← Q8×#14523248 #30652311
Subgroups — Q8×#14523248 #30652311
Order 16256.
1068 conjugacy classes of subgroups.
The subgroup lattice has 258 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | Q8×Dic254 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22550604 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #22550586 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #22550581 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | Q8×D508 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | Q8×C22⋊D127 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22550433 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #22550430 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | C4×#14523248 | C2 | 1 | 3 | yes | maximal |
| 10 | C4×Q8×D127 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | C2×C4×Q8×C127 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#14523248 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×Q8×D127 | C22 | 1 | 2 | yes | — |
| 14 | #16861209 | C22 | 1 | 3 | yes | — |
| 15 | #16861209 | C22 | 1 | 3 | yes | — |
| 16 | #16861208 | C22 | 1 | 6 | yes | — |
| 17 | #16861207 | C22 | 1 | 6 | yes | — |
| 18 | #16861201 | C22 | 1 | 6 | yes | — |
| 19 | #16861199 | C22 | 1 | 12 | yes | — |
| 20 | #16861189 | C22 | 1 | 6 | yes | — |
| 21 | #16861188 | C22 | 1 | 6 | yes | — |
| 22 | #16861188 | C22 | 1 | 6 | yes | — |
| 23 | #16861187 | C22 | 1 | 12 | yes | — |
| 24 | #16861185 | C22 | 1 | 6 | yes | — |
| 25 | Q8×Dic127 | C22 | 1 | 2 | yes | — |
| 26 | C4×Dic254 | C22 | 1 | 6 | yes | — |
| 27 | C4×Dic254 | C22 | 1 | 3 | yes | — |
| 28 | C4×Dic254 | C22 | 1 | 3 | yes | — |
| 29 | C4×Dic254 | C22 | 1 | 6 | yes | — |
| 30 | #16861174 | C22 | 1 | 6 | yes | — |
| 31 | C4×D508 | C22 | 1 | 3 | yes | — |
| 32 | C4×D508 | C22 | 1 | 3 | yes | — |
| 33 | C4×C22⋊D127 | C22 | 1 | 6 | yes | — |
| 34 | C4×C22⋊D127 | C22 | 1 | 6 | yes | — |
| 35 | #16861168 | C22 | 1 | 3 | yes | — |
| 36 | #16861166 | C22 | 1 | 6 | yes | — |
| 37 | C4⋊C4×D127 | C22 | 1 | 6 | yes | — |
| 38 | C42×D127 | C22 | 1 | 6 | yes | — |
| 39 | C22×Q8×C127 | C22 | 1 | 1 | yes | characteristic |
| 40 | C4×Q8×C127 | C22 | 1 | 1 | yes | characteristic |
| 41 | C4×Q8×C127 | C22 | 1 | 3 | yes | — |
| 42 | C4×Q8×C127 | C22 | 1 | 1 | yes | characteristic |
| 43 | C4×Q8×C127 | C22 | 1 | 3 | yes | — |
| 44 | C2×C4⋊C4×C127 | C22 | 1 | 3 | yes | — |
| 45 | C2×C42×C127 | C22 | 1 | 3 | yes | — |
| 46 | #14523248 | Q8 | 1 | 8 | yes | direct product factor |
| 47 | C2×Dic254 | C23 | 1 | 6 | yes | — |
| 48 | C2×Dic254 | C23 | 1 | 6 | yes | — |
| 49 | C2×Dic254 | C23 | 1 | 1 | yes | characteristic |
| 50 | Q8×D127 | — | 2 | 8 | no | — |
| 51 | C2×D508 | C23 | 1 | 1 | yes | characteristic |
| 52 | C2×C22⋊D127 | C23 | 1 | 2 | yes | — |
| 53 | C2×C4×D127 | C23 | 1 | 6 | yes | — |
| 54 | C2×C4×D127 | C23 | 1 | 6 | yes | — |
| 55 | C2×C4×D127 | C23 | 1 | 2 | yes | — |
| 56 | C4⋊Dic127 | C23 | 1 | 6 | yes | — |
| 57 | C4⋊Dic127 | C23 | 1 | 6 | yes | — |
| 58 | C4⋊Dic127 | C23 | 1 | 3 | yes | — |
| 59 | C4⋊Dic127 | C23 | 1 | 3 | yes | — |
| 60 | Dic127⋊C4 | C23 | 1 | 12 | yes | — |
| 61 | Dic127⋊C4 | C23 | 1 | 12 | yes | — |
| 62 | Dic127⋊C4 | C23 | 1 | 6 | yes | — |
| 63 | Dic127⋊C4 | C23 | 1 | 6 | yes | — |
| 64 | C22⋊Dic127 | C23 | 1 | 6 | yes | — |
| 65 | (C2×D127)⋊C4 | C23 | 1 | 6 | yes | — |
| 66 | (C2×D127)⋊C4 | C23 | 1 | 6 | yes | — |
| 67 | C4×Dic127 | C23 | 1 | 6 | yes | — |
| 68 | C4×Dic127 | C23 | 1 | 6 | yes | — |
| 69 | C4×Dic127 | C23 | 1 | 6 | yes | — |
| 70 | C2×Q8×C127 | — | 2 | 4 | no | — |
| 71 | C2×Q8×C127 | C23 | 1 | 3 | yes | — |
| 72 | C2×Q8×C127 | C23 | 1 | 1 | yes | characteristic |
| 73 | C22×C4×C127 | C23 | 1 | 3 | yes | — |
| 74 | C22×C4×C127 | C23 | 1 | 3 | yes | — |
| 75 | C22×C4×C127 | C23 | 1 | 1 | yes | characteristic |
| 76 | C4⋊C4×C127 | C23 | 1 | 6 | yes | — |
| 77 | C4⋊C4×C127 | C23 | 1 | 3 | yes | — |
| 78 | C4⋊C4×C127 | C23 | 1 | 3 | yes | — |
| 79 | C42×C127 | C23 | 1 | 3 | yes | — |
| 80 | C42×C127 | C23 | 1 | 3 | yes | — |
| 81 | C42×C127 | C23 | 1 | 3 | yes | — |
| 82 | C42×C127 | C23 | 1 | 3 | yes | — |
| 83 | C22×D127 | C24 | 1 | 2 | yes | — |
| 84 | Dic254 | — | 2 | 12 | no | — |
| 85 | Dic254 | — | 4 | 6 | no | — |
| 86 | Dic254 | C2×Q8 | 1 | 4 | yes | complemented |
| 87 | D508 | C2×Q8 | 1 | 4 | yes | complemented |
| 88 | C22⋊D127 | C2×Q8 | 1 | 8 | yes | complemented |
| 89 | C2×Dic127 | C24 | 1 | 6 | yes | — |
| 90 | C2×Dic127 | C24 | 1 | 6 | yes | — |
| 91 | C2×Dic127 | C24 | 1 | 2 | yes | — |
| 92 | C4×D127 | — | 2 | 12 | no | — |
| 93 | C4×D127 | — | 4 | 6 | no | — |
| 94 | C4×D127 | C2×Q8 | 1 | 8 | yes | complemented |
| 95 | C23×C127 | C24 | 1 | 1 | yes | characteristic |
| 96 | Q8×C127 | — | 2 | 6 | no | — |
| 97 | Q8×C127 | #41 | 1 | 4 | yes | complemented |
| 98 | C2×C4×C127 | — | 2 | 6 | no | — |
| 99 | C2×C4×C127 | — | 4 | 3 | no | — |
| 100 | C2×C4×C127 | C2×Q8 | 1 | 4 | yes | complemented |
| 101 | C2×C4×C127 | C24 | 1 | 3 | yes | — |
| 102 | C2×C4×C127 | C24 | 1 | 3 | yes | — |
| 103 | C2×C4×C127 | C24 | 1 | 3 | yes | — |
| 104 | C2×C4×C127 | C24 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C127 | C24 | 1 | 3 | yes | — |
| 106 | C2×C4×C127 | C24 | 1 | 1 | yes | characteristic |
| 107 | C22×C127 | — | 2 | 1 | no | — |
| 108 | C22×C127 | — | 2 | 1 | no | — |
| 109 | C22×C127 | C22×Q8 | 1 | 2 | yes | — |
| 110 | C22×C127 | C25 | 1 | 1 | yes | characteristic |
| 111 | C2×D127 | — | 2 | 2 | no | — |
| 112 | C2×D127 | — | 2 | 2 | no | — |
| 113 | C2×D127 | C22×Q8 | 1 | 4 | yes | — |
| 114 | C4×C127 | — | 2 | 3 | no | — |
| 115 | C4×C127 | — | 2 | 3 | no | — |
| 116 | C4×C127 | — | 2 | 3 | no | — |
| 117 | C4×C127 | C22×Q8 | 1 | 2 | yes | — |
| 118 | C4×C127 | C22×Q8 | 1 | 2 | yes | — |
| 119 | C4×C127 | C2×#41 | 1 | 6 | yes | — |
| 120 | Dic127 | — | 2 | 6 | no | — |
| 121 | Dic127 | — | 2 | 6 | no | — |
| 122 | Dic127 | C22×Q8 | 1 | 4 | yes | — |
| 123 | C2×C127 | — | 2 | 2 | no | — |
| 124 | C2×C127 | C23×Q8 | 1 | 1 | yes | characteristic |
| 125 | C2×C127 | C22×#41 | 1 | 1 | yes | characteristic |
| 126 | C2×C127 | #585 | 1 | 1 | yes | characteristic |
| 127 | D127 | — | 2 | 4 | no | — |
| 128 | Q8×#41 | — | 127 | 1 | no | maximal, Sylow |
| 129 | C127 | Q8×#41 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 130 | Q82 | — | 127 | 1 | no | — |
| 131 | #557 | — | 127 | 3 | no | — |
| 132 | #554 | — | 127 | 3 | no | — |
| 133 | #554 | — | 127 | 6 | no | — |
| 134 | D4×Q8 | — | 127 | 2 | no | — |
| 135 | D4×Q8 | — | 127 | 1 | no | — |
| 136 | #533 | — | 127 | 6 | no | — |
| 137 | #533 | — | 127 | 3 | no | — |
| 138 | C4×#41 | — | 127 | 3 | no | — |
| 139 | C2×C4×Q8 | — | 127 | 2 | no | — |
| 140 | C2×C4×Q8 | D127 | 1 | 1 | yes | characteristic, complemented |
| 141 | C2×#41 | — | 127 | 1 | no | — |
| 142 | C22×Q8 | — | 127 | 2 | no | — |
| 143 | C22×Q8 | C2×D127 | 1 | 1 | yes | characteristic |
| 144 | C4⋊Q8 | — | 127 | 6 | no | — |
| 145 | C4⋊Q8 | — | 127 | 6 | no | — |
| 146 | C4⋊Q8 | — | 127 | 3 | no | — |
| 147 | C4⋊Q8 | — | 127 | 3 | no | — |
| 148 | #125 | — | 127 | 12 | no | — |
| 149 | #125 | — | 127 | 6 | no | — |
| 150 | C22⋊Q8 | — | 127 | 6 | no | — |
| 151 | C22⋊Q8 | — | 127 | 6 | no | — |
| 152 | C22⋊Q8 | — | 127 | 6 | no | — |
| 153 | C22⋊Q8 | — | 127 | 12 | no | — |
| 154 | C22⋊Q8 | — | 127 | 6 | no | — |
| 155 | C4×Q8 | — | 127 | 2 | no | — |
| 156 | C4×Q8 | — | 127 | 6 | no | — |
| 157 | C4×Q8 | — | 127 | 6 | no | — |
| 158 | C4×Q8 | — | 127 | 3 | no | — |
| 159 | C4×Q8 | — | 127 | 3 | no | — |
| 160 | C4×Q8 | — | 127 | 6 | no | — |
| 161 | C4×Q8 | C2×D127 | 1 | 1 | yes | characteristic |
| 162 | C4×Q8 | C2×D127 | 1 | 3 | yes | — |
| 163 | C4×Q8 | C2×D127 | 1 | 1 | yes | characteristic |
| 164 | C4×Q8 | C2×D127 | 1 | 3 | yes | — |
| 165 | C4×D4 | — | 127 | 6 | no | — |
| 166 | C4×D4 | — | 127 | 6 | no | — |
| 167 | C4×D4 | — | 127 | 3 | no | — |
| 168 | C4×D4 | — | 127 | 3 | no | — |
| 169 | #117 | — | 127 | 3 | no | — |
| 170 | #117 | — | 127 | 6 | no | — |
| 171 | C2×C4⋊C4 | — | 127 | 6 | no | — |
| 172 | C2×C4⋊C4 | C2×D127 | 1 | 3 | yes | — |
| 173 | C2×C42 | — | 127 | 6 | no | — |
| 174 | C2×C42 | C2×D127 | 1 | 3 | yes | — |
| 175 | #41 | — | 127 | 8 | no | — |
| 176 | C2×Q8 | — | 2 | 4 | no | — |
| 177 | C2×Q8 | — | 127 | 1 | no | — |
| 178 | C2×Q8 | — | 127 | 6 | no | — |
| 179 | C2×Q8 | — | 127 | 6 | no | — |
| 180 | C2×Q8 | — | 254 | 8 | no | — |
| 181 | C2×Q8 | C22×D127 | 1 | 3 | yes | — |
| 182 | C2×Q8 | C22×D127 | 1 | 1 | yes | characteristic |
| 183 | C2×D4 | — | 127 | 2 | no | — |
| 184 | C2×D4 | — | 127 | 1 | no | — |
| 185 | C22×C4 | — | 127 | 2 | no | — |
| 186 | C22×C4 | — | 127 | 6 | no | — |
| 187 | C22×C4 | — | 127 | 6 | no | — |
| 188 | C22×C4 | C22×D127 | 1 | 3 | yes | — |
| 189 | C22×C4 | C22×D127 | 1 | 3 | yes | — |
| 190 | C22×C4 | C22×D127 | 1 | 1 | yes | characteristic |
| 191 | C4⋊C4 | — | 127 | 12 | no | — |
| 192 | C4⋊C4 | — | 127 | 6 | no | — |
| 193 | C4⋊C4 | — | 127 | 6 | no | — |
| 194 | C4⋊C4 | — | 127 | 12 | no | — |
| 195 | C4⋊C4 | — | 127 | 6 | no | — |
| 196 | C4⋊C4 | — | 127 | 3 | no | — |
| 197 | C4⋊C4 | — | 127 | 3 | no | — |
| 198 | C4⋊C4 | — | 127 | 6 | no | — |
| 199 | C4⋊C4 | C22×D127 | 1 | 3 | yes | — |
| 200 | C4⋊C4 | C22×D127 | 1 | 6 | yes | — |
| 201 | C4⋊C4 | C22×D127 | 1 | 3 | yes | — |
| 202 | C22⋊C4 | — | 127 | 6 | no | — |
| 203 | C22⋊C4 | — | 127 | 6 | no | — |
| 204 | C22⋊C4 | — | 127 | 6 | no | — |
| 205 | C42 | — | 127 | 6 | no | — |
| 206 | C42 | — | 127 | 6 | no | — |
| 207 | C42 | — | 127 | 6 | no | — |
| 208 | C42 | C22×D127 | 1 | 3 | yes | — |
| 209 | C42 | C22×D127 | 1 | 3 | yes | — |
| 210 | C42 | C22×D127 | 1 | 3 | yes | — |
| 211 | C42 | C22×D127 | 1 | 3 | yes | — |
| 212 | C23 | — | 127 | 2 | no | — |
| 213 | C23 | C23×D127 | 1 | 1 | yes | characteristic |
| 214 | Q8 | — | 2 | 6 | no | — |
| 215 | Q8 | — | 127 | 4 | no | — |
| 216 | Q8 | — | 508 | 6 | no | — |
| 217 | Q8 | — | 254 | 12 | no | — |
| 218 | Q8 | #14523248 | 1 | 4 | yes | direct product factor |
| 219 | D4 | — | 127 | 8 | no | — |
| 220 | D4 | — | 127 | 4 | no | — |
| 221 | C2×C4 | — | 2 | 6 | no | — |
| 222 | C2×C4 | — | 4 | 3 | no | — |
| 223 | C2×C4 | — | 127 | 2 | no | — |
| 224 | C2×C4 | — | 127 | 8 | no | — |
| 225 | C2×C4 | — | 127 | 6 | no | — |
| 226 | C2×C4 | — | 127 | 6 | no | — |
| 227 | C2×C4 | — | 508 | 6 | no | — |
| 228 | C2×C4 | — | 254 | 12 | no | — |
| 229 | C2×C4 | Q8×D127 | 1 | 4 | yes | complemented |
| 230 | C2×C4 | C23×D127 | 1 | 3 | yes | — |
| 231 | C2×C4 | C23×D127 | 1 | 3 | yes | — |
| 232 | C2×C4 | C23×D127 | 1 | 3 | yes | — |
| 233 | C2×C4 | C23×D127 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | C23×D127 | 1 | 3 | yes | — |
| 235 | C2×C4 | C23×D127 | 1 | 1 | yes | characteristic |
| 236 | C22 | — | 2 | 1 | no | — |
| 237 | C22 | — | 2 | 1 | no | — |
| 238 | C22 | — | 127 | 4 | no | — |
| 239 | C22 | — | 254 | 2 | no | — |
| 240 | C22 | — | 254 | 2 | no | — |
| 241 | C22 | C2×Q8×D127 | 1 | 2 | yes | — |
| 242 | C22 | C24×D127 | 1 | 1 | yes | characteristic |
| 243 | C4 | — | 2 | 3 | no | — |
| 244 | C4 | — | 2 | 3 | no | — |
| 245 | C4 | — | 2 | 3 | no | — |
| 246 | C4 | — | 127 | 4 | no | — |
| 247 | C4 | — | 254 | 6 | no | — |
| 248 | C4 | — | 254 | 6 | no | — |
| 249 | C4 | C2×Q8×D127 | 1 | 2 | yes | — |
| 250 | C4 | C2×Q8×D127 | 1 | 2 | yes | — |
| 251 | C4 | C2×#14523248 | 1 | 6 | yes | — |
| 252 | C2 | — | 2 | 2 | no | — |
| 253 | C2 | — | 254 | 4 | no | — |
| 254 | C2 | C22×Q8×D127 | 1 | 1 | yes | characteristic |
| 255 | C2 | C22×#14523248 | 1 | 1 | yes | characteristic |
| 256 | C2 | #22550712 | 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