Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
366 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 57 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 | C4×#983 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2982⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×SL(2,𝔽3)⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #73020 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #73018 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #73010 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #73009 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #17874 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×#983 | C4 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 11 | #5420 | C4 | 1 | 1 | yes | characteristic |
| 12 | #5394 | C4 | 1 | 1 | yes | characteristic |
| 13 | C4×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 14 | #5388 | C4 | 1 | 1 | yes | characteristic |
| 15 | C4×#226 | C22 | 1 | 1 | yes | characteristic |
| 16 | #4730 | — | 4 | 1 | no | maximal |
| 17 | C8×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 18 | C8×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 19 | SL(2,𝔽3)⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 20 | SL(2,𝔽3)⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2989 | — | 3 | 1 | no | — |
| 22 | #2982 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | C4×#137 | — | 3 | 1 | no | — |
| 24 | #1588 | — | 3 | 1 | no | — |
| 25 | #1588 | — | 3 | 1 | no | — |
| 26 | #1587 | — | 3 | 1 | no | — |
| 27 | #1587 | — | 3 | 1 | no | — |
| 28 | #1584 | — | 3 | 1 | no | — |
| 29 | #1584 | — | 3 | 1 | no | — |
| 30 | #1583 | — | 3 | 1 | no | — |
| 31 | #1583 | — | 3 | 1 | no | — |
| 32 | #1565 | — | 3 | 1 | no | — |
| 33 | #1476 | — | 3 | 1 | no | — |
| 34 | C2×Q8⋊C8 | — | 3 | 1 | no | — |
| 35 | #1451 | — | 3 | 1 | no | — |
| 36 | C22×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | #983 | — | 2 | 2 | no | — |
| 38 | C2×GL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C2×#226 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C4×#67 | — | 4 | 1 | no | — |
| 41 | C22×C3⋊C8 | — | 4 | 1 | no | — |
| 42 | C2×ΓL(1,𝔽25) | — | 4 | 1 | no | — |
| 43 | C4×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 45 | #859 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | #859 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C3×C22⋊C8 | — | 4 | 1 | no | — |
| 48 | (C2×D3)⋊C8 | — | 4 | 1 | no | — |
| 49 | #815 | — | 4 | 1 | no | — |
| 50 | Dic3⋊C8 | — | 4 | 1 | no | — |
| 51 | C2×#137 | — | 3 | 1 | no | — |
| 52 | C4×#41 | — | 3 | 1 | no | — |
| 53 | C2×C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 54 | #446 | — | 3 | 1 | no | — |
| 55 | #446 | — | 3 | 1 | no | — |
| 56 | C8×Q8 | — | 3 | 1 | no | — |
| 57 | C8×Q8 | — | 3 | 1 | no | — |
| 58 | C8×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | C8×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 60 | #441 | — | 3 | 1 | no | — |
| 61 | #441 | — | 3 | 1 | no | — |
| 62 | #440 | — | 3 | 1 | no | — |
| 63 | #440 | — | 3 | 1 | no | — |
| 64 | C4×Q16 | — | 3 | 1 | no | — |
| 65 | C4×Q16 | — | 3 | 1 | no | — |
| 66 | C4×SD16 | — | 3 | 1 | no | — |
| 67 | C4×SD16 | — | 3 | 1 | no | — |
| 68 | #436 | — | 3 | 1 | no | — |
| 69 | #436 | — | 3 | 1 | no | — |
| 70 | #435 | — | 6 | 1 | no | — |
| 71 | #435 | — | 6 | 1 | no | — |
| 72 | #432 | — | 6 | 1 | no | — |
| 73 | #431 | — | 3 | 1 | no | — |
| 74 | #431 | — | 3 | 1 | no | — |
| 75 | #428 | — | 3 | 1 | no | — |
| 76 | #423 | — | 6 | 1 | no | — |
| 77 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 78 | #419 | — | 3 | 1 | no | — |
| 79 | #417 | — | 3 | 1 | no | — |
| 80 | C4×M16 | — | 3 | 1 | no | — |
| 81 | C2×C4×C8 | — | 3 | 1 | no | — |
| 82 | #335 | — | 3 | 1 | no | — |
| 83 | #335 | — | 3 | 1 | no | — |
| 84 | #334 | — | 3 | 1 | no | — |
| 85 | #334 | — | 3 | 1 | no | — |
| 86 | Q8⋊C8 | — | 3 | 1 | no | — |
| 87 | Q8⋊C8 | — | 3 | 1 | no | — |
| 88 | Q8⋊C8 | — | 3 | 1 | no | — |
| 89 | Q8⋊C8 | — | 3 | 1 | no | — |
| 90 | Q8⋊C8 | — | 3 | 1 | no | — |
| 91 | Q8⋊C8 | — | 3 | 1 | no | — |
| 92 | D4⋊C8 | — | 3 | 1 | no | — |
| 93 | D4⋊C8 | — | 3 | 1 | no | — |
| 94 | #322 | — | 3 | 1 | no | — |
| 95 | #322 | — | 3 | 1 | no | — |
| 96 | #322 | — | 3 | 1 | no | — |
| 97 | #322 | — | 3 | 1 | no | — |
| 98 | C22×C3×C4 | — | 4 | 1 | no | — |
| 99 | C2×#67 | — | 4 | 1 | no | — |
| 100 | C2×C4×D3 | — | 4 | 1 | no | — |
| 101 | C2×SL(2,𝔽3) | M16 | 1 | 2 | yes | complemented |
| 102 | C2×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 103 | GL(2,𝔽3) | — | 2 | 1 | no | — |
| 104 | #226 | — | 2 | 1 | no | — |
| 105 | C2×C3×C8 | — | 4 | 1 | no | — |
| 106 | C2×C3×C8 | — | 4 | 1 | no | — |
| 107 | #217 | — | 4 | 1 | no | — |
| 108 | (C2×D3)⋊C4 | — | 4 | 1 | no | — |
| 109 | Dic3⋊C4 | — | 4 | 1 | no | — |
| 110 | C4×Dic3 | — | 4 | 1 | no | — |
| 111 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 112 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 113 | C2×C3⋊C8 | — | 8 | 2 | no | — |
| 114 | ΓL(1,𝔽25) | — | 8 | 2 | no | — |
| 115 | C2×#41 | — | 3 | 1 | no | — |
| 116 | C22×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 117 | #137 | — | 6 | 2 | no | — |
| 118 | #137 | — | 6 | 2 | no | — |
| 119 | C2×Q16 | — | 3 | 1 | no | — |
| 120 | C2×Q16 | — | 3 | 1 | no | — |
| 121 | C2×SD16 | — | 3 | 1 | no | — |
| 122 | C2×SD16 | — | 3 | 1 | no | — |
| 123 | C2×M16 | — | 3 | 1 | no | — |
| 124 | C2×M16 | — | 3 | 1 | no | — |
| 125 | C2×M16 | — | 6 | 1 | no | — |
| 126 | C2×M16 | — | 6 | 1 | no | — |
| 127 | C22×C8 | — | 6 | 1 | no | — |
| 128 | C22×C8 | — | 6 | 1 | no | — |
| 129 | C4×Q8 | — | 3 | 1 | no | — |
| 130 | C4×Q8 | — | 3 | 1 | no | — |
| 131 | C4×Q8 | — | 3 | 1 | no | — |
| 132 | C4×Q8 | — | 3 | 1 | no | — |
| 133 | C4×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 134 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 135 | C4×D4 | — | 3 | 1 | no | — |
| 136 | C4×D4 | — | 3 | 1 | no | — |
| 137 | C4×D4 | — | 6 | 1 | no | — |
| 138 | C4×D4 | — | 6 | 1 | no | — |
| 139 | #117 | — | 3 | 1 | no | — |
| 140 | #117 | — | 6 | 1 | no | — |
| 141 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 142 | C2×C42 | — | 3 | 1 | no | — |
| 143 | C2×C42 | — | 6 | 1 | no | — |
| 144 | #107 | — | 3 | 1 | no | — |
| 145 | #107 | — | 3 | 1 | no | — |
| 146 | #106 | — | 3 | 1 | no | — |
| 147 | #106 | — | 3 | 1 | no | — |
| 148 | C4⋊C8 | — | 3 | 1 | no | — |
| 149 | C4⋊C8 | — | 3 | 1 | no | — |
| 150 | C4⋊C8 | — | 3 | 1 | no | — |
| 151 | C4⋊C8 | — | 3 | 1 | no | — |
| 152 | C4⋊C8 | — | 3 | 1 | no | — |
| 153 | C4⋊C8 | — | 3 | 1 | no | — |
| 154 | C4⋊C8 | — | 6 | 1 | no | — |
| 155 | C4⋊C8 | — | 6 | 1 | no | — |
| 156 | C4⋊C8 | — | 6 | 1 | no | — |
| 157 | #103 | — | 3 | 1 | no | — |
| 158 | #103 | — | 3 | 1 | no | — |
| 159 | #103 | — | 3 | 1 | no | — |
| 160 | #103 | — | 3 | 1 | no | — |
| 161 | #103 | — | 3 | 1 | no | — |
| 162 | #103 | — | 3 | 1 | no | — |
| 163 | #102 | — | 3 | 1 | no | — |
| 164 | #102 | — | 3 | 1 | no | — |
| 165 | C22⋊C8 | — | 3 | 1 | no | — |
| 166 | C22⋊C8 | — | 6 | 1 | no | — |
| 167 | C22⋊C8 | — | 6 | 1 | no | — |
| 168 | C22⋊C8 | S4 | 1 | 1 | yes | characteristic |
| 169 | #97 | — | 3 | 1 | no | — |
| 170 | #97 | — | 3 | 1 | no | — |
| 171 | #97 | — | 6 | 1 | no | — |
| 172 | #97 | — | 6 | 1 | no | — |
| 173 | C4×C8 | — | 3 | 1 | no | — |
| 174 | C4×C8 | — | 3 | 1 | no | — |
| 175 | C4×C8 | — | 3 | 1 | no | — |
| 176 | C4×C8 | — | 3 | 1 | no | — |
| 177 | C4×C8 | — | 3 | 1 | no | — |
| 178 | C4×C8 | — | 3 | 1 | no | — |
| 179 | C4×C8 | — | 6 | 1 | no | — |
| 180 | C4×C8 | — | 6 | 1 | no | — |
| 181 | C23×C3 | — | 4 | 1 | no | — |
| 182 | C22×D3 | — | 4 | 1 | no | — |
| 183 | C2×C3×C4 | — | 4 | 1 | no | — |
| 184 | C2×C3×C4 | — | 4 | 1 | no | — |
| 185 | C2×C3×C4 | — | 8 | 1 | no | — |
| 186 | C2×C3×C4 | — | 8 | 1 | no | — |
| 187 | #67 | — | 8 | 2 | no | — |
| 188 | C2×Dic3 | — | 4 | 1 | no | — |
| 189 | C2×Dic3 | — | 4 | 1 | no | — |
| 190 | C2×Dic3 | — | 4 | 1 | no | — |
| 191 | C4×D3 | — | 8 | 1 | no | — |
| 192 | C4×D3 | — | 8 | 1 | no | — |
| 193 | SL(2,𝔽3) | C2×M16 | 1 | 1 | yes | characteristic |
| 194 | C3×C8 | — | 8 | 1 | no | — |
| 195 | C3×C8 | — | 8 | 1 | no | — |
| 196 | C3⋊C8 | — | 4 | 2 | no | — |
| 197 | C3⋊C8 | — | 8 | 1 | no | — |
| 198 | #41 | — | 6 | 2 | no | — |
| 199 | #41 | — | 6 | 2 | no | — |
| 200 | C2×Q8 | — | 3 | 1 | no | — |
| 201 | C2×Q8 | — | 3 | 2 | no | — |
| 202 | C2×Q8 | — | 3 | 1 | no | — |
| 203 | C2×Q8 | ΓL(1,𝔽25) | 1 | 2 | yes | complemented |
| 204 | C2×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 205 | C2×D4 | — | 3 | 1 | no | — |
| 206 | C2×D4 | — | 6 | 1 | no | — |
| 207 | C22×C4 | — | 3 | 1 | no | — |
| 208 | C22×C4 | — | 3 | 1 | no | — |
| 209 | C22×C4 | — | 6 | 1 | no | — |
| 210 | C22×C4 | — | 6 | 1 | no | — |
| 211 | C22×C4 | — | 6 | 1 | no | — |
| 212 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 213 | Q16 | — | 6 | 2 | no | — |
| 214 | Q16 | — | 6 | 1 | no | — |
| 215 | Q16 | — | 6 | 1 | no | — |
| 216 | SD16 | — | 6 | 2 | no | — |
| 217 | SD16 | — | 6 | 1 | no | — |
| 218 | SD16 | — | 6 | 1 | no | — |
| 219 | M16 | — | 6 | 2 | no | — |
| 220 | M16 | — | 6 | 2 | no | — |
| 221 | M16 | — | 24 | 1 | no | — |
| 222 | M16 | — | 24 | 1 | no | — |
| 223 | C2×C8 | — | 3 | 1 | no | — |
| 224 | C2×C8 | — | 3 | 1 | no | — |
| 225 | C2×C8 | — | 3 | 1 | no | — |
| 226 | C2×C8 | — | 3 | 1 | no | — |
| 227 | C2×C8 | — | 3 | 1 | no | — |
| 228 | C2×C8 | — | 3 | 1 | no | — |
| 229 | C2×C8 | — | 6 | 1 | no | — |
| 230 | C2×C8 | — | 6 | 1 | no | — |
| 231 | C2×C8 | — | 6 | 1 | no | — |
| 232 | C2×C8 | — | 6 | 1 | no | — |
| 233 | C2×C8 | — | 12 | 2 | no | — |
| 234 | C2×C8 | — | 24 | 1 | no | — |
| 235 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 237 | C4⋊C4 | — | 3 | 1 | no | — |
| 238 | C4⋊C4 | — | 3 | 1 | no | — |
| 239 | C4⋊C4 | — | 3 | 1 | no | — |
| 240 | C4⋊C4 | — | 3 | 1 | no | — |
| 241 | C4⋊C4 | — | 6 | 1 | no | — |
| 242 | C4⋊C4 | — | 6 | 1 | no | — |
| 243 | C4⋊C4 | — | 6 | 1 | no | — |
| 244 | C22⋊C4 | — | 6 | 1 | no | — |
| 245 | C22⋊C4 | — | 6 | 1 | no | — |
| 246 | C22⋊C4 | — | 6 | 1 | no | — |
| 247 | C42 | — | 3 | 1 | no | — |
| 248 | C42 | — | 3 | 1 | no | — |
| 249 | C42 | — | 3 | 1 | no | — |
| 250 | C42 | — | 3 | 1 | no | — |
| 251 | C42 | — | 6 | 1 | no | — |
| 252 | C42 | — | 6 | 1 | no | — |
| 253 | C42 | — | 6 | 1 | no | — |
| 254 | C22×C3 | — | 4 | 1 | no | — |
| 255 | C22×C3 | — | 4 | 2 | no | — |
| 256 | C22×C3 | — | 8 | 1 | no | — |
| 257 | C22×C3 | — | 8 | 1 | no | — |
| 258 | C2×D3 | — | 8 | 1 | no | — |
| 259 | C2×D3 | — | 8 | 1 | no | — |
| 260 | C2×D3 | — | 8 | 1 | no | — |
| 261 | C3×C4 | — | 4 | 1 | no | — |
| 262 | C3×C4 | — | 4 | 1 | no | — |
| 263 | C3×C4 | — | 8 | 1 | no | — |
| 264 | Dic3 | — | 8 | 1 | no | — |
| 265 | Dic3 | — | 8 | 1 | no | — |
| 266 | Dic3 | — | 8 | 1 | no | — |
| 267 | C23 | — | 6 | 1 | no | — |
| 268 | C23 | C4×S4 | 1 | 1 | yes | characteristic |
| 269 | Q8 | — | 3 | 2 | no | — |
| 270 | Q8 | — | 3 | 1 | no | — |
| 271 | Q8 | — | 6 | 1 | no | — |
| 272 | Q8 | — | 6 | 1 | no | — |
| 273 | Q8 | — | 6 | 1 | no | — |
| 274 | Q8 | C2×ΓL(1,𝔽25) | 1 | 1 | yes | characteristic |
| 275 | D4 | — | 6 | 1 | no | — |
| 276 | D4 | — | 6 | 1 | no | — |
| 277 | D4 | — | 12 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 1 | no | — |
| 279 | C2×C4 | — | 2 | 1 | no | — |
| 280 | C2×C4 | — | 3 | 1 | no | — |
| 281 | C2×C4 | — | 3 | 1 | no | — |
| 282 | C2×C4 | — | 3 | 2 | no | — |
| 283 | C2×C4 | — | 3 | 2 | no | — |
| 284 | C2×C4 | — | 3 | 1 | no | — |
| 285 | C2×C4 | — | 3 | 1 | no | — |
| 286 | C2×C4 | — | 6 | 1 | no | — |
| 287 | C2×C4 | — | 6 | 1 | no | — |
| 288 | C2×C4 | — | 6 | 1 | no | — |
| 289 | C2×C4 | — | 12 | 2 | no | — |
| 290 | C2×C4 | — | 12 | 1 | no | — |
| 291 | C2×C4 | — | 12 | 1 | no | — |
| 292 | C2×C4 | — | 12 | 1 | no | — |
| 293 | C2×C4 | — | 24 | 1 | no | — |
| 294 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 296 | C8 | — | 2 | 1 | no | — |
| 297 | C8 | — | 2 | 1 | no | — |
| 298 | C8 | — | 6 | 1 | no | — |
| 299 | C8 | — | 6 | 1 | no | — |
| 300 | C8 | — | 6 | 1 | no | — |
| 301 | C8 | — | 6 | 1 | no | — |
| 302 | C8 | — | 6 | 2 | no | — |
| 303 | C8 | — | 6 | 1 | no | — |
| 304 | C8 | — | 6 | 1 | no | — |
| 305 | C8 | — | 12 | 1 | no | — |
| 306 | C8 | — | 12 | 1 | no | — |
| 307 | C8 | — | 12 | 1 | no | — |
| 308 | C2×C3 | — | 4 | 1 | no | — |
| 309 | C2×C3 | — | 4 | 1 | no | — |
| 310 | C2×C3 | — | 4 | 1 | no | — |
| 311 | C2×C3 | — | 8 | 2 | no | — |
| 312 | D3 | — | 16 | 1 | no | — |
| 313 | C22 | — | 2 | 1 | no | — |
| 314 | C22 | — | 2 | 1 | no | — |
| 315 | C22 | — | 12 | 1 | no | — |
| 316 | C22 | — | 12 | 1 | no | — |
| 317 | C22 | — | 12 | 1 | no | — |
| 318 | C22 | #5400 | 1 | 2 | yes | — |
| 319 | C22 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 2 | 1 | no | — |
| 321 | C4 | — | 3 | 1 | no | — |
| 322 | C4 | — | 3 | 1 | no | — |
| 323 | C4 | — | 3 | 2 | no | — |
| 324 | C4 | — | 6 | 1 | no | — |
| 325 | C4 | — | 6 | 1 | no | — |
| 326 | C4 | — | 12 | 1 | no | — |
| 327 | C4 | — | 12 | 1 | no | — |
| 328 | C4 | — | 12 | 1 | no | — |
| 329 | C4 | #5926 | 1 | 1 | yes | characteristic |
| 330 | C4 | #5927 | 1 | 1 | yes | characteristic |
| 331 | C3 | — | 4 | 1 | no | Sylow |
| 332 | C2 | — | 2 | 2 | no | — |
| 333 | C2 | — | 24 | 1 | no | — |
| 334 | C2 | #85354 | 1 | 1 | yes | characteristic |
| 335 | C2 | C2×#5400 | 1 | 1 | yes | characteristic |
| 336 | C2 | #85453 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2982⋊C3 | C2 | semidirect |
| 3. C2×SL(2,𝔽3)⋊C8 | C2 | semidirect |
| 4. #73020 | C2 | semidirect |
| 5. #73018 | C2 | semidirect |
| 6. #73010 | C2 | semidirect |
| 7. #73009 | C2 | semidirect |
| 22. #2982 | D3 | semidirect |
| 101. C2×SL(2,𝔽3) | M16 | semidirect |
| 203. C2×Q8 | ΓL(1,𝔽25) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#983 | 2 |
| #2982⋊C3 | 2 |
| C2×SL(2,𝔽3)⋊C8 | 2 |
| #73020 | 2 |
| #73018 | 2 |
| #73010 | 2 |
| #73009 | 2 |
| #17874 | 3 |
| #4730 | 4 |