← #25116399
Subgroups — #25116399
Order 11008.
922 conjugacy classes of subgroups.
The subgroup lattice has 330 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4×D344 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #19137546 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #19135469 | C2 | 1 | 2 | yes | maximal |
| 4 | #19135221 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #19134592 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #19134495 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #19134462 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C43×#1844 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C22×D344 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×D172 | C22 | 1 | 2 | yes | — |
| 11 | C2×#13063283 | — | 2 | 2 | no | — |
| 12 | C4×D344 | C4 | 1 | 8 | yes | complemented |
| 13 | C2×#13063259 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C4⋊(C43⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#13063247 | C22 | 1 | 2 | yes | — |
| 16 | C2×#13063245 | C22 | 1 | 4 | yes | — |
| 17 | C2×#13063235 | C22 | 1 | 2 | yes | — |
| 18 | #14582800 | C22 | 1 | 4 | yes | — |
| 19 | #14582789 | — | 2 | 4 | no | — |
| 20 | #14582756 | — | 2 | 2 | no | — |
| 21 | C4⋊C4×Dic43 | C22 | 1 | 4 | yes | — |
| 22 | C4×C4⋊Dic43 | C22 | 1 | 1 | yes | characteristic |
| 23 | #14582661 | C22 | 1 | 4 | yes | — |
| 24 | #14582653 | C22 | 1 | 2 | yes | — |
| 25 | C2×#106×C43 | C22 | 1 | 2 | yes | — |
| 26 | C2×#106×C43 | C22 | 1 | 2 | yes | — |
| 27 | C2×C4×C8×C43 | C22 | 1 | 1 | yes | characteristic |
| 28 | C43×#384 | C22 | 1 | 2 | yes | — |
| 29 | C22×D172 | C23 | 1 | 2 | yes | — |
| 30 | C22×C4×D43 | — | 2 | 2 | no | — |
| 31 | C2×D344 | — | 2 | 2 | no | — |
| 32 | C2×D344 | — | 2 | 2 | no | — |
| 33 | C2×D344 | C2×C4 | 1 | 4 | yes | — |
| 34 | C22×C43⋊C8 | C23 | 1 | 2 | yes | — |
| 35 | C4×D172 | — | 2 | 4 | no | — |
| 36 | C4×D172 | C2×C4 | 1 | 8 | yes | — |
| 37 | #13063283 | — | 4 | 8 | no | — |
| 38 | C2×C4⋊Dic43 | D4 | 1 | 2 | yes | — |
| 39 | C2×C4⋊Dic43 | C23 | 1 | 2 | yes | — |
| 40 | C2×(C2×D43)⋊C4 | — | 2 | 2 | no | — |
| 41 | C2×(C2×D43)⋊C4 | — | 2 | 4 | no | — |
| 42 | C2×(C2×D43)⋊C4 | — | 2 | 4 | no | — |
| 43 | C2×C4×Dic43 | — | 2 | 4 | no | — |
| 44 | C2×C4×Dic43 | C23 | 1 | 2 | yes | — |
| 45 | #13063259 | C2×C4 | 1 | 4 | yes | — |
| 46 | C4⋊(C43⋊C8) | — | 2 | 2 | no | — |
| 47 | #13063247 | C2×C4 | 1 | 8 | yes | — |
| 48 | #13063245 | — | 4 | 4 | no | — |
| 49 | #13063235 | — | 2 | 4 | no | — |
| 50 | C43⋊#95 | — | 2 | 4 | no | — |
| 51 | C43⋊#95 | D4 | 1 | 2 | yes | — |
| 52 | C22×C8×C43 | C23 | 1 | 1 | yes | characteristic |
| 53 | C22×C8×C43 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×C4⋊C4×C43 | — | 2 | 2 | no | — |
| 55 | C2×C4⋊C4×C43 | — | 2 | 2 | no | — |
| 56 | C2×C4⋊C4×C43 | C23 | 1 | 4 | yes | — |
| 57 | C2×C42×C43 | C23 | 1 | 1 | yes | characteristic |
| 58 | #106×C43 | — | 2 | 4 | no | — |
| 59 | #106×C43 | — | 4 | 2 | no | — |
| 60 | C4×C8×C43 | C2×C4 | 1 | 4 | yes | — |
| 61 | C23×D43 | — | 2 | 2 | no | — |
| 62 | C22×C4×C43 | — | 2 | 4 | no | — |
| 63 | C22×C4×C43 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C22×C4×C43 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C22×C4×C43 | C24 | 1 | 1 | yes | characteristic |
| 66 | C22×Dic43 | — | 2 | 2 | no | — |
| 67 | C22×Dic43 | C2×D4 | 1 | 4 | yes | — |
| 68 | C2×D172 | — | 2 | 2 | no | — |
| 69 | C2×D172 | — | 2 | 2 | no | — |
| 70 | C2×D172 | — | 2 | 2 | no | — |
| 71 | C2×D172 | — | 2 | 2 | no | — |
| 72 | C2×D172 | — | 2 | 2 | no | — |
| 73 | C2×D172 | C22×C4 | 1 | 4 | yes | — |
| 74 | C2×C4×D43 | — | 2 | 8 | no | — |
| 75 | C2×C4×D43 | — | 4 | 4 | no | — |
| 76 | C2×C4×D43 | — | 4 | 4 | no | — |
| 77 | C2×C8×C43 | — | 2 | 1 | no | — |
| 78 | C2×C8×C43 | — | 2 | 1 | no | — |
| 79 | C2×C8×C43 | C22×C4 | 1 | 2 | yes | — |
| 80 | C2×C8×C43 | C22×C4 | 1 | 2 | yes | — |
| 81 | C2×C8×C43 | #41 | 1 | 2 | yes | — |
| 82 | C2×C8×C43 | #41 | 1 | 2 | yes | — |
| 83 | C4⋊C4×C43 | — | 2 | 4 | no | — |
| 84 | C4⋊C4×C43 | — | 2 | 4 | no | — |
| 85 | C4⋊C4×C43 | — | 4 | 4 | no | — |
| 86 | C4⋊C4×C43 | — | 8 | 2 | no | — |
| 87 | C42×C43 | C22×C4 | 1 | 2 | yes | — |
| 88 | C42×C43 | #41 | 1 | 2 | yes | — |
| 89 | (C2×D43)⋊C4 | — | 2 | 8 | no | — |
| 90 | (C2×D43)⋊C4 | — | 4 | 8 | no | — |
| 91 | (C2×D43)⋊C4 | — | 8 | 4 | no | — |
| 92 | C4⋊Dic43 | — | 2 | 4 | no | — |
| 93 | C4⋊Dic43 | — | 2 | 2 | no | — |
| 94 | C4⋊Dic43 | C22×C4 | 1 | 4 | yes | — |
| 95 | C4×Dic43 | — | 2 | 2 | no | — |
| 96 | C4×Dic43 | — | 2 | 2 | no | — |
| 97 | C4×Dic43 | — | 4 | 8 | no | — |
| 98 | C2×C43⋊C8 | — | 2 | 2 | no | — |
| 99 | C2×C43⋊C8 | — | 2 | 2 | no | — |
| 100 | C2×C43⋊C8 | — | 2 | 2 | no | — |
| 101 | D344 | — | 2 | 8 | no | — |
| 102 | C23×C43 | C22×D4 | 1 | 1 | yes | characteristic |
| 103 | C22×D43 | — | 2 | 4 | no | — |
| 104 | C22×D43 | — | 4 | 2 | no | — |
| 105 | C22×D43 | — | 4 | 2 | no | — |
| 106 | C22×D43 | — | 4 | 2 | no | — |
| 107 | C22×D43 | — | 4 | 2 | no | — |
| 108 | C22×D43 | — | 4 | 2 | no | — |
| 109 | C22×D43 | — | 4 | 2 | no | — |
| 110 | C2×C4×C43 | — | 2 | 2 | no | — |
| 111 | C2×C4×C43 | — | 4 | 4 | no | — |
| 112 | C2×C4×C43 | — | 4 | 4 | no | — |
| 113 | C2×C4×C43 | — | 4 | 4 | no | — |
| 114 | C2×C4×C43 | C4×D4 | 1 | 2 | yes | — |
| 115 | C2×C4×C43 | C4×D4 | 1 | 2 | yes | — |
| 116 | C2×C4×C43 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 117 | C2×C4×C43 | C23×C4 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C43 | C2×#41 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C43 | C2×#41 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C43 | C2×#41 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C43 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C43 | 2−1+4 | 1 | 1 | yes | characteristic |
| 123 | C2×Dic43 | — | 2 | 4 | no | — |
| 124 | C2×Dic43 | — | 2 | 4 | no | — |
| 125 | C2×Dic43 | — | 2 | 4 | no | — |
| 126 | C2×Dic43 | — | 2 | 4 | no | — |
| 127 | C2×Dic43 | — | 4 | 2 | no | — |
| 128 | C2×Dic43 | — | 4 | 2 | no | — |
| 129 | D172 | — | 2 | 4 | no | — |
| 130 | D172 | — | 2 | 4 | no | — |
| 131 | D172 | — | 4 | 2 | no | — |
| 132 | D172 | — | 4 | 2 | no | — |
| 133 | C4×D43 | — | 4 | 16 | no | — |
| 134 | C8×C43 | — | 2 | 2 | no | — |
| 135 | C8×C43 | #117 | 1 | 4 | yes | complemented |
| 136 | C43⋊C8 | — | 4 | 2 | no | — |
| 137 | #22813 | — | 43 | 1 | no | maximal, Sylow |
| 138 | C22×C43 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 139 | C22×C43 | #547 | 1 | 1 | yes | characteristic |
| 140 | C22×C43 | #550 | 1 | 1 | yes | characteristic |
| 141 | C22×C43 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 142 | C22×C43 | #577 | 1 | 2 | yes | — |
| 143 | C2×D43 | — | 4 | 4 | no | — |
| 144 | C2×D43 | — | 4 | 4 | no | — |
| 145 | C2×D43 | — | 4 | 4 | no | — |
| 146 | C2×D43 | — | 8 | 2 | no | — |
| 147 | C2×D43 | — | 8 | 2 | no | — |
| 148 | C2×D43 | — | 8 | 2 | no | — |
| 149 | C2×D43 | — | 8 | 2 | no | — |
| 150 | C4×C43 | — | 2 | 2 | no | — |
| 151 | C4×C43 | — | 8 | 4 | no | — |
| 152 | C4×C43 | #442 | 1 | 4 | yes | — |
| 153 | C4×C43 | C2×#117 | 1 | 1 | yes | characteristic |
| 154 | C4×C43 | #520 | 1 | 1 | yes | characteristic |
| 155 | C4×C43 | #555 | 1 | 1 | yes | characteristic |
| 156 | C4×C43 | #556 | 1 | 1 | yes | characteristic |
| 157 | Dic43 | — | 4 | 4 | no | — |
| 158 | Dic43 | — | 4 | 4 | no | — |
| 159 | C2×C4×D8 | — | 43 | 1 | no | — |
| 160 | #2354 | — | 43 | 2 | no | — |
| 161 | #1972 | — | 43 | 2 | no | — |
| 162 | #1926 | — | 43 | 4 | no | — |
| 163 | #1844 | D43 | 1 | 1 | yes | characteristic, complemented |
| 164 | #1802 | — | 43 | 2 | no | — |
| 165 | #1776 | — | 43 | 2 | no | — |
| 166 | #1770 | — | 43 | 1 | no | — |
| 167 | C2×C43 | #2354 | 1 | 1 | yes | characteristic |
| 168 | C2×C43 | C2×#442 | 1 | 1 | yes | characteristic |
| 169 | C2×C43 | #2952 | 1 | 1 | yes | characteristic |
| 170 | C2×C43 | #3354 | 1 | 2 | yes | — |
| 171 | C2×C43 | #3397 | 1 | 2 | yes | — |
| 172 | D43 | — | 8 | 4 | no | — |
| 173 | C22×D8 | — | 43 | 1 | no | — |
| 174 | C2×C4×D4 | — | 43 | 2 | no | — |
| 175 | C2×#117 | — | 86 | 2 | no | — |
| 176 | C4×D8 | — | 43 | 8 | no | — |
| 177 | C2×#107 | — | 43 | 1 | no | — |
| 178 | C2×#106 | C2×D43 | 1 | 2 | yes | — |
| 179 | C2×#106 | C2×D43 | 1 | 2 | yes | — |
| 180 | C2×C4⋊C8 | — | 43 | 1 | no | — |
| 181 | C2×#102 | — | 43 | 2 | no | — |
| 182 | C2×#102 | — | 43 | 4 | no | — |
| 183 | C2×#97 | — | 43 | 2 | no | — |
| 184 | C2×C4×C8 | C2×D43 | 1 | 1 | yes | characteristic |
| 185 | #390 | — | 43 | 4 | no | — |
| 186 | #388 | — | 86 | 4 | no | — |
| 187 | #384 | C2×D43 | 1 | 2 | yes | — |
| 188 | #380 | — | 86 | 2 | no | — |
| 189 | C4×C4⋊C4 | — | 43 | 1 | no | — |
| 190 | C4×C4⋊C4 | — | 43 | 4 | no | — |
| 191 | #340 | — | 43 | 4 | no | — |
| 192 | #336 | — | 43 | 2 | no | — |
| 193 | C43 | #22813 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 194 | C22×D4 | — | 43 | 2 | no | — |
| 195 | C23×C4 | — | 86 | 2 | no | — |
| 196 | C2×D8 | — | 43 | 4 | no | — |
| 197 | C2×D8 | — | 86 | 2 | no | — |
| 198 | C2×D8 | — | 86 | 2 | no | — |
| 199 | C22×C8 | — | 43 | 2 | no | — |
| 200 | C22×C8 | C22×D43 | 1 | 1 | yes | characteristic |
| 201 | C22×C8 | C22×D43 | 1 | 1 | yes | characteristic |
| 202 | C4×D4 | — | 43 | 8 | no | — |
| 203 | C4×D4 | — | 86 | 4 | no | — |
| 204 | #117 | — | 172 | 8 | no | — |
| 205 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 206 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 207 | C2×C4⋊C4 | — | 43 | 2 | no | — |
| 208 | C2×C4⋊C4 | — | 43 | 2 | no | — |
| 209 | C2×C4⋊C4 | C22×D43 | 1 | 4 | yes | — |
| 210 | C2×C22⋊C4 | — | 86 | 2 | no | — |
| 211 | C2×C22⋊C4 | — | 86 | 4 | no | — |
| 212 | C2×C22⋊C4 | — | 86 | 4 | no | — |
| 213 | C2×C42 | — | 43 | 2 | no | — |
| 214 | C2×C42 | — | 86 | 4 | no | — |
| 215 | C2×C42 | C22×D43 | 1 | 1 | yes | characteristic |
| 216 | #107 | — | 43 | 4 | no | — |
| 217 | #106 | — | 2 | 4 | no | — |
| 218 | #106 | — | 4 | 2 | no | — |
| 219 | C4⋊C8 | — | 86 | 2 | no | — |
| 220 | #102 | — | 43 | 8 | no | — |
| 221 | #102 | — | 172 | 4 | no | — |
| 222 | #97 | — | 86 | 4 | no | — |
| 223 | C4×C8 | C4×D43 | 1 | 4 | yes | — |
| 224 | #95 | — | 43 | 2 | no | — |
| 225 | #95 | — | 86 | 4 | no | — |
| 226 | C24 | — | 86 | 2 | no | — |
| 227 | C2×D4 | — | 43 | 4 | no | — |
| 228 | C2×D4 | — | 86 | 2 | no | — |
| 229 | C2×D4 | — | 86 | 2 | no | — |
| 230 | C2×D4 | — | 86 | 2 | no | — |
| 231 | C2×D4 | — | 86 | 2 | no | — |
| 232 | C2×D4 | — | 86 | 2 | no | — |
| 233 | C22×C4 | — | 2 | 4 | no | — |
| 234 | C22×C4 | — | 43 | 4 | no | — |
| 235 | C22×C4 | — | 86 | 2 | no | — |
| 236 | C22×C4 | — | 86 | 8 | no | — |
| 237 | C22×C4 | — | 172 | 4 | no | — |
| 238 | C22×C4 | — | 172 | 4 | no | — |
| 239 | C22×C4 | D4×D43 | 1 | 1 | yes | characteristic |
| 240 | C22×C4 | D4×D43 | 1 | 1 | yes | characteristic |
| 241 | C22×C4 | C23×D43 | 1 | 1 | yes | characteristic |
| 242 | D8 | — | 86 | 8 | no | — |
| 243 | C2×C8 | — | 2 | 1 | no | — |
| 244 | C2×C8 | — | 2 | 1 | no | — |
| 245 | C2×C8 | — | 86 | 2 | no | — |
| 246 | C2×C8 | — | 86 | 2 | no | — |
| 247 | C2×C8 | — | 86 | 2 | no | — |
| 248 | C2×C8 | C2×C4×D43 | 1 | 2 | yes | — |
| 249 | C2×C8 | C2×C4×D43 | 1 | 2 | yes | — |
| 250 | C2×C8 | #10621900 | 1 | 2 | yes | — |
| 251 | C2×C8 | #10621902 | 1 | 2 | yes | — |
| 252 | C4⋊C4 | — | 2 | 4 | no | — |
| 253 | C4⋊C4 | — | 2 | 4 | no | — |
| 254 | C4⋊C4 | — | 4 | 4 | no | — |
| 255 | C4⋊C4 | — | 8 | 2 | no | — |
| 256 | C4⋊C4 | — | 43 | 4 | no | — |
| 257 | C4⋊C4 | — | 86 | 4 | no | — |
| 258 | C4⋊C4 | — | 86 | 2 | no | — |
| 259 | C22⋊C4 | — | 86 | 8 | no | — |
| 260 | C22⋊C4 | — | 344 | 4 | no | — |
| 261 | C22⋊C4 | — | 172 | 8 | no | — |
| 262 | C42 | — | 86 | 2 | no | — |
| 263 | C42 | — | 86 | 2 | no | — |
| 264 | C42 | — | 172 | 8 | no | — |
| 265 | C42 | C2×C4×D43 | 1 | 2 | yes | — |
| 266 | C42 | #10621902 | 1 | 2 | yes | — |
| 267 | C23 | — | 86 | 4 | no | — |
| 268 | C23 | — | 172 | 2 | no | — |
| 269 | C23 | — | 172 | 2 | no | — |
| 270 | C23 | — | 172 | 2 | no | — |
| 271 | C23 | — | 172 | 2 | no | — |
| 272 | C23 | — | 172 | 2 | no | — |
| 273 | C23 | — | 172 | 2 | no | — |
| 274 | C23 | C2×D4×D43 | 1 | 1 | yes | characteristic |
| 275 | D4 | — | 86 | 4 | no | — |
| 276 | D4 | — | 86 | 4 | no | — |
| 277 | D4 | — | 172 | 2 | no | — |
| 278 | D4 | — | 172 | 2 | no | — |
| 279 | C2×C4 | — | 2 | 2 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 4 | 4 | no | — |
| 282 | C2×C4 | — | 4 | 4 | no | — |
| 283 | C2×C4 | — | 86 | 4 | no | — |
| 284 | C2×C4 | — | 86 | 4 | no | — |
| 285 | C2×C4 | — | 86 | 4 | no | — |
| 286 | C2×C4 | — | 86 | 4 | no | — |
| 287 | C2×C4 | — | 172 | 2 | no | — |
| 288 | C2×C4 | — | 172 | 2 | no | — |
| 289 | C2×C4 | — | 172 | 16 | no | — |
| 290 | C2×C4 | #13063287 | 1 | 2 | yes | — |
| 291 | C2×C4 | #13063287 | 1 | 2 | yes | — |
| 292 | C2×C4 | #13063352 | 1 | 4 | yes | — |
| 293 | C2×C4 | C22×C4×D43 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×#10621900 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | C2×#10621902 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C2×#10621902 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #13063370 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #13063373 | 1 | 1 | yes | characteristic |
| 299 | C8 | — | 2 | 2 | no | — |
| 300 | C8 | — | 172 | 2 | no | — |
| 301 | C8 | #13063283 | 1 | 4 | yes | complemented |
| 302 | C22 | — | 344 | 2 | no | — |
| 303 | C22 | — | 344 | 2 | no | — |
| 304 | C22 | — | 344 | 2 | no | — |
| 305 | C22 | — | 344 | 2 | no | — |
| 306 | C22 | — | 172 | 4 | no | — |
| 307 | C22 | — | 172 | 4 | no | — |
| 308 | C22 | — | 172 | 4 | no | — |
| 309 | C22 | C2×#13063287 | 1 | 1 | yes | characteristic |
| 310 | C22 | #14583552 | 1 | 1 | yes | characteristic |
| 311 | C22 | #14583569 | 1 | 1 | yes | characteristic |
| 312 | C22 | C2×#13063352 | 1 | 2 | yes | — |
| 313 | C22 | #14583705 | 1 | 2 | yes | — |
| 314 | C4 | — | 2 | 2 | no | — |
| 315 | C4 | — | 8 | 4 | no | — |
| 316 | C4 | — | 172 | 4 | no | — |
| 317 | C4 | — | 172 | 4 | no | — |
| 318 | C4 | #14583010 | 1 | 4 | yes | — |
| 319 | C4 | C2×#13063283 | 1 | 1 | yes | characteristic |
| 320 | C4 | #14583380 | 1 | 1 | yes | characteristic |
| 321 | C4 | #14583615 | 1 | 1 | yes | characteristic |
| 322 | C4 | #14583623 | 1 | 1 | yes | characteristic |
| 323 | C2 | — | 344 | 4 | no | — |
| 324 | C2 | #19137546 | 1 | 1 | yes | characteristic |
| 325 | C2 | C2×#14583010 | 1 | 1 | yes | characteristic |
| 326 | C2 | #19143478 | 1 | 1 | yes | characteristic |
| 327 | C2 | #19147189 | 1 | 2 | yes | — |
| 328 | C2 | #19147620 | 1 | 2 | yes | — |
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