← #10686408
Subgroups — #10686408
Order 768.
394 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×#1767 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#4479 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#4480 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C4×C4⋊(C3⋊C8) | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #69388 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #67703 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67703 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #8279 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×#97 | — | 2 | 1 | no | — |
| 12 | C3×C43 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C4×C3⋊C8 | — | 2 | 1 | no | — |
| 16 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 18 | C3×#336 | — | 2 | 1 | no | — |
| 19 | #4524 | — | 2 | 1 | no | — |
| 20 | #4524 | C4 | 1 | 4 | yes | — |
| 21 | #4480 | C4 | 1 | 4 | yes | complemented |
| 22 | #4479 | C4 | 1 | 4 | yes | complemented |
| 23 | #1767 | D3 | 1 | 1 | yes | characteristic |
| 24 | C4×C4⋊C8 | — | 3 | 1 | no | — |
| 25 | #1744 | — | 3 | 1 | no | — |
| 26 | C2×#335 | — | 3 | 1 | no | — |
| 27 | C2×#334 | — | 3 | 1 | no | — |
| 28 | #1277 | — | 3 | 1 | no | — |
| 29 | #1277 | — | 3 | 1 | no | — |
| 30 | C22×C3×C8 | — | 2 | 1 | no | — |
| 31 | C22×C3×C8 | — | 2 | 1 | no | — |
| 32 | C2×C3×C42 | C2×C4 | 1 | 2 | yes | — |
| 33 | C2×C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×C3×C42 | D4 | 1 | 1 | yes | characteristic |
| 35 | C2×C3×C42 | Q8 | 1 | 2 | yes | — |
| 36 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 37 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 38 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 39 | C22×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 40 | C22×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 41 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 42 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 43 | C3×C4⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 44 | C3×C4⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 45 | C3×#97 | — | 4 | 2 | no | — |
| 46 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 47 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 48 | C4⋊(C3⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 49 | C4⋊(C3⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 50 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 51 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 52 | C4×C3⋊C8 | — | 4 | 2 | no | — |
| 53 | C4×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 54 | C4×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 55 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 56 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 57 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 58 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | C2×#97 | — | 2 | 1 | no | — |
| 60 | C2×C4×C8 | — | 3 | 1 | no | — |
| 61 | C2×C4×C8 | — | 3 | 1 | no | — |
| 62 | C2×C4×C8 | — | 6 | 1 | no | — |
| 63 | C43 | C2×D3 | 1 | 1 | yes | characteristic |
| 64 | #336 | — | 2 | 1 | no | — |
| 65 | #336 | — | 3 | 4 | no | — |
| 66 | #336 | — | 6 | 1 | no | — |
| 67 | #335 | — | 3 | 4 | no | — |
| 68 | #334 | — | 3 | 4 | no | — |
| 69 | C22×C3×C4 | C4⋊C4 | 1 | 2 | yes | — |
| 70 | C22×C3×C4 | C4⋊C4 | 1 | 2 | yes | — |
| 71 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C22×C3×C4 | C2×Q8 | 1 | 1 | yes | characteristic |
| 74 | C2×C3×C8 | — | 2 | 2 | no | — |
| 75 | C2×C3×C8 | — | 2 | 2 | no | — |
| 76 | C2×C3×C8 | — | 4 | 1 | no | — |
| 77 | C2×C3×C8 | — | 4 | 1 | no | — |
| 78 | C2×C3×C8 | — | 4 | 1 | no | — |
| 79 | C2×C3×C8 | — | 4 | 1 | no | — |
| 80 | C3×C42 | — | 2 | 4 | no | — |
| 81 | C3×C42 | — | 2 | 4 | no | — |
| 82 | C3×C42 | — | 2 | 1 | no | — |
| 83 | C3×C42 | — | 2 | 1 | no | — |
| 84 | C3×C42 | — | 2 | 1 | no | — |
| 85 | C3×C42 | — | 2 | 1 | no | — |
| 86 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 87 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 89 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 90 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 91 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 92 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 93 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 94 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 95 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 96 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 97 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 98 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 99 | C2×C3⋊C8 | C42 | 1 | 4 | yes | — |
| 100 | C22×C8 | — | 2 | 1 | no | — |
| 101 | C22×C8 | — | 2 | 1 | no | — |
| 102 | C22×C8 | — | 3 | 2 | no | — |
| 103 | C22×C8 | — | 3 | 2 | no | — |
| 104 | C22×C8 | — | 6 | 1 | no | — |
| 105 | C22×C8 | — | 6 | 1 | no | — |
| 106 | C2×C42 | Dic6 | 1 | 2 | yes | — |
| 107 | C2×C42 | C4×D3 | 1 | 2 | yes | — |
| 108 | C2×C42 | #67 | 1 | 1 | yes | characteristic |
| 109 | C2×C42 | #67 | 1 | 1 | yes | characteristic |
| 110 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 111 | C4⋊C8 | — | 2 | 1 | no | — |
| 112 | C4⋊C8 | — | 2 | 1 | no | — |
| 113 | C4⋊C8 | — | 3 | 2 | no | — |
| 114 | C4⋊C8 | — | 3 | 2 | no | — |
| 115 | C4⋊C8 | — | 6 | 1 | no | — |
| 116 | C4⋊C8 | — | 6 | 1 | no | — |
| 117 | C4⋊C8 | C4×D3 | 1 | 2 | yes | — |
| 118 | C4⋊C8 | C4×D3 | 1 | 2 | yes | — |
| 119 | #97 | — | 4 | 2 | no | — |
| 120 | C4×C8 | — | 3 | 2 | no | — |
| 121 | C4×C8 | — | 3 | 2 | no | — |
| 122 | C4×C8 | — | 6 | 1 | no | — |
| 123 | C4×C8 | — | 6 | 1 | no | — |
| 124 | C4×C8 | — | 12 | 2 | no | — |
| 125 | C23×C3 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C2×C3×C4 | — | 2 | 2 | no | — |
| 127 | C2×C3×C4 | — | 2 | 2 | no | — |
| 128 | C2×C3×C4 | — | 2 | 2 | no | — |
| 129 | C2×C3×C4 | — | 2 | 2 | no | — |
| 130 | C2×C3×C4 | — | 2 | 2 | no | — |
| 131 | C2×C3×C4 | — | 2 | 2 | no | — |
| 132 | C2×C3×C4 | #108 | 1 | 4 | yes | — |
| 133 | C2×C3×C4 | C2×C42 | 1 | 1 | yes | characteristic |
| 134 | C2×C3×C4 | #117 | 1 | 1 | yes | characteristic |
| 135 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 136 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 137 | C2×C3×C4 | C4×Q8 | 1 | 1 | yes | characteristic |
| 138 | C2×C3×C4 | C4×Q8 | 1 | 1 | yes | characteristic |
| 139 | C2×C3×C4 | #131 | 1 | 4 | yes | — |
| 140 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 141 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 142 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 143 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 144 | C3×C8 | — | 4 | 2 | no | — |
| 145 | C3×C8 | — | 4 | 2 | no | — |
| 146 | C3⋊C8 | — | 2 | 4 | no | — |
| 147 | C3⋊C8 | — | 4 | 2 | no | — |
| 148 | C3⋊C8 | — | 4 | 2 | no | — |
| 149 | C3⋊C8 | — | 4 | 2 | no | — |
| 150 | C22×C4 | Dic3⋊C4 | 1 | 2 | yes | — |
| 151 | C22×C4 | Dic3⋊C4 | 1 | 2 | yes | — |
| 152 | C22×C4 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 153 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 154 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 155 | C2×C8 | — | 2 | 2 | no | — |
| 156 | C2×C8 | — | 2 | 2 | no | — |
| 157 | C2×C8 | — | 3 | 4 | no | — |
| 158 | C2×C8 | — | 4 | 1 | no | — |
| 159 | C2×C8 | — | 4 | 1 | no | — |
| 160 | C2×C8 | — | 4 | 1 | no | — |
| 161 | C2×C8 | — | 4 | 1 | no | — |
| 162 | C2×C8 | — | 6 | 4 | no | — |
| 163 | C2×C8 | — | 6 | 2 | no | — |
| 164 | C2×C8 | — | 6 | 4 | no | — |
| 165 | C2×C8 | — | 6 | 2 | no | — |
| 166 | C2×C8 | — | 6 | 2 | no | — |
| 167 | C2×C8 | — | 12 | 1 | no | — |
| 168 | C2×C8 | — | 12 | 1 | no | — |
| 169 | C2×C8 | — | 12 | 1 | no | — |
| 170 | C2×C8 | — | 12 | 1 | no | — |
| 171 | C42 | — | 2 | 4 | no | — |
| 172 | C42 | — | 2 | 4 | no | — |
| 173 | C42 | — | 2 | 1 | no | — |
| 174 | C42 | — | 2 | 1 | no | — |
| 175 | C42 | — | 2 | 1 | no | — |
| 176 | C42 | — | 2 | 1 | no | — |
| 177 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 178 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 179 | C42 | #235 | 1 | 1 | yes | characteristic |
| 180 | C42 | #235 | 1 | 1 | yes | characteristic |
| 181 | C22×C3 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 182 | C22×C3 | #424 | 1 | 2 | yes | — |
| 183 | C22×C3 | #428 | 1 | 1 | yes | characteristic |
| 184 | C22×C3 | #428 | 1 | 1 | yes | characteristic |
| 185 | C22×C3 | C2×#108 | 1 | 2 | yes | — |
| 186 | C3×C4 | — | 2 | 4 | no | — |
| 187 | C3×C4 | — | 2 | 1 | no | — |
| 188 | C3×C4 | — | 2 | 1 | no | — |
| 189 | C3×C4 | — | 4 | 2 | no | — |
| 190 | C3×C4 | #405 | 1 | 2 | yes | — |
| 191 | C3×C4 | #433 | 1 | 2 | yes | — |
| 192 | C3×C4 | #440 | 1 | 1 | yes | characteristic |
| 193 | C3×C4 | #440 | 1 | 1 | yes | characteristic |
| 194 | C3×C4 | #441 | 1 | 1 | yes | characteristic |
| 195 | C3×C4 | #442 | 1 | 1 | yes | characteristic |
| 196 | C23 | C2×Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 197 | C2×C4 | — | 2 | 2 | no | — |
| 198 | C2×C4 | — | 2 | 2 | no | — |
| 199 | C2×C4 | — | 2 | 2 | no | — |
| 200 | C2×C4 | — | 2 | 2 | no | — |
| 201 | C2×C4 | — | 2 | 2 | no | — |
| 202 | C2×C4 | — | 2 | 2 | no | — |
| 203 | C2×C4 | #822 | 1 | 4 | yes | — |
| 204 | C2×C4 | C4×Dic6 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | C4×Dic6 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | C42×D3 | 1 | 1 | yes | characteristic |
| 207 | C2×C4 | #872 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | #907 | 1 | 4 | yes | — |
| 209 | C2×C4 | C4×#67 | 1 | 1 | yes | characteristic |
| 210 | C2×C4 | C4×#67 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | #932 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | #942 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #949 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | #951 | 1 | 1 | yes | characteristic |
| 215 | C8 | — | 4 | 2 | no | — |
| 216 | C8 | — | 4 | 2 | no | — |
| 217 | C8 | — | 6 | 4 | no | — |
| 218 | C8 | — | 12 | 2 | no | — |
| 219 | C8 | — | 12 | 2 | no | — |
| 220 | C8 | — | 12 | 2 | no | — |
| 221 | C2×C3 | #1573 | 1 | 2 | yes | — |
| 222 | C2×C3 | #1574 | 1 | 2 | yes | — |
| 223 | C2×C3 | #1768 | 1 | 1 | yes | characteristic |
| 224 | C2×C3 | #1776 | 1 | 1 | yes | characteristic |
| 225 | C2×C3 | C4×#108 | 1 | 1 | yes | characteristic |
| 226 | C22 | C4×Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 227 | C22 | #4964 | 1 | 1 | yes | characteristic |
| 228 | C22 | #4995 | 1 | 1 | yes | characteristic |
| 229 | C22 | #5119 | 1 | 2 | yes | — |
| 230 | C22 | C2×#822 | 1 | 2 | yes | — |
| 231 | C4 | — | 2 | 4 | no | — |
| 232 | C4 | — | 2 | 1 | no | — |
| 233 | C4 | — | 2 | 1 | no | — |
| 234 | C4 | — | 4 | 2 | no | — |
| 235 | C4 | #4708 | 1 | 2 | yes | — |
| 236 | C4 | #4709 | 1 | 2 | yes | — |
| 237 | C4 | #5014 | 1 | 1 | yes | characteristic |
| 238 | C4 | #5018 | 1 | 1 | yes | characteristic |
| 239 | C4 | #5026 | 1 | 1 | yes | characteristic |
| 240 | C4 | #5030 | 1 | 1 | yes | characteristic |
| 241 | C3 | #8279 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 242 | C2 | #69481 | 1 | 1 | yes | characteristic |
| 243 | C2 | #69832 | 1 | 1 | yes | characteristic |
| 244 | C2 | C4×#822 | 1 | 1 | yes | characteristic |
| 245 | C2 | #70495 | 1 | 2 | yes | — |
| 246 | C2 | #70797 | 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