Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
399 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 67 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×#985 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C8×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #73068 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | GL(2,𝔽3)⋊C8 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #226⋊C8 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #17849 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#985 | C4 | 1 | 1 | yes | characteristic |
| 8 | C4×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | #5424 | C4 | 1 | 1 | yes | characteristic |
| 10 | C4×GL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic |
| 11 | C4×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 12 | C4×#226 | C4 | 1 | 1 | yes | characteristic |
| 13 | C4×#226 | C22 | 1 | 1 | yes | characteristic |
| 14 | #4685 | — | 4 | 1 | no | maximal |
| 15 | C8×SL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 16 | SL(2,𝔽3)⋊C8 | C22 | 1 | 2 | yes | — |
| 17 | C8×#41 | — | 3 | 1 | no | — |
| 18 | C8×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C4×#135 | — | 3 | 1 | no | — |
| 20 | #1580 | — | 3 | 2 | no | — |
| 21 | #1579 | — | 3 | 2 | no | — |
| 22 | SD16⋊C8 | — | 3 | 2 | no | — |
| 23 | SD16⋊C8 | — | 3 | 2 | no | — |
| 24 | #1564 | — | 3 | 1 | no | — |
| 25 | #1476 | — | 3 | 1 | no | — |
| 26 | #1476 | — | 3 | 1 | no | — |
| 27 | C2×#322 | — | 3 | 1 | no | — |
| 28 | C2×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #985 | C8 | 1 | 4 | yes | complemented |
| 30 | C2×GL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C2×#226 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C42×D3 | — | 4 | 1 | no | — |
| 33 | C4×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 35 | #859 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | #859 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C3×C4×C8 | — | 4 | 1 | no | — |
| 38 | (C2×D3)⋊C8 | — | 4 | 2 | no | — |
| 39 | Dic3⋊C8 | — | 4 | 2 | no | — |
| 40 | C4×C3⋊C8 | — | 4 | 1 | no | — |
| 41 | C2×#135 | — | 3 | 1 | no | — |
| 42 | C4×#41 | — | 3 | 1 | no | — |
| 43 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 44 | C8×Q8 | — | 3 | 2 | no | — |
| 45 | C8×Q8 | C2×D3 | 1 | 2 | yes | — |
| 46 | C4×Q16 | — | 3 | 1 | no | — |
| 47 | C4×Q16 | — | 3 | 1 | no | — |
| 48 | C4×SD16 | — | 3 | 1 | no | — |
| 49 | C4×SD16 | — | 3 | 1 | no | — |
| 50 | C4×SD16 | — | 3 | 1 | no | — |
| 51 | C4×SD16 | — | 3 | 1 | no | — |
| 52 | C4×D8 | — | 3 | 1 | no | — |
| 53 | C4×D8 | — | 3 | 1 | no | — |
| 54 | C8×D4 | — | 3 | 2 | no | — |
| 55 | C8×D4 | — | 3 | 2 | no | — |
| 56 | C8×D4 | — | 6 | 2 | no | — |
| 57 | #431 | — | 3 | 1 | no | — |
| 58 | #431 | — | 3 | 1 | no | — |
| 59 | #431 | — | 6 | 1 | no | — |
| 60 | #427 | — | 3 | 1 | no | — |
| 61 | #416 | — | 3 | 1 | no | — |
| 62 | #416 | — | 3 | 1 | no | — |
| 63 | C2×C4×C8 | — | 3 | 1 | no | — |
| 64 | C2×C4×C8 | — | 3 | 1 | no | — |
| 65 | C2×C4×C8 | — | 3 | 1 | no | — |
| 66 | C2×C4×C8 | — | 6 | 1 | no | — |
| 67 | #335 | — | 3 | 2 | no | — |
| 68 | #334 | — | 3 | 2 | no | — |
| 69 | Q8⋊C8 | — | 3 | 2 | no | — |
| 70 | Q8⋊C8 | — | 3 | 2 | no | — |
| 71 | D4⋊C8 | — | 3 | 2 | no | — |
| 72 | D4⋊C8 | — | 3 | 2 | no | — |
| 73 | #322 | — | 3 | 2 | no | — |
| 74 | #322 | — | 3 | 2 | no | — |
| 75 | C2×C4×D3 | — | 4 | 1 | no | — |
| 76 | C2×C4×D3 | — | 4 | 1 | no | — |
| 77 | C2×C4×D3 | — | 4 | 1 | no | — |
| 78 | #41⋊C3 | C2×C8 | 1 | 2 | yes | complemented |
| 79 | C2×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 80 | GL(2,𝔽3) | C2×C8 | 1 | 2 | yes | complemented |
| 81 | #226 | C2×C8 | 1 | 2 | yes | complemented |
| 82 | C2×C3×C8 | — | 4 | 2 | no | — |
| 83 | C3×C42 | — | 4 | 1 | no | — |
| 84 | C4×Dic3 | — | 4 | 1 | no | — |
| 85 | C4×Dic3 | — | 4 | 1 | no | — |
| 86 | C4×Dic3 | — | 4 | 1 | no | — |
| 87 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 88 | C2×#41 | — | 3 | 1 | no | — |
| 89 | C2×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 90 | #135 | — | 3 | 4 | no | — |
| 91 | #135 | — | 3 | 4 | no | — |
| 92 | C2×Q16 | — | 3 | 1 | no | — |
| 93 | C2×SD16 | — | 3 | 1 | no | — |
| 94 | C2×SD16 | — | 3 | 1 | no | — |
| 95 | C2×D8 | — | 3 | 1 | no | — |
| 96 | C22×C8 | — | 3 | 1 | no | — |
| 97 | C22×C8 | — | 3 | 1 | no | — |
| 98 | C22×C8 | — | 3 | 2 | no | — |
| 99 | C22×C8 | — | 6 | 2 | no | — |
| 100 | C22×C8 | — | 6 | 1 | no | — |
| 101 | C4×Q8 | — | 3 | 1 | no | — |
| 102 | C4×Q8 | — | 3 | 1 | no | — |
| 103 | C4×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 104 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 105 | C4×D4 | — | 3 | 1 | no | — |
| 106 | C4×D4 | — | 3 | 1 | no | — |
| 107 | C4×D4 | — | 3 | 1 | no | — |
| 108 | C4×D4 | — | 3 | 1 | no | — |
| 109 | C4×D4 | — | 6 | 1 | no | — |
| 110 | C4×D4 | — | 6 | 1 | no | — |
| 111 | #117 | — | 3 | 1 | no | — |
| 112 | #117 | — | 3 | 1 | no | — |
| 113 | #117 | — | 6 | 1 | no | — |
| 114 | C2×C42 | — | 3 | 1 | no | — |
| 115 | C2×C42 | — | 6 | 1 | no | — |
| 116 | #107 | — | 3 | 1 | no | — |
| 117 | #107 | — | 3 | 1 | no | — |
| 118 | #106 | — | 3 | 1 | no | — |
| 119 | #106 | — | 3 | 1 | no | — |
| 120 | C4⋊C8 | — | 3 | 2 | no | — |
| 121 | C4⋊C8 | — | 3 | 2 | no | — |
| 122 | C4⋊C8 | — | 6 | 2 | no | — |
| 123 | #103 | — | 3 | 1 | no | — |
| 124 | #103 | — | 3 | 1 | no | — |
| 125 | #103 | — | 3 | 1 | no | — |
| 126 | #103 | — | 3 | 1 | no | — |
| 127 | #102 | — | 3 | 1 | no | — |
| 128 | #102 | — | 3 | 1 | no | — |
| 129 | #102 | — | 3 | 1 | no | — |
| 130 | #102 | — | 3 | 1 | no | — |
| 131 | C22⋊C8 | — | 6 | 1 | no | — |
| 132 | C22⋊C8 | — | 6 | 1 | no | — |
| 133 | C22⋊C8 | — | 6 | 2 | no | — |
| 134 | C4×C8 | — | 3 | 1 | no | — |
| 135 | C4×C8 | — | 3 | 1 | no | — |
| 136 | C4×C8 | — | 3 | 1 | no | — |
| 137 | C4×C8 | — | 3 | 1 | no | — |
| 138 | C4×C8 | — | 3 | 2 | no | — |
| 139 | C4×C8 | — | 3 | 2 | no | — |
| 140 | C4×C8 | — | 3 | 1 | no | — |
| 141 | C4×C8 | — | 6 | 1 | no | — |
| 142 | C4×C8 | — | 6 | 2 | no | — |
| 143 | C4×C8 | — | 6 | 1 | no | — |
| 144 | C4×C8 | S4 | 1 | 1 | yes | characteristic |
| 145 | C22×D3 | — | 4 | 1 | no | — |
| 146 | C2×C3×C4 | — | 4 | 1 | no | — |
| 147 | C2×C3×C4 | — | 4 | 1 | no | — |
| 148 | C2×C3×C4 | — | 4 | 1 | no | — |
| 149 | C2×Dic3 | — | 4 | 1 | no | — |
| 150 | C2×Dic3 | — | 4 | 1 | no | — |
| 151 | C2×Dic3 | — | 4 | 1 | no | — |
| 152 | C4×D3 | — | 4 | 4 | no | — |
| 153 | C4×D3 | — | 8 | 2 | no | — |
| 154 | C4×D3 | — | 8 | 2 | no | — |
| 155 | SL(2,𝔽3) | C22×C8 | 1 | 1 | yes | characteristic |
| 156 | C3×C8 | — | 8 | 2 | no | — |
| 157 | C3⋊C8 | — | 8 | 2 | no | — |
| 158 | #41 | — | 3 | 2 | no | — |
| 159 | #41 | — | 3 | 4 | no | — |
| 160 | #41 | — | 6 | 2 | no | — |
| 161 | #41 | D3×C8 | 1 | 2 | yes | — |
| 162 | C2×Q8 | — | 3 | 1 | no | — |
| 163 | C2×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 164 | C2×D4 | — | 3 | 1 | no | — |
| 165 | C2×D4 | — | 3 | 1 | no | — |
| 166 | C2×D4 | — | 6 | 1 | no | — |
| 167 | C22×C4 | — | 3 | 1 | no | — |
| 168 | C22×C4 | — | 3 | 1 | no | — |
| 169 | C22×C4 | — | 3 | 1 | no | — |
| 170 | C22×C4 | — | 6 | 1 | no | — |
| 171 | C22×C4 | — | 6 | 1 | no | — |
| 172 | C22×C4 | — | 6 | 1 | no | — |
| 173 | Q16 | — | 3 | 2 | no | — |
| 174 | Q16 | — | 3 | 2 | no | — |
| 175 | SD16 | — | 3 | 4 | no | — |
| 176 | SD16 | — | 3 | 2 | no | — |
| 177 | SD16 | — | 3 | 2 | no | — |
| 178 | D8 | — | 3 | 4 | no | — |
| 179 | C2×C8 | — | 3 | 1 | no | — |
| 180 | C2×C8 | — | 3 | 1 | no | — |
| 181 | C2×C8 | — | 3 | 4 | no | — |
| 182 | C2×C8 | — | 3 | 1 | no | — |
| 183 | C2×C8 | — | 3 | 1 | no | — |
| 184 | C2×C8 | — | 3 | 2 | no | — |
| 185 | C2×C8 | — | 6 | 2 | no | — |
| 186 | C2×C8 | — | 6 | 2 | no | — |
| 187 | C2×C8 | — | 6 | 2 | no | — |
| 188 | C2×C8 | — | 12 | 2 | no | — |
| 189 | C2×C8 | — | 12 | 4 | no | — |
| 190 | C2×C8 | — | 12 | 2 | no | — |
| 191 | C2×C8 | C2×S4 | 1 | 2 | yes | — |
| 192 | C4⋊C4 | — | 3 | 1 | no | — |
| 193 | C4⋊C4 | — | 3 | 1 | no | — |
| 194 | C4⋊C4 | — | 3 | 1 | no | — |
| 195 | C4⋊C4 | — | 3 | 1 | no | — |
| 196 | C4⋊C4 | — | 6 | 1 | no | — |
| 197 | C4⋊C4 | — | 6 | 1 | no | — |
| 198 | C22⋊C4 | — | 6 | 1 | no | — |
| 199 | C22⋊C4 | — | 6 | 1 | no | — |
| 200 | C22⋊C4 | — | 6 | 1 | no | — |
| 201 | C22⋊C4 | — | 6 | 1 | no | — |
| 202 | C42 | — | 3 | 1 | no | — |
| 203 | C42 | — | 3 | 1 | no | — |
| 204 | C42 | — | 3 | 1 | no | — |
| 205 | C42 | — | 6 | 1 | no | — |
| 206 | C42 | — | 6 | 1 | no | — |
| 207 | C42 | — | 6 | 1 | no | — |
| 208 | C42 | C2×S4 | 1 | 1 | yes | characteristic |
| 209 | C22×C3 | — | 4 | 1 | no | — |
| 210 | C2×D3 | — | 4 | 2 | no | — |
| 211 | C2×D3 | — | 8 | 1 | no | — |
| 212 | C2×D3 | — | 8 | 1 | no | — |
| 213 | C3×C4 | — | 4 | 2 | no | — |
| 214 | C3×C4 | — | 4 | 2 | no | — |
| 215 | C3×C4 | — | 4 | 2 | no | — |
| 216 | Dic3 | — | 4 | 2 | no | — |
| 217 | Dic3 | — | 8 | 1 | no | — |
| 218 | Dic3 | — | 8 | 1 | no | — |
| 219 | C23 | — | 3 | 1 | no | — |
| 220 | C23 | — | 6 | 1 | no | — |
| 221 | Q8 | — | 3 | 1 | no | — |
| 222 | Q8 | — | 3 | 2 | no | — |
| 223 | Q8 | — | 6 | 1 | no | — |
| 224 | Q8 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 225 | D4 | — | 3 | 2 | no | — |
| 226 | D4 | — | 3 | 2 | no | — |
| 227 | D4 | — | 6 | 1 | no | — |
| 228 | D4 | — | 6 | 1 | no | — |
| 229 | D4 | — | 6 | 4 | no | — |
| 230 | C2×C4 | — | 3 | 1 | no | — |
| 231 | C2×C4 | — | 3 | 2 | no | — |
| 232 | C2×C4 | — | 3 | 2 | no | — |
| 233 | C2×C4 | — | 3 | 1 | no | — |
| 234 | C2×C4 | — | 3 | 1 | no | — |
| 235 | C2×C4 | — | 6 | 1 | no | — |
| 236 | C2×C4 | — | 6 | 4 | no | — |
| 237 | C2×C4 | — | 6 | 2 | no | — |
| 238 | C2×C4 | — | 6 | 2 | no | — |
| 239 | C2×C4 | — | 6 | 1 | no | — |
| 240 | C2×C4 | — | 6 | 1 | no | — |
| 241 | C2×C4 | — | 12 | 2 | no | — |
| 242 | C2×C4 | — | 12 | 2 | no | — |
| 243 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 244 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 245 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 246 | C8 | — | 2 | 2 | no | — |
| 247 | C8 | — | 3 | 2 | no | — |
| 248 | C8 | — | 3 | 2 | no | — |
| 249 | C8 | — | 6 | 1 | no | — |
| 250 | C8 | — | 6 | 1 | no | — |
| 251 | C8 | — | 6 | 2 | no | — |
| 252 | C8 | — | 12 | 2 | no | — |
| 253 | C8 | — | 12 | 2 | no | — |
| 254 | C2×C3 | — | 4 | 1 | no | — |
| 255 | C2×C3 | — | 4 | 1 | no | — |
| 256 | C2×C3 | — | 4 | 1 | no | — |
| 257 | D3 | — | 8 | 2 | no | — |
| 258 | C22 | — | 3 | 2 | no | — |
| 259 | C22 | — | 6 | 1 | no | — |
| 260 | C22 | — | 6 | 1 | no | — |
| 261 | C22 | — | 6 | 2 | no | — |
| 262 | C22 | — | 12 | 1 | no | — |
| 263 | C22 | — | 12 | 1 | no | — |
| 264 | C22 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 265 | C4 | — | 3 | 1 | no | — |
| 266 | C4 | — | 3 | 1 | no | — |
| 267 | C4 | — | 6 | 1 | no | — |
| 268 | C4 | — | 6 | 2 | no | — |
| 269 | C4 | — | 6 | 1 | no | — |
| 270 | C4 | — | 12 | 1 | no | — |
| 271 | C4 | — | 12 | 1 | no | — |
| 272 | C4 | C8×S4 | 1 | 2 | yes | — |
| 273 | C4 | #5405 | 1 | 2 | yes | — |
| 274 | C4 | #5923 | 1 | 2 | yes | — |
| 275 | C3 | — | 4 | 1 | no | Sylow |
| 276 | C2 | — | 6 | 2 | no | — |
| 277 | C2 | — | 12 | 2 | no | — |
| 278 | C2 | #85341 | 1 | 1 | yes | characteristic |
| 279 | C2 | C2×C8×S4 | 1 | 1 | yes | characteristic |
| 280 | C2 | C2×#5405 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C8×#41⋊C3 | C2 | semidirect |
| 3. #73068 | C2 | semidirect |
| 4. GL(2,𝔽3)⋊C8 | C2 | semidirect |
| 5. #226⋊C8 | C2 | semidirect |
| 18. C8×#41 | D3 | semidirect |
| 29. #985 | C8 | semidirect |
| 78. #41⋊C3 | C2×C8 | semidirect |
| 80. GL(2,𝔽3) | C2×C8 | semidirect |
| 81. #226 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#985 | 2 |
| C8×#41⋊C3 | 2 |
| #73068 | 2 |
| GL(2,𝔽3)⋊C8 | 2 |
| #226⋊C8 | 2 |
| #17849 | 3 |
| #4685 | 4 |