← #30747010
Subgroups — #30747010
Order 17152.
342 conjugacy classes of subgroups.
The subgroup lattice has 236 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22559954 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22558957 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22558502 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #22558493 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22558041 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #22558041 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C67×#1564 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16863449 | C22 | 1 | 1 | yes | characteristic |
| 9 | #16863449 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C42×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16863150 | C4 | 1 | 2 | yes | — |
| 13 | #16863150 | C4 | 1 | 2 | yes | — |
| 14 | #16863149 | C4 | 1 | 2 | yes | complemented |
| 15 | #16863147 | C4 | 1 | 2 | yes | complemented |
| 16 | #16863118 | — | 2 | 1 | no | — |
| 17 | #16863118 | — | 2 | 1 | no | — |
| 18 | #16863117 | — | 2 | 1 | no | — |
| 19 | C8×C67⋊C8 | — | 2 | 1 | no | — |
| 20 | #431×C67 | C22 | 1 | 1 | yes | characteristic |
| 21 | #431×C67 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4×C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 23 | #335×C67 | C4 | 1 | 2 | yes | complemented |
| 24 | #334×C67 | C4 | 1 | 2 | yes | complemented |
| 25 | C22×C67⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 26 | C2×C4×Dic67 | — | 2 | 1 | no | — |
| 27 | C2×C4×Dic67 | — | 2 | 1 | no | — |
| 28 | C2×C4×Dic67 | C2×C4 | 1 | 2 | yes | — |
| 29 | C4⋊(C67⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 30 | C4⋊(C67⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 31 | C22⋊(C67⋊C8) | — | 2 | 1 | no | — |
| 32 | C22⋊(C67⋊C8) | — | 2 | 1 | no | — |
| 33 | C4×C67⋊C8 | — | 2 | 1 | no | — |
| 34 | C4×C67⋊C8 | — | 2 | 1 | no | — |
| 35 | C4×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 36 | C4×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 37 | C4×C67⋊C8 | Q8 | 1 | 2 | yes | — |
| 38 | C22×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C22×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 41 | C4⋊C8×C67 | C2×C4 | 1 | 2 | yes | — |
| 42 | C4⋊C8×C67 | C2×C4 | 1 | 2 | yes | — |
| 43 | C22⋊C8×C67 | — | 2 | 1 | no | — |
| 44 | C22⋊C8×C67 | — | 2 | 1 | no | — |
| 45 | C4×C8×C67 | — | 2 | 1 | no | — |
| 46 | C4×C8×C67 | — | 2 | 1 | no | — |
| 47 | C4×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C4×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C4×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C4×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C22×Dic67 | — | 2 | 2 | no | — |
| 52 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 53 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 54 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 55 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 56 | C2×C67⋊C8 | C4⋊C4 | 1 | 2 | yes | — |
| 57 | C2×C67⋊C8 | C4⋊C4 | 1 | 2 | yes | — |
| 58 | C4×Dic67 | — | 2 | 4 | no | — |
| 59 | C4×Dic67 | — | 2 | 2 | no | — |
| 60 | C4×Dic67 | — | 2 | 2 | no | — |
| 61 | C4×Dic67 | — | 4 | 2 | no | — |
| 62 | C22×C4×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 63 | C22×C4×C67 | #41 | 1 | 1 | yes | characteristic |
| 64 | C22×C4×C67 | #41 | 1 | 1 | yes | characteristic |
| 65 | C2×C8×C67 | — | 2 | 2 | no | — |
| 66 | C2×C8×C67 | — | 2 | 2 | no | — |
| 67 | C2×C8×C67 | — | 2 | 2 | no | — |
| 68 | C2×C8×C67 | — | 2 | 2 | no | — |
| 69 | C2×C8×C67 | C42 | 1 | 1 | yes | characteristic |
| 70 | C2×C8×C67 | C42 | 1 | 1 | yes | characteristic |
| 71 | C2×C8×C67 | C42 | 1 | 1 | yes | characteristic |
| 72 | C2×C8×C67 | C42 | 1 | 1 | yes | characteristic |
| 73 | C42×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C42×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C42×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 77 | C23×C67 | #117 | 1 | 1 | yes | characteristic |
| 78 | C2×C4×C67 | — | 2 | 2 | no | — |
| 79 | C2×C4×C67 | — | 2 | 2 | no | — |
| 80 | C2×C4×C67 | C2×C42 | 1 | 1 | yes | characteristic |
| 81 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C4×C67 | C4×Q8 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C67 | C4×Q8 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C67 | #131 | 1 | 2 | yes | — |
| 87 | C2×C4×C67 | #131 | 1 | 2 | yes | — |
| 88 | C2×Dic67 | — | 2 | 4 | no | — |
| 89 | C2×Dic67 | — | 4 | 2 | no | — |
| 90 | C2×Dic67 | — | 4 | 2 | no | — |
| 91 | C8×C67 | — | 2 | 1 | no | — |
| 92 | C8×C67 | — | 2 | 1 | no | — |
| 93 | C8×C67 | — | 2 | 1 | no | — |
| 94 | C8×C67 | — | 2 | 1 | no | — |
| 95 | C8×C67 | — | 4 | 2 | no | — |
| 96 | C8×C67 | — | 4 | 2 | no | — |
| 97 | C67⋊C8 | — | 2 | 2 | no | — |
| 98 | C67⋊C8 | — | 2 | 2 | no | — |
| 99 | C67⋊C8 | — | 4 | 2 | no | — |
| 100 | C67⋊C8 | — | 4 | 2 | no | — |
| 101 | C22×C67 | — | 2 | 1 | no | — |
| 102 | C22×C67 | — | 2 | 1 | no | — |
| 103 | C22×C67 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C22×C67 | #433 | 1 | 2 | yes | — |
| 105 | C4×C67 | — | 2 | 1 | no | — |
| 106 | C4×C67 | — | 2 | 1 | no | — |
| 107 | C4×C67 | #405 | 1 | 1 | yes | characteristic |
| 108 | C4×C67 | #405 | 1 | 1 | yes | characteristic |
| 109 | C4×C67 | #424 | 1 | 2 | yes | — |
| 110 | C4×C67 | #443 | 1 | 2 | yes | — |
| 111 | C4×C67 | #444 | 1 | 2 | yes | — |
| 112 | Dic67 | — | 4 | 4 | no | — |
| 113 | #8270 | — | 67 | 1 | no | maximal, Sylow |
| 114 | C2×C67 | — | 2 | 2 | no | — |
| 115 | C2×C67 | #1768 | 1 | 1 | yes | characteristic |
| 116 | C2×C67 | #1772 | 1 | 1 | yes | characteristic |
| 117 | C2×C67 | #1773 | 1 | 1 | yes | characteristic |
| 118 | #1744 | — | 67 | 1 | no | — |
| 119 | #1564 | — | 67 | 1 | no | — |
| 120 | #1564 | D67 | 1 | 1 | yes | characteristic |
| 121 | #1452 | — | 67 | 1 | no | — |
| 122 | #1450 | — | 67 | 1 | no | — |
| 123 | #1276 | — | 67 | 1 | no | — |
| 124 | #1276 | — | 67 | 1 | no | — |
| 125 | C67 | #8270 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 126 | #431 | — | 67 | 1 | no | — |
| 127 | #431 | — | 67 | 1 | no | — |
| 128 | #431 | C2×D67 | 1 | 1 | yes | characteristic |
| 129 | #431 | C2×D67 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C8 | — | 67 | 1 | no | — |
| 131 | C2×C4×C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 132 | C43 | — | 67 | 1 | no | — |
| 133 | #336 | — | 67 | 2 | no | — |
| 134 | #336 | — | 67 | 2 | no | — |
| 135 | #335 | — | 67 | 2 | no | — |
| 136 | #335 | Dic67 | 1 | 2 | yes | complemented |
| 137 | #334 | — | 67 | 2 | no | — |
| 138 | #334 | Dic67 | 1 | 2 | yes | complemented |
| 139 | #322 | — | 134 | 1 | no | — |
| 140 | #322 | — | 134 | 1 | no | — |
| 141 | #322 | — | 134 | 1 | no | — |
| 142 | C82 | — | 134 | 1 | no | — |
| 143 | C22×C8 | — | 67 | 2 | no | — |
| 144 | C22×C8 | C4×D67 | 1 | 1 | yes | characteristic |
| 145 | C22×C8 | C4×D67 | 1 | 1 | yes | characteristic |
| 146 | C2×C42 | — | 67 | 2 | no | — |
| 147 | C2×C42 | — | 134 | 1 | no | — |
| 148 | C2×C42 | — | 134 | 1 | no | — |
| 149 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 150 | C4⋊C8 | — | 67 | 2 | no | — |
| 151 | C4⋊C8 | — | 67 | 2 | no | — |
| 152 | C4⋊C8 | C2×Dic67 | 1 | 2 | yes | — |
| 153 | C4⋊C8 | C2×Dic67 | 1 | 2 | yes | — |
| 154 | C22⋊C8 | — | 2 | 1 | no | — |
| 155 | C22⋊C8 | — | 2 | 1 | no | — |
| 156 | C22⋊C8 | — | 134 | 1 | no | — |
| 157 | C22⋊C8 | — | 134 | 1 | no | — |
| 158 | C4×C8 | — | 2 | 1 | no | — |
| 159 | C4×C8 | — | 2 | 1 | no | — |
| 160 | C4×C8 | — | 67 | 1 | no | — |
| 161 | C4×C8 | — | 67 | 1 | no | — |
| 162 | C4×C8 | — | 67 | 2 | no | — |
| 163 | C4×C8 | — | 134 | 1 | no | — |
| 164 | C4×C8 | — | 134 | 1 | no | — |
| 165 | C4×C8 | C4×D67 | 1 | 1 | yes | characteristic |
| 166 | C4×C8 | C4×D67 | 1 | 1 | yes | characteristic |
| 167 | C4×C8 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 168 | C4×C8 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | — | 134 | 2 | no | — |
| 170 | C22×C4 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 171 | C22×C4 | #11761627 | 1 | 1 | yes | characteristic |
| 172 | C22×C4 | #11761629 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | — | 2 | 2 | no | — |
| 174 | C2×C8 | — | 2 | 2 | no | — |
| 175 | C2×C8 | — | 2 | 2 | no | — |
| 176 | C2×C8 | — | 2 | 2 | no | — |
| 177 | C2×C8 | — | 67 | 2 | no | — |
| 178 | C2×C8 | — | 67 | 2 | no | — |
| 179 | C2×C8 | — | 134 | 2 | no | — |
| 180 | C2×C8 | — | 134 | 2 | no | — |
| 181 | C2×C8 | — | 134 | 2 | no | — |
| 182 | C2×C8 | — | 134 | 2 | no | — |
| 183 | C2×C8 | C4×Dic67 | 1 | 1 | yes | characteristic |
| 184 | C2×C8 | C4×Dic67 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | C4×Dic67 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | C4×Dic67 | 1 | 1 | yes | characteristic |
| 187 | C42 | — | 268 | 2 | no | — |
| 188 | C42 | — | 134 | 4 | no | — |
| 189 | C42 | — | 134 | 2 | no | — |
| 190 | C42 | — | 134 | 2 | no | — |
| 191 | C42 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 192 | C42 | C22×Dic67 | 1 | 1 | yes | characteristic |
| 193 | C42 | D4×D67 | 1 | 1 | yes | characteristic |
| 194 | C42 | Q8×D67 | 1 | 1 | yes | characteristic |
| 195 | C23 | #14524001 | 1 | 1 | yes | characteristic |
| 196 | C2×C4 | — | 2 | 2 | no | — |
| 197 | C2×C4 | — | 2 | 2 | no | — |
| 198 | C2×C4 | — | 268 | 2 | no | — |
| 199 | C2×C4 | — | 268 | 2 | no | — |
| 200 | C2×C4 | — | 134 | 4 | no | — |
| 201 | C2×C4 | C2×C4×Dic67 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | #14524005 | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | #14524008 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | Q8×Dic67 | 1 | 1 | yes | characteristic |
| 207 | C2×C4 | #14524049 | 1 | 2 | yes | — |
| 208 | C2×C4 | #14524051 | 1 | 2 | yes | — |
| 209 | C8 | — | 2 | 1 | no | — |
| 210 | C8 | — | 2 | 1 | no | — |
| 211 | C8 | — | 2 | 1 | no | — |
| 212 | C8 | — | 2 | 1 | no | — |
| 213 | C8 | — | 4 | 2 | no | — |
| 214 | C8 | — | 4 | 2 | no | — |
| 215 | C8 | — | 268 | 2 | no | — |
| 216 | C8 | — | 268 | 2 | no | — |
| 217 | C8 | — | 134 | 2 | no | — |
| 218 | C8 | — | 134 | 2 | no | — |
| 219 | C22 | — | 2 | 1 | no | — |
| 220 | C22 | — | 2 | 1 | no | — |
| 221 | C22 | C4⋊C4×Dic67 | 1 | 1 | yes | characteristic |
| 222 | C22 | #16863456 | 1 | 2 | yes | — |
| 223 | C4 | — | 2 | 1 | no | — |
| 224 | C4 | — | 2 | 1 | no | — |
| 225 | C4 | — | 268 | 4 | no | — |
| 226 | C4 | #16863351 | 1 | 1 | yes | characteristic |
| 227 | C4 | #16863354 | 1 | 1 | yes | characteristic |
| 228 | C4 | #16863421 | 1 | 2 | yes | — |
| 229 | C4 | #16863512 | 1 | 2 | yes | — |
| 230 | C4 | #16863518 | 1 | 2 | yes | — |
| 231 | C2 | — | 2 | 2 | no | — |
| 232 | C2 | #22560068 | 1 | 1 | yes | characteristic |
| 233 | C2 | #22560091 | 1 | 1 | yes | characteristic |
| 234 | C2 | #22560095 | 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