Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
950 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 111 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 | #83302 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #73318 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#1962 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #70502 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #67611 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10280 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C3×#532 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#874 | — | 2 | 4 | no | — |
| 9 | C2×#874 | — | 2 | 2 | no | — |
| 10 | C2×#874 | C22 | 1 | 2 | yes | — |
| 11 | C3×#423 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C3×#104 | — | 2 | 2 | no | — |
| 13 | C3×C43 | C22 | 1 | 1 | yes | characteristic |
| 14 | #5088 | — | 2 | 2 | no | — |
| 15 | #5073 | — | 2 | 2 | no | — |
| 16 | #5036 | — | 2 | 4 | no | — |
| 17 | #5015 | — | 2 | 4 | no | — |
| 18 | #4925 | C22 | 1 | 2 | yes | — |
| 19 | #4485 | — | 2 | 4 | no | — |
| 20 | #3205 | — | 3 | 2 | no | — |
| 21 | #2867 | — | 3 | 1 | no | — |
| 22 | #1962 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | #1545 | — | 3 | 2 | no | — |
| 24 | #1276 | — | 3 | 1 | no | — |
| 25 | C2×C3×#41 | — | 2 | 1 | no | — |
| 26 | C22×D12 | — | 2 | 8 | no | — |
| 27 | C22×D12 | — | 2 | 2 | no | — |
| 28 | C22×D12 | — | 2 | 2 | no | — |
| 29 | C22×D12 | D4 | 1 | 4 | yes | — |
| 30 | C2×C3×M16 | — | 2 | 1 | no | — |
| 31 | C3×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 32 | C3×#127 | D4 | 1 | 1 | yes | characteristic, complemented |
| 33 | C3×#124 | — | 2 | 1 | no | — |
| 34 | C3×#121 | — | 2 | 2 | no | — |
| 35 | C3×C4×D4 | — | 2 | 2 | no | — |
| 36 | C2×C3×C42 | — | 2 | 2 | no | — |
| 37 | C2×C3×C42 | D4 | 1 | 2 | yes | — |
| 38 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×#213 | — | 2 | 8 | no | — |
| 40 | #874 | — | 2 | 2 | no | — |
| 41 | #874 | — | 2 | 2 | no | — |
| 42 | #874 | — | 2 | 2 | no | — |
| 43 | #874 | — | 2 | 2 | no | — |
| 44 | #874 | — | 4 | 16 | no | — |
| 45 | #874 | — | 4 | 4 | no | — |
| 46 | #874 | — | 4 | 4 | no | — |
| 47 | C3×C4⋊C8 | — | 2 | 2 | no | — |
| 48 | C3×#104 | — | 4 | 4 | no | — |
| 49 | #830 | — | 2 | 2 | no | — |
| 50 | #809 | — | 2 | 8 | no | — |
| 51 | C4⋊(C3⋊C8) | D4 | 1 | 4 | yes | complemented |
| 52 | C4×C3⋊C8 | — | 2 | 2 | no | — |
| 53 | #532 | C2×D3 | 1 | 1 | yes | characteristic |
| 54 | C2×#127 | — | 3 | 2 | no | — |
| 55 | C2×#127 | — | 6 | 4 | no | — |
| 56 | C2×#127 | — | 6 | 2 | no | — |
| 57 | #493 | — | 6 | 2 | no | — |
| 58 | #486 | — | 6 | 2 | no | — |
| 59 | #459 | — | 6 | 4 | no | — |
| 60 | #447 | — | 6 | 4 | no | — |
| 61 | #431 | — | 3 | 2 | no | — |
| 62 | #423 | C2×D3 | 1 | 1 | yes | characteristic |
| 63 | C2×#104 | — | 2 | 2 | no | — |
| 64 | C43 | C2×D3 | 1 | 1 | yes | characteristic |
| 65 | #331 | — | 6 | 4 | no | — |
| 66 | C23×D3 | — | 2 | 4 | no | — |
| 67 | C3×#41 | — | 4 | 2 | no | — |
| 68 | C3×#41 | — | 4 | 2 | no | — |
| 69 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 70 | C2×C3×D4 | — | 2 | 2 | no | — |
| 71 | C2×C3×D4 | — | 2 | 1 | no | — |
| 72 | C2×C3×D4 | — | 4 | 1 | no | — |
| 73 | C22×C3×C4 | — | 2 | 2 | no | — |
| 74 | C22×C3×C4 | — | 4 | 1 | no | — |
| 75 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 76 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C2×D12 | — | 2 | 4 | no | — |
| 78 | C2×D12 | — | 2 | 4 | no | — |
| 79 | C2×D12 | — | 4 | 8 | no | — |
| 80 | C2×D12 | — | 4 | 32 | no | — |
| 81 | C2×D12 | — | 4 | 16 | no | — |
| 82 | C2×D12 | — | 4 | 8 | no | — |
| 83 | C2×D12 | — | 4 | 2 | no | — |
| 84 | C2×D12 | — | 4 | 2 | no | — |
| 85 | C2×D12 | — | 4 | 2 | no | — |
| 86 | C2×D12 | — | 4 | 2 | no | — |
| 87 | C2×D12 | — | 4 | 4 | no | — |
| 88 | C2×D12 | — | 4 | 4 | no | — |
| 89 | C3×M16 | — | 4 | 2 | no | — |
| 90 | C2×C3×C8 | — | 4 | 1 | no | — |
| 91 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 92 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 93 | C3×C42 | — | 2 | 8 | no | — |
| 94 | C3×C42 | — | 2 | 2 | no | — |
| 95 | C3×C42 | D8 | 1 | 4 | yes | complemented |
| 96 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 98 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 99 | #213 | — | 8 | 8 | no | — |
| 100 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 101 | C2×#41 | — | 2 | 1 | no | — |
| 102 | C22×D4 | — | 3 | 4 | no | — |
| 103 | C22×D4 | — | 6 | 8 | no | — |
| 104 | C22×D4 | — | 6 | 2 | no | — |
| 105 | C22×D4 | — | 6 | 2 | no | — |
| 106 | C2×D8 | — | 6 | 8 | no | — |
| 107 | C2×M16 | — | 2 | 1 | no | — |
| 108 | C4⋊Q8 | D12 | 1 | 1 | yes | characteristic, complemented |
| 109 | #127 | — | 6 | 2 | no | — |
| 110 | #127 | — | 6 | 2 | no | — |
| 111 | #127 | — | 6 | 2 | no | — |
| 112 | #127 | — | 6 | 2 | no | — |
| 113 | #127 | — | 12 | 16 | no | — |
| 114 | #127 | — | 12 | 4 | no | — |
| 115 | #127 | — | 12 | 4 | no | — |
| 116 | #127 | D12 | 1 | 1 | yes | characteristic, complemented |
| 117 | #124 | — | 2 | 1 | no | — |
| 118 | #121 | — | 2 | 2 | no | — |
| 119 | C4×D4 | — | 2 | 2 | no | — |
| 120 | C2×C42 | — | 2 | 2 | no | — |
| 121 | C2×C42 | #67 | 1 | 2 | yes | — |
| 122 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 123 | C4⋊C8 | — | 2 | 2 | no | — |
| 124 | C4⋊C8 | — | 3 | 4 | no | — |
| 125 | #104 | — | 4 | 4 | no | — |
| 126 | #102 | — | 6 | 8 | no | — |
| 127 | C22⋊C8 | — | 6 | 2 | no | — |
| 128 | C4×C8 | — | 6 | 2 | no | — |
| 129 | C23×C3 | — | 4 | 1 | no | — |
| 130 | C23×C3 | #120 | 1 | 1 | yes | characteristic |
| 131 | C22×D3 | — | 4 | 8 | no | — |
| 132 | C22×D3 | — | 4 | 8 | no | — |
| 133 | C22×D3 | — | 4 | 4 | no | — |
| 134 | C22×D3 | — | 4 | 8 | no | — |
| 135 | C3×Q8 | — | 4 | 2 | no | — |
| 136 | C3×D4 | — | 4 | 4 | no | — |
| 137 | C3×D4 | — | 4 | 2 | no | — |
| 138 | C3×D4 | — | 8 | 2 | no | — |
| 139 | C2×C3×C4 | — | 2 | 8 | no | — |
| 140 | C2×C3×C4 | — | 2 | 4 | no | — |
| 141 | C2×C3×C4 | — | 2 | 4 | no | — |
| 142 | C2×C3×C4 | — | 4 | 1 | no | — |
| 143 | C2×C3×C4 | — | 8 | 1 | no | — |
| 144 | C2×C3×C4 | — | 8 | 1 | no | — |
| 145 | C2×C3×C4 | #120 | 1 | 2 | yes | — |
| 146 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 147 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 148 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 149 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 150 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 151 | D12 | — | 4 | 8 | no | — |
| 152 | D12 | — | 8 | 4 | no | — |
| 153 | D12 | — | 8 | 32 | no | — |
| 154 | D12 | — | 8 | 4 | no | — |
| 155 | D12 | — | 8 | 4 | no | — |
| 156 | D12 | — | 8 | 4 | no | — |
| 157 | D12 | — | 8 | 2 | no | — |
| 158 | D12 | — | 8 | 2 | no | — |
| 159 | C3×C8 | — | 8 | 1 | no | — |
| 160 | C3⋊C8 | — | 4 | 4 | no | — |
| 161 | C24 | — | 6 | 4 | no | — |
| 162 | #41 | — | 4 | 2 | no | — |
| 163 | #41 | — | 4 | 2 | no | — |
| 164 | C2×Q8 | — | 2 | 1 | no | — |
| 165 | C2×D4 | — | 2 | 2 | no | — |
| 166 | C2×D4 | — | 2 | 1 | no | — |
| 167 | C2×D4 | — | 4 | 1 | no | — |
| 168 | C2×D4 | — | 6 | 4 | no | — |
| 169 | C2×D4 | — | 6 | 4 | no | — |
| 170 | C2×D4 | — | 12 | 8 | no | — |
| 171 | C2×D4 | — | 12 | 32 | no | — |
| 172 | C2×D4 | — | 12 | 16 | no | — |
| 173 | C2×D4 | — | 12 | 8 | no | — |
| 174 | C2×D4 | — | 12 | 2 | no | — |
| 175 | C2×D4 | — | 12 | 2 | no | — |
| 176 | C2×D4 | — | 12 | 2 | no | — |
| 177 | C2×D4 | — | 12 | 4 | no | — |
| 178 | C2×D4 | — | 12 | 2 | no | — |
| 179 | C2×D4 | — | 12 | 4 | no | — |
| 180 | C22×C4 | — | 2 | 2 | no | — |
| 181 | C22×C4 | — | 4 | 1 | no | — |
| 182 | C22×C4 | D3×D4 | 1 | 2 | yes | — |
| 183 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 184 | D8 | — | 24 | 8 | no | — |
| 185 | M16 | — | 4 | 2 | no | — |
| 186 | C2×C8 | — | 4 | 1 | no | — |
| 187 | C2×C8 | — | 6 | 4 | no | — |
| 188 | C4⋊C4 | — | 2 | 2 | no | — |
| 189 | C22⋊C4 | — | 4 | 2 | no | — |
| 190 | C42 | — | 2 | 8 | no | — |
| 191 | C42 | — | 2 | 2 | no | — |
| 192 | C42 | #213 | 1 | 4 | yes | complemented |
| 193 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 194 | C42 | #235 | 1 | 1 | yes | characteristic |
| 195 | C42 | D3×D4 | 1 | 2 | yes | — |
| 196 | C22×C3 | — | 2 | 2 | no | — |
| 197 | C22×C3 | — | 8 | 2 | no | — |
| 198 | C22×C3 | — | 8 | 1 | no | — |
| 199 | C22×C3 | #392 | 1 | 1 | yes | characteristic |
| 200 | C22×C3 | #447 | 1 | 2 | yes | — |
| 201 | C2×D3 | — | 8 | 16 | no | — |
| 202 | C2×D3 | — | 8 | 4 | no | — |
| 203 | C2×D3 | — | 8 | 4 | no | — |
| 204 | C2×D3 | — | 8 | 4 | no | — |
| 205 | C3×C4 | — | 2 | 8 | no | — |
| 206 | C3×C4 | — | 2 | 2 | no | — |
| 207 | C3×C4 | — | 8 | 1 | no | — |
| 208 | C3×C4 | #453 | 1 | 4 | yes | — |
| 209 | C3×C4 | #459 | 1 | 2 | yes | — |
| 210 | C3×C4 | #493 | 1 | 1 | yes | characteristic |
| 211 | C3×C4 | #496 | 1 | 1 | yes | characteristic |
| 212 | C23 | — | 4 | 1 | no | — |
| 213 | C23 | — | 12 | 8 | no | — |
| 214 | C23 | — | 12 | 8 | no | — |
| 215 | C23 | — | 12 | 4 | no | — |
| 216 | C23 | — | 12 | 8 | no | — |
| 217 | C23 | #937 | 1 | 1 | yes | characteristic |
| 218 | Q8 | — | 4 | 2 | no | — |
| 219 | D4 | — | 4 | 2 | no | — |
| 220 | D4 | — | 4 | 4 | no | — |
| 221 | D4 | — | 8 | 2 | no | — |
| 222 | D4 | — | 12 | 8 | no | — |
| 223 | D4 | — | 24 | 4 | no | — |
| 224 | D4 | — | 24 | 32 | no | — |
| 225 | D4 | — | 24 | 4 | no | — |
| 226 | D4 | — | 24 | 4 | no | — |
| 227 | D4 | — | 24 | 2 | no | — |
| 228 | D4 | — | 24 | 2 | no | — |
| 229 | D4 | — | 24 | 4 | no | — |
| 230 | C2×C4 | — | 2 | 8 | no | — |
| 231 | C2×C4 | — | 2 | 4 | no | — |
| 232 | C2×C4 | — | 2 | 4 | no | — |
| 233 | C2×C4 | — | 4 | 1 | no | — |
| 234 | C2×C4 | — | 8 | 1 | no | — |
| 235 | C2×C4 | — | 8 | 1 | no | — |
| 236 | C2×C4 | #882 | 1 | 2 | yes | — |
| 237 | C2×C4 | #884 | 1 | 2 | yes | — |
| 238 | C2×C4 | #930 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | C2×#213 | 1 | 2 | yes | — |
| 240 | C2×C4 | #940 | 1 | 1 | yes | characteristic |
| 241 | C2×C4 | #949 | 1 | 2 | yes | — |
| 242 | C8 | — | 8 | 1 | no | — |
| 243 | C8 | — | 12 | 4 | no | — |
| 244 | C2×C3 | — | 2 | 2 | no | — |
| 245 | C2×C3 | — | 16 | 1 | no | — |
| 246 | C2×C3 | #2002 | 1 | 2 | yes | — |
| 247 | C2×C3 | #2011 | 1 | 1 | yes | characteristic |
| 248 | D3 | — | 16 | 4 | no | — |
| 249 | C22 | — | 2 | 2 | no | — |
| 250 | C22 | — | 8 | 1 | no | — |
| 251 | C22 | — | 8 | 2 | no | — |
| 252 | C22 | — | 24 | 16 | no | — |
| 253 | C22 | — | 24 | 4 | no | — |
| 254 | C22 | — | 24 | 4 | no | — |
| 255 | C22 | — | 24 | 4 | no | — |
| 256 | C22 | #4960 | 1 | 1 | yes | characteristic |
| 257 | C22 | #5036 | 1 | 2 | yes | — |
| 258 | C4 | — | 2 | 8 | no | — |
| 259 | C4 | — | 2 | 2 | no | — |
| 260 | C4 | — | 8 | 1 | no | — |
| 261 | C4 | #4822 | 1 | 4 | yes | — |
| 262 | C4 | #5015 | 1 | 2 | yes | — |
| 263 | C4 | #5058 | 1 | 1 | yes | characteristic |
| 264 | C4 | #5073 | 1 | 1 | yes | characteristic |
| 265 | C3 | #10280 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 266 | C2 | — | 2 | 2 | no | — |
| 267 | C2 | — | 16 | 1 | no | — |
| 268 | C2 | — | 48 | 4 | no | — |
| 269 | C2 | #69658 | 1 | 1 | yes | characteristic |
| 270 | C2 | #70385 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #83302 | C2 | semidirect |
| 2. #73318 | C2 | semidirect |
| 3. C3×#1962 | C2 | semidirect |
| 4. #70502 | C2 | semidirect |
| 5. #67611 | C2 | semidirect |
| 22. #1962 | D3 | semidirect |
| 31. C3×C4⋊Q8 | D4 | semidirect |
| 32. C3×#127 | D4 | semidirect |
| 51. C4⋊(C3⋊C8) | D4 | semidirect |
| 95. C3×C42 | D8 | semidirect |
| 108. C4⋊Q8 | D12 | semidirect |
| 116. #127 | D12 | semidirect |
| 192. C42 | #213 | semidirect |
| 265. C3 | #10280 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #83302 | 2 |
| #73318 | 2 |
| C3×#1962 | 2 |
| #70502 | 2 |
| #67611 | 2 |
| #10280 | 3 |