← #30647674
Subgroups — #30647674
Order 16256.
356 conjugacy classes of subgroups.
The subgroup lattice has 234 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 | #22550006 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22550004 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22550004 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C8×Dic254 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22549903 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #22549902 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×Dic127⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #22549834 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | M16×Dic127 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #405×C127 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×#14523232 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×C127⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16861208 | C4 | 1 | 1 | yes | characteristic |
| 15 | #16861208 | C4 | 1 | 1 | yes | characteristic |
| 16 | #16861200 | C4 | 1 | 1 | yes | characteristic |
| 17 | #16861200 | C4 | 1 | 1 | yes | characteristic |
| 18 | #16861186 | C4 | 1 | 2 | yes | — |
| 19 | #16861186 | C4 | 1 | 2 | yes | — |
| 20 | C4×Dic254 | C22 | 1 | 2 | yes | — |
| 21 | C4×Dic254 | C22 | 1 | 2 | yes | — |
| 22 | #16861168 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×C4×Dic127 | C22 | 1 | 1 | yes | characteristic |
| 24 | C4⋊(C127⋊C8) | C22 | 1 | 2 | yes | — |
| 25 | C4⋊(C127⋊C8) | C22 | 1 | 2 | yes | — |
| 26 | Dic127⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 27 | Dic127⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 28 | Dic127⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 29 | Dic127⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 30 | Dic127⋊C8 | C22 | 1 | 2 | yes | — |
| 31 | Dic127⋊C8 | C22 | 1 | 2 | yes | — |
| 32 | #16861119 | C22 | 1 | 2 | yes | — |
| 33 | #16861119 | C22 | 1 | 2 | yes | — |
| 34 | C8×Dic127 | C22 | 1 | 2 | yes | — |
| 35 | C8×Dic127 | C22 | 1 | 2 | yes | — |
| 36 | C2×M16×C127 | C22 | 1 | 1 | yes | characteristic |
| 37 | C22×C8×C127 | C22 | 1 | 1 | yes | characteristic |
| 38 | #117×C127 | C22 | 1 | 1 | yes | characteristic |
| 39 | #97×C127 | C22 | 1 | 2 | yes | — |
| 40 | C4×C8×C127 | C22 | 1 | 2 | yes | — |
| 41 | C2×Dic254 | C2×C4 | 1 | 2 | yes | — |
| 42 | C2×Dic254 | C2×C4 | 1 | 2 | yes | — |
| 43 | C22×Dic127 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C22×Dic127 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | #14523232 | — | 2 | 2 | no | — |
| 46 | C2×C127⋊C8 | — | 2 | 2 | no | — |
| 47 | C2×C127⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×C127⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C127⋊C8 | C23 | 1 | 2 | yes | — |
| 50 | C4⋊Dic127 | C2×C4 | 1 | 2 | yes | — |
| 51 | C4⋊Dic127 | C2×C4 | 1 | 2 | yes | — |
| 52 | C4⋊Dic127 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C4⋊Dic127 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | Dic127⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 55 | Dic127⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 56 | Dic127⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 57 | Dic127⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 58 | C22⋊Dic127 | C2×C4 | 1 | 2 | yes | — |
| 59 | C4×Dic127 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | C4×Dic127 | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | C4×Dic127 | C23 | 1 | 1 | yes | characteristic |
| 62 | C4×Dic127 | C23 | 1 | 2 | yes | — |
| 63 | C4×Dic127 | C23 | 1 | 1 | yes | characteristic |
| 64 | C22×C4×C127 | C23 | 1 | 1 | yes | characteristic |
| 65 | M16×C127 | Q8 | 1 | 4 | yes | complemented |
| 66 | C2×C8×C127 | — | 2 | 2 | no | — |
| 67 | C2×C8×C127 | C23 | 1 | 1 | yes | characteristic |
| 68 | C2×C8×C127 | C23 | 1 | 1 | yes | characteristic |
| 69 | C2×C8×C127 | C23 | 1 | 2 | yes | — |
| 70 | C4⋊C4×C127 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | C4⋊C4×C127 | C2×C4 | 1 | 1 | yes | characteristic |
| 72 | C22⋊C4×C127 | C2×C4 | 1 | 2 | yes | — |
| 73 | C42×C127 | C23 | 1 | 2 | yes | — |
| 74 | Dic254 | — | 2 | 4 | no | — |
| 75 | Dic254 | — | 4 | 2 | no | — |
| 76 | C2×Dic127 | — | 2 | 2 | no | — |
| 77 | C2×Dic127 | — | 4 | 1 | no | — |
| 78 | C2×Dic127 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C2×Dic127 | C22×C4 | 1 | 1 | yes | characteristic |
| 80 | C2×Dic127 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C2×Dic127 | C22×C4 | 1 | 2 | yes | — |
| 82 | C2×Dic127 | C22×C4 | 1 | 2 | yes | — |
| 83 | C2×Dic127 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C127⋊C8 | — | 2 | 1 | no | — |
| 85 | C127⋊C8 | — | 2 | 1 | no | — |
| 86 | C127⋊C8 | — | 2 | 2 | no | — |
| 87 | C23×C127 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C127 | C22×C4 | 1 | 2 | yes | — |
| 89 | C2×C4×C127 | C22×C4 | 1 | 2 | yes | — |
| 90 | C2×C4×C127 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C127 | C2×Q8 | 1 | 2 | yes | — |
| 92 | C2×C4×C127 | #41 | 1 | 2 | yes | — |
| 93 | C2×C4×C127 | C24 | 1 | 1 | yes | characteristic |
| 94 | C8×C127 | — | 2 | 1 | no | — |
| 95 | C8×C127 | — | 2 | 1 | no | — |
| 96 | C8×C127 | C2×Q8 | 1 | 4 | yes | — |
| 97 | C22×C127 | — | 2 | 1 | no | — |
| 98 | C22×C127 | — | 2 | 1 | no | — |
| 99 | C22×C127 | C4×Q8 | 1 | 2 | yes | — |
| 100 | C22×C127 | C23×C4 | 1 | 1 | yes | characteristic |
| 101 | C4×C127 | — | 2 | 2 | no | — |
| 102 | C4×C127 | — | 2 | 2 | no | — |
| 103 | C4×C127 | C4×Q8 | 1 | 2 | yes | — |
| 104 | C4×C127 | C22×Q8 | 1 | 1 | yes | characteristic |
| 105 | C4×C127 | C2×#41 | 1 | 1 | yes | characteristic |
| 106 | Dic127 | — | 2 | 1 | no | — |
| 107 | Dic127 | — | 2 | 2 | no | — |
| 108 | Dic127 | — | 2 | 1 | no | — |
| 109 | Dic127 | — | 2 | 1 | no | — |
| 110 | Dic127 | — | 2 | 2 | no | — |
| 111 | Dic127 | #131 | 1 | 2 | yes | — |
| 112 | C2×C127 | — | 2 | 2 | no | — |
| 113 | C2×C127 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 114 | C2×C127 | C2×#131 | 1 | 1 | yes | characteristic |
| 115 | C2×C127 | #568 | 1 | 1 | yes | characteristic |
| 116 | #2962 | — | 127 | 1 | no | maximal, Sylow |
| 117 | C127 | #2962 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 118 | #533 | — | 127 | 1 | no | — |
| 119 | #446 | — | 127 | 2 | no | — |
| 120 | #446 | — | 127 | 2 | no | — |
| 121 | #446 | — | 127 | 2 | no | — |
| 122 | C8×Q8 | — | 127 | 2 | no | — |
| 123 | #424 | — | 127 | 1 | no | — |
| 124 | #424 | — | 127 | 1 | no | — |
| 125 | C2×C4⋊C8 | — | 127 | 1 | no | — |
| 126 | #405 | — | 127 | 1 | no | — |
| 127 | #405 | D127 | 1 | 1 | yes | characteristic |
| 128 | C4×M16 | — | 127 | 1 | no | — |
| 129 | C2×M16 | — | 127 | 1 | no | — |
| 130 | C2×M16 | C2×D127 | 1 | 1 | yes | characteristic |
| 131 | C22×C8 | — | 127 | 1 | no | — |
| 132 | C22×C8 | C2×D127 | 1 | 1 | yes | characteristic |
| 133 | C4⋊Q8 | — | 127 | 1 | no | — |
| 134 | C4⋊Q8 | — | 127 | 1 | no | — |
| 135 | #125 | — | 127 | 1 | no | — |
| 136 | #125 | — | 127 | 1 | no | — |
| 137 | C22⋊Q8 | — | 127 | 2 | no | — |
| 138 | C22⋊Q8 | — | 127 | 2 | no | — |
| 139 | C4×Q8 | — | 127 | 2 | no | — |
| 140 | C4×Q8 | — | 127 | 2 | no | — |
| 141 | #117 | — | 127 | 1 | no | — |
| 142 | #117 | C2×D127 | 1 | 1 | yes | characteristic |
| 143 | C2×C42 | — | 127 | 1 | no | — |
| 144 | C4⋊C8 | — | 127 | 1 | no | — |
| 145 | C4⋊C8 | — | 127 | 1 | no | — |
| 146 | C4⋊C8 | — | 127 | 1 | no | — |
| 147 | C4⋊C8 | — | 127 | 1 | no | — |
| 148 | C4⋊C8 | — | 127 | 2 | no | — |
| 149 | C4⋊C8 | — | 127 | 2 | no | — |
| 150 | C4⋊C8 | — | 127 | 2 | no | — |
| 151 | C4⋊C8 | — | 127 | 2 | no | — |
| 152 | #97 | — | 127 | 2 | no | — |
| 153 | #97 | — | 127 | 2 | no | — |
| 154 | #97 | C2×D127 | 1 | 2 | yes | — |
| 155 | C4×C8 | — | 127 | 2 | no | — |
| 156 | C4×C8 | — | 127 | 2 | no | — |
| 157 | C4×C8 | C2×D127 | 1 | 2 | yes | — |
| 158 | C2×Q8 | — | 127 | 2 | no | — |
| 159 | C2×Q8 | — | 127 | 2 | no | — |
| 160 | C22×C4 | — | 127 | 1 | no | — |
| 161 | C22×C4 | — | 127 | 1 | no | — |
| 162 | C22×C4 | C22×D127 | 1 | 1 | yes | characteristic |
| 163 | M16 | — | 254 | 2 | no | — |
| 164 | M16 | Dic254 | 1 | 4 | yes | complemented |
| 165 | C2×C8 | — | 2 | 2 | no | — |
| 166 | C2×C8 | — | 127 | 1 | no | — |
| 167 | C2×C8 | — | 127 | 2 | no | — |
| 168 | C2×C8 | — | 127 | 1 | no | — |
| 169 | C2×C8 | — | 254 | 2 | no | — |
| 170 | C2×C8 | C22×D127 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | C22×D127 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | C22×D127 | 1 | 2 | yes | — |
| 173 | C4⋊C4 | — | 127 | 2 | no | — |
| 174 | C4⋊C4 | — | 127 | 2 | no | — |
| 175 | C4⋊C4 | — | 127 | 2 | no | — |
| 176 | C4⋊C4 | — | 127 | 2 | no | — |
| 177 | C4⋊C4 | — | 127 | 1 | no | — |
| 178 | C4⋊C4 | — | 127 | 1 | no | — |
| 179 | C4⋊C4 | — | 127 | 2 | no | — |
| 180 | C4⋊C4 | — | 127 | 2 | no | — |
| 181 | C4⋊C4 | C4×D127 | 1 | 1 | yes | characteristic |
| 182 | C4⋊C4 | C4×D127 | 1 | 1 | yes | characteristic |
| 183 | C22⋊C4 | — | 127 | 2 | no | — |
| 184 | C22⋊C4 | C4×D127 | 1 | 2 | yes | — |
| 185 | C42 | — | 127 | 1 | no | — |
| 186 | C42 | — | 127 | 1 | no | — |
| 187 | C42 | — | 127 | 1 | no | — |
| 188 | C42 | — | 127 | 2 | no | — |
| 189 | C42 | — | 127 | 1 | no | — |
| 190 | C42 | C22×D127 | 1 | 2 | yes | — |
| 191 | C23 | C2×C4×D127 | 1 | 1 | yes | characteristic |
| 192 | Q8 | — | 508 | 2 | no | — |
| 193 | Q8 | — | 254 | 4 | no | — |
| 194 | C2×C4 | — | 127 | 1 | no | — |
| 195 | C2×C4 | — | 127 | 1 | no | — |
| 196 | C2×C4 | — | 127 | 1 | no | — |
| 197 | C2×C4 | — | 127 | 1 | no | — |
| 198 | C2×C4 | — | 127 | 2 | no | — |
| 199 | C2×C4 | — | 127 | 2 | no | — |
| 200 | C2×C4 | — | 508 | 1 | no | — |
| 201 | C2×C4 | — | 254 | 2 | no | — |
| 202 | C2×C4 | C2×C4×D127 | 1 | 2 | yes | — |
| 203 | C2×C4 | C2×C4×D127 | 1 | 2 | yes | — |
| 204 | C2×C4 | C2×C4×D127 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | C2×Dic254 | 1 | 2 | yes | — |
| 206 | C2×C4 | #14523248 | 1 | 2 | yes | — |
| 207 | C2×C4 | C23×D127 | 1 | 1 | yes | characteristic |
| 208 | C8 | — | 2 | 1 | no | — |
| 209 | C8 | — | 2 | 1 | no | — |
| 210 | C8 | — | 254 | 1 | no | — |
| 211 | C8 | — | 254 | 1 | no | — |
| 212 | C8 | — | 254 | 2 | no | — |
| 213 | C8 | C2×Dic254 | 1 | 4 | yes | — |
| 214 | C22 | — | 2 | 1 | no | — |
| 215 | C22 | — | 2 | 1 | no | — |
| 216 | C22 | C4×Dic254 | 1 | 2 | yes | — |
| 217 | C22 | C22×C4×D127 | 1 | 1 | yes | characteristic |
| 218 | C4 | — | 2 | 2 | no | — |
| 219 | C4 | — | 2 | 2 | no | — |
| 220 | C4 | — | 127 | 2 | no | — |
| 221 | C4 | — | 254 | 1 | no | — |
| 222 | C4 | — | 254 | 2 | no | — |
| 223 | C4 | — | 254 | 1 | no | — |
| 224 | C4 | — | 254 | 1 | no | — |
| 225 | C4 | — | 254 | 2 | no | — |
| 226 | C4 | C4×Dic254 | 1 | 2 | yes | — |
| 227 | C4 | C22×Dic254 | 1 | 1 | yes | characteristic |
| 228 | C4 | C2×#14523248 | 1 | 1 | yes | characteristic |
| 229 | C2 | — | 2 | 2 | no | — |
| 230 | C2 | C2×C4×Dic254 | 1 | 1 | yes | characteristic |
| 231 | C2 | #131×D127 | 1 | 1 | yes | characteristic |
| 232 | C2 | #22550641 | 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