Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1006 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 157 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 | #41×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #85428 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Q8×#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #85393 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | D4×#226 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #85322 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×C4×#226 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | Q16×#41 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×#226 | C22 | 1 | 3 | yes | — |
| 10 | Dic3×#41 | — | 4 | 1 | no | maximal |
| 11 | Q8×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 12 | D4×SL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 13 | C2×C4×SL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 14 | #5419 | C22 | 1 | 6 | yes | — |
| 15 | C2×#859 | C22 | 1 | 3 | yes | — |
| 16 | #5391 | C22 | 1 | 3 | yes | — |
| 17 | C4⋊#226 | C22 | 1 | 3 | yes | — |
| 18 | C4×#226 | C22 | 1 | 1 | yes | characteristic |
| 19 | C4×#226 | C22 | 1 | 3 | yes | — |
| 20 | C4×#226 | C22 | 1 | 3 | yes | — |
| 21 | C4×#226 | C22 | 1 | 3 | yes | — |
| 22 | Q8×#41 | — | 3 | 1 | no | — |
| 23 | Q8×#41 | D3 | 1 | 1 | yes | characteristic |
| 24 | #3395 | — | 3 | 1 | no | — |
| 25 | Q8×Q16 | — | 3 | 1 | no | — |
| 26 | #3363 | — | 3 | 1 | no | — |
| 27 | #3363 | — | 3 | 1 | no | — |
| 28 | #3338 | — | 3 | 3 | no | — |
| 29 | #3299 | — | 3 | 3 | no | — |
| 30 | #3299 | — | 3 | 3 | no | — |
| 31 | D4×Q16 | — | 3 | 3 | no | — |
| 32 | #3170 | — | 3 | 3 | no | — |
| 33 | #3104 | — | 3 | 3 | no | — |
| 34 | #3104 | — | 3 | 3 | no | — |
| 35 | C8×#41 | — | 3 | 1 | no | — |
| 36 | C2×C4×Q16 | — | 3 | 3 | no | — |
| 37 | C2×C3×#41 | — | 4 | 1 | no | — |
| 38 | C22×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 39 | C2×#226 | — | 2 | 6 | no | — |
| 40 | C2×#226 | C23 | 1 | 3 | yes | — |
| 41 | C2×#226 | C23 | 1 | 1 | yes | characteristic |
| 42 | Q8×Dic3 | — | 4 | 1 | no | — |
| 43 | Q8×Dic3 | — | 4 | 1 | no | — |
| 44 | D4×Dic3 | — | 4 | 3 | no | — |
| 45 | D4×Dic3 | — | 4 | 3 | no | — |
| 46 | #926 | — | 4 | 3 | no | — |
| 47 | C2×C4×Dic3 | — | 4 | 3 | no | — |
| 48 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 49 | C4×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 50 | #859 | C23 | 1 | 1 | yes | characteristic |
| 51 | #859 | C23 | 1 | 3 | yes | — |
| 52 | C22×Q16 | — | 3 | 3 | no | — |
| 53 | Q82 | — | 3 | 1 | no | — |
| 54 | Q82 | C2×D3 | 1 | 1 | yes | characteristic |
| 55 | #557 | — | 3 | 1 | no | — |
| 56 | #557 | — | 3 | 1 | no | — |
| 57 | #557 | — | 6 | 1 | no | — |
| 58 | #554 | — | 3 | 3 | no | — |
| 59 | #554 | — | 3 | 3 | no | — |
| 60 | #554 | — | 6 | 3 | no | — |
| 61 | D4×Q8 | — | 3 | 3 | no | — |
| 62 | D4×Q8 | C2×D3 | 1 | 3 | yes | — |
| 63 | #533 | — | 3 | 3 | no | — |
| 64 | #533 | — | 3 | 3 | no | — |
| 65 | #533 | — | 6 | 3 | no | — |
| 66 | C4×#41 | — | 3 | 1 | no | — |
| 67 | C4×#41 | — | 6 | 1 | no | — |
| 68 | C2×C4×Q8 | — | 3 | 3 | no | — |
| 69 | C2×C4×Q8 | C2×D3 | 1 | 3 | yes | — |
| 70 | #500 | — | 3 | 3 | no | — |
| 71 | #494 | — | 3 | 3 | no | — |
| 72 | #487 | — | 3 | 3 | no | — |
| 73 | #487 | — | 3 | 3 | no | — |
| 74 | #484 | — | 3 | 6 | no | — |
| 75 | #484 | — | 3 | 6 | no | — |
| 76 | #477 | — | 3 | 3 | no | — |
| 77 | #477 | — | 3 | 3 | no | — |
| 78 | #477 | — | 3 | 3 | no | — |
| 79 | #477 | — | 3 | 3 | no | — |
| 80 | #467 | — | 3 | 6 | no | — |
| 81 | #467 | — | 3 | 6 | no | — |
| 82 | #462 | — | 3 | 3 | no | — |
| 83 | #462 | — | 3 | 3 | no | — |
| 84 | #462 | — | 3 | 3 | no | — |
| 85 | #462 | — | 3 | 3 | no | — |
| 86 | #451 | — | 3 | 6 | no | — |
| 87 | #451 | — | 3 | 6 | no | — |
| 88 | C8×Q8 | — | 3 | 1 | no | — |
| 89 | C8×Q8 | — | 3 | 1 | no | — |
| 90 | C4×Q16 | — | 3 | 3 | no | — |
| 91 | C4×Q16 | — | 3 | 6 | no | — |
| 92 | C4×Q16 | — | 3 | 3 | no | — |
| 93 | C4×Q16 | — | 3 | 3 | no | — |
| 94 | C4×Q16 | — | 3 | 3 | no | — |
| 95 | C4×Q16 | — | 3 | 3 | no | — |
| 96 | C4×Q16 | — | 3 | 3 | no | — |
| 97 | C4×Q16 | — | 3 | 3 | no | — |
| 98 | C4×Q16 | — | 3 | 1 | no | — |
| 99 | C8×D4 | — | 3 | 3 | no | — |
| 100 | C8×D4 | — | 3 | 3 | no | — |
| 101 | #431 | — | 3 | 3 | no | — |
| 102 | C2×#107 | — | 3 | 3 | no | — |
| 103 | C2×#103 | — | 3 | 3 | no | — |
| 104 | C2×#103 | — | 3 | 3 | no | — |
| 105 | C2×C4×C8 | — | 3 | 3 | no | — |
| 106 | C3×#41 | — | 4 | 8 | no | — |
| 107 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 108 | C2×C3×D4 | — | 4 | 3 | no | — |
| 109 | C22×C3×C4 | — | 4 | 3 | no | — |
| 110 | C22×Dic3 | — | 4 | 3 | no | — |
| 111 | C22×Dic3 | — | 4 | 3 | no | — |
| 112 | C2×SL(2,𝔽3) | — | 2 | 3 | no | — |
| 113 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 114 | #226 | — | 2 | 3 | no | — |
| 115 | #226 | #41 | 1 | 2 | yes | direct product factor |
| 116 | #217 | — | 4 | 6 | no | — |
| 117 | C4⋊Dic3 | — | 4 | 3 | no | — |
| 118 | C4⋊Dic3 | — | 4 | 3 | no | — |
| 119 | C4×Dic3 | — | 4 | 1 | no | — |
| 120 | C4×Dic3 | — | 4 | 3 | no | — |
| 121 | C4×Dic3 | — | 4 | 3 | no | — |
| 122 | C4×Dic3 | — | 4 | 3 | no | — |
| 123 | C2×#41 | S4 | 1 | 1 | yes | characteristic |
| 124 | C22×Q8 | — | 3 | 3 | no | — |
| 125 | C22×Q8 | C22×D3 | 1 | 3 | yes | — |
| 126 | C2×Q16 | — | 3 | 1 | no | — |
| 127 | C2×Q16 | — | 3 | 3 | no | — |
| 128 | C2×Q16 | — | 3 | 3 | no | — |
| 129 | C2×Q16 | — | 3 | 3 | no | — |
| 130 | C2×Q16 | — | 3 | 6 | no | — |
| 131 | C2×Q16 | — | 6 | 6 | no | — |
| 132 | C2×Q16 | — | 6 | 6 | no | — |
| 133 | C22×C8 | — | 3 | 3 | no | — |
| 134 | C22×C8 | — | 3 | 3 | no | — |
| 135 | C4⋊Q8 | — | 3 | 3 | no | — |
| 136 | C4⋊Q8 | — | 3 | 3 | no | — |
| 137 | C4⋊Q8 | — | 3 | 3 | no | — |
| 138 | C4⋊Q8 | — | 3 | 3 | no | — |
| 139 | C4⋊Q8 | — | 6 | 3 | no | — |
| 140 | C4⋊Q8 | — | 6 | 3 | no | — |
| 141 | #125 | — | 6 | 3 | no | — |
| 142 | #125 | — | 6 | 3 | no | — |
| 143 | #125 | — | 6 | 3 | no | — |
| 144 | #125 | — | 6 | 3 | no | — |
| 145 | C22⋊Q8 | — | 3 | 6 | no | — |
| 146 | C22⋊Q8 | — | 3 | 6 | no | — |
| 147 | C22⋊Q8 | — | 6 | 3 | no | — |
| 148 | C22⋊Q8 | — | 6 | 6 | no | — |
| 149 | C22⋊Q8 | — | 6 | 6 | no | — |
| 150 | C22⋊Q8 | — | 6 | 3 | no | — |
| 151 | C4×Q8 | — | 3 | 1 | no | — |
| 152 | C4×Q8 | — | 3 | 1 | no | — |
| 153 | C4×Q8 | — | 3 | 3 | no | — |
| 154 | C4×Q8 | — | 3 | 3 | no | — |
| 155 | C4×Q8 | — | 3 | 1 | no | — |
| 156 | C4×Q8 | — | 3 | 3 | no | — |
| 157 | C4×Q8 | — | 3 | 3 | no | — |
| 158 | C4×Q8 | — | 3 | 3 | no | — |
| 159 | C4×Q8 | — | 6 | 1 | no | — |
| 160 | C4×Q8 | — | 6 | 3 | no | — |
| 161 | C4×Q8 | — | 6 | 3 | no | — |
| 162 | C4×Q8 | — | 6 | 3 | no | — |
| 163 | C4×Q8 | — | 6 | 1 | no | — |
| 164 | C4×Q8 | — | 6 | 1 | no | — |
| 165 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 166 | C4×Q8 | C22×D3 | 1 | 3 | yes | — |
| 167 | C4×D4 | — | 3 | 3 | no | — |
| 168 | C4×D4 | — | 3 | 3 | no | — |
| 169 | C4×D4 | — | 6 | 3 | no | — |
| 170 | C4×D4 | — | 6 | 3 | no | — |
| 171 | #117 | — | 3 | 3 | no | — |
| 172 | #117 | — | 6 | 3 | no | — |
| 173 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 174 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 175 | C2×C4⋊C4 | — | 6 | 3 | no | — |
| 176 | C2×C42 | — | 3 | 3 | no | — |
| 177 | C2×C42 | — | 6 | 3 | no | — |
| 178 | #107 | — | 3 | 1 | no | — |
| 179 | #107 | — | 3 | 3 | no | — |
| 180 | #107 | — | 3 | 3 | no | — |
| 181 | #107 | — | 3 | 3 | no | — |
| 182 | #107 | — | 3 | 6 | no | — |
| 183 | C4⋊C8 | — | 3 | 3 | no | — |
| 184 | C4⋊C8 | — | 3 | 3 | no | — |
| 185 | #103 | — | 3 | 1 | no | — |
| 186 | #103 | — | 3 | 3 | no | — |
| 187 | #103 | — | 3 | 3 | no | — |
| 188 | #103 | — | 3 | 3 | no | — |
| 189 | #103 | — | 3 | 3 | no | — |
| 190 | #103 | — | 3 | 3 | no | — |
| 191 | #103 | — | 3 | 6 | no | — |
| 192 | #103 | — | 3 | 6 | no | — |
| 193 | #103 | — | 3 | 1 | no | — |
| 194 | #103 | — | 3 | 3 | no | — |
| 195 | C22⋊C8 | — | 3 | 6 | no | — |
| 196 | C4×C8 | — | 3 | 1 | no | — |
| 197 | C4×C8 | — | 3 | 3 | no | — |
| 198 | C4×C8 | — | 3 | 3 | no | — |
| 199 | C4×C8 | — | 3 | 3 | no | — |
| 200 | C23×C3 | — | 4 | 3 | no | — |
| 201 | C3×Q8 | — | 4 | 4 | no | — |
| 202 | C3×D4 | — | 4 | 12 | no | — |
| 203 | C2×C3×C4 | — | 4 | 12 | no | — |
| 204 | C2×C3×C4 | — | 4 | 3 | no | — |
| 205 | C2×C3×C4 | — | 4 | 1 | no | — |
| 206 | C2×Dic3 | — | 4 | 1 | no | — |
| 207 | C2×Dic3 | — | 4 | 1 | no | — |
| 208 | C2×Dic3 | — | 4 | 3 | no | — |
| 209 | C2×Dic3 | — | 4 | 3 | no | — |
| 210 | C2×Dic3 | — | 8 | 6 | no | — |
| 211 | C2×Dic3 | — | 8 | 6 | no | — |
| 212 | SL(2,𝔽3) | C2×#41 | 1 | 1 | yes | characteristic |
| 213 | #41 | #226 | 1 | 8 | yes | direct product factor |
| 214 | C2×Q8 | — | 2 | 3 | no | — |
| 215 | C2×Q8 | — | 3 | 3 | no | — |
| 216 | C2×Q8 | — | 3 | 1 | no | — |
| 217 | C2×Q8 | — | 3 | 3 | no | — |
| 218 | C2×Q8 | — | 6 | 1 | no | — |
| 219 | C2×Q8 | — | 6 | 3 | no | — |
| 220 | C2×Q8 | — | 6 | 3 | no | — |
| 221 | C2×Q8 | — | 6 | 3 | no | — |
| 222 | C2×Q8 | — | 6 | 6 | no | — |
| 223 | C2×Q8 | — | 12 | 3 | no | — |
| 224 | C2×Q8 | C2×S4 | 1 | 1 | yes | characteristic |
| 225 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 226 | C2×D4 | C2×S4 | 1 | 3 | yes | — |
| 227 | C22×C4 | — | 3 | 3 | no | — |
| 228 | C22×C4 | — | 3 | 3 | no | — |
| 229 | C22×C4 | — | 6 | 3 | no | — |
| 230 | C22×C4 | — | 6 | 3 | no | — |
| 231 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 232 | Q16 | — | 3 | 2 | no | — |
| 233 | Q16 | — | 3 | 2 | no | — |
| 234 | Q16 | — | 6 | 6 | no | — |
| 235 | Q16 | — | 6 | 3 | no | — |
| 236 | Q16 | — | 6 | 6 | no | — |
| 237 | Q16 | — | 6 | 3 | no | — |
| 238 | Q16 | — | 12 | 6 | no | — |
| 239 | C2×C8 | — | 3 | 1 | no | — |
| 240 | C2×C8 | — | 3 | 1 | no | — |
| 241 | C2×C8 | — | 3 | 3 | no | — |
| 242 | C2×C8 | — | 3 | 3 | no | — |
| 243 | C2×C8 | — | 6 | 6 | no | — |
| 244 | C2×C8 | — | 12 | 3 | no | — |
| 245 | C4⋊C4 | — | 3 | 1 | no | — |
| 246 | C4⋊C4 | — | 3 | 3 | no | — |
| 247 | C4⋊C4 | — | 3 | 3 | no | — |
| 248 | C4⋊C4 | — | 3 | 3 | no | — |
| 249 | C4⋊C4 | — | 3 | 1 | no | — |
| 250 | C4⋊C4 | — | 3 | 3 | no | — |
| 251 | C4⋊C4 | — | 6 | 3 | no | — |
| 252 | C4⋊C4 | — | 6 | 3 | no | — |
| 253 | C4⋊C4 | — | 6 | 3 | no | — |
| 254 | C4⋊C4 | — | 6 | 1 | no | — |
| 255 | C4⋊C4 | — | 6 | 3 | no | — |
| 256 | C4⋊C4 | — | 6 | 3 | no | — |
| 257 | C4⋊C4 | — | 6 | 3 | no | — |
| 258 | C4⋊C4 | — | 6 | 6 | no | — |
| 259 | C4⋊C4 | — | 6 | 3 | no | — |
| 260 | C4⋊C4 | — | 6 | 3 | no | — |
| 261 | C4⋊C4 | — | 6 | 3 | no | — |
| 262 | C22⋊C4 | — | 3 | 6 | no | — |
| 263 | C22⋊C4 | — | 6 | 6 | no | — |
| 264 | C42 | — | 3 | 3 | no | — |
| 265 | C42 | — | 3 | 3 | no | — |
| 266 | C42 | — | 3 | 1 | no | — |
| 267 | C42 | — | 3 | 3 | no | — |
| 268 | C42 | — | 6 | 3 | no | — |
| 269 | C42 | — | 6 | 3 | no | — |
| 270 | C42 | — | 6 | 1 | no | — |
| 271 | C42 | — | 6 | 3 | no | — |
| 272 | C22×C3 | — | 4 | 6 | no | — |
| 273 | C22×C3 | — | 4 | 1 | no | — |
| 274 | C22×C3 | — | 8 | 3 | no | — |
| 275 | C22×C3 | — | 8 | 3 | no | — |
| 276 | C3×C4 | — | 4 | 2 | no | — |
| 277 | C3×C4 | — | 4 | 6 | no | — |
| 278 | Dic3 | — | 4 | 2 | no | — |
| 279 | Dic3 | — | 4 | 2 | no | — |
| 280 | Dic3 | — | 8 | 3 | no | — |
| 281 | Dic3 | — | 8 | 3 | no | — |
| 282 | C23 | C22×S4 | 1 | 3 | yes | — |
| 283 | Q8 | — | 3 | 1 | no | — |
| 284 | Q8 | — | 3 | 2 | no | — |
| 285 | Q8 | — | 6 | 3 | no | — |
| 286 | Q8 | — | 6 | 3 | no | — |
| 287 | Q8 | — | 6 | 1 | no | — |
| 288 | Q8 | — | 6 | 3 | no | — |
| 289 | Q8 | — | 6 | 3 | no | — |
| 290 | Q8 | — | 12 | 6 | no | — |
| 291 | Q8 | — | 24 | 1 | no | — |
| 292 | Q8 | — | 24 | 3 | no | — |
| 293 | Q8 | C2×#226 | 1 | 4 | yes | complemented |
| 294 | Q8 | D3×#41 | 1 | 1 | yes | characteristic |
| 295 | D4 | C2×#226 | 1 | 12 | yes | complemented |
| 296 | C2×C4 | — | 3 | 1 | no | — |
| 297 | C2×C4 | — | 3 | 1 | no | — |
| 298 | C2×C4 | — | 3 | 3 | no | — |
| 299 | C2×C4 | — | 3 | 3 | no | — |
| 300 | C2×C4 | — | 6 | 3 | no | — |
| 301 | C2×C4 | — | 6 | 3 | no | — |
| 302 | C2×C4 | — | 6 | 1 | no | — |
| 303 | C2×C4 | — | 6 | 1 | no | — |
| 304 | C2×C4 | — | 6 | 3 | no | — |
| 305 | C2×C4 | — | 6 | 3 | no | — |
| 306 | C2×C4 | — | 12 | 3 | no | — |
| 307 | C2×C4 | — | 12 | 6 | no | — |
| 308 | C2×C4 | — | 24 | 3 | no | — |
| 309 | C2×C4 | C2×#226 | 1 | 12 | yes | complemented |
| 310 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 311 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 312 | C8 | — | 3 | 2 | no | — |
| 313 | C8 | — | 6 | 3 | no | — |
| 314 | C8 | — | 6 | 1 | no | — |
| 315 | C8 | — | 6 | 3 | no | — |
| 316 | C2×C3 | — | 4 | 1 | no | — |
| 317 | C2×C3 | — | 4 | 1 | no | — |
| 318 | C2×C3 | — | 4 | 1 | no | — |
| 319 | C2×C3 | — | 8 | 6 | no | — |
| 320 | C22 | — | 2 | 3 | no | — |
| 321 | C22 | — | 2 | 3 | no | — |
| 322 | C22 | C22×#226 | 1 | 6 | yes | — |
| 323 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 324 | C4 | — | 3 | 1 | no | — |
| 325 | C4 | — | 3 | 1 | no | — |
| 326 | C4 | — | 6 | 3 | no | — |
| 327 | C4 | — | 6 | 2 | no | — |
| 328 | C4 | — | 6 | 1 | no | — |
| 329 | C4 | — | 6 | 3 | no | — |
| 330 | C4 | — | 12 | 3 | no | — |
| 331 | C4 | — | 12 | 1 | no | — |
| 332 | C4 | — | 12 | 3 | no | — |
| 333 | C4 | C22×#226 | 1 | 2 | yes | — |
| 334 | C4 | C22×#226 | 1 | 6 | yes | — |
| 335 | C3 | — | 4 | 1 | no | Sylow |
| 336 | C2 | — | 2 | 6 | no | — |
| 337 | C2 | C23×#226 | 1 | 1 | yes | characteristic |
| 338 | C2 | #41×S4 | 1 | 1 | yes | characteristic |
| 339 | C2 | #87497 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #85428 | C2 | semidirect |
| 3. Q8×#226 | C2 | semidirect |
| 4. #85393 | C2 | semidirect |
| 5. D4×#226 | C2 | semidirect |
| 6. #85322 | C2 | semidirect |
| 7. C2×C4×#226 | C2 | semidirect |
| 115. #226 | #41 | direct (N × H) |
| 213. #41 | #226 | direct (N × H) |
| 293. Q8 | C2×#226 | semidirect |
| 295. D4 | C2×#226 | semidirect |
| 309. C2×C4 | C2×#226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×SL(2,𝔽3) | 2 |
| #85428 | 2 |
| Q8×#226 | 2 |
| #85393 | 2 |
| D4×#226 | 2 |
| #85322 | 2 |
| C2×C4×#226 | 2 |
| Q16×#41 | 3 |
| Dic3×#41 | 4 |