← #11659535
Subgroups — #11659535
Order 768.
2604 conjugacy classes of subgroups.
The subgroup lattice has 340 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #87163 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #87160 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C3×#2875 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #83789 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #82536 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #78580 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #78579 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #78541 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | C2×#4717 | C2 | 1 | 6 | yes | maximal, complemented |
| 10 | #60184 | — | 3 | 1 | no | maximal, Sylow |
| 11 | #5964 | C22 | 1 | 2 | yes | — |
| 12 | C2×C3×#131 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C3×#131 | C22 | 1 | 1 | yes | characteristic |
| 14 | C3×C4×#41 | C22 | 1 | 1 | yes | characteristic |
| 15 | #5819 | C22 | 1 | 2 | yes | — |
| 16 | #5753 | C4 | 1 | 4 | yes | complemented |
| 17 | #5752 | C4 | 1 | 4 | yes | complemented |
| 18 | #5751 | C4 | 1 | 8 | yes | complemented |
| 19 | #5750 | — | 2 | 8 | no | — |
| 20 | D3×#131 | — | 2 | 8 | no | — |
| 21 | #5748 | C22 | 1 | 6 | yes | — |
| 22 | #5745 | C22 | 1 | 12 | yes | complemented |
| 23 | C2×#904 | C22 | 1 | 6 | yes | — |
| 24 | C2×#901 | C22 | 1 | 12 | yes | complemented |
| 25 | C3×#405 | C22 | 1 | 3 | yes | complemented |
| 26 | C3×#405 | C22 | 1 | 3 | yes | complemented |
| 27 | C3×C4×M16 | C22 | 1 | 3 | yes | complemented |
| 28 | C2×C3×#97 | C22 | 1 | 3 | yes | complemented |
| 29 | #5125 | C22 | 1 | 6 | yes | complemented |
| 30 | C2×#819 | C22 | 1 | 6 | yes | complemented |
| 31 | #5005 | C22 | 1 | 12 | yes | complemented |
| 32 | C2×#805 | C22 | 1 | 12 | yes | complemented |
| 33 | #4717 | C22 | 1 | 4 | yes | complemented |
| 34 | #4717 | C22 | 1 | 12 | yes | complemented |
| 35 | #4717 | C22 | 1 | 12 | yes | complemented |
| 36 | #4717 | C22 | 1 | 12 | yes | complemented |
| 37 | #4700 | C22 | 1 | 24 | yes | complemented |
| 38 | #3584 | — | 3 | 1 | no | — |
| 39 | #3573 | — | 3 | 2 | no | — |
| 40 | #2957 | — | 3 | 6 | no | — |
| 41 | #2956 | — | 3 | 6 | no | — |
| 42 | #2955 | — | 3 | 6 | no | — |
| 43 | C2×#444 | — | 3 | 6 | no | — |
| 44 | #2916 | — | 3 | 1 | no | — |
| 45 | #2901 | — | 3 | 2 | no | — |
| 46 | #2875 | D3 | 1 | 1 | yes | characteristic, complemented |
| 47 | C2×C3×#41 | C23 | 1 | 1 | yes | characteristic |
| 48 | #1010 | C2×C4 | 1 | 8 | yes | — |
| 49 | #1009 | C2×C4 | 1 | 8 | yes | — |
| 50 | D3×#41 | — | 2 | 8 | no | — |
| 51 | #1007 | — | 2 | 2 | no | — |
| 52 | #1004 | — | 2 | 6 | no | — |
| 53 | C2×#235 | C23 | 1 | 12 | yes | — |
| 54 | C3×#131 | C2×C4 | 1 | 4 | yes | — |
| 55 | C3×#131 | C2×C4 | 1 | 4 | yes | — |
| 56 | C3×#131 | D4 | 1 | 8 | yes | — |
| 57 | C2×C3×M16 | C23 | 1 | 3 | yes | — |
| 58 | C2×C3×M16 | C23 | 1 | 3 | yes | — |
| 59 | C22×C3×C8 | C23 | 1 | 3 | yes | — |
| 60 | C22×C3×C8 | C23 | 1 | 3 | yes | — |
| 61 | C3×C4×Q8 | — | 2 | 1 | no | — |
| 62 | C3×C4×D4 | — | 2 | 3 | no | — |
| 63 | C3×#117 | C23 | 1 | 3 | yes | — |
| 64 | C2×C3×C42 | C23 | 1 | 3 | yes | — |
| 65 | #948 | — | 2 | 8 | no | — |
| 66 | C2×#208 | C23 | 1 | 12 | yes | — |
| 67 | #909 | C2×C4 | 1 | 24 | yes | complemented |
| 68 | #908 | C2×C4 | 1 | 24 | yes | complemented |
| 69 | #907 | — | 2 | 24 | no | — |
| 70 | D3×M16 | — | 2 | 24 | no | — |
| 71 | C2×Dic12 | C2×C4 | 1 | 6 | yes | complemented |
| 72 | #904 | C2×C4 | 1 | 12 | yes | complemented |
| 73 | #904 | C2×C4 | 1 | 12 | yes | complemented |
| 74 | #904 | C23 | 1 | 12 | yes | — |
| 75 | #904 | C23 | 1 | 4 | yes | — |
| 76 | C2×D24 | C2×C4 | 1 | 6 | yes | complemented |
| 77 | C2×#204 | C2×C4 | 1 | 12 | yes | complemented |
| 78 | #901 | — | 2 | 12 | no | — |
| 79 | #901 | — | 2 | 12 | no | — |
| 80 | #901 | C23 | 1 | 24 | yes | — |
| 81 | #901 | C23 | 1 | 8 | yes | — |
| 82 | C2×ΓL(1,𝔽25) | — | 2 | 12 | no | — |
| 83 | C2×D3×C8 | — | 2 | 12 | no | — |
| 84 | C3×#97 | C23 | 1 | 1 | yes | characteristic |
| 85 | C3×#97 | C23 | 1 | 3 | yes | — |
| 86 | C3×#97 | C23 | 1 | 3 | yes | — |
| 87 | C3×#97 | C23 | 1 | 3 | yes | — |
| 88 | C3×C4×C8 | C23 | 1 | 6 | yes | — |
| 89 | #819 | C23 | 1 | 12 | yes | — |
| 90 | #819 | C23 | 1 | 4 | yes | — |
| 91 | #805 | C23 | 1 | 8 | yes | — |
| 92 | #805 | C23 | 1 | 24 | yes | — |
| 93 | #585 | — | 3 | 2 | no | — |
| 94 | #578 | — | 3 | 4 | no | — |
| 95 | #577 | — | 3 | 8 | no | — |
| 96 | #576 | — | 3 | 4 | no | — |
| 97 | #575 | — | 3 | 6 | no | — |
| 98 | C2×#135 | — | 3 | 6 | no | — |
| 99 | #568 | — | 3 | 2 | no | — |
| 100 | #568 | — | 3 | 12 | no | — |
| 101 | #568 | — | 6 | 8 | no | — |
| 102 | C2×#131 | — | 3 | 12 | no | — |
| 103 | C2×#131 | — | 6 | 8 | no | — |
| 104 | C2×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 105 | C2×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 106 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 107 | #444 | — | 3 | 4 | no | — |
| 108 | #444 | — | 3 | 12 | no | — |
| 109 | #444 | — | 3 | 12 | no | — |
| 110 | #444 | — | 3 | 12 | no | — |
| 111 | #443 | — | 3 | 24 | no | — |
| 112 | #430 | — | 3 | 6 | no | — |
| 113 | C2×#108 | — | 3 | 6 | no | — |
| 114 | #421 | — | 3 | 12 | no | — |
| 115 | C2×#104 | — | 3 | 12 | no | — |
| 116 | #405 | C2×D3 | 1 | 3 | yes | complemented |
| 117 | #405 | C2×D3 | 1 | 3 | yes | complemented |
| 118 | C4×M16 | C2×D3 | 1 | 3 | yes | complemented |
| 119 | C2×#97 | C2×D3 | 1 | 3 | yes | complemented |
| 120 | C3×#41 | C22×C4 | 1 | 4 | yes | — |
| 121 | C3×#41 | C2×D4 | 1 | 4 | yes | — |
| 122 | C2×C3×Q8 | #41 | 1 | 1 | yes | characteristic |
| 123 | C2×C3×D4 | #41 | 1 | 3 | yes | — |
| 124 | C22×C3×C4 | — | 2 | 3 | no | — |
| 125 | C22×C3×C4 | C24 | 1 | 3 | yes | — |
| 126 | C2×#67 | — | 2 | 12 | no | — |
| 127 | #239 | — | 2 | 8 | no | — |
| 128 | D3×Q8 | — | 2 | 8 | no | — |
| 129 | #237 | — | 2 | 24 | no | — |
| 130 | D3×D4 | — | 2 | 24 | no | — |
| 131 | #235 | — | 2 | 12 | no | — |
| 132 | #235 | — | 2 | 12 | no | — |
| 133 | #235 | C22×C4 | 1 | 24 | yes | — |
| 134 | #235 | C24 | 1 | 8 | yes | — |
| 135 | C2×D12 | C22×C4 | 1 | 12 | yes | — |
| 136 | C2×C4×D3 | — | 2 | 12 | no | — |
| 137 | C2×Dic6 | C22×C4 | 1 | 12 | yes | — |
| 138 | C3×M16 | C22×C4 | 1 | 12 | yes | — |
| 139 | C3×M16 | C2×D4 | 1 | 12 | yes | — |
| 140 | C2×C3×C8 | C22×C4 | 1 | 6 | yes | — |
| 141 | C2×C3×C8 | C22×C4 | 1 | 6 | yes | — |
| 142 | C2×C3×C8 | C2×D4 | 1 | 12 | yes | — |
| 143 | C2×C3×C8 | C24 | 1 | 3 | yes | — |
| 144 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 145 | C2×C3×C8 | C24 | 1 | 3 | yes | — |
| 146 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 147 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 148 | C3×C22⋊C4 | — | 2 | 3 | no | — |
| 149 | C3×C42 | — | 2 | 3 | no | — |
| 150 | C3×C42 | C24 | 1 | 3 | yes | — |
| 151 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 152 | #208 | — | 2 | 12 | no | — |
| 153 | #208 | C24 | 1 | 8 | yes | — |
| 154 | C2×C3⋊C8 | — | 2 | 12 | no | — |
| 155 | Dic12 | C22×C4 | 1 | 4 | yes | — |
| 156 | Dic12 | C22×C4 | 1 | 12 | yes | — |
| 157 | D24 | C22×C4 | 1 | 4 | yes | — |
| 158 | D24 | C22×C4 | 1 | 12 | yes | — |
| 159 | #204 | C22×C4 | 1 | 8 | yes | — |
| 160 | #204 | C22×C4 | 1 | 24 | yes | — |
| 161 | ΓL(1,𝔽25) | — | 2 | 8 | no | — |
| 162 | ΓL(1,𝔽25) | — | 2 | 24 | no | — |
| 163 | D3×C8 | — | 2 | 8 | no | — |
| 164 | D3×C8 | — | 2 | 24 | no | — |
| 165 | 2−1+4 | — | 3 | 8 | no | — |
| 166 | 2−1+4 | — | 6 | 2 | no | — |
| 167 | H(5,𝔽2) | — | 3 | 8 | no | — |
| 168 | H(5,𝔽2) | — | 6 | 6 | no | — |
| 169 | C2×#41 | — | 3 | 12 | no | — |
| 170 | C2×#41 | — | 6 | 8 | no | — |
| 171 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 172 | #137 | — | 3 | 24 | no | — |
| 173 | AGL(1,ℤ/8) | — | 3 | 24 | no | — |
| 174 | #135 | — | 3 | 12 | no | — |
| 175 | #135 | — | 3 | 12 | no | — |
| 176 | #135 | — | 3 | 4 | no | — |
| 177 | #135 | — | 3 | 12 | no | — |
| 178 | C2×Q16 | — | 3 | 6 | no | — |
| 179 | C2×SD16 | — | 3 | 12 | no | — |
| 180 | C2×D8 | — | 3 | 6 | no | — |
| 181 | #131 | — | 3 | 8 | no | — |
| 182 | #131 | — | 3 | 24 | no | — |
| 183 | #131 | — | 6 | 8 | no | — |
| 184 | #131 | — | 6 | 24 | no | — |
| 185 | #131 | — | 6 | 12 | no | — |
| 186 | #131 | — | 6 | 12 | no | — |
| 187 | #131 | C4×D3 | 1 | 4 | yes | — |
| 188 | #131 | C4×D3 | 1 | 4 | yes | — |
| 189 | #131 | D12 | 1 | 8 | yes | — |
| 190 | C2×M16 | — | 3 | 12 | no | — |
| 191 | C2×M16 | — | 6 | 24 | no | — |
| 192 | C2×M16 | — | 6 | 12 | no | — |
| 193 | C2×M16 | C22×D3 | 1 | 3 | yes | — |
| 194 | C2×M16 | C22×D3 | 1 | 3 | yes | — |
| 195 | C22×C8 | — | 6 | 12 | no | — |
| 196 | C22×C8 | C22×D3 | 1 | 3 | yes | — |
| 197 | C22×C8 | C22×D3 | 1 | 3 | yes | — |
| 198 | C4×Q8 | — | 2 | 1 | no | — |
| 199 | C4×D4 | — | 2 | 3 | no | — |
| 200 | #117 | C22×D3 | 1 | 3 | yes | — |
| 201 | C2×C42 | C22×D3 | 1 | 3 | yes | — |
| 202 | #108 | — | 3 | 4 | no | — |
| 203 | #108 | — | 3 | 12 | no | — |
| 204 | #104 | — | 3 | 8 | no | — |
| 205 | #104 | — | 3 | 24 | no | — |
| 206 | #97 | C22×D3 | 1 | 1 | yes | characteristic |
| 207 | #97 | C22×D3 | 1 | 3 | yes | — |
| 208 | #97 | C22×D3 | 1 | 3 | yes | — |
| 209 | #97 | C22×D3 | 1 | 3 | yes | — |
| 210 | C4×C8 | C22×D3 | 1 | 6 | yes | — |
| 211 | C23×C3 | C2×#41 | 1 | 3 | yes | — |
| 212 | C22×D3 | — | 2 | 12 | no | — |
| 213 | C3×Q8 | C4×D4 | 1 | 4 | yes | — |
| 214 | C3×D4 | C4×D4 | 1 | 12 | yes | — |
| 215 | C2×C3×C4 | — | 2 | 1 | no | — |
| 216 | C2×C3×C4 | — | 2 | 3 | no | — |
| 217 | C2×C3×C4 | — | 8 | 3 | no | — |
| 218 | C2×C3×C4 | C23×C4 | 1 | 6 | yes | — |
| 219 | C2×C3×C4 | C22×D4 | 1 | 6 | yes | — |
| 220 | C2×C3×C4 | C2×#41 | 1 | 3 | yes | — |
| 221 | C2×C3×C4 | C25 | 1 | 1 | yes | characteristic |
| 222 | #67 | — | 2 | 24 | no | — |
| 223 | #67 | — | 2 | 8 | no | — |
| 224 | C2×Dic3 | — | 2 | 12 | no | — |
| 225 | D12 | — | 2 | 12 | no | — |
| 226 | D12 | C23×C4 | 1 | 8 | yes | — |
| 227 | C4×D3 | — | 2 | 24 | no | — |
| 228 | C4×D3 | — | 2 | 8 | no | — |
| 229 | Dic6 | — | 2 | 12 | no | — |
| 230 | Dic6 | C23×C4 | 1 | 8 | yes | — |
| 231 | C3×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 232 | C3×C8 | C23×C4 | 1 | 6 | yes | — |
| 233 | C3×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 234 | C3×C8 | C22×D4 | 1 | 2 | yes | — |
| 235 | C3×C8 | C22×D4 | 1 | 6 | yes | — |
| 236 | C3⋊C8 | — | 2 | 8 | no | — |
| 237 | #41 | — | 3 | 24 | no | — |
| 238 | #41 | — | 3 | 8 | no | — |
| 239 | #41 | — | 6 | 8 | no | — |
| 240 | #41 | — | 6 | 24 | no | — |
| 241 | #41 | — | 6 | 12 | no | — |
| 242 | #41 | — | 6 | 12 | no | — |
| 243 | #41 | C2×C4×D3 | 1 | 4 | yes | — |
| 244 | #41 | C2×D12 | 1 | 4 | yes | — |
| 245 | C2×Q8 | — | 3 | 12 | no | — |
| 246 | C2×Q8 | — | 6 | 8 | no | — |
| 247 | C2×Q8 | #235 | 1 | 1 | yes | characteristic |
| 248 | C2×D4 | — | 3 | 12 | no | — |
| 249 | C2×D4 | — | 6 | 12 | no | — |
| 250 | C2×D4 | — | 6 | 24 | no | — |
| 251 | C2×D4 | #235 | 1 | 3 | yes | — |
| 252 | C22×C4 | — | 2 | 3 | no | — |
| 253 | C22×C4 | — | 6 | 12 | no | — |
| 254 | C22×C4 | C23×D3 | 1 | 3 | yes | — |
| 255 | Q16 | — | 3 | 4 | no | — |
| 256 | Q16 | — | 3 | 12 | no | — |
| 257 | SD16 | — | 3 | 8 | no | — |
| 258 | SD16 | — | 3 | 24 | no | — |
| 259 | D8 | — | 3 | 4 | no | — |
| 260 | D8 | — | 3 | 12 | no | — |
| 261 | M16 | — | 3 | 8 | no | — |
| 262 | M16 | — | 6 | 12 | no | — |
| 263 | M16 | — | 6 | 8 | no | — |
| 264 | M16 | — | 6 | 24 | no | — |
| 265 | M16 | C2×C4×D3 | 1 | 12 | yes | — |
| 266 | M16 | C2×D12 | 1 | 12 | yes | — |
| 267 | C2×C8 | — | 6 | 12 | no | — |
| 268 | C2×C8 | — | 6 | 8 | no | — |
| 269 | C2×C8 | — | 6 | 24 | no | — |
| 270 | C2×C8 | C2×C4×D3 | 1 | 6 | yes | — |
| 271 | C2×C8 | C2×C4×D3 | 1 | 6 | yes | — |
| 272 | C2×C8 | C2×D12 | 1 | 12 | yes | — |
| 273 | C2×C8 | C23×D3 | 1 | 3 | yes | — |
| 274 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 275 | C2×C8 | C23×D3 | 1 | 3 | yes | — |
| 276 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 277 | C4⋊C4 | — | 2 | 3 | no | — |
| 278 | C22⋊C4 | — | 2 | 3 | no | — |
| 279 | C42 | — | 2 | 3 | no | — |
| 280 | C42 | C23×D3 | 1 | 3 | yes | — |
| 281 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 282 | C22×C3 | — | 4 | 3 | no | — |
| 283 | C22×C3 | C2×C4×D4 | 1 | 6 | yes | — |
| 284 | C22×C3 | C22×#41 | 1 | 1 | yes | characteristic |
| 285 | C2×D3 | — | 2 | 8 | no | — |
| 286 | C2×D3 | — | 8 | 12 | no | — |
| 287 | C3×C4 | — | 4 | 3 | no | — |
| 288 | C3×C4 | — | 4 | 1 | no | — |
| 289 | C3×C4 | C2×C4×D4 | 1 | 6 | yes | — |
| 290 | C3×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 291 | C3×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 292 | Dic3 | — | 2 | 8 | no | — |
| 293 | C23 | — | 6 | 12 | no | — |
| 294 | C23 | C2×#235 | 1 | 3 | yes | — |
| 295 | Q8 | — | 3 | 8 | no | — |
| 296 | Q8 | — | 6 | 12 | no | — |
| 297 | Q8 | C4×D12 | 1 | 4 | yes | — |
| 298 | D4 | — | 3 | 8 | no | — |
| 299 | D4 | — | 6 | 8 | no | — |
| 300 | D4 | — | 6 | 24 | no | — |
| 301 | D4 | — | 6 | 12 | no | — |
| 302 | D4 | C4×D12 | 1 | 12 | yes | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 3 | no | — |
| 305 | C2×C4 | — | 6 | 12 | no | — |
| 306 | C2×C4 | — | 6 | 8 | no | — |
| 307 | C2×C4 | — | 6 | 24 | no | — |
| 308 | C2×C4 | — | 8 | 3 | no | — |
| 309 | C2×C4 | C22×C4×D3 | 1 | 6 | yes | — |
| 310 | C2×C4 | C22×D12 | 1 | 6 | yes | — |
| 311 | C2×C4 | C2×#235 | 1 | 3 | yes | — |
| 312 | C2×C4 | C24×D3 | 1 | 1 | yes | characteristic |
| 313 | C8 | — | 6 | 8 | no | — |
| 314 | C8 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 315 | C8 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 316 | C8 | C22×C4×D3 | 1 | 6 | yes | — |
| 317 | C8 | C22×D12 | 1 | 2 | yes | — |
| 318 | C8 | C22×D12 | 1 | 6 | yes | — |
| 319 | C2×C3 | — | 2 | 1 | no | — |
| 320 | C2×C3 | — | 2 | 6 | no | — |
| 321 | C2×C3 | C22×C4×D4 | 1 | 1 | yes | characteristic |
| 322 | D3 | — | 4 | 8 | no | — |
| 323 | C22 | — | 4 | 3 | no | — |
| 324 | C22 | — | 6 | 8 | no | — |
| 325 | C22 | — | 24 | 12 | no | — |
| 326 | C22 | C2×C4×D12 | 1 | 6 | yes | — |
| 327 | C22 | C22×#235 | 1 | 1 | yes | characteristic |
| 328 | C4 | — | 4 | 1 | no | — |
| 329 | C4 | — | 4 | 3 | no | — |
| 330 | C4 | — | 6 | 8 | no | — |
| 331 | C4 | C2×C4×D12 | 1 | 6 | yes | — |
| 332 | C4 | C23×C4×D3 | 1 | 1 | yes | characteristic |
| 333 | C4 | C23×D12 | 1 | 1 | yes | characteristic |
| 334 | C3 | #60184 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 335 | C2 | — | 2 | 1 | no | — |
| 336 | C2 | — | 2 | 6 | no | — |
| 337 | C2 | — | 12 | 8 | no | — |
| 338 | C2 | C22×C4×D12 | 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