Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
939 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 939 conjugacy classes of subgroups — too many to draw. Only its 109 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 | #3480⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #85419 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #85414 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #85412 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #85404 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #85386 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #85378 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #85349 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C4×#984 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C4×#983 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #85319 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #61467 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C2×#984 | C22 | 1 | 2 | yes | — |
| 14 | C2×#983 | C22 | 1 | 1 | yes | characteristic |
| 15 | #5533 | — | 4 | 1 | no | maximal |
| 16 | #557⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4⋊(#41⋊C3) | C22 | 1 | 1 | yes | characteristic |
| 18 | #548⋊C3 | C22 | 1 | 2 | yes | — |
| 19 | C4×#41⋊C3 | C22 | 1 | 2 | yes | — |
| 20 | C2×C4×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 21 | #5427 | C22 | 1 | 2 | yes | — |
| 22 | #5426 | C22 | 1 | 2 | yes | — |
| 23 | #5425 | C22 | 1 | 2 | yes | — |
| 24 | #5420 | C22 | 1 | 1 | yes | characteristic |
| 25 | #5419 | C22 | 1 | 2 | yes | — |
| 26 | #5395 | C22 | 1 | 2 | yes | — |
| 27 | #5394 | C22 | 1 | 1 | yes | characteristic |
| 28 | C4×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 29 | C4⋊#226 | C22 | 1 | 1 | yes | characteristic |
| 30 | #5389 | C22 | 1 | 2 | yes | — |
| 31 | #5388 | C22 | 1 | 2 | yes | — |
| 32 | #5388 | C22 | 1 | 1 | yes | characteristic |
| 33 | C4×#226 | C22 | 1 | 1 | yes | characteristic |
| 34 | C4×#226 | C22 | 1 | 2 | yes | — |
| 35 | C4×#226 | C22 | 1 | 2 | yes | — |
| 36 | #5386 | C22 | 1 | 1 | yes | characteristic |
| 37 | #3482 | — | 3 | 1 | no | — |
| 38 | #3480 | D3 | 1 | 1 | yes | characteristic, complemented |
| 39 | #3394 | — | 3 | 1 | no | — |
| 40 | #3383 | — | 3 | 1 | no | — |
| 41 | #3365 | — | 3 | 1 | no | — |
| 42 | #3364 | — | 3 | 1 | no | — |
| 43 | #3353 | — | 3 | 1 | no | — |
| 44 | #3351 | — | 3 | 2 | no | — |
| 45 | #3332 | — | 3 | 1 | no | — |
| 46 | #3328 | — | 3 | 1 | no | — |
| 47 | #3327 | — | 3 | 2 | no | — |
| 48 | #3326 | — | 3 | 2 | no | — |
| 49 | #3278 | — | 3 | 1 | no | — |
| 50 | #3277 | — | 3 | 2 | no | — |
| 51 | #3183 | — | 3 | 2 | no | — |
| 52 | #3182 | — | 3 | 1 | no | — |
| 53 | #3117 | — | 3 | 2 | no | — |
| 54 | #3115 | — | 3 | 1 | no | — |
| 55 | #3113 | — | 3 | 1 | no | — |
| 56 | #3111 | — | 3 | 2 | no | — |
| 57 | #2971 | — | 3 | 1 | no | — |
| 58 | C4×#137 | — | 3 | 2 | no | — |
| 59 | C4×#137 | — | 3 | 1 | no | — |
| 60 | C2×#41⋊C3 | C23 | 1 | 2 | yes | — |
| 61 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 62 | #984 | — | 2 | 4 | no | — |
| 63 | #983 | — | 2 | 2 | no | — |
| 64 | C2×GL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×#226 | C23 | 1 | 2 | yes | — |
| 66 | C2×#226 | C23 | 1 | 1 | yes | characteristic |
| 67 | C3×#117 | — | 4 | 1 | no | — |
| 68 | C4×#67 | — | 4 | 1 | no | — |
| 69 | #924 | — | 4 | 1 | no | — |
| 70 | #897 | — | 4 | 1 | no | — |
| 71 | #892 | — | 4 | 1 | no | — |
| 72 | #888 | — | 4 | 1 | no | — |
| 73 | #887 | — | 4 | 1 | no | — |
| 74 | #885 | — | 4 | 2 | no | — |
| 75 | #879 | — | 4 | 2 | no | — |
| 76 | #876 | — | 4 | 2 | no | — |
| 77 | C4×Dic6 | — | 4 | 2 | no | — |
| 78 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 79 | C4×SL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 80 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 81 | #859 | C23 | 1 | 2 | yes | — |
| 82 | #859 | C23 | 1 | 1 | yes | characteristic |
| 83 | #859 | C23 | 1 | 1 | yes | characteristic |
| 84 | C2×#137 | — | 3 | 2 | no | — |
| 85 | C2×#137 | — | 3 | 1 | no | — |
| 86 | #559 | — | 3 | 1 | no | — |
| 87 | #557 | — | 3 | 1 | no | — |
| 88 | #557 | C2×D3 | 1 | 1 | yes | characteristic |
| 89 | #556 | — | 3 | 1 | no | — |
| 90 | #556 | — | 3 | 1 | no | — |
| 91 | #556 | — | 6 | 1 | no | — |
| 92 | #553 | — | 3 | 2 | no | — |
| 93 | #552 | — | 3 | 2 | no | — |
| 94 | #552 | — | 6 | 2 | no | — |
| 95 | #550 | C2×D3 | 1 | 1 | yes | characteristic |
| 96 | D4×Q8 | — | 3 | 1 | no | — |
| 97 | #548 | — | 3 | 2 | no | — |
| 98 | #548 | C2×D3 | 1 | 2 | yes | — |
| 99 | #547 | — | 3 | 1 | no | — |
| 100 | #547 | — | 6 | 1 | no | — |
| 101 | #533 | — | 3 | 1 | no | — |
| 102 | #533 | — | 6 | 1 | no | — |
| 103 | #532 | — | 3 | 1 | no | — |
| 104 | #529 | — | 3 | 2 | no | — |
| 105 | #529 | — | 3 | 2 | no | — |
| 106 | #529 | — | 6 | 2 | no | — |
| 107 | C4×#41 | — | 3 | 1 | no | — |
| 108 | C4×#41 | — | 3 | 1 | no | — |
| 109 | C4×#41 | — | 6 | 1 | no | — |
| 110 | C4×#41 | C2×D3 | 1 | 2 | yes | — |
| 111 | C2×C4×Q8 | — | 3 | 2 | no | — |
| 112 | C2×C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 113 | #500 | — | 3 | 1 | no | — |
| 114 | #499 | — | 3 | 2 | no | — |
| 115 | #495 | — | 3 | 2 | no | — |
| 116 | #492 | — | 3 | 1 | no | — |
| 117 | #488 | — | 3 | 2 | no | — |
| 118 | #488 | — | 3 | 2 | no | — |
| 119 | #487 | — | 3 | 1 | no | — |
| 120 | #486 | — | 3 | 1 | no | — |
| 121 | #485 | — | 3 | 2 | no | — |
| 122 | #484 | — | 3 | 2 | no | — |
| 123 | #483 | — | 3 | 2 | no | — |
| 124 | #482 | — | 3 | 2 | no | — |
| 125 | #482 | — | 3 | 2 | no | — |
| 126 | #481 | — | 3 | 2 | no | — |
| 127 | #479 | — | 3 | 2 | no | — |
| 128 | #479 | — | 3 | 2 | no | — |
| 129 | #479 | — | 3 | 2 | no | — |
| 130 | #478 | — | 3 | 2 | no | — |
| 131 | #475 | — | 3 | 1 | no | — |
| 132 | #475 | — | 3 | 1 | no | — |
| 133 | #475 | — | 3 | 1 | no | — |
| 134 | #474 | — | 3 | 1 | no | — |
| 135 | #470 | — | 3 | 2 | no | — |
| 136 | #469 | — | 3 | 2 | no | — |
| 137 | #468 | — | 3 | 2 | no | — |
| 138 | #468 | — | 3 | 2 | no | — |
| 139 | #468 | — | 3 | 2 | no | — |
| 140 | #468 | — | 3 | 2 | no | — |
| 141 | #464 | — | 3 | 2 | no | — |
| 142 | #464 | — | 3 | 2 | no | — |
| 143 | #464 | — | 3 | 2 | no | — |
| 144 | #463 | — | 3 | 2 | no | — |
| 145 | #462 | — | 3 | 1 | no | — |
| 146 | #462 | — | 3 | 1 | no | — |
| 147 | #460 | — | 3 | 1 | no | — |
| 148 | #459 | — | 3 | 1 | no | — |
| 149 | #452 | — | 3 | 2 | no | — |
| 150 | #451 | — | 3 | 2 | no | — |
| 151 | #451 | — | 3 | 2 | no | — |
| 152 | #449 | — | 3 | 2 | no | — |
| 153 | #449 | — | 3 | 2 | no | — |
| 154 | #448 | — | 3 | 2 | no | — |
| 155 | C8×Q8 | — | 3 | 1 | no | — |
| 156 | C8×Q8 | — | 3 | 1 | no | — |
| 157 | #442 | — | 3 | 2 | no | — |
| 158 | #441 | — | 3 | 2 | no | — |
| 159 | #441 | — | 3 | 1 | no | — |
| 160 | #441 | — | 3 | 2 | no | — |
| 161 | #441 | — | 3 | 1 | no | — |
| 162 | #440 | — | 3 | 2 | no | — |
| 163 | #440 | — | 3 | 2 | no | — |
| 164 | #440 | — | 3 | 2 | no | — |
| 165 | #440 | — | 3 | 1 | no | — |
| 166 | #440 | — | 3 | 1 | no | — |
| 167 | C4×Q16 | — | 3 | 1 | no | — |
| 168 | C4×Q16 | — | 3 | 2 | no | — |
| 169 | C4×Q16 | — | 3 | 2 | no | — |
| 170 | C4×Q16 | — | 3 | 1 | no | — |
| 171 | C4×SD16 | — | 3 | 2 | no | — |
| 172 | C4×SD16 | — | 3 | 1 | no | — |
| 173 | C4×SD16 | — | 3 | 2 | no | — |
| 174 | C4×SD16 | — | 3 | 1 | no | — |
| 175 | #436 | — | 3 | 1 | no | — |
| 176 | #436 | — | 3 | 2 | no | — |
| 177 | #436 | — | 3 | 1 | no | — |
| 178 | #436 | — | 3 | 2 | no | — |
| 179 | #432 | — | 3 | 2 | no | — |
| 180 | #432 | — | 3 | 1 | no | — |
| 181 | #428 | — | 3 | 1 | no | — |
| 182 | #428 | — | 3 | 2 | no | — |
| 183 | #419 | — | 3 | 2 | no | — |
| 184 | #419 | — | 3 | 1 | no | — |
| 185 | #417 | — | 3 | 1 | no | — |
| 186 | #417 | — | 3 | 2 | no | — |
| 187 | C4×M16 | — | 3 | 1 | no | — |
| 188 | C4×M16 | — | 3 | 2 | no | — |
| 189 | C22×C3×C4 | — | 4 | 1 | no | — |
| 190 | C2×#67 | — | 4 | 1 | no | — |
| 191 | C2×C4×D3 | — | 4 | 1 | no | — |
| 192 | C2×Dic6 | — | 4 | 1 | no | — |
| 193 | C2×Dic6 | — | 4 | 2 | no | — |
| 194 | #41⋊C3 | — | 2 | 2 | no | — |
| 195 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 196 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 197 | GL(2,𝔽3) | — | 2 | 1 | no | — |
| 198 | #226 | — | 2 | 2 | no | — |
| 199 | #226 | #41 | 1 | 2 | yes | complemented |
| 200 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 201 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 202 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 203 | C3×C42 | — | 4 | 2 | no | — |
| 204 | #217 | — | 4 | 2 | no | — |
| 205 | #217 | — | 4 | 1 | no | — |
| 206 | (C2×D3)⋊C4 | — | 4 | 1 | no | — |
| 207 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 208 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 209 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 210 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 211 | Dic3⋊C4 | — | 4 | 1 | no | — |
| 212 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 213 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 214 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 215 | C4×Dic3 | — | 4 | 2 | no | — |
| 216 | C4×Dic3 | — | 4 | 2 | no | — |
| 217 | C4×Dic3 | — | 4 | 1 | no | — |
| 218 | C2×#41 | — | 3 | 1 | no | — |
| 219 | C2×#41 | — | 3 | 1 | no | — |
| 220 | C2×#41 | — | 6 | 1 | no | — |
| 221 | C2×#41 | C22×D3 | 1 | 2 | yes | — |
| 222 | C22×Q8 | — | 3 | 2 | no | — |
| 223 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 224 | #137 | — | 6 | 2 | no | — |
| 225 | #137 | — | 6 | 4 | no | — |
| 226 | #137 | — | 6 | 2 | no | — |
| 227 | #137 | — | 6 | 4 | no | — |
| 228 | C2×Q16 | — | 3 | 1 | no | — |
| 229 | C2×Q16 | — | 3 | 1 | no | — |
| 230 | C2×Q16 | — | 3 | 2 | no | — |
| 231 | C2×Q16 | — | 3 | 2 | no | — |
| 232 | C2×SD16 | — | 3 | 2 | no | — |
| 233 | C2×SD16 | — | 3 | 2 | no | — |
| 234 | C2×SD16 | — | 3 | 1 | no | — |
| 235 | C2×SD16 | — | 3 | 1 | no | — |
| 236 | C2×SD16 | — | 3 | 2 | no | — |
| 237 | C2×D8 | — | 3 | 2 | no | — |
| 238 | C2×M16 | — | 3 | 1 | no | — |
| 239 | C2×M16 | — | 3 | 2 | no | — |
| 240 | C2×M16 | — | 3 | 1 | no | — |
| 241 | C2×M16 | — | 3 | 2 | no | — |
| 242 | C4⋊Q8 | — | 3 | 1 | no | — |
| 243 | C4⋊Q8 | — | 3 | 1 | no | — |
| 244 | C4⋊Q8 | — | 3 | 1 | no | — |
| 245 | C4⋊Q8 | — | 6 | 1 | no | — |
| 246 | C4⋊Q8 | — | 6 | 1 | no | — |
| 247 | #127 | — | 3 | 1 | no | — |
| 248 | #126 | — | 6 | 2 | no | — |
| 249 | #126 | — | 6 | 2 | no | — |
| 250 | #126 | — | 6 | 2 | no | — |
| 251 | #126 | — | 6 | 2 | no | — |
| 252 | #125 | — | 3 | 2 | no | — |
| 253 | #125 | — | 3 | 2 | no | — |
| 254 | #125 | — | 6 | 1 | no | — |
| 255 | #125 | — | 6 | 2 | no | — |
| 256 | #125 | — | 6 | 2 | no | — |
| 257 | #124 | — | 3 | 2 | no | — |
| 258 | #124 | — | 3 | 2 | no | — |
| 259 | #124 | — | 6 | 1 | no | — |
| 260 | #124 | — | 6 | 2 | no | — |
| 261 | #123 | — | 6 | 2 | no | — |
| 262 | #123 | — | 6 | 2 | no | — |
| 263 | #123 | — | 6 | 2 | no | — |
| 264 | #123 | — | 6 | 2 | no | — |
| 265 | C22⋊Q8 | — | 3 | 2 | no | — |
| 266 | C22⋊Q8 | — | 3 | 2 | no | — |
| 267 | C22⋊Q8 | — | 3 | 2 | no | — |
| 268 | C22⋊Q8 | — | 6 | 2 | no | — |
| 269 | C22⋊Q8 | — | 6 | 2 | no | — |
| 270 | C22⋊Q8 | — | 6 | 1 | no | — |
| 271 | C22⋊Q8 | — | 6 | 1 | no | — |
| 272 | C22⋊Q8 | — | 6 | 1 | no | — |
| 273 | #121 | — | 3 | 2 | no | — |
| 274 | #121 | — | 3 | 2 | no | — |
| 275 | #121 | — | 3 | 2 | no | — |
| 276 | #121 | — | 6 | 1 | no | — |
| 277 | #121 | — | 6 | 2 | no | — |
| 278 | C4×Q8 | — | 3 | 1 | no | — |
| 279 | C4×Q8 | — | 3 | 1 | no | — |
| 280 | C4×Q8 | — | 3 | 1 | no | — |
| 281 | C4×Q8 | — | 3 | 1 | no | — |
| 282 | C4×Q8 | — | 3 | 1 | no | — |
| 283 | C4×Q8 | — | 3 | 2 | no | — |
| 284 | C4×Q8 | — | 3 | 2 | no | — |
| 285 | C4×Q8 | — | 3 | 2 | no | — |
| 286 | C4×Q8 | — | 3 | 1 | no | — |
| 287 | C4×Q8 | — | 6 | 2 | no | — |
| 288 | C4×Q8 | — | 6 | 1 | no | — |
| 289 | C4×Q8 | — | 6 | 1 | no | — |
| 290 | C4×Q8 | — | 6 | 2 | no | — |
| 291 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 292 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 293 | C4×Q8 | C22×D3 | 1 | 2 | yes | — |
| 294 | C4×D4 | — | 3 | 1 | no | — |
| 295 | C4×D4 | — | 3 | 1 | no | — |
| 296 | C4×D4 | — | 3 | 2 | no | — |
| 297 | C4×D4 | — | 3 | 2 | no | — |
| 298 | C4×D4 | — | 3 | 2 | no | — |
| 299 | C4×D4 | — | 3 | 2 | no | — |
| 300 | C4×D4 | — | 3 | 1 | no | — |
| 301 | C4×D4 | — | 3 | 1 | no | — |
| 302 | C4×D4 | — | 6 | 1 | no | — |
| 303 | C4×D4 | — | 6 | 2 | no | — |
| 304 | C4×D4 | — | 6 | 1 | no | — |
| 305 | C4×D4 | — | 6 | 2 | no | — |
| 306 | C4×D4 | — | 6 | 1 | no | — |
| 307 | C4×D4 | — | 6 | 1 | no | — |
| 308 | #117 | — | 3 | 2 | no | — |
| 309 | #117 | — | 3 | 2 | no | — |
| 310 | #117 | — | 3 | 1 | no | — |
| 311 | #117 | — | 6 | 1 | no | — |
| 312 | #117 | — | 6 | 2 | no | — |
| 313 | #117 | — | 6 | 2 | no | — |
| 314 | #117 | — | 6 | 1 | no | — |
| 315 | #117 | — | 6 | 1 | no | — |
| 316 | #117 | S4 | 1 | 1 | yes | characteristic |
| 317 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 318 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 319 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 320 | C2×C42 | — | 3 | 1 | no | — |
| 321 | C2×C42 | — | 3 | 2 | no | — |
| 322 | C2×C42 | — | 6 | 1 | no | — |
| 323 | C2×C42 | — | 6 | 2 | no | — |
| 324 | #107 | — | 3 | 1 | no | — |
| 325 | #107 | — | 3 | 2 | no | — |
| 326 | #107 | — | 3 | 2 | no | — |
| 327 | #107 | — | 3 | 1 | no | — |
| 328 | #107 | — | 3 | 2 | no | — |
| 329 | #106 | — | 3 | 1 | no | — |
| 330 | #106 | — | 3 | 2 | no | — |
| 331 | #106 | — | 3 | 1 | no | — |
| 332 | #106 | — | 3 | 2 | no | — |
| 333 | #106 | — | 3 | 2 | no | — |
| 334 | C4⋊C8 | — | 3 | 1 | no | — |
| 335 | C4⋊C8 | — | 3 | 1 | no | — |
| 336 | C4⋊C8 | — | 3 | 2 | no | — |
| 337 | C4⋊C8 | — | 3 | 2 | no | — |
| 338 | #103 | — | 3 | 1 | no | — |
| 339 | #103 | — | 3 | 1 | no | — |
| 340 | #103 | — | 3 | 2 | no | — |
| 341 | #103 | — | 3 | 1 | no | — |
| 342 | #103 | — | 3 | 2 | no | — |
| 343 | #103 | — | 3 | 2 | no | — |
| 344 | #103 | — | 3 | 1 | no | — |
| 345 | #103 | — | 3 | 2 | no | — |
| 346 | #103 | — | 3 | 2 | no | — |
| 347 | #103 | — | 3 | 2 | no | — |
| 348 | #103 | — | 3 | 1 | no | — |
| 349 | #103 | — | 3 | 2 | no | — |
| 350 | #103 | — | 3 | 1 | no | — |
| 351 | #102 | — | 3 | 2 | no | — |
| 352 | #102 | — | 3 | 1 | no | — |
| 353 | #102 | — | 3 | 2 | no | — |
| 354 | #102 | — | 3 | 1 | no | — |
| 355 | #102 | — | 3 | 2 | no | — |
| 356 | #102 | — | 3 | 2 | no | — |
| 357 | #102 | — | 3 | 2 | no | — |
| 358 | C22⋊C8 | — | 3 | 2 | no | — |
| 359 | C22⋊C8 | — | 3 | 2 | no | — |
| 360 | C22⋊C8 | — | 3 | 2 | no | — |
| 361 | #97 | — | 3 | 1 | no | — |
| 362 | #97 | — | 3 | 1 | no | — |
| 363 | #97 | — | 3 | 2 | no | — |
| 364 | C4×C8 | — | 3 | 1 | no | — |
| 365 | C4×C8 | — | 3 | 2 | no | — |
| 366 | C4×C8 | — | 3 | 1 | no | — |
| 367 | C4×C8 | — | 3 | 2 | no | — |
| 368 | C23×C3 | — | 4 | 1 | no | — |
| 369 | C22×D3 | — | 4 | 1 | no | — |
| 370 | C2×C3×C4 | — | 4 | 1 | no | — |
| 371 | C2×C3×C4 | — | 4 | 2 | no | — |
| 372 | C2×C3×C4 | — | 4 | 1 | no | — |
| 373 | C2×C3×C4 | — | 4 | 2 | no | — |
| 374 | C2×C3×C4 | — | 8 | 2 | no | — |
| 375 | #67 | — | 8 | 2 | no | — |
| 376 | C2×Dic3 | — | 4 | 1 | no | — |
| 377 | C2×Dic3 | — | 4 | 1 | no | — |
| 378 | C2×Dic3 | — | 4 | 1 | no | — |
| 379 | C2×Dic3 | — | 4 | 2 | no | — |
| 380 | C2×Dic3 | — | 4 | 2 | no | — |
| 381 | C4×D3 | — | 8 | 2 | no | — |
| 382 | Dic6 | — | 8 | 4 | no | — |
| 383 | Dic6 | — | 8 | 2 | no | — |
| 384 | SL(2,𝔽3) | C2×#41 | 1 | 1 | yes | characteristic |
| 385 | #41 | — | 2 | 2 | no | — |
| 386 | #41 | — | 6 | 2 | no | — |
| 387 | #41 | — | 6 | 2 | no | — |
| 388 | #41 | — | 12 | 2 | no | — |
| 389 | #41 | — | 12 | 1 | no | — |
| 390 | #41 | — | 12 | 4 | no | — |
| 391 | C2×Q8 | — | 2 | 1 | no | — |
| 392 | C2×Q8 | — | 3 | 1 | no | — |
| 393 | C2×Q8 | — | 3 | 1 | no | — |
| 394 | C2×Q8 | — | 3 | 1 | no | — |
| 395 | C2×Q8 | — | 3 | 2 | no | — |
| 396 | C2×Q8 | — | 6 | 2 | no | — |
| 397 | C2×Q8 | — | 6 | 1 | no | — |
| 398 | C2×Q8 | — | 6 | 1 | no | — |
| 399 | C2×Q8 | — | 6 | 4 | no | — |
| 400 | C2×Q8 | — | 12 | 2 | no | — |
| 401 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 402 | C2×D4 | — | 3 | 2 | no | — |
| 403 | C2×D4 | — | 3 | 1 | no | — |
| 404 | C2×D4 | — | 3 | 2 | no | — |
| 405 | C2×D4 | — | 3 | 1 | no | — |
| 406 | C2×D4 | — | 6 | 1 | no | — |
| 407 | C2×D4 | — | 6 | 2 | no | — |
| 408 | C2×D4 | — | 6 | 1 | no | — |
| 409 | C22×C4 | — | 3 | 1 | no | — |
| 410 | C22×C4 | — | 3 | 1 | no | — |
| 411 | C22×C4 | — | 3 | 2 | no | — |
| 412 | C22×C4 | — | 3 | 2 | no | — |
| 413 | C22×C4 | — | 3 | 2 | no | — |
| 414 | C22×C4 | — | 6 | 1 | no | — |
| 415 | C22×C4 | — | 6 | 1 | no | — |
| 416 | C22×C4 | — | 6 | 2 | no | — |
| 417 | C22×C4 | — | 6 | 2 | no | — |
| 418 | C22×C4 | — | 6 | 1 | no | — |
| 419 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 420 | Q16 | — | 3 | 2 | no | — |
| 421 | Q16 | — | 3 | 2 | no | — |
| 422 | Q16 | — | 6 | 2 | no | — |
| 423 | Q16 | — | 6 | 2 | no | — |
| 424 | Q16 | — | 6 | 2 | no | — |
| 425 | Q16 | — | 12 | 2 | no | — |
| 426 | SD16 | — | 6 | 4 | no | — |
| 427 | SD16 | — | 6 | 2 | no | — |
| 428 | SD16 | — | 6 | 4 | no | — |
| 429 | SD16 | — | 6 | 1 | no | — |
| 430 | SD16 | — | 6 | 1 | no | — |
| 431 | SD16 | — | 12 | 2 | no | — |
| 432 | D8 | — | 12 | 2 | no | — |
| 433 | M16 | — | 6 | 4 | no | — |
| 434 | M16 | — | 6 | 2 | no | — |
| 435 | M16 | — | 12 | 2 | no | — |
| 436 | M16 | — | 12 | 1 | no | — |
| 437 | C2×C8 | — | 3 | 1 | no | — |
| 438 | C2×C8 | — | 3 | 1 | no | — |
| 439 | C2×C8 | — | 3 | 1 | no | — |
| 440 | C2×C8 | — | 3 | 2 | no | — |
| 441 | C2×C8 | — | 3 | 2 | no | — |
| 442 | C2×C8 | — | 3 | 1 | no | — |
| 443 | C4⋊C4 | — | 3 | 1 | no | — |
| 444 | C4⋊C4 | — | 3 | 1 | no | — |
| 445 | C4⋊C4 | — | 3 | 2 | no | — |
| 446 | C4⋊C4 | — | 3 | 2 | no | — |
| 447 | C4⋊C4 | — | 3 | 2 | no | — |
| 448 | C4⋊C4 | — | 3 | 1 | no | — |
| 449 | C4⋊C4 | — | 3 | 2 | no | — |
| 450 | C4⋊C4 | — | 3 | 1 | no | — |
| 451 | C4⋊C4 | — | 6 | 1 | no | — |
| 452 | C4⋊C4 | — | 6 | 1 | no | — |
| 453 | C4⋊C4 | — | 6 | 2 | no | — |
| 454 | C4⋊C4 | — | 6 | 2 | no | — |
| 455 | C4⋊C4 | — | 6 | 1 | no | — |
| 456 | C4⋊C4 | — | 6 | 2 | no | — |
| 457 | C4⋊C4 | — | 6 | 1 | no | — |
| 458 | C4⋊C4 | — | 6 | 2 | no | — |
| 459 | C4⋊C4 | — | 6 | 1 | no | — |
| 460 | C4⋊C4 | — | 6 | 2 | no | — |
| 461 | C4⋊C4 | — | 6 | 1 | no | — |
| 462 | C4⋊C4 | — | 6 | 2 | no | — |
| 463 | C4⋊C4 | — | 6 | 1 | no | — |
| 464 | C4⋊C4 | — | 6 | 2 | no | — |
| 465 | C4⋊C4 | — | 6 | 2 | no | — |
| 466 | C4⋊C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 467 | C4⋊C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 468 | C22⋊C4 | — | 3 | 2 | no | — |
| 469 | C22⋊C4 | — | 3 | 2 | no | — |
| 470 | C22⋊C4 | — | 6 | 2 | no | — |
| 471 | C22⋊C4 | — | 6 | 2 | no | — |
| 472 | C22⋊C4 | — | 6 | 1 | no | — |
| 473 | C22⋊C4 | — | 6 | 2 | no | — |
| 474 | C22⋊C4 | — | 6 | 2 | no | — |
| 475 | C22⋊C4 | — | 6 | 1 | no | — |
| 476 | C22⋊C4 | — | 6 | 2 | no | — |
| 477 | C22⋊C4 | — | 6 | 1 | no | — |
| 478 | C22⋊C4 | — | 6 | 2 | no | — |
| 479 | C22⋊C4 | — | 6 | 2 | no | — |
| 480 | C22⋊C4 | C2×S4 | 1 | 2 | yes | — |
| 481 | C42 | — | 3 | 2 | no | — |
| 482 | C42 | — | 3 | 1 | no | — |
| 483 | C42 | — | 3 | 1 | no | — |
| 484 | C42 | — | 3 | 1 | no | — |
| 485 | C42 | — | 3 | 1 | no | — |
| 486 | C42 | — | 3 | 2 | no | — |
| 487 | C42 | — | 6 | 1 | no | — |
| 488 | C42 | — | 6 | 1 | no | — |
| 489 | C42 | — | 6 | 2 | no | — |
| 490 | C42 | — | 6 | 2 | no | — |
| 491 | C42 | — | 6 | 1 | no | — |
| 492 | C42 | — | 6 | 2 | no | — |
| 493 | C42 | C2×S4 | 1 | 2 | yes | — |
| 494 | C22×C3 | — | 4 | 1 | no | — |
| 495 | C22×C3 | — | 8 | 1 | no | — |
| 496 | C22×C3 | — | 8 | 1 | no | — |
| 497 | C22×C3 | — | 8 | 1 | no | — |
| 498 | C2×D3 | — | 8 | 1 | no | — |
| 499 | C2×D3 | — | 8 | 1 | no | — |
| 500 | C2×D3 | — | 8 | 1 | no | — |
| 501 | C3×C4 | — | 4 | 2 | no | — |
| 502 | C3×C4 | — | 8 | 2 | no | — |
| 503 | C3×C4 | — | 8 | 2 | no | — |
| 504 | C3×C4 | — | 8 | 1 | no | — |
| 505 | Dic3 | — | 4 | 2 | no | — |
| 506 | Dic3 | — | 8 | 2 | no | — |
| 507 | Dic3 | — | 8 | 2 | no | — |
| 508 | Dic3 | — | 8 | 1 | no | — |
| 509 | Dic3 | — | 8 | 1 | no | — |
| 510 | C23 | — | 3 | 2 | no | — |
| 511 | C23 | — | 6 | 1 | no | — |
| 512 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 513 | Q8 | — | 3 | 1 | no | — |
| 514 | Q8 | — | 3 | 2 | no | — |
| 515 | Q8 | — | 6 | 1 | no | — |
| 516 | Q8 | — | 6 | 1 | no | — |
| 517 | Q8 | — | 6 | 1 | no | — |
| 518 | Q8 | — | 6 | 2 | no | — |
| 519 | Q8 | — | 6 | 2 | no | — |
| 520 | Q8 | — | 6 | 2 | no | — |
| 521 | Q8 | — | 12 | 4 | no | — |
| 522 | Q8 | — | 12 | 2 | no | — |
| 523 | Q8 | D3×#41 | 1 | 1 | yes | characteristic |
| 524 | D4 | — | 6 | 2 | no | — |
| 525 | D4 | — | 6 | 2 | no | — |
| 526 | D4 | — | 6 | 1 | no | — |
| 527 | D4 | — | 6 | 1 | no | — |
| 528 | D4 | — | 6 | 2 | no | — |
| 529 | D4 | — | 12 | 2 | no | — |
| 530 | D4 | — | 12 | 2 | no | — |
| 531 | D4 | — | 12 | 4 | no | — |
| 532 | D4 | — | 24 | 1 | no | — |
| 533 | C2×C4 | — | 2 | 2 | no | — |
| 534 | C2×C4 | — | 3 | 1 | no | — |
| 535 | C2×C4 | — | 3 | 1 | no | — |
| 536 | C2×C4 | — | 3 | 1 | no | — |
| 537 | C2×C4 | — | 3 | 1 | no | — |
| 538 | C2×C4 | — | 3 | 2 | no | — |
| 539 | C2×C4 | — | 6 | 2 | no | — |
| 540 | C2×C4 | — | 6 | 2 | no | — |
| 541 | C2×C4 | — | 6 | 1 | no | — |
| 542 | C2×C4 | — | 6 | 1 | no | — |
| 543 | C2×C4 | — | 6 | 1 | no | — |
| 544 | C2×C4 | — | 6 | 1 | no | — |
| 545 | C2×C4 | — | 6 | 1 | no | — |
| 546 | C2×C4 | — | 6 | 2 | no | — |
| 547 | C2×C4 | — | 6 | 2 | no | — |
| 548 | C2×C4 | — | 12 | 2 | no | — |
| 549 | C2×C4 | — | 12 | 2 | no | — |
| 550 | C2×C4 | — | 12 | 1 | no | — |
| 551 | C2×C4 | — | 12 | 2 | no | — |
| 552 | C2×C4 | — | 12 | 4 | no | — |
| 553 | C2×C4 | — | 12 | 2 | no | — |
| 554 | C2×C4 | — | 12 | 4 | no | — |
| 555 | C2×C4 | — | 12 | 2 | no | — |
| 556 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 557 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 558 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 559 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 560 | C8 | — | 3 | 2 | no | — |
| 561 | C8 | — | 6 | 2 | no | — |
| 562 | C8 | — | 6 | 1 | no | — |
| 563 | C8 | — | 6 | 1 | no | — |
| 564 | C8 | — | 6 | 2 | no | — |
| 565 | C8 | — | 6 | 1 | no | — |
| 566 | C2×C3 | — | 4 | 1 | no | — |
| 567 | C2×C3 | — | 4 | 1 | no | — |
| 568 | C2×C3 | — | 4 | 1 | no | — |
| 569 | C2×C3 | — | 16 | 1 | no | — |
| 570 | D3 | — | 16 | 1 | no | — |
| 571 | C22 | — | 2 | 1 | no | — |
| 572 | C22 | — | 2 | 1 | no | — |
| 573 | C22 | — | 2 | 1 | no | — |
| 574 | C22 | — | 6 | 2 | no | — |
| 575 | C22 | — | 6 | 2 | no | — |
| 576 | C22 | — | 6 | 2 | no | — |
| 577 | C22 | — | 12 | 1 | no | — |
| 578 | C22 | — | 12 | 1 | no | — |
| 579 | C22 | — | 12 | 1 | no | — |
| 580 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 581 | C4 | — | 2 | 2 | no | — |
| 582 | C4 | — | 2 | 1 | no | — |
| 583 | C4 | — | 2 | 2 | no | — |
| 584 | C4 | — | 3 | 1 | no | — |
| 585 | C4 | — | 3 | 1 | no | — |
| 586 | C4 | — | 6 | 1 | no | — |
| 587 | C4 | — | 6 | 2 | no | — |
| 588 | C4 | — | 6 | 2 | no | — |
| 589 | C4 | — | 6 | 1 | no | — |
| 590 | C4 | — | 6 | 1 | no | — |
| 591 | C4 | — | 12 | 2 | no | — |
| 592 | C4 | — | 12 | 1 | no | — |
| 593 | C4 | — | 12 | 2 | no | — |
| 594 | C4 | — | 12 | 1 | no | — |
| 595 | C4 | #5923 | 1 | 2 | yes | — |
| 596 | C3 | — | 4 | 1 | no | Sylow |
| 597 | C2 | — | 4 | 1 | no | — |
| 598 | C2 | — | 12 | 2 | no | — |
| 599 | C2 | — | 24 | 1 | no | — |
| 600 | C2 | #41×S4 | 1 | 1 | yes | characteristic |
| 601 | C2 | C2×#5923 | 1 | 1 | yes | characteristic |
| 602 | C2 | #87499 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Fitting | #3480 | D3 |
| hypercenter | #117 | S4 |
| hypocenter | SL(2,𝔽3) | C2×#41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3480⋊C3 | C2 | semidirect |
| 2. #85419 | C2 | semidirect |
| 3. #85414 | C2 | semidirect |
| 4. #85412 | C2 | semidirect |
| 5. #85404 | C2 | semidirect |
| 6. #85386 | C2 | semidirect |
| 7. #85378 | C2 | semidirect |
| 8. #85349 | C2 | semidirect |
| 9. C4×#984 | C2 | semidirect |
| 10. C4×#983 | C2 | semidirect |
| 11. #85319 | C2 | semidirect |
| 38. #3480 | D3 | semidirect |
| 199. #226 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3480⋊C3 | 2 |
| #85419 | 2 |
| #85414 | 2 |
| #85412 | 2 |
| #85404 | 2 |
| #85386 | 2 |
| #85378 | 2 |
| #85349 | 2 |
| C4×#984 | 2 |
| C4×#983 | 2 |
| #85319 | 2 |
| #61467 | 3 |
| #5533 | 4 |