4 | — | 254 | 8 | no | — |
| 259 | C2×C4 | C23×D127 | 1 | 2 | yes | — |
| 260 | C2×C4 | C23×D127 | 1 | 4 | yes | — |
| 261 | C2×C4 | C23×D127 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | C23×D127 | 1 | 2 | yes | — |
| 263 | C2×C4 | C23×D127 | 1 | 1 | yes | characteristic |
| 264 | C22 | — | 2 | 2 | no | — |
| 265 | C22 | — | 2 | 8 | no | — |
| 266 | C22 | — | 2 | 1 | no | — |
| 267 | C22 | — | 2 | 4 | no | — |
| 268 | C22 | — | 254 | 4 | no | — |
| 269 | C22 | — | 254 | 4 | no | — |
| 270 | C22 | — | 254 | 4 | no | — |
| 271 | C22 | C24×D127 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 2 | no | — |
| 273 | C4 | — | 2 | 1 | no | — |
| 274 | C4 | — | 2 | 4 | no | — |
| 275 | C4 | — | 127 | 8 | no | — |
| 276 | C4 | — | 254 | 4 | no | — |
| 277 | C4 | — | 254 | 4 | no | — |
| 278 | C4 | #41×D127 | 1 | 4 | yes | — |
| 279 | C4 | #16861254 | 1 | 2 | yes | — |
| 280 | C2 | — | 4 | 1 | no | — |
| 281 | C2 | — | 4 | 4 | no | — |
| 282 | C2 | — | 508 | 4 | no | — |
| 283 | C2 | C2×#41×D127 | 1 | 2 | yes | — |
| 284 | C2 | C2×#16861254 | 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