← #27155816
Subgroups — #27155816
Order 12416.
1104 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 | #20261471 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | Q8×Dic194 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20261442 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20261392 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20261385 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #20261352 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #20261283 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #20261267 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #20261218 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C4×Q8×D97 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #550×C97 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#13095510 | C22 | 1 | 6 | yes | — |
| 13 | C2×Q8×D97 | C22 | 1 | 1 | yes | characteristic |
| 14 | #15690810 | C22 | 1 | 3 | yes | — |
| 15 | #15690809 | C22 | 1 | 6 | yes | — |
| 16 | #15690803 | C22 | 1 | 6 | yes | — |
| 17 | #15690801 | C22 | 1 | 3 | yes | — |
| 18 | #15690797 | C22 | 1 | 6 | yes | — |
| 19 | #15690797 | C22 | 1 | 3 | yes | — |
| 20 | #15690797 | C22 | 1 | 6 | yes | — |
| 21 | #15690792 | C22 | 1 | 6 | yes | — |
| 22 | #15690791 | C22 | 1 | 12 | yes | — |
| 23 | #15690790 | C22 | 1 | 1 | yes | characteristic |
| 24 | #15690789 | C22 | 1 | 6 | yes | — |
| 25 | #15690787 | C22 | 1 | 12 | yes | — |
| 26 | #15690781 | C22 | 1 | 6 | yes | — |
| 27 | #15690781 | C22 | 1 | 3 | yes | — |
| 28 | Q8×Dic97 | C22 | 1 | 2 | yes | — |
| 29 | Q8×Dic97 | C22 | 1 | 1 | yes | characteristic |
| 30 | C4×Dic194 | C22 | 1 | 3 | yes | — |
| 31 | C4×Dic194 | C22 | 1 | 3 | yes | — |
| 32 | #15690775 | C22 | 1 | 3 | yes | — |
| 33 | D4×Dic97 | C22 | 1 | 6 | yes | — |
| 34 | D4×Dic97 | C22 | 1 | 6 | yes | — |
| 35 | D4×Dic97 | C22 | 1 | 6 | yes | — |
| 36 | C4×C22⋊D97 | C22 | 1 | 6 | yes | — |
| 37 | #15690769 | C22 | 1 | 6 | yes | — |
| 38 | C2×C4⋊Dic97 | C22 | 1 | 6 | yes | — |
| 39 | C4⋊C4×D97 | C22 | 1 | 3 | yes | — |
| 40 | C42×D97 | C22 | 1 | 3 | yes | — |
| 41 | C2×#41×C97 | C22 | 1 | 2 | yes | — |
| 42 | #127×C97 | C22 | 1 | 3 | yes | — |
| 43 | #121×C97 | C22 | 1 | 6 | yes | — |
| 44 | C4×Q8×C97 | C22 | 1 | 1 | yes | characteristic |
| 45 | C4×D4×C97 | C22 | 1 | 3 | yes | — |
| 46 | #13095510 | — | 4 | 12 | no | — |
| 47 | C2×Dic194 | C23 | 1 | 3 | yes | — |
| 48 | C2×Dic194 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×Dic194 | C23 | 1 | 6 | yes | — |
| 50 | Q8×D97 | — | 2 | 4 | no | — |
| 51 | C2×C22⋊D97 | C23 | 1 | 6 | yes | — |
| 52 | C22×Dic97 | C23 | 1 | 12 | yes | — |
| 53 | C2×C4×D97 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×C4×D97 | C23 | 1 | 3 | yes | — |
| 55 | C2×C4×D97 | C23 | 1 | 3 | yes | — |
| 56 | C4⋊Dic97 | C23 | 1 | 3 | yes | — |
| 57 | C4⋊Dic97 | C23 | 1 | 6 | yes | — |
| 58 | C4⋊Dic97 | C23 | 1 | 6 | yes | — |
| 59 | C4⋊Dic97 | C23 | 1 | 2 | yes | — |
| 60 | C4⋊Dic97 | C23 | 1 | 3 | yes | — |
| 61 | #13095490 | C23 | 1 | 6 | yes | — |
| 62 | #13095490 | C23 | 1 | 6 | yes | — |
| 63 | #13095490 | C23 | 1 | 6 | yes | — |
| 64 | C22⋊Dic97 | C23 | 1 | 6 | yes | — |
| 65 | C22⋊Dic97 | C23 | 1 | 6 | yes | — |
| 66 | C22⋊Dic97 | C23 | 1 | 6 | yes | — |
| 67 | C22⋊Dic97 | C23 | 1 | 12 | yes | — |
| 68 | #13095488 | C23 | 1 | 2 | yes | — |
| 69 | C4×Dic97 | C23 | 1 | 3 | yes | — |
| 70 | C4×Dic97 | C23 | 1 | 3 | yes | — |
| 71 | C4×Dic97 | C23 | 1 | 3 | yes | — |
| 72 | C4×Dic97 | C23 | 1 | 6 | yes | — |
| 73 | #41×C97 | — | 2 | 8 | no | — |
| 74 | C2×Q8×C97 | C23 | 1 | 1 | yes | characteristic |
| 75 | C2×D4×C97 | C23 | 1 | 3 | yes | — |
| 76 | C2×D4×C97 | C23 | 1 | 6 | yes | — |
| 77 | C2×D4×C97 | C23 | 1 | 6 | yes | — |
| 78 | C22×C4×C97 | C23 | 1 | 6 | yes | — |
| 79 | C4⋊C4×C97 | C23 | 1 | 3 | yes | — |
| 80 | C4⋊C4×C97 | C23 | 1 | 1 | yes | characteristic |
| 81 | C22⋊C4×C97 | C23 | 1 | 6 | yes | — |
| 82 | C42×C97 | C23 | 1 | 3 | yes | — |
| 83 | C23×C97 | C24 | 1 | 6 | yes | — |
| 84 | C22×D97 | C24 | 1 | 1 | yes | characteristic |
| 85 | Q8×C97 | #41 | 1 | 4 | yes | complemented |
| 86 | D4×C97 | — | 2 | 12 | no | — |
| 87 | D4×C97 | — | 2 | 6 | no | — |
| 88 | D4×C97 | — | 4 | 6 | no | — |
| 89 | C2×C4×C97 | — | 2 | 12 | no | — |
| 90 | C2×C4×C97 | C24 | 1 | 3 | yes | — |
| 91 | C2×C4×C97 | C24 | 1 | 2 | yes | — |
| 92 | C2×C4×C97 | C24 | 1 | 3 | yes | — |
| 93 | C2×C4×C97 | C24 | 1 | 1 | yes | characteristic |
| 94 | C22⋊D97 | — | 2 | 12 | no | — |
| 95 | C2×Dic97 | — | 4 | 12 | no | — |
| 96 | C2×Dic97 | C24 | 1 | 1 | yes | characteristic |
| 97 | C2×Dic97 | C24 | 1 | 2 | yes | — |
| 98 | C2×Dic97 | C24 | 1 | 3 | yes | — |
| 99 | C2×Dic97 | C24 | 1 | 3 | yes | — |
| 100 | C2×Dic97 | C24 | 1 | 6 | yes | — |
| 101 | C4×D97 | — | 2 | 6 | no | — |
| 102 | C4×D97 | — | 2 | 6 | no | — |
| 103 | C4×D97 | — | 2 | 2 | no | — |
| 104 | Dic194 | — | 2 | 6 | no | — |
| 105 | Dic194 | — | 2 | 2 | no | — |
| 106 | Dic194 | — | 4 | 6 | no | — |
| 107 | C22×C97 | — | 2 | 6 | no | — |
| 108 | C22×C97 | — | 2 | 6 | no | — |
| 109 | C22×C97 | — | 2 | 6 | no | — |
| 110 | C22×C97 | C25 | 1 | 1 | yes | characteristic |
| 111 | C2×D97 | — | 2 | 1 | no | — |
| 112 | C2×D97 | — | 2 | 1 | no | — |
| 113 | C2×D97 | H(5,𝔽2) | 1 | 2 | yes | — |
| 114 | C4×C97 | — | 2 | 2 | no | — |
| 115 | C4×C97 | — | 2 | 3 | no | — |
| 116 | C4×C97 | C2×#41 | 1 | 6 | yes | — |
| 117 | C4×C97 | 2−1+4 | 1 | 2 | yes | — |
| 118 | Dic97 | — | 2 | 3 | no | — |
| 119 | Dic97 | — | 2 | 6 | no | — |
| 120 | Dic97 | — | 2 | 3 | no | — |
| 121 | Dic97 | — | 2 | 2 | no | — |
| 122 | Dic97 | — | 2 | 1 | no | — |
| 123 | C2×C97 | — | 4 | 6 | no | — |
| 124 | C2×C97 | C22×#41 | 1 | 1 | yes | characteristic |
| 125 | C2×C97 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 126 | C2×C97 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 127 | D97 | — | 2 | 2 | no | — |
| 128 | #3522 | — | 97 | 1 | no | maximal, Sylow |
| 129 | C97 | #3522 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 130 | #559 | — | 97 | 3 | no | — |
| 131 | Q82 | — | 97 | 1 | no | — |
| 132 | #554 | — | 97 | 3 | no | — |
| 133 | #550 | D97 | 1 | 1 | yes | characteristic, complemented |
| 134 | #548 | — | 97 | 2 | no | — |
| 135 | #547 | — | 97 | 6 | no | — |
| 136 | #542 | — | 97 | 6 | no | — |
| 137 | #532 | — | 97 | 3 | no | — |
| 138 | #529 | — | 97 | 3 | no | — |
| 139 | #520 | — | 97 | 2 | no | — |
| 140 | C2×C4×Q8 | — | 97 | 1 | no | — |
| 141 | C2×#41 | — | 97 | 6 | no | — |
| 142 | C2×#41 | C2×D97 | 1 | 2 | yes | — |
| 143 | C22×Q8 | — | 97 | 1 | no | — |
| 144 | C4⋊Q8 | — | 97 | 6 | no | — |
| 145 | C4⋊Q8 | — | 97 | 3 | no | — |
| 146 | #127 | C2×D97 | 1 | 3 | yes | — |
| 147 | #126 | — | 97 | 6 | no | — |
| 148 | #125 | — | 97 | 3 | no | — |
| 149 | #124 | — | 97 | 3 | no | — |
| 150 | #124 | — | 97 | 6 | no | — |
| 151 | #124 | — | 97 | 6 | no | — |
| 152 | #123 | — | 97 | 12 | no | — |
| 153 | #123 | — | 97 | 6 | no | — |
| 154 | C22⋊Q8 | — | 97 | 6 | no | — |
| 155 | C22⋊Q8 | — | 97 | 1 | no | — |
| 156 | C22⋊Q8 | — | 97 | 12 | no | — |
| 157 | #121 | — | 97 | 3 | no | — |
| 158 | #121 | — | 97 | 6 | no | — |
| 159 | #121 | C2×D97 | 1 | 6 | yes | — |
| 160 | C4×Q8 | — | 97 | 1 | no | — |
| 161 | C4×Q8 | — | 97 | 2 | no | — |
| 162 | C4×Q8 | — | 97 | 3 | no | — |
| 163 | C4×Q8 | — | 97 | 3 | no | — |
| 164 | C4×Q8 | — | 97 | 3 | no | — |
| 165 | C4×Q8 | C2×D97 | 1 | 1 | yes | characteristic |
| 166 | C4×D4 | — | 97 | 6 | no | — |
| 167 | C4×D4 | — | 97 | 6 | no | — |
| 168 | C4×D4 | — | 97 | 6 | no | — |
| 169 | C4×D4 | — | 97 | 6 | no | — |
| 170 | C4×D4 | C2×D97 | 1 | 3 | yes | — |
| 171 | #117 | — | 97 | 6 | no | — |
| 172 | C2×C4⋊C4 | — | 97 | 6 | no | — |
| 173 | C2×C4⋊C4 | — | 97 | 3 | no | — |
| 174 | C2×C42 | — | 97 | 3 | no | — |
| 175 | #41 | — | 2 | 8 | no | — |
| 176 | #41 | — | 388 | 12 | no | — |
| 177 | C2×Q8 | — | 97 | 6 | no | — |
| 178 | C2×Q8 | — | 97 | 1 | no | — |
| 179 | C2×Q8 | — | 97 | 3 | no | — |
| 180 | C2×Q8 | — | 194 | 4 | no | — |
| 181 | C2×Q8 | C22×D97 | 1 | 1 | yes | characteristic |
| 182 | C2×D4 | — | 97 | 6 | no | — |
| 183 | C2×D4 | C22×D97 | 1 | 3 | yes | — |
| 184 | C2×D4 | C22×D97 | 1 | 6 | yes | — |
| 185 | C2×D4 | C22×D97 | 1 | 6 | yes | — |
| 186 | C22×C4 | — | 97 | 12 | no | — |
| 187 | C22×C4 | — | 97 | 1 | no | — |
| 188 | C22×C4 | — | 97 | 3 | no | — |
| 189 | C22×C4 | — | 97 | 3 | no | — |
| 190 | C22×C4 | C22×D97 | 1 | 6 | yes | — |
| 191 | C4⋊C4 | — | 97 | 6 | no | — |
| 192 | C4⋊C4 | — | 97 | 6 | no | — |
| 193 | C4⋊C4 | — | 97 | 6 | no | — |
| 194 | C4⋊C4 | — | 97 | 3 | no | — |
| 195 | C4⋊C4 | — | 97 | 6 | no | — |
| 196 | C4⋊C4 | — | 97 | 6 | no | — |
| 197 | C4⋊C4 | — | 97 | 2 | no | — |
| 198 | C4⋊C4 | — | 97 | 3 | no | — |
| 199 | C4⋊C4 | C22×D97 | 1 | 3 | yes | — |
| 200 | C4⋊C4 | C22×D97 | 1 | 1 | yes | characteristic |
| 201 | C22⋊C4 | — | 97 | 6 | no | — |
| 202 | C22⋊C4 | — | 97 | 6 | no | — |
| 203 | C22⋊C4 | — | 97 | 12 | no | — |
| 204 | C22⋊C4 | — | 97 | 6 | no | — |
| 205 | C22⋊C4 | — | 97 | 2 | no | — |
| 206 | C22⋊C4 | C22×D97 | 1 | 6 | yes | — |
| 207 | C42 | — | 97 | 3 | no | — |
| 208 | C42 | — | 97 | 3 | no | — |
| 209 | C42 | — | 97 | 3 | no | — |
| 210 | C42 | — | 97 | 6 | no | — |
| 211 | C42 | C22×D97 | 1 | 3 | yes | — |
| 212 | C23 | — | 97 | 1 | no | — |
| 213 | C23 | C23×D97 | 1 | 6 | yes | — |
| 214 | Q8 | — | 388 | 6 | no | — |
| 215 | Q8 | — | 194 | 6 | no | — |
| 216 | Q8 | — | 194 | 2 | no | — |
| 217 | Q8 | #13095510 | 1 | 4 | yes | complemented |
| 218 | D4 | — | 2 | 12 | no | — |
| 219 | D4 | — | 2 | 6 | no | — |
| 220 | D4 | — | 4 | 6 | no | — |
| 221 | D4 | — | 194 | 12 | no | — |
| 222 | C2×C4 | — | 2 | 12 | no | — |
| 223 | C2×C4 | — | 97 | 1 | no | — |
| 224 | C2×C4 | — | 97 | 2 | no | — |
| 225 | C2×C4 | — | 97 | 3 | no | — |
| 226 | C2×C4 | — | 97 | 3 | no | — |
| 227 | C2×C4 | — | 97 | 6 | no | — |
| 228 | C2×C4 | — | 388 | 12 | no | — |
| 229 | C2×C4 | — | 194 | 6 | no | — |
| 230 | C2×C4 | — | 194 | 6 | no | — |
| 231 | C2×C4 | — | 194 | 2 | no | — |
| 232 | C2×C4 | C23×D97 | 1 | 3 | yes | — |
| 233 | C2×C4 | C23×D97 | 1 | 2 | yes | — |
| 234 | C2×C4 | C23×D97 | 1 | 3 | yes | — |
| 235 | C2×C4 | C23×D97 | 1 | 1 | yes | characteristic |
| 236 | C22 | — | 2 | 6 | no | — |
| 237 | C22 | — | 2 | 6 | no | — |
| 238 | C22 | — | 2 | 6 | no | — |
| 239 | C22 | — | 97 | 2 | no | — |
| 240 | C22 | — | 194 | 1 | no | — |
| 241 | C22 | — | 194 | 1 | no | — |
| 242 | C22 | C24×D97 | 1 | 1 | yes | characteristic |
| 243 | C4 | — | 2 | 2 | no | — |
| 244 | C4 | — | 2 | 3 | no | — |
| 245 | C4 | — | 194 | 3 | no | — |
| 246 | C4 | — | 194 | 3 | no | — |
| 247 | C4 | — | 194 | 6 | no | — |
| 248 | C4 | — | 194 | 1 | no | — |
| 249 | C4 | — | 194 | 2 | no | — |
| 250 | C4 | C2×#13095510 | 1 | 6 | yes | — |
| 251 | C4 | #15690857 | 1 | 2 | yes | — |
| 252 | C2 | — | 4 | 6 | no | — |
| 253 | C2 | — | 194 | 2 | no | — |
| 254 | C2 | C22×#13095510 | 1 | 1 | yes | characteristic |
| 255 | C2 | H(5,𝔽2)×D97 | 1 | 1 | yes | characteristic |
| 256 | C2 | C2×#15690857 | 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