← #30649274
Subgroups — #30649274
Order 16256.
380 conjugacy classes of subgroups.
The subgroup lattice has 256 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22550429 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16861208 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22550284 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22550283 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22550280 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22550273 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22549919 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #22549918 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #22549916 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | M16×Dic127 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #428×C127 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C22×Dic254 | — | 2 | 1 | no | — |
| 13 | C2×#14523232 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16861208 | — | 2 | 1 | no | — |
| 15 | #16861208 | — | 2 | 1 | no | — |
| 16 | #16861208 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16861208 | C22 | 1 | 2 | yes | — |
| 18 | #16861208 | C22 | 1 | 1 | yes | characteristic |
| 19 | #16861208 | C22 | 1 | 2 | yes | — |
| 20 | #16861200 | C22 | 1 | 1 | yes | characteristic |
| 21 | #16861200 | C22 | 1 | 1 | yes | characteristic |
| 22 | #16861186 | — | 2 | 1 | no | — |
| 23 | #16861186 | — | 2 | 1 | no | — |
| 24 | C4×Dic254 | — | 2 | 1 | no | — |
| 25 | C4×Dic254 | — | 2 | 1 | no | — |
| 26 | #16861168 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×C4⋊Dic127 | C22 | 1 | 1 | yes | characteristic |
| 28 | C2×Dic127⋊C4 | — | 2 | 1 | no | — |
| 29 | C2×C4×Dic127 | C22 | 1 | 1 | yes | characteristic |
| 30 | #16861143 | C22 | 1 | 2 | yes | — |
| 31 | #16861142 | C22 | 1 | 2 | yes | — |
| 32 | #16861142 | C22 | 1 | 2 | yes | — |
| 33 | #16861141 | C22 | 1 | 2 | yes | — |
| 34 | #16861140 | C22 | 1 | 2 | yes | — |
| 35 | #16861140 | C22 | 1 | 2 | yes | — |
| 36 | #16861119 | C22 | 1 | 2 | yes | — |
| 37 | C8×Dic127 | C22 | 1 | 2 | yes | — |
| 38 | C2×M16×C127 | C22 | 1 | 1 | yes | characteristic |
| 39 | #117×C127 | C22 | 1 | 1 | yes | characteristic |
| 40 | C2×C4⋊C4×C127 | C22 | 1 | 1 | yes | characteristic |
| 41 | #107×C127 | C22 | 1 | 2 | yes | — |
| 42 | #106×C127 | C22 | 1 | 2 | yes | — |
| 43 | C2×Dic254 | — | 2 | 1 | no | — |
| 44 | C2×Dic254 | — | 2 | 1 | no | — |
| 45 | C2×Dic254 | — | 2 | 2 | no | — |
| 46 | C2×Dic254 | — | 2 | 2 | no | — |
| 47 | C2×Dic254 | — | 4 | 4 | no | — |
| 48 | C22×Dic127 | — | 2 | 1 | no | — |
| 49 | C22×Dic127 | D4 | 1 | 1 | yes | characteristic |
| 50 | C22×Dic127 | D4 | 1 | 1 | yes | characteristic |
| 51 | #14523232 | Q8 | 1 | 4 | yes | complemented |
| 52 | C2×C127⋊C8 | C23 | 1 | 2 | yes | — |
| 53 | C4⋊Dic127 | — | 2 | 1 | no | — |
| 54 | C4⋊Dic127 | — | 2 | 1 | no | — |
| 55 | C4⋊Dic127 | — | 2 | 1 | no | — |
| 56 | C4⋊Dic127 | C23 | 1 | 1 | yes | characteristic |
| 57 | C4⋊Dic127 | C23 | 1 | 1 | yes | characteristic |
| 58 | C4⋊Dic127 | C23 | 1 | 2 | yes | — |
| 59 | Dic127⋊C4 | — | 2 | 1 | no | — |
| 60 | Dic127⋊C4 | — | 2 | 1 | no | — |
| 61 | Dic127⋊C4 | — | 2 | 1 | no | — |
| 62 | Dic127⋊C4 | — | 2 | 1 | no | — |
| 63 | Dic127⋊C4 | — | 2 | 2 | no | — |
| 64 | Dic127⋊C4 | — | 2 | 2 | no | — |
| 65 | C22⋊Dic127 | — | 2 | 1 | no | — |
| 66 | C4×Dic127 | — | 2 | 1 | no | — |
| 67 | C4×Dic127 | D4 | 1 | 1 | yes | characteristic |
| 68 | C4×Dic127 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×Dic127 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4×Dic127 | C23 | 1 | 1 | yes | characteristic |
| 71 | C22×C4×C127 | — | 2 | 1 | no | — |
| 72 | C22×C4×C127 | C23 | 1 | 1 | yes | characteristic |
| 73 | M16×C127 | Q8 | 1 | 4 | yes | complemented |
| 74 | C2×C8×C127 | C23 | 1 | 2 | yes | — |
| 75 | C4⋊C4×C127 | — | 2 | 1 | no | — |
| 76 | C4⋊C4×C127 | C23 | 1 | 1 | yes | characteristic |
| 77 | C4⋊C4×C127 | C23 | 1 | 1 | yes | characteristic |
| 78 | C4⋊C4×C127 | C23 | 1 | 2 | yes | — |
| 79 | C22⋊C4×C127 | — | 2 | 1 | no | — |
| 80 | C42×C127 | — | 2 | 1 | no | — |
| 81 | Dic254 | — | 4 | 4 | no | — |
| 82 | Dic254 | — | 4 | 2 | no | — |
| 83 | Dic254 | — | 4 | 4 | no | — |
| 84 | Dic254 | — | 4 | 2 | no | — |
| 85 | C2×Dic127 | — | 2 | 2 | no | — |
| 86 | C2×Dic127 | — | 2 | 2 | no | — |
| 87 | C2×Dic127 | — | 2 | 1 | no | — |
| 88 | C2×Dic127 | — | 2 | 1 | no | — |
| 89 | C2×Dic127 | — | 2 | 2 | no | — |
| 90 | C2×Dic127 | — | 4 | 2 | no | — |
| 91 | C2×Dic127 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×Dic127 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×Dic127 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×Dic127 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C127⋊C8 | C2×Q8 | 1 | 4 | yes | complemented |
| 96 | C23×C127 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C127 | — | 2 | 1 | no | — |
| 98 | C2×C4×C127 | — | 2 | 1 | no | — |
| 99 | C2×C4×C127 | — | 2 | 2 | no | — |
| 100 | C2×C4×C127 | — | 4 | 2 | no | — |
| 101 | C2×C4×C127 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C127 | C2×Q8 | 1 | 2 | yes | — |
| 103 | C2×C4×C127 | C2×Q8 | 1 | 2 | yes | — |
| 104 | C2×C4×C127 | C24 | 1 | 1 | yes | characteristic |
| 105 | C8×C127 | C2×Q8 | 1 | 4 | yes | complemented |
| 106 | C22×C127 | — | 2 | 1 | no | — |
| 107 | C22×C127 | — | 2 | 1 | no | — |
| 108 | C22×C127 | C4⋊Q8 | 1 | 2 | yes | — |
| 109 | C22×C127 | C22×D4 | 1 | 1 | yes | characteristic |
| 110 | C4×C127 | — | 4 | 2 | no | — |
| 111 | C4×C127 | — | 4 | 1 | no | — |
| 112 | C4×C127 | — | 4 | 1 | no | — |
| 113 | C4×C127 | C4⋊Q8 | 1 | 2 | yes | — |
| 114 | C4×C127 | C22×Q8 | 1 | 1 | yes | characteristic |
| 115 | C4×C127 | C22×Q8 | 1 | 1 | yes | characteristic |
| 116 | Dic127 | — | 2 | 1 | no | — |
| 117 | Dic127 | — | 2 | 1 | no | — |
| 118 | Dic127 | — | 4 | 2 | no | — |
| 119 | Dic127 | — | 4 | 1 | no | — |
| 120 | Dic127 | — | 4 | 1 | no | — |
| 121 | Dic127 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 122 | Dic127 | #137 | 1 | 2 | yes | — |
| 123 | C2×C127 | — | 2 | 2 | no | — |
| 124 | C2×C127 | C2×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 125 | C2×C127 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 126 | C2×C127 | C2×#137 | 1 | 1 | yes | characteristic |
| 127 | #3165 | — | 127 | 1 | no | maximal, Sylow |
| 128 | C127 | #3165 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 129 | #533 | — | 127 | 1 | no | — |
| 130 | C2×C4⋊Q8 | — | 127 | 1 | no | — |
| 131 | #501 | — | 127 | 2 | no | — |
| 132 | #501 | — | 127 | 2 | no | — |
| 133 | #500 | — | 127 | 2 | no | — |
| 134 | #498 | — | 127 | 2 | no | — |
| 135 | #428 | — | 127 | 1 | no | — |
| 136 | #428 | — | 127 | 1 | no | — |
| 137 | #428 | — | 127 | 1 | no | — |
| 138 | #428 | D127 | 1 | 1 | yes | characteristic |
| 139 | C4×M16 | — | 127 | 1 | no | — |
| 140 | C22×Q8 | — | 254 | 1 | no | — |
| 141 | C2×M16 | — | 127 | 1 | no | — |
| 142 | C2×M16 | C2×D127 | 1 | 1 | yes | characteristic |
| 143 | C4⋊Q8 | — | 127 | 2 | no | — |
| 144 | C4⋊Q8 | — | 127 | 2 | no | — |
| 145 | C4⋊Q8 | — | 127 | 1 | no | — |
| 146 | C4⋊Q8 | — | 127 | 1 | no | — |
| 147 | C4⋊Q8 | — | 254 | 1 | no | — |
| 148 | C4⋊Q8 | — | 254 | 1 | no | — |
| 149 | #125 | — | 127 | 1 | no | — |
| 150 | #125 | — | 127 | 1 | no | — |
| 151 | C22⋊Q8 | — | 254 | 1 | no | — |
| 152 | C22⋊Q8 | — | 254 | 1 | no | — |
| 153 | C4×Q8 | — | 254 | 1 | no | — |
| 154 | C4×Q8 | — | 254 | 1 | no | — |
| 155 | #117 | — | 127 | 1 | no | — |
| 156 | #117 | C2×D127 | 1 | 1 | yes | characteristic |
| 157 | C2×C4⋊C4 | — | 127 | 1 | no | — |
| 158 | C2×C4⋊C4 | — | 254 | 1 | no | — |
| 159 | C2×C4⋊C4 | C2×D127 | 1 | 1 | yes | characteristic |
| 160 | C2×C42 | — | 127 | 1 | no | — |
| 161 | #107 | — | 127 | 2 | no | — |
| 162 | #107 | — | 127 | 2 | no | — |
| 163 | #107 | — | 127 | 2 | no | — |
| 164 | #107 | C2×D127 | 1 | 2 | yes | — |
| 165 | #106 | — | 127 | 2 | no | — |
| 166 | #106 | — | 127 | 2 | no | — |
| 167 | #106 | — | 127 | 2 | no | — |
| 168 | #106 | C2×D127 | 1 | 2 | yes | — |
| 169 | #97 | — | 127 | 2 | no | — |
| 170 | C4×C8 | — | 127 | 2 | no | — |
| 171 | C2×Q8 | — | 508 | 4 | no | — |
| 172 | C2×Q8 | — | 254 | 1 | no | — |
| 173 | C2×Q8 | — | 254 | 1 | no | — |
| 174 | C2×Q8 | — | 254 | 2 | no | — |
| 175 | C2×Q8 | — | 254 | 2 | no | — |
| 176 | C22×C4 | — | 2 | 1 | no | — |
| 177 | C22×C4 | — | 127 | 1 | no | — |
| 178 | C22×C4 | — | 127 | 1 | no | — |
| 179 | C22×C4 | — | 254 | 1 | no | — |
| 180 | C22×C4 | C22×D127 | 1 | 1 | yes | characteristic |
| 181 | M16 | — | 127 | 4 | no | — |
| 182 | M16 | Dic254 | 1 | 4 | yes | complemented |
| 183 | C2×C8 | — | 127 | 2 | no | — |
| 184 | C2×C8 | C22×D127 | 1 | 2 | yes | — |
| 185 | C4⋊C4 | — | 2 | 1 | no | — |
| 186 | C4⋊C4 | — | 127 | 1 | no | — |
| 187 | C4⋊C4 | — | 127 | 2 | no | — |
| 188 | C4⋊C4 | — | 127 | 1 | no | — |
| 189 | C4⋊C4 | — | 254 | 2 | no | — |
| 190 | C4⋊C4 | — | 254 | 2 | no | — |
| 191 | C4⋊C4 | — | 254 | 1 | no | — |
| 192 | C4⋊C4 | — | 254 | 1 | no | — |
| 193 | C4⋊C4 | — | 254 | 1 | no | — |
| 194 | C4⋊C4 | — | 254 | 1 | no | — |
| 195 | C4⋊C4 | — | 254 | 1 | no | — |
| 196 | C4⋊C4 | — | 254 | 1 | no | — |
| 197 | C4⋊C4 | — | 254 | 1 | no | — |
| 198 | C4⋊C4 | C22×D127 | 1 | 1 | yes | characteristic |
| 199 | C4⋊C4 | C22×D127 | 1 | 2 | yes | — |
| 200 | C4⋊C4 | C22×D127 | 1 | 1 | yes | characteristic |
| 201 | C22⋊C4 | — | 2 | 1 | no | — |
| 202 | C22⋊C4 | — | 254 | 1 | no | — |
| 203 | C42 | — | 2 | 1 | no | — |
| 204 | C42 | — | 127 | 1 | no | — |
| 205 | C42 | — | 127 | 1 | no | — |
| 206 | C42 | — | 127 | 1 | no | — |
| 207 | C42 | — | 127 | 1 | no | — |
| 208 | C42 | — | 254 | 1 | no | — |
| 209 | C23 | D4×D127 | 1 | 1 | yes | characteristic |
| 210 | Q8 | — | 508 | 4 | no | — |
| 211 | Q8 | — | 508 | 2 | no | — |
| 212 | Q8 | — | 508 | 2 | no | — |
| 213 | Q8 | — | 508 | 4 | no | — |
| 214 | C2×C4 | — | 2 | 1 | no | — |
| 215 | C2×C4 | — | 2 | 1 | no | — |
| 216 | C2×C4 | — | 2 | 2 | no | — |
| 217 | C2×C4 | — | 4 | 2 | no | — |
| 218 | C2×C4 | — | 127 | 1 | no | — |
| 219 | C2×C4 | — | 127 | 1 | no | — |
| 220 | C2×C4 | — | 127 | 1 | no | — |
| 221 | C2×C4 | — | 127 | 1 | no | — |
| 222 | C2×C4 | — | 508 | 2 | no | — |
| 223 | C2×C4 | — | 254 | 2 | no | — |
| 224 | C2×C4 | — | 254 | 2 | no | — |
| 225 | C2×C4 | — | 254 | 1 | no | — |
| 226 | C2×C4 | — | 254 | 1 | no | — |
| 227 | C2×C4 | — | 254 | 2 | no | — |
| 228 | C2×C4 | D4×D127 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | Q8×D127 | 1 | 2 | yes | — |
| 230 | C2×C4 | C2×Dic254 | 1 | 2 | yes | — |
| 231 | C2×C4 | C23×D127 | 1 | 1 | yes | characteristic |
| 232 | C8 | — | 127 | 4 | no | — |
| 233 | C8 | C2×Dic254 | 1 | 4 | yes | complemented |
| 234 | C22 | — | 2 | 1 | no | — |
| 235 | C22 | — | 2 | 1 | no | — |
| 236 | C22 | #16861208 | 1 | 2 | yes | — |
| 237 | C22 | C2×D4×D127 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 4 | 2 | no | — |
| 239 | C4 | — | 4 | 1 | no | — |
| 240 | C4 | — | 4 | 1 | no | — |
| 241 | C4 | — | 127 | 2 | no | — |
| 242 | C4 | — | 127 | 2 | no | — |
| 243 | C4 | — | 508 | 2 | no | — |
| 244 | C4 | — | 508 | 1 | no | — |
| 245 | C4 | — | 508 | 1 | no | — |
| 246 | C4 | — | 254 | 1 | no | — |
| 247 | C4 | — | 254 | 1 | no | — |
| 248 | C4 | #16861208 | 1 | 2 | yes | — |
| 249 | C4 | C2×Q8×D127 | 1 | 1 | yes | characteristic |
| 250 | C4 | C22×Dic254 | 1 | 1 | yes | characteristic |
| 251 | C2 | — | 2 | 2 | no | — |
| 252 | C2 | C2×#16861208 | 1 | 1 | yes | characteristic |
| 253 | C2 | AGL(1,ℤ/8)×D127 | 1 | 1 | yes | characteristic |
| 254 | C2 | #137×D127 | 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