← #11709055
Subgroups — #11709055
Order 768.
10796 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #87574 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #87572 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#3573 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #87168 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | #87167 | C2 | 1 | 15 | yes | maximal, complemented |
| 6 | #87163 | C2 | 1 | 15 | yes | maximal, complemented |
| 7 | C2×#5751 | C2 | 1 | 15 | yes | maximal, complemented |
| 8 | #63092 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×#585 | C22 | 1 | 1 | yes | characteristic |
| 10 | #5968 | — | 2 | 6 | no | — |
| 11 | D3×2−1+4 | — | 2 | 6 | no | — |
| 12 | #5966 | — | 2 | 10 | no | — |
| 13 | D3×H(5,𝔽2) | — | 2 | 10 | no | — |
| 14 | #5964 | C22 | 1 | 15 | yes | complemented |
| 15 | #5964 | C22 | 1 | 15 | yes | complemented |
| 16 | C2×#1010 | C22 | 1 | 15 | yes | complemented |
| 17 | C2×#1009 | C22 | 1 | 15 | yes | complemented |
| 18 | C3×#568 | C22 | 1 | 15 | yes | complemented |
| 19 | C2×C3×#131 | C22 | 1 | 15 | yes | complemented |
| 20 | #5753 | C22 | 1 | 60 | yes | complemented |
| 21 | #5752 | C22 | 1 | 60 | yes | complemented |
| 22 | #5751 | C22 | 1 | 60 | yes | complemented |
| 23 | #5751 | C22 | 1 | 120 | yes | complemented |
| 24 | #5751 | C22 | 1 | 20 | yes | complemented |
| 25 | #5748 | C22 | 1 | 90 | yes | complemented |
| 26 | C2×#909 | C22 | 1 | 45 | yes | complemented |
| 27 | C2×#908 | C22 | 1 | 45 | yes | complemented |
| 28 | C2×#904 | C22 | 1 | 45 | yes | complemented |
| 29 | C22×#204 | C22 | 1 | 15 | yes | complemented |
| 30 | 2−1+6 | — | 3 | 1 | no | — |
| 31 | H(7,𝔽2) | — | 3 | 1 | no | — |
| 32 | #3586 | — | 3 | 15 | no | — |
| 33 | #3585 | — | 3 | 15 | no | — |
| 34 | #3584 | — | 3 | 15 | no | — |
| 35 | C2×#577 | — | 3 | 15 | no | — |
| 36 | #3573 | D3 | 1 | 1 | yes | characteristic, complemented |
| 37 | C3×2−1+4 | D4 | 1 | 6 | yes | — |
| 38 | C3×H(5,𝔽2) | D4 | 1 | 10 | yes | — |
| 39 | C2×C3×#41 | C23 | 1 | 15 | yes | — |
| 40 | #1010 | — | 2 | 60 | no | — |
| 41 | #1010 | C23 | 1 | 20 | yes | — |
| 42 | #1010 | C23 | 1 | 60 | yes | complemented |
| 43 | #1009 | — | 2 | 60 | no | — |
| 44 | #1009 | C23 | 1 | 60 | yes | — |
| 45 | #1009 | C23 | 1 | 20 | yes | complemented |
| 46 | D3×#41 | — | 2 | 60 | no | — |
| 47 | D3×#41 | — | 2 | 60 | no | — |
| 48 | #1007 | — | 2 | 15 | no | — |
| 49 | #1007 | — | 2 | 15 | no | — |
| 50 | C2×#239 | — | 2 | 15 | no | — |
| 51 | C2×D3×Q8 | — | 2 | 15 | no | — |
| 52 | #1004 | — | 2 | 45 | no | — |
| 53 | #1004 | — | 2 | 45 | no | — |
| 54 | C2×#237 | — | 2 | 45 | no | — |
| 55 | C2×D3×D4 | — | 2 | 45 | no | — |
| 56 | C2×#235 | C23 | 1 | 45 | yes | — |
| 57 | C2×#235 | C23 | 1 | 45 | yes | complemented |
| 58 | C22×D12 | C23 | 1 | 15 | yes | — |
| 59 | C22×Dic6 | C23 | 1 | 15 | yes | complemented |
| 60 | C3×#131 | C23 | 1 | 20 | yes | complemented |
| 61 | C3×#131 | C23 | 1 | 60 | yes | complemented |
| 62 | C2×C3×M16 | C23 | 1 | 45 | yes | complemented |
| 63 | C22×C3×C8 | C23 | 1 | 15 | yes | complemented |
| 64 | #909 | C23 | 1 | 180 | yes | complemented |
| 65 | #909 | C23 | 1 | 60 | yes | complemented |
| 66 | #908 | C23 | 1 | 180 | yes | complemented |
| 67 | #908 | C23 | 1 | 60 | yes | complemented |
| 68 | C2×Dic12 | C23 | 1 | 45 | yes | complemented |
| 69 | #904 | C23 | 1 | 60 | yes | complemented |
| 70 | #904 | C23 | 1 | 180 | yes | complemented |
| 71 | C2×D24 | C23 | 1 | 45 | yes | complemented |
| 72 | C2×#204 | C23 | 1 | 45 | yes | complemented |
| 73 | C2×#204 | C23 | 1 | 90 | yes | complemented |
| 74 | C2×#204 | C23 | 1 | 15 | yes | complemented |
| 75 | #585 | — | 3 | 15 | no | — |
| 76 | #585 | — | 3 | 15 | no | — |
| 77 | #585 | — | 6 | 6 | no | — |
| 78 | #585 | — | 6 | 10 | no | — |
| 79 | #585 | C2×D3 | 1 | 1 | yes | characteristic |
| 80 | C2×2−1+4 | — | 3 | 15 | no | — |
| 81 | C2×2−1+4 | — | 6 | 6 | no | — |
| 82 | C2×H(5,𝔽2) | — | 3 | 15 | no | — |
| 83 | C2×H(5,𝔽2) | — | 6 | 10 | no | — |
| 84 | #578 | — | 3 | 60 | no | — |
| 85 | #577 | — | 3 | 20 | no | — |
| 86 | #577 | — | 3 | 60 | no | — |
| 87 | #577 | — | 3 | 120 | no | — |
| 88 | #576 | — | 3 | 60 | no | — |
| 89 | #575 | — | 3 | 90 | no | — |
| 90 | C2×#137 | — | 3 | 45 | no | — |
| 91 | C2×AGL(1,ℤ/8) | — | 3 | 45 | no | — |
| 92 | C2×#135 | — | 3 | 45 | no | — |
| 93 | C22×SD16 | — | 3 | 15 | no | — |
| 94 | #568 | C2×D3 | 1 | 15 | yes | complemented |
| 95 | C2×#131 | C2×D3 | 1 | 15 | yes | complemented |
| 96 | C23×D3 | — | 2 | 15 | no | — |
| 97 | C3×#41 | C2×D4 | 1 | 60 | yes | — |
| 98 | C3×#41 | C24 | 1 | 20 | yes | — |
| 99 | C2×C3×Q8 | C2×D4 | 1 | 15 | yes | — |
| 100 | C2×C3×D4 | C2×D4 | 1 | 45 | yes | — |
| 101 | C22×C3×C4 | C24 | 1 | 15 | yes | — |
| 102 | C2×#67 | — | 2 | 45 | no | — |
| 103 | C2×#67 | — | 2 | 45 | no | — |
| 104 | C22×Dic3 | — | 2 | 15 | no | — |
| 105 | #239 | — | 2 | 60 | no | — |
| 106 | #239 | — | 2 | 20 | no | — |
| 107 | D3×Q8 | — | 2 | 60 | no | — |
| 108 | D3×Q8 | — | 2 | 20 | no | — |
| 109 | #237 | — | 2 | 180 | no | — |
| 110 | #237 | — | 2 | 60 | no | — |
| 111 | D3×D4 | — | 2 | 180 | no | — |
| 112 | D3×D4 | — | 2 | 60 | no | — |
| 113 | #235 | — | 2 | 90 | no | — |
| 114 | #235 | — | 2 | 90 | no | — |
| 115 | #235 | C24 | 1 | 60 | yes | — |
| 116 | #235 | C24 | 1 | 60 | yes | — |
| 117 | C2×D12 | — | 2 | 45 | no | — |
| 118 | C2×D12 | C24 | 1 | 45 | yes | — |
| 119 | C2×D12 | C24 | 1 | 15 | yes | — |
| 120 | C2×C4×D3 | — | 2 | 45 | no | — |
| 121 | C2×C4×D3 | — | 2 | 45 | no | — |
| 122 | C2×Dic6 | — | 2 | 45 | no | — |
| 123 | C2×Dic6 | C24 | 1 | 45 | yes | — |
| 124 | C2×Dic6 | C24 | 1 | 15 | yes | — |
| 125 | C3×M16 | C24 | 1 | 60 | yes | — |
| 126 | C2×C3×C8 | C24 | 1 | 15 | yes | — |
| 127 | C2×C3×C8 | C24 | 1 | 45 | yes | — |
| 128 | Dic12 | C24 | 1 | 60 | yes | — |
| 129 | D24 | C24 | 1 | 60 | yes | — |
| 130 | #204 | C24 | 1 | 10 | yes | — |
| 131 | #204 | C24 | 1 | 6 | yes | — |
| 132 | #204 | C24 | 1 | 90 | yes | — |
| 133 | #204 | C24 | 1 | 30 | yes | — |
| 134 | 2−1+4 | — | 3 | 60 | no | — |
| 135 | 2−1+4 | — | 3 | 20 | no | — |
| 136 | 2−1+4 | — | 6 | 60 | no | — |
| 137 | 2−1+4 | — | 6 | 15 | no | — |
| 138 | 2−1+4 | — | 6 | 15 | no | — |
| 139 | 2−1+4 | D12 | 1 | 6 | yes | — |
| 140 | H(5,𝔽2) | — | 3 | 20 | no | — |
| 141 | H(5,𝔽2) | — | 3 | 60 | no | — |
| 142 | H(5,𝔽2) | — | 6 | 60 | no | — |
| 143 | H(5,𝔽2) | — | 6 | 45 | no | — |
| 144 | H(5,𝔽2) | — | 6 | 45 | no | — |
| 145 | H(5,𝔽2) | D12 | 1 | 10 | yes | — |
| 146 | C2×#41 | — | 3 | 45 | no | — |
| 147 | C2×#41 | — | 3 | 45 | no | — |
| 148 | C2×#41 | — | 6 | 60 | no | — |
| 149 | C2×#41 | — | 6 | 60 | no | — |
| 150 | C2×#41 | — | 6 | 15 | no | — |
| 151 | C2×#41 | — | 6 | 45 | no | — |
| 152 | C2×#41 | C22×D3 | 1 | 15 | yes | — |
| 153 | C22×Q8 | — | 3 | 15 | no | — |
| 154 | C22×Q8 | — | 6 | 15 | no | — |
| 155 | C22×D4 | — | 3 | 15 | no | — |
| 156 | C22×D4 | — | 6 | 45 | no | — |
| 157 | #137 | — | 3 | 180 | no | — |
| 158 | #137 | — | 3 | 60 | no | — |
| 159 | AGL(1,ℤ/8) | — | 3 | 180 | no | — |
| 160 | AGL(1,ℤ/8) | — | 3 | 60 | no | — |
| 161 | #135 | — | 3 | 60 | no | — |
| 162 | #135 | — | 3 | 180 | no | — |
| 163 | C2×Q16 | — | 3 | 45 | no | — |
| 164 | C2×SD16 | — | 3 | 45 | no | — |
| 165 | C2×SD16 | — | 3 | 90 | no | — |
| 166 | C2×SD16 | — | 3 | 15 | no | — |
| 167 | C2×D8 | — | 3 | 45 | no | — |
| 168 | #131 | C22×D3 | 1 | 20 | yes | complemented |
| 169 | #131 | C22×D3 | 1 | 60 | yes | complemented |
| 170 | C2×M16 | C22×D3 | 1 | 45 | yes | complemented |
| 171 | C22×C8 | C22×D3 | 1 | 15 | yes | complemented |
| 172 | C23×C3 | C22×D4 | 1 | 15 | yes | — |
| 173 | C22×D3 | — | 2 | 15 | no | — |
| 174 | C22×D3 | — | 2 | 45 | no | — |
| 175 | C22×D3 | — | 16 | 15 | no | — |
| 176 | C3×Q8 | C22×D4 | 1 | 20 | yes | — |
| 177 | C3×D4 | C22×D4 | 1 | 60 | yes | — |
| 178 | C2×C3×C4 | C22×D4 | 1 | 45 | yes | — |
| 179 | C2×C3×C4 | C25 | 1 | 15 | yes | — |
| 180 | #67 | — | 2 | 60 | no | — |
| 181 | #67 | — | 2 | 60 | no | — |
| 182 | C2×Dic3 | — | 2 | 45 | no | — |
| 183 | C2×Dic3 | — | 2 | 15 | no | — |
| 184 | D12 | — | 2 | 45 | no | — |
| 185 | D12 | — | 2 | 15 | no | — |
| 186 | D12 | C25 | 1 | 10 | yes | — |
| 187 | D12 | C25 | 1 | 6 | yes | — |
| 188 | C4×D3 | — | 2 | 60 | no | — |
| 189 | C4×D3 | — | 2 | 60 | no | — |
| 190 | Dic6 | — | 2 | 45 | no | — |
| 191 | Dic6 | — | 2 | 15 | no | — |
| 192 | Dic6 | C25 | 1 | 10 | yes | — |
| 193 | Dic6 | C25 | 1 | 6 | yes | — |
| 194 | C3×C8 | C25 | 1 | 1 | yes | characteristic |
| 195 | C3×C8 | C25 | 1 | 15 | yes | — |
| 196 | C24 | — | 6 | 15 | no | — |
| 197 | #41 | — | 3 | 60 | no | — |
| 198 | #41 | — | 3 | 60 | no | — |
| 199 | #41 | — | 6 | 60 | no | — |
| 200 | #41 | — | 6 | 20 | no | — |
| 201 | #41 | — | 6 | 60 | no | — |
| 202 | #41 | — | 6 | 180 | no | — |
| 203 | #41 | — | 6 | 90 | no | — |
| 204 | #41 | — | 6 | 90 | no | — |
| 205 | #41 | C2×D12 | 1 | 60 | yes | — |
| 206 | #41 | C23×D3 | 1 | 20 | yes | — |
| 207 | C2×Q8 | — | 3 | 45 | no | — |
| 208 | C2×Q8 | — | 3 | 15 | no | — |
| 209 | C2×Q8 | — | 6 | 60 | no | — |
| 210 | C2×Q8 | — | 6 | 20 | no | — |
| 211 | C2×Q8 | — | 6 | 45 | no | — |
| 212 | C2×Q8 | C2×D12 | 1 | 15 | yes | — |
| 213 | C2×D4 | — | 3 | 15 | no | — |
| 214 | C2×D4 | — | 3 | 45 | no | — |
| 215 | C2×D4 | — | 6 | 45 | no | — |
| 216 | C2×D4 | — | 6 | 45 | no | — |
| 217 | C2×D4 | — | 6 | 180 | no | — |
| 218 | C2×D4 | — | 6 | 60 | no | — |
| 219 | C2×D4 | — | 6 | 45 | no | — |
| 220 | C2×D4 | C2×D12 | 1 | 45 | yes | — |
| 221 | C22×C4 | — | 6 | 15 | no | — |
| 222 | C22×C4 | — | 6 | 45 | no | — |
| 223 | C22×C4 | — | 6 | 45 | no | — |
| 224 | C22×C4 | C23×D3 | 1 | 15 | yes | — |
| 225 | Q16 | — | 3 | 60 | no | — |
| 226 | SD16 | — | 3 | 10 | no | — |
| 227 | SD16 | — | 3 | 6 | no | — |
| 228 | SD16 | — | 3 | 90 | no | — |
| 229 | SD16 | — | 3 | 30 | no | — |
| 230 | D8 | — | 3 | 60 | no | — |
| 231 | M16 | C23×D3 | 1 | 60 | yes | — |
| 232 | C2×C8 | C23×D3 | 1 | 15 | yes | — |
| 233 | C2×C8 | C23×D3 | 1 | 45 | yes | — |
| 234 | C22×C3 | — | 4 | 15 | no | — |
| 235 | C22×C3 | C23×D4 | 1 | 15 | yes | — |
| 236 | C2×D3 | — | 2 | 6 | no | — |
| 237 | C2×D3 | — | 2 | 10 | no | — |
| 238 | C2×D3 | — | 8 | 15 | no | — |
| 239 | C2×D3 | — | 8 | 45 | no | — |
| 240 | C3×C4 | C23×D4 | 1 | 15 | yes | — |
| 241 | C3×C4 | C26 | 1 | 1 | yes | characteristic |
| 242 | Dic3 | — | 2 | 10 | no | — |
| 243 | Dic3 | — | 2 | 6 | no | — |
| 244 | C23 | — | 6 | 15 | no | — |
| 245 | C23 | — | 6 | 45 | no | — |
| 246 | C23 | — | 48 | 15 | no | — |
| 247 | C23 | C22×D12 | 1 | 15 | yes | — |
| 248 | Q8 | — | 3 | 10 | no | — |
| 249 | Q8 | — | 3 | 6 | no | — |
| 250 | Q8 | — | 6 | 45 | no | — |
| 251 | Q8 | — | 6 | 15 | no | — |
| 252 | Q8 | C22×D12 | 1 | 20 | yes | — |
| 253 | D4 | — | 3 | 6 | no | — |
| 254 | D4 | — | 3 | 10 | no | — |
| 255 | D4 | — | 6 | 60 | no | — |
| 256 | D4 | — | 6 | 60 | no | — |
| 257 | D4 | — | 6 | 45 | no | — |
| 258 | D4 | — | 6 | 15 | no | — |
| 259 | D4 | C22×D12 | 1 | 60 | yes | — |
| 260 | C2×C4 | — | 6 | 15 | no | — |
| 261 | C2×C4 | — | 6 | 45 | no | — |
| 262 | C2×C4 | — | 6 | 60 | no | — |
| 263 | C2×C4 | — | 6 | 60 | no | — |
| 264 | C2×C4 | C22×D12 | 1 | 45 | yes | — |
| 265 | C2×C4 | C24×D3 | 1 | 15 | yes | — |
| 266 | C8 | C24×D3 | 1 | 1 | yes | characteristic |
| 267 | C8 | C24×D3 | 1 | 15 | yes | — |
| 268 | C2×C3 | — | 2 | 15 | no | — |
| 269 | C2×C3 | C24×D4 | 1 | 1 | yes | characteristic |
| 270 | D3 | — | 4 | 6 | no | — |
| 271 | D3 | — | 4 | 10 | no | — |
| 272 | C22 | — | 4 | 15 | no | — |
| 273 | C22 | — | 6 | 6 | no | — |
| 274 | C22 | — | 6 | 10 | no | — |
| 275 | C22 | — | 24 | 45 | no | — |
| 276 | C22 | — | 24 | 15 | no | — |
| 277 | C22 | C23×D12 | 1 | 15 | yes | — |
| 278 | C4 | — | 6 | 6 | no | — |
| 279 | C4 | — | 6 | 10 | no | — |
| 280 | C4 | C23×D12 | 1 | 15 | yes | — |
| 281 | C4 | C25×D3 | 1 | 1 | yes | characteristic |
| 282 | C3 | #63092 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 283 | C2 | — | 2 | 15 | no | — |
| 284 | C2 | — | 12 | 10 | no | — |
| 285 | C2 | — | 12 | 6 | no | — |
| 286 | C2 | C24×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