Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
650 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 105 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 | C3×#3150 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #83302 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #71814 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #69823 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #67661 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10218 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×C3×#135 | — | 2 | 2 | no | — |
| 8 | C3×#532 | C22 | 1 | 2 | yes | — |
| 9 | C2×#874 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#497 | — | 2 | 2 | no | — |
| 11 | C3×#496 | — | 2 | 2 | no | — |
| 12 | C2×C3×#97 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#821 | — | 2 | 2 | no | — |
| 14 | #5088 | — | 2 | 2 | no | — |
| 15 | #5073 | — | 2 | 2 | no | — |
| 16 | #5036 | — | 2 | 4 | no | — |
| 17 | #5015 | — | 2 | 4 | no | — |
| 18 | #4936 | C22 | 1 | 1 | yes | characteristic |
| 19 | #4925 | C22 | 1 | 2 | yes | — |
| 20 | #4537 | — | 2 | 4 | no | — |
| 21 | #3205 | — | 3 | 2 | no | — |
| 22 | #3150 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | #1961 | — | 3 | 1 | no | — |
| 24 | #1517 | — | 3 | 2 | no | — |
| 25 | #1281 | — | 3 | 1 | no | — |
| 26 | C2×C3×#41 | — | 2 | 2 | no | — |
| 27 | C22×D12 | — | 2 | 1 | no | — |
| 28 | C22×D12 | — | 2 | 2 | no | — |
| 29 | C3×#135 | — | 4 | 4 | no | — |
| 30 | C3×#135 | — | 4 | 4 | no | — |
| 31 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 32 | C2×C3×SD16 | — | 2 | 4 | no | — |
| 33 | C2×C3×D8 | — | 2 | 2 | no | — |
| 34 | C22×C3×C8 | D4 | 1 | 2 | yes | — |
| 35 | C3×C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 36 | C3×#127 | D4 | 1 | 2 | yes | — |
| 37 | C3×#124 | — | 2 | 2 | no | — |
| 38 | C3×#121 | — | 2 | 2 | no | — |
| 39 | C3×#121 | — | 2 | 2 | no | — |
| 40 | C3×C4×D4 | — | 2 | 2 | no | — |
| 41 | C3×C4×D4 | — | 2 | 2 | no | — |
| 42 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×#213 | — | 2 | 8 | no | — |
| 44 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 45 | #874 | — | 2 | 2 | no | — |
| 46 | #874 | — | 2 | 1 | no | — |
| 47 | #874 | D4 | 1 | 2 | yes | — |
| 48 | C3×#97 | — | 2 | 2 | no | — |
| 49 | C3⋊#95 | — | 2 | 2 | no | — |
| 50 | #830 | D4 | 1 | 4 | yes | — |
| 51 | #821 | — | 4 | 4 | no | — |
| 52 | #809 | — | 2 | 8 | no | — |
| 53 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 54 | C4×C3⋊C8 | — | 2 | 2 | no | — |
| 55 | C2×#135 | — | 2 | 2 | no | — |
| 56 | #532 | C2×D3 | 1 | 2 | yes | — |
| 57 | C2×#127 | — | 3 | 1 | no | — |
| 58 | #497 | — | 2 | 2 | no | — |
| 59 | #496 | — | 2 | 2 | no | — |
| 60 | #493 | — | 6 | 2 | no | — |
| 61 | #486 | — | 6 | 2 | no | — |
| 62 | #459 | — | 6 | 4 | no | — |
| 63 | #447 | — | 6 | 4 | no | — |
| 64 | #431 | — | 3 | 2 | no | — |
| 65 | C2×#102 | — | 6 | 2 | no | — |
| 66 | C2×#97 | C2×D3 | 1 | 1 | yes | characteristic |
| 67 | #382 | — | 3 | 1 | no | — |
| 68 | #327 | — | 6 | 4 | no | — |
| 69 | C23×D3 | — | 4 | 1 | no | — |
| 70 | C3×#41 | — | 4 | 4 | no | — |
| 71 | C3×#41 | — | 4 | 4 | no | — |
| 72 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 73 | C2×C3×D4 | — | 2 | 2 | no | — |
| 74 | C2×C3×D4 | — | 2 | 2 | no | — |
| 75 | C2×C3×D4 | — | 2 | 2 | no | — |
| 76 | C2×C3×D4 | — | 4 | 2 | no | — |
| 77 | C22×C3×C4 | — | 4 | 2 | no | — |
| 78 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 79 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C22×Dic3 | — | 4 | 1 | no | — |
| 81 | C2×D12 | — | 2 | 4 | no | — |
| 82 | C2×D12 | — | 2 | 2 | no | — |
| 83 | C2×D12 | — | 4 | 2 | no | — |
| 84 | C2×D12 | — | 4 | 2 | no | — |
| 85 | C2×D12 | — | 4 | 1 | no | — |
| 86 | C2×D12 | — | 4 | 2 | no | — |
| 87 | C2×D12 | — | 8 | 4 | no | — |
| 88 | C3×Q16 | — | 4 | 4 | no | — |
| 89 | C3×SD16 | — | 4 | 8 | no | — |
| 90 | C3×D8 | — | 4 | 4 | no | — |
| 91 | C2×C3×C8 | — | 2 | 2 | no | — |
| 92 | C2×C3×C8 | — | 2 | 2 | no | — |
| 93 | C2×C3×C8 | D8 | 1 | 4 | yes | complemented |
| 94 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 95 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 96 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 97 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 98 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 100 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 101 | #213 | — | 8 | 8 | no | — |
| 102 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 103 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 104 | C2×#41 | — | 2 | 2 | no | — |
| 105 | C22×D4 | — | 6 | 2 | no | — |
| 106 | C22×D4 | — | 6 | 1 | no | — |
| 107 | #135 | — | 4 | 4 | no | — |
| 108 | #135 | — | 4 | 4 | no | — |
| 109 | C2×Q16 | — | 2 | 2 | no | — |
| 110 | C2×SD16 | — | 2 | 4 | no | — |
| 111 | C2×D8 | — | 2 | 2 | no | — |
| 112 | C2×D8 | — | 6 | 8 | no | — |
| 113 | C22×C8 | #67 | 1 | 2 | yes | — |
| 114 | C4⋊Q8 | #67 | 1 | 2 | yes | — |
| 115 | #127 | — | 3 | 2 | no | — |
| 116 | #127 | — | 6 | 1 | no | — |
| 117 | #127 | — | 6 | 2 | no | — |
| 118 | #127 | #67 | 1 | 2 | yes | — |
| 119 | #124 | — | 2 | 2 | no | — |
| 120 | #121 | — | 2 | 2 | no | — |
| 121 | #121 | — | 2 | 2 | no | — |
| 122 | C4×D4 | — | 2 | 2 | no | — |
| 123 | C4×D4 | — | 2 | 2 | no | — |
| 124 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 125 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 126 | C4⋊C8 | — | 6 | 2 | no | — |
| 127 | #102 | — | 6 | 8 | no | — |
| 128 | #102 | — | 12 | 4 | no | — |
| 129 | C22⋊C8 | — | 3 | 4 | no | — |
| 130 | #97 | — | 2 | 2 | no | — |
| 131 | C4×C8 | — | 6 | 2 | no | — |
| 132 | #95 | — | 6 | 2 | no | — |
| 133 | C23×C3 | — | 4 | 2 | no | — |
| 134 | C23×C3 | #127 | 1 | 1 | yes | characteristic |
| 135 | C22×D3 | — | 4 | 2 | no | — |
| 136 | C22×D3 | — | 8 | 4 | no | — |
| 137 | C22×D3 | — | 8 | 1 | no | — |
| 138 | C22×D3 | — | 8 | 1 | no | — |
| 139 | C3×Q8 | — | 4 | 4 | no | — |
| 140 | C3×D4 | — | 4 | 4 | no | — |
| 141 | C3×D4 | — | 4 | 4 | no | — |
| 142 | C3×D4 | — | 8 | 2 | no | — |
| 143 | C3×D4 | — | 8 | 2 | no | — |
| 144 | C3×D4 | — | 8 | 2 | no | — |
| 145 | C2×C3×C4 | — | 4 | 2 | no | — |
| 146 | C2×C3×C4 | — | 4 | 2 | no | — |
| 147 | C2×C3×C4 | — | 8 | 4 | no | — |
| 148 | C2×C3×C4 | #120 | 1 | 2 | yes | — |
| 149 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 150 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 151 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 152 | C2×C3×C4 | C2×D8 | 1 | 2 | yes | — |
| 153 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 154 | C2×Dic3 | — | 8 | 2 | no | — |
| 155 | C2×Dic3 | — | 8 | 1 | no | — |
| 156 | D12 | — | 4 | 8 | no | — |
| 157 | D12 | — | 4 | 4 | no | — |
| 158 | D12 | — | 8 | 4 | no | — |
| 159 | D12 | — | 8 | 2 | no | — |
| 160 | C3×C8 | — | 2 | 4 | no | — |
| 161 | C3⋊C8 | — | 4 | 4 | no | — |
| 162 | C24 | — | 12 | 1 | no | — |
| 163 | #41 | — | 4 | 4 | no | — |
| 164 | #41 | — | 4 | 4 | no | — |
| 165 | C2×Q8 | — | 2 | 2 | no | — |
| 166 | C2×D4 | — | 2 | 2 | no | — |
| 167 | C2×D4 | — | 2 | 2 | no | — |
| 168 | C2×D4 | — | 2 | 2 | no | — |
| 169 | C2×D4 | — | 4 | 2 | no | — |
| 170 | C2×D4 | — | 6 | 4 | no | — |
| 171 | C2×D4 | — | 6 | 2 | no | — |
| 172 | C2×D4 | — | 12 | 2 | no | — |
| 173 | C2×D4 | — | 12 | 2 | no | — |
| 174 | C2×D4 | — | 12 | 2 | no | — |
| 175 | C2×D4 | — | 12 | 1 | no | — |
| 176 | C2×D4 | — | 24 | 4 | no | — |
| 177 | C22×C4 | — | 4 | 2 | no | — |
| 178 | C22×C4 | — | 12 | 1 | no | — |
| 179 | C22×C4 | D3×D4 | 1 | 2 | yes | — |
| 180 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 181 | Q16 | — | 4 | 4 | no | — |
| 182 | SD16 | — | 4 | 8 | no | — |
| 183 | D8 | — | 4 | 4 | no | — |
| 184 | D8 | — | 24 | 8 | no | — |
| 185 | C2×C8 | — | 2 | 2 | no | — |
| 186 | C2×C8 | — | 2 | 2 | no | — |
| 187 | C2×C8 | — | 6 | 4 | no | — |
| 188 | C2×C8 | #213 | 1 | 4 | yes | complemented |
| 189 | C4⋊C4 | — | 2 | 2 | no | — |
| 190 | C4⋊C4 | — | 2 | 2 | no | — |
| 191 | C4⋊C4 | — | 12 | 2 | no | — |
| 192 | C22⋊C4 | — | 4 | 2 | no | — |
| 193 | C22⋊C4 | — | 4 | 2 | no | — |
| 194 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 195 | C42 | #237 | 1 | 1 | yes | characteristic |
| 196 | C42 | C2×#67 | 1 | 2 | yes | — |
| 197 | C22×C3 | — | 2 | 2 | no | — |
| 198 | C22×C3 | — | 8 | 2 | no | — |
| 199 | C22×C3 | — | 8 | 2 | no | — |
| 200 | C22×C3 | — | 8 | 2 | no | — |
| 201 | C22×C3 | #392 | 1 | 1 | yes | characteristic |
| 202 | C22×C3 | #493 | 1 | 1 | yes | characteristic |
| 203 | C22×C3 | #496 | 1 | 1 | yes | characteristic |
| 204 | C2×D3 | — | 8 | 4 | no | — |
| 205 | C2×D3 | — | 8 | 2 | no | — |
| 206 | C2×D3 | — | 16 | 2 | no | — |
| 207 | C2×D3 | — | 16 | 2 | no | — |
| 208 | C3×C4 | — | 2 | 2 | no | — |
| 209 | C3×C4 | — | 2 | 2 | no | — |
| 210 | C3×C4 | — | 8 | 2 | no | — |
| 211 | C3×C4 | #447 | 1 | 2 | yes | — |
| 212 | C3×C4 | #459 | 1 | 2 | yes | — |
| 213 | Dic3 | — | 16 | 1 | no | — |
| 214 | C23 | — | 4 | 2 | no | — |
| 215 | C23 | — | 12 | 2 | no | — |
| 216 | C23 | — | 24 | 4 | no | — |
| 217 | C23 | — | 24 | 1 | no | — |
| 218 | C23 | — | 24 | 1 | no | — |
| 219 | C23 | #940 | 1 | 1 | yes | characteristic |
| 220 | Q8 | — | 4 | 4 | no | — |
| 221 | D4 | — | 4 | 4 | no | — |
| 222 | D4 | — | 4 | 4 | no | — |
| 223 | D4 | — | 8 | 2 | no | — |
| 224 | D4 | — | 8 | 2 | no | — |
| 225 | D4 | — | 8 | 2 | no | — |
| 226 | D4 | — | 12 | 8 | no | — |
| 227 | D4 | — | 12 | 4 | no | — |
| 228 | D4 | — | 24 | 4 | no | — |
| 229 | D4 | — | 24 | 2 | no | — |
| 230 | C2×C4 | — | 4 | 2 | no | — |
| 231 | C2×C4 | — | 4 | 2 | no | — |
| 232 | C2×C4 | — | 8 | 4 | no | — |
| 233 | C2×C4 | — | 24 | 2 | no | — |
| 234 | C2×C4 | — | 24 | 1 | no | — |
| 235 | C2×C4 | C2×#213 | 1 | 2 | yes | — |
| 236 | C2×C4 | #932 | 1 | 2 | yes | — |
| 237 | C2×C4 | #937 | 1 | 2 | yes | — |
| 238 | C2×C4 | #938 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | #939 | 1 | 2 | yes | — |
| 240 | C2×C4 | #953 | 1 | 1 | yes | characteristic |
| 241 | C8 | — | 2 | 4 | no | — |
| 242 | C8 | — | 12 | 4 | no | — |
| 243 | C2×C3 | — | 2 | 1 | no | — |
| 244 | C2×C3 | — | 2 | 1 | no | — |
| 245 | C2×C3 | — | 16 | 2 | no | — |
| 246 | C2×C3 | #2008 | 1 | 2 | yes | — |
| 247 | C2×C3 | #2011 | 1 | 1 | yes | characteristic |
| 248 | D3 | — | 16 | 2 | no | — |
| 249 | C22 | — | 2 | 2 | no | — |
| 250 | C22 | — | 8 | 2 | no | — |
| 251 | C22 | — | 8 | 2 | no | — |
| 252 | C22 | — | 8 | 2 | no | — |
| 253 | C22 | — | 24 | 2 | no | — |
| 254 | C22 | — | 24 | 4 | no | — |
| 255 | C22 | — | 48 | 2 | no | — |
| 256 | C22 | — | 48 | 2 | no | — |
| 257 | C22 | #5073 | 1 | 1 | yes | characteristic |
| 258 | C22 | #5074 | 1 | 1 | yes | characteristic |
| 259 | C22 | #5223 | 1 | 1 | yes | characteristic |
| 260 | C4 | — | 2 | 2 | no | — |
| 261 | C4 | — | 2 | 2 | no | — |
| 262 | C4 | — | 8 | 2 | no | — |
| 263 | C4 | — | 48 | 1 | no | — |
| 264 | C4 | #5072 | 1 | 2 | yes | — |
| 265 | C4 | #5217 | 1 | 2 | yes | — |
| 266 | C3 | #10218 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 267 | C2 | — | 2 | 1 | no | — |
| 268 | C2 | — | 2 | 1 | no | — |
| 269 | C2 | — | 16 | 2 | no | — |
| 270 | C2 | — | 48 | 2 | no | — |
| 271 | C2 | #71767 | 1 | 1 | yes | characteristic |
| 272 | C2 | #72119 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#3150 | C2 | semidirect |
| 2. #83302 | C2 | semidirect |
| 3. #71814 | C2 | semidirect |
| 4. #69823 | C2 | semidirect |
| 5. #67661 | C2 | semidirect |
| 22. #3150 | D3 | semidirect |
| 93. C2×C3×C8 | D8 | semidirect |
| 188. C2×C8 | #213 | semidirect |
| 266. C3 | #10218 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#3150 | 2 |
| #83302 | 2 |
| #71814 | 2 |
| #69823 | 2 |
| #67661 | 2 |
| #10218 | 3 |