Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
572 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #22573261 | C2 | 1 | 2 | yes | maximal |
| 2 | #22573137 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22559795 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #22559718 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22559679 | C2 | 1 | 2 | yes | maximal |
| 6 | #22559489 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22559328 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #22559103 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C67×#1623 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #16864132 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16864124 | C22 | 1 | 2 | yes | — |
| 12 | #16863802 | — | 2 | 2 | no | — |
| 13 | #16863794 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16863792 | C22 | 1 | 2 | yes | — |
| 15 | #16863787 | — | 2 | 2 | no | — |
| 16 | #16863768 | C22 | 1 | 2 | yes | — |
| 17 | #16863703 | — | 2 | 2 | no | — |
| 18 | #16863688 | C22 | 1 | 2 | yes | — |
| 19 | #16863681 | C22 | 1 | 2 | yes | — |
| 20 | #16863678 | — | 2 | 2 | no | — |
| 21 | #16863617 | — | 2 | 2 | no | — |
| 22 | #16863607 | C22 | 1 | 2 | yes | — |
| 23 | #16863599 | — | 2 | 2 | no | — |
| 24 | #16863527 | C22 | 1 | 2 | yes | — |
| 25 | #16863523 | C22 | 1 | 2 | yes | — |
| 26 | #16863501 | — | 2 | 2 | no | — |
| 27 | #16863493 | — | 2 | 2 | no | — |
| 28 | C4×#11761618 | C22 | 1 | 2 | yes | — |
| 29 | #16863148 | C22 | 1 | 2 | yes | — |
| 30 | #16863141 | C22 | 1 | 2 | yes | — |
| 31 | #16863140 | C22 | 1 | 1 | yes | characteristic |
| 32 | #16863128 | C22 | 1 | 2 | yes | — |
| 33 | C8×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 34 | Q82×C67 | C22 | 1 | 1 | yes | characteristic |
| 35 | #492×C67 | C22 | 1 | 1 | yes | characteristic |
| 36 | #460×C67 | C22 | 1 | 2 | yes | — |
| 37 | #333×C67 | C22 | 1 | 1 | yes | characteristic |
| 38 | Q8⋊C8×C67 | C22 | 1 | 2 | yes | — |
| 39 | C2×Dic268 | — | 2 | 2 | no | — |
| 40 | C2×#11761618 | D4 | 1 | 2 | yes | — |
| 41 | C2×#11761616 | — | 2 | 2 | no | — |
| 42 | C2×#11761616 | — | 2 | 2 | no | — |
| 43 | #14524043 | C23 | 1 | 2 | yes | — |
| 44 | #14524041 | — | 2 | 2 | no | — |
| 45 | #14524035 | — | 2 | 2 | no | — |
| 46 | #14524033 | — | 4 | 2 | no | — |
| 47 | #14524030 | — | 2 | 2 | no | — |
| 48 | #14524024 | — | 4 | 1 | no | — |
| 49 | C4×Dic134 | — | 2 | 2 | no | — |
| 50 | C4×Dic134 | C23 | 1 | 2 | yes | — |
| 51 | #14524008 | — | 2 | 2 | no | — |
| 52 | D4×Dic67 | — | 2 | 2 | no | — |
| 53 | #14523976 | — | 2 | 2 | no | — |
| 54 | #14523976 | D4 | 1 | 2 | yes | — |
| 55 | #14523975 | — | 4 | 1 | no | — |
| 56 | #14523974 | — | 2 | 2 | no | — |
| 57 | #14523974 | — | 2 | 2 | no | — |
| 58 | #14523974 | — | 2 | 2 | no | — |
| 59 | C4⋊(C67⋊C8) | — | 2 | 2 | no | — |
| 60 | C4⋊(C67⋊C8) | C23 | 1 | 2 | yes | — |
| 61 | Dic67⋊C8 | — | 2 | 2 | no | — |
| 62 | #14523968 | — | 2 | 2 | no | — |
| 63 | #14523967 | — | 2 | 2 | no | — |
| 64 | #14523967 | D4 | 1 | 2 | yes | — |
| 65 | #14523966 | — | 2 | 2 | no | — |
| 66 | #14523966 | D4 | 1 | 2 | yes | — |
| 67 | #14523954 | — | 2 | 2 | no | — |
| 68 | #14523953 | — | 2 | 2 | no | — |
| 69 | C4×C67⋊C8 | C23 | 1 | 2 | yes | — |
| 70 | C2×SD16×C67 | — | 2 | 2 | no | — |
| 71 | C2×SD16×C67 | — | 4 | 1 | no | — |
| 72 | C4⋊Q8×C67 | — | 2 | 2 | no | — |
| 73 | C4⋊Q8×C67 | — | 4 | 1 | no | — |
| 74 | C4⋊Q8×C67 | C23 | 1 | 1 | yes | characteristic |
| 75 | #127×C67 | C23 | 1 | 1 | yes | characteristic |
| 76 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 77 | C4×Q8×C67 | C23 | 1 | 2 | yes | — |
| 78 | C4⋊C8×C67 | C23 | 1 | 2 | yes | — |
| 79 | #102×C67 | — | 2 | 2 | no | — |
| 80 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 81 | C2×Dic134 | — | 2 | 2 | no | — |
| 82 | C2×Dic134 | C2×D4 | 1 | 2 | yes | — |
| 83 | C22×Dic67 | — | 4 | 1 | no | — |
| 84 | Dic268 | — | 8 | 2 | no | — |
| 85 | #11761618 | — | 2 | 4 | no | — |
| 86 | #11761616 | — | 4 | 4 | no | — |
| 87 | #11761616 | — | 8 | 2 | no | — |
| 88 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 89 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 90 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 91 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 92 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 93 | C4⋊Dic67 | C2×D4 | 1 | 2 | yes | — |
| 94 | Dic67⋊C4 | — | 2 | 2 | no | — |
| 95 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 96 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 97 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 98 | C22⋊Dic67 | — | 4 | 2 | no | — |
| 99 | C22⋊Dic67 | — | 4 | 1 | no | — |
| 100 | C4×Dic67 | — | 2 | 2 | no | — |
| 101 | C4×Dic67 | — | 4 | 2 | no | — |
| 102 | C2×Q8×C67 | — | 2 | 1 | no | — |
| 103 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 104 | C2×Q8×C67 | C2×D4 | 1 | 2 | yes | — |
| 105 | C2×D4×C67 | — | 2 | 2 | no | — |
| 106 | C2×D4×C67 | — | 2 | 1 | no | — |
| 107 | SD16×C67 | — | 8 | 2 | no | — |
| 108 | SD16×C67 | — | 8 | 2 | no | — |
| 109 | C2×C8×C67 | — | 2 | 1 | no | — |
| 110 | C2×C8×C67 | — | 2 | 2 | no | — |
| 111 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 112 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 113 | C4⋊C4×C67 | — | 4 | 2 | no | — |
| 114 | C4⋊C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 115 | C42×C67 | — | 2 | 2 | no | — |
| 116 | C42×C67 | — | 4 | 1 | no | — |
| 117 | C42×C67 | C24 | 1 | 1 | yes | characteristic |
| 118 | C23×C67 | — | 4 | 1 | no | — |
| 119 | Q8×C67 | — | 2 | 2 | no | — |
| 120 | Q8×C67 | — | 2 | 2 | no | — |
| 121 | Q8×C67 | — | 8 | 1 | no | — |
| 122 | Q8×C67 | — | 16 | 1 | no | — |
| 123 | D4×C67 | — | 4 | 2 | no | — |
| 124 | D4×C67 | — | 4 | 2 | no | — |
| 125 | D4×C67 | — | 8 | 1 | no | — |
| 126 | C2×C4×C67 | — | 2 | 2 | no | — |
| 127 | C2×C4×C67 | — | 2 | 2 | no | — |
| 128 | C2×C4×C67 | — | 4 | 1 | no | — |
| 129 | C2×C4×C67 | C22×D4 | 1 | 2 | yes | — |
| 130 | C2×C4×C67 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 131 | C2×Dic67 | — | 2 | 2 | no | — |
| 132 | C2×Dic67 | — | 2 | 2 | no | — |
| 133 | C2×Dic67 | — | 4 | 2 | no | — |
| 134 | C2×Dic67 | — | 16 | 1 | no | — |
| 135 | Dic134 | — | 2 | 2 | no | — |
| 136 | Dic134 | — | 2 | 2 | no | — |
| 137 | Dic134 | — | 4 | 2 | no | — |
| 138 | Dic134 | — | 4 | 2 | no | — |
| 139 | C8×C67 | — | 2 | 2 | no | — |
| 140 | C8×C67 | — | 4 | 2 | no | — |
| 141 | C67⋊C8 | — | 2 | 2 | no | — |
| 142 | C67⋊C8 | — | 4 | 2 | no | — |
| 143 | C67⋊C8 | #137 | 1 | 4 | yes | — |
| 144 | C22×C67 | — | 8 | 2 | no | — |
| 145 | C22×C67 | — | 8 | 1 | no | — |
| 146 | C22×C67 | D42 | 1 | 1 | yes | characteristic |
| 147 | C4×C67 | — | 2 | 1 | no | — |
| 148 | C4×C67 | — | 4 | 2 | no | — |
| 149 | C4×C67 | — | 4 | 2 | no | — |
| 150 | C4×C67 | — | 8 | 1 | no | — |
| 151 | C4×C67 | C2×#137 | 1 | 2 | yes | — |
| 152 | C4×C67 | #577 | 1 | 2 | yes | — |
| 153 | Dic67 | — | 4 | 2 | no | — |
| 154 | Dic67 | — | 4 | 2 | no | — |
| 155 | Dic67 | — | 8 | 2 | no | — |
| 156 | #20323 | — | 67 | 1 | no | maximal, Sylow |
| 157 | C2×C67 | — | 16 | 1 | no | — |
| 158 | C2×C67 | #3278 | 1 | 2 | yes | — |
| 159 | C2×C67 | #3291 | 1 | 1 | yes | characteristic |
| 160 | #3400 | — | 67 | 2 | no | — |
| 161 | #3387 | — | 67 | 2 | no | — |
| 162 | #1705 | — | 67 | 1 | no | — |
| 163 | #1692 | — | 67 | 2 | no | — |
| 164 | #1686 | — | 67 | 2 | no | — |
| 165 | #1653 | — | 67 | 2 | no | — |
| 166 | #1630 | — | 67 | 1 | no | — |
| 167 | #1623 | D67 | 1 | 1 | yes | characteristic |
| 168 | #1591 | — | 67 | 2 | no | — |
| 169 | C67 | #20323 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 170 | #559 | — | 67 | 1 | no | — |
| 171 | Q82 | C2×D67 | 1 | 1 | yes | characteristic |
| 172 | #557 | — | 67 | 2 | no | — |
| 173 | #501 | — | 134 | 2 | no | — |
| 174 | #498 | — | 67 | 2 | no | — |
| 175 | #498 | — | 67 | 1 | no | — |
| 176 | #497 | — | 134 | 2 | no | — |
| 177 | #492 | — | 67 | 2 | no | — |
| 178 | #492 | C2×D67 | 1 | 1 | yes | characteristic |
| 179 | #479 | — | 134 | 2 | no | — |
| 180 | #477 | — | 67 | 2 | no | — |
| 181 | #475 | — | 67 | 2 | no | — |
| 182 | #475 | — | 134 | 2 | no | — |
| 183 | #464 | — | 134 | 2 | no | — |
| 184 | #462 | — | 67 | 2 | no | — |
| 185 | #461 | — | 134 | 2 | no | — |
| 186 | #460 | C2×D67 | 1 | 2 | yes | — |
| 187 | #446 | — | 67 | 2 | no | — |
| 188 | #446 | — | 67 | 2 | no | — |
| 189 | #441 | — | 134 | 2 | no | — |
| 190 | #440 | — | 134 | 2 | no | — |
| 191 | C4×Q16 | — | 67 | 2 | no | — |
| 192 | #335 | — | 67 | 2 | no | — |
| 193 | #333 | C2×D67 | 1 | 1 | yes | characteristic |
| 194 | #332 | — | 67 | 2 | no | — |
| 195 | #331 | — | 67 | 1 | no | — |
| 196 | Q8⋊C8 | — | 67 | 2 | no | — |
| 197 | Q8⋊C8 | C2×D67 | 1 | 2 | yes | — |
| 198 | C82 | — | 67 | 1 | no | — |
| 199 | C2×Q16 | — | 67 | 2 | no | — |
| 200 | C2×Q16 | — | 134 | 2 | no | — |
| 201 | C2×SD16 | — | 2 | 2 | no | — |
| 202 | C2×SD16 | — | 4 | 1 | no | — |
| 203 | C2×SD16 | — | 134 | 2 | no | — |
| 204 | C2×SD16 | — | 134 | 2 | no | — |
| 205 | C4⋊Q8 | — | 2 | 2 | no | — |
| 206 | C4⋊Q8 | — | 4 | 1 | no | — |
| 207 | C4⋊Q8 | — | 67 | 2 | no | — |
| 208 | C4⋊Q8 | — | 134 | 2 | no | — |
| 209 | C4⋊Q8 | C22×D67 | 1 | 1 | yes | characteristic |
| 210 | #127 | C22×D67 | 1 | 1 | yes | characteristic |
| 211 | #125 | — | 268 | 2 | no | — |
| 212 | #125 | — | 134 | 2 | no | — |
| 213 | #124 | — | 134 | 2 | no | — |
| 214 | #123 | — | 268 | 1 | no | — |
| 215 | C4×Q8 | — | 2 | 2 | no | — |
| 216 | C4×Q8 | — | 67 | 2 | no | — |
| 217 | C4×Q8 | — | 134 | 2 | no | — |
| 218 | C4×Q8 | — | 134 | 2 | no | — |
| 219 | C4×Q8 | C22×D67 | 1 | 2 | yes | — |
| 220 | C4×D4 | — | 134 | 2 | no | — |
| 221 | #107 | — | 67 | 2 | no | — |
| 222 | #107 | — | 134 | 2 | no | — |
| 223 | #106 | — | 268 | 1 | no | — |
| 224 | #106 | — | 134 | 2 | no | — |
| 225 | #106 | — | 134 | 2 | no | — |
| 226 | #106 | — | 134 | 2 | no | — |
| 227 | C4⋊C8 | — | 67 | 2 | no | — |
| 228 | C4⋊C8 | — | 134 | 2 | no | — |
| 229 | C4⋊C8 | — | 134 | 2 | no | — |
| 230 | C4⋊C8 | C22×D67 | 1 | 2 | yes | — |
| 231 | #103 | — | 67 | 2 | no | — |
| 232 | #103 | — | 67 | 2 | no | — |
| 233 | #103 | — | 134 | 2 | no | — |
| 234 | #103 | — | 134 | 2 | no | — |
| 235 | #103 | — | 134 | 2 | no | — |
| 236 | #102 | — | 2 | 2 | no | — |
| 237 | #97 | — | 134 | 2 | no | — |
| 238 | #97 | — | 134 | 2 | no | — |
| 239 | C4×C8 | — | 67 | 2 | no | — |
| 240 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 241 | C2×Q8 | — | 2 | 1 | no | — |
| 242 | C2×Q8 | — | 4 | 1 | no | — |
| 243 | C2×Q8 | — | 67 | 2 | no | — |
| 244 | C2×Q8 | — | 134 | 2 | no | — |
| 245 | C2×Q8 | D4×D67 | 1 | 2 | yes | — |
| 246 | C2×D4 | — | 2 | 2 | no | — |
| 247 | C2×D4 | — | 2 | 1 | no | — |
| 248 | C22×C4 | — | 268 | 1 | no | — |
| 249 | Q16 | — | 536 | 2 | no | — |
| 250 | Q16 | — | 134 | 4 | no | — |
| 251 | SD16 | — | 8 | 2 | no | — |
| 252 | SD16 | — | 8 | 2 | no | — |
| 253 | SD16 | — | 268 | 4 | no | — |
| 254 | SD16 | — | 536 | 2 | no | — |
| 255 | C2×C8 | — | 2 | 1 | no | — |
| 256 | C2×C8 | — | 2 | 2 | no | — |
| 257 | C2×C8 | — | 67 | 2 | no | — |
| 258 | C2×C8 | — | 67 | 2 | no | — |
| 259 | C2×C8 | — | 134 | 2 | no | — |
| 260 | C4⋊C4 | — | 2 | 2 | no | — |
| 261 | C4⋊C4 | — | 2 | 2 | no | — |
| 262 | C4⋊C4 | — | 4 | 2 | no | — |
| 263 | C4⋊C4 | — | 268 | 2 | no | — |
| 264 | C4⋊C4 | — | 268 | 2 | no | — |
| 265 | C4⋊C4 | — | 268 | 2 | no | — |
| 266 | C4⋊C4 | — | 67 | 2 | no | — |
| 267 | C4⋊C4 | — | 134 | 2 | no | — |
| 268 | C4⋊C4 | — | 134 | 2 | no | — |
| 269 | C4⋊C4 | — | 134 | 2 | no | — |
| 270 | C4⋊C4 | D4×D67 | 1 | 2 | yes | — |
| 271 | C22⋊C4 | — | 268 | 2 | no | — |
| 272 | C22⋊C4 | — | 268 | 1 | no | — |
| 273 | C42 | — | 2 | 2 | no | — |
| 274 | C42 | — | 4 | 1 | no | — |
| 275 | C42 | — | 268 | 2 | no | — |
| 276 | C42 | — | 134 | 2 | no | — |
| 277 | C42 | C23×D67 | 1 | 1 | yes | characteristic |
| 278 | C23 | — | 4 | 1 | no | — |
| 279 | Q8 | — | 2 | 2 | no | — |
| 280 | Q8 | — | 2 | 2 | no | — |
| 281 | Q8 | — | 8 | 1 | no | — |
| 282 | Q8 | — | 268 | 2 | no | — |
| 283 | Q8 | — | 268 | 2 | no | — |
| 284 | Q8 | — | 16 | 1 | no | — |
| 285 | Q8 | — | 134 | 2 | no | — |
| 286 | Q8 | — | 134 | 2 | no | — |
| 287 | D4 | — | 4 | 2 | no | — |
| 288 | D4 | — | 4 | 2 | no | — |
| 289 | D4 | — | 8 | 1 | no | — |
| 290 | C2×C4 | — | 2 | 2 | no | — |
| 291 | C2×C4 | — | 2 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 1 | no | — |
| 293 | C2×C4 | — | 268 | 2 | no | — |
| 294 | C2×C4 | — | 1072 | 1 | no | — |
| 295 | C2×C4 | — | 134 | 2 | no | — |
| 296 | C2×C4 | — | 134 | 2 | no | — |
| 297 | C2×C4 | C2×D4×D67 | 1 | 2 | yes | — |
| 298 | C2×C4 | #14524088 | 1 | 1 | yes | characteristic |
| 299 | C8 | — | 2 | 2 | no | — |
| 300 | C8 | — | 4 | 2 | no | — |
| 301 | C8 | — | 268 | 2 | no | — |
| 302 | C8 | — | 67 | 4 | no | — |
| 303 | C8 | — | 134 | 2 | no | — |
| 304 | C22 | — | 8 | 2 | no | — |
| 305 | C22 | — | 8 | 1 | no | — |
| 306 | C22 | #16864028 | 1 | 1 | yes | characteristic |
| 307 | C4 | — | 2 | 1 | no | — |
| 308 | C4 | — | 4 | 2 | no | — |
| 309 | C4 | — | 4 | 2 | no | — |
| 310 | C4 | — | 8 | 1 | no | — |
| 311 | C4 | — | 268 | 2 | no | — |
| 312 | C4 | — | 268 | 2 | no | — |
| 313 | C4 | — | 536 | 2 | no | — |
| 314 | C4 | #137×D67 | 1 | 2 | yes | — |
| 315 | C4 | #16864202 | 1 | 2 | yes | — |
| 316 | C2 | — | 16 | 1 | no | — |
| 317 | C2 | #22571873 | 1 | 2 | yes | — |
| 318 | C2 | #22572004 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22573137 | C2 | semidirect |
| 4. #22559718 | C2 | semidirect |
| 6. #22559489 | C2 | semidirect |
| 8. #22559103 | C2 | semidirect |
| 169. C67 | #20323 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22573261 | 2 |
| #22573137 | 2 |
| #22559795 | 2 |
| #22559718 | 2 |
| #22559679 | 2 |
| #22559489 | 2 |
| #22559328 | 2 |
| #22559103 | 2 |
| C67×#1623 | 2 |
| #20323 | 67 |