Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
502 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2980 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #83286 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #71805 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #70801 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #67751 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #9007 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C3×#532 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#950 | — | 2 | 2 | no | — |
| 9 | #5477 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#435 | — | 2 | 4 | no | — |
| 11 | C3×#432 | C22 | 1 | 2 | yes | — |
| 12 | C3×#408 | — | 2 | 2 | no | — |
| 13 | #5086 | — | 2 | 1 | no | — |
| 14 | #5074 | — | 2 | 1 | no | — |
| 15 | #5060 | C4 | 1 | 2 | yes | — |
| 16 | #5058 | C4 | 1 | 2 | yes | — |
| 17 | #4925 | C22 | 1 | 2 | yes | — |
| 18 | C2×#803 | C22 | 1 | 1 | yes | characteristic |
| 19 | #4546 | — | 2 | 1 | no | — |
| 20 | #4544 | — | 2 | 1 | no | — |
| 21 | #4483 | — | 2 | 4 | no | — |
| 22 | #3150 | — | 3 | 1 | no | — |
| 23 | #2980 | D3 | 1 | 1 | yes | characteristic, complemented |
| 24 | #1543 | — | 3 | 1 | no | — |
| 25 | #1496 | — | 3 | 2 | no | — |
| 26 | #1302 | — | 3 | 2 | no | — |
| 27 | C2×C3×#41 | D4 | 1 | 2 | yes | — |
| 28 | C2×#235 | — | 2 | 1 | no | — |
| 29 | C2×C3×M16 | — | 2 | 2 | no | — |
| 30 | C22×C3×C8 | — | 2 | 2 | no | — |
| 31 | C3×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 32 | C3×#127 | D4 | 1 | 1 | yes | characteristic |
| 33 | C3×#124 | C2×C4 | 1 | 2 | yes | — |
| 34 | C3×#121 | — | 2 | 2 | no | — |
| 35 | C3×C4×D4 | — | 2 | 2 | no | — |
| 36 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 37 | #950 | — | 4 | 4 | no | — |
| 38 | #950 | — | 4 | 4 | no | — |
| 39 | C2×#216 | — | 2 | 2 | no | — |
| 40 | C2×#215 | — | 2 | 2 | no | — |
| 41 | C2×#214 | — | 2 | 2 | no | — |
| 42 | C2×#213 | — | 2 | 2 | no | — |
| 43 | #930 | — | 2 | 2 | no | — |
| 44 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 45 | #875 | — | 2 | 1 | no | — |
| 46 | #874 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C4×D12 | — | 2 | 2 | no | — |
| 48 | #869 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C3×C4⋊C8 | D4 | 1 | 4 | yes | — |
| 50 | C3×C22⋊C8 | — | 2 | 4 | no | — |
| 51 | C3×C22⋊C8 | — | 2 | 2 | no | — |
| 52 | C3×#97 | — | 2 | 2 | no | — |
| 53 | #830 | — | 2 | 2 | no | — |
| 54 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 55 | #803 | — | 2 | 1 | no | — |
| 56 | #803 | C2×C4 | 1 | 2 | yes | — |
| 57 | C4×C3⋊C8 | — | 2 | 2 | no | — |
| 58 | C2×#135 | — | 6 | 2 | no | — |
| 59 | #532 | — | 3 | 1 | no | — |
| 60 | #532 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | #497 | — | 3 | 2 | no | — |
| 62 | #497 | — | 6 | 1 | no | — |
| 63 | #496 | — | 3 | 2 | no | — |
| 64 | #496 | — | 6 | 1 | no | — |
| 65 | #435 | — | 2 | 4 | no | — |
| 66 | #432 | C2×D3 | 1 | 2 | yes | — |
| 67 | #431 | — | 3 | 2 | no | — |
| 68 | #408 | — | 2 | 2 | no | — |
| 69 | C2×#97 | — | 3 | 1 | no | — |
| 70 | #333 | — | 6 | 1 | no | — |
| 71 | #331 | — | 6 | 1 | no | — |
| 72 | D4⋊C8 | — | 6 | 4 | no | — |
| 73 | C3×#41 | — | 4 | 2 | no | — |
| 74 | C3×#41 | — | 4 | 2 | no | — |
| 75 | C2×C3×Q8 | C22⋊C4 | 1 | 2 | yes | — |
| 76 | C2×C3×D4 | — | 2 | 2 | no | — |
| 77 | C2×C3×D4 | — | 2 | 2 | no | — |
| 78 | C2×C3×D4 | C22⋊C4 | 1 | 2 | yes | — |
| 79 | C22×C3×C4 | — | 2 | 2 | no | — |
| 80 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C22×C3×C4 | #41 | 1 | 2 | yes | — |
| 82 | C2×#67 | — | 4 | 1 | no | — |
| 83 | #235 | — | 4 | 2 | no | — |
| 84 | #235 | — | 4 | 2 | no | — |
| 85 | C2×D12 | — | 2 | 1 | no | — |
| 86 | C2×D12 | — | 2 | 2 | no | — |
| 87 | C2×C4×D3 | — | 4 | 1 | no | — |
| 88 | C2×Dic6 | — | 2 | 1 | no | — |
| 89 | C3×M16 | — | 8 | 2 | no | — |
| 90 | C2×C3×C8 | — | 2 | 4 | no | — |
| 91 | C2×C3×C8 | — | 8 | 2 | no | — |
| 92 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 93 | C3×C22⋊C4 | — | 2 | 4 | no | — |
| 94 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 95 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 97 | #216 | — | 4 | 4 | no | — |
| 98 | #215 | — | 4 | 4 | no | — |
| 99 | #214 | — | 4 | 4 | no | — |
| 100 | #213 | — | 4 | 4 | no | — |
| 101 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 102 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 103 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 104 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 105 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 106 | C2×#41 | — | 6 | 1 | no | — |
| 107 | C2×#41 | #67 | 1 | 2 | yes | — |
| 108 | #135 | — | 12 | 4 | no | — |
| 109 | #135 | — | 12 | 4 | no | — |
| 110 | C2×Q16 | — | 6 | 2 | no | — |
| 111 | C2×SD16 | — | 6 | 2 | no | — |
| 112 | C2×SD16 | — | 6 | 2 | no | — |
| 113 | C2×D8 | — | 6 | 2 | no | — |
| 114 | C2×M16 | — | 2 | 2 | no | — |
| 115 | C22×C8 | — | 2 | 2 | no | — |
| 116 | C22×C8 | — | 6 | 1 | no | — |
| 117 | C4⋊Q8 | — | 3 | 1 | no | — |
| 118 | C4⋊Q8 | D12 | 1 | 1 | yes | characteristic |
| 119 | #127 | — | 3 | 1 | no | — |
| 120 | #127 | D12 | 1 | 1 | yes | characteristic |
| 121 | #124 | — | 6 | 1 | no | — |
| 122 | #124 | C4×D3 | 1 | 2 | yes | — |
| 123 | #121 | — | 2 | 2 | no | — |
| 124 | #121 | — | 6 | 2 | no | — |
| 125 | C4×D4 | — | 2 | 2 | no | — |
| 126 | C4×D4 | — | 6 | 2 | no | — |
| 127 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 128 | C4⋊C8 | — | 6 | 2 | no | — |
| 129 | C4⋊C8 | #67 | 1 | 4 | yes | — |
| 130 | C22⋊C8 | — | 2 | 4 | no | — |
| 131 | C22⋊C8 | — | 2 | 2 | no | — |
| 132 | C22⋊C8 | — | 6 | 2 | no | — |
| 133 | #97 | — | 2 | 2 | no | — |
| 134 | #97 | — | 3 | 2 | no | — |
| 135 | #97 | — | 6 | 1 | no | — |
| 136 | C4×C8 | — | 6 | 2 | no | — |
| 137 | C23×C3 | — | 2 | 2 | no | — |
| 138 | C23×C3 | #123 | 1 | 1 | yes | characteristic |
| 139 | C22×D3 | — | 4 | 1 | no | — |
| 140 | C3×Q8 | — | 4 | 2 | no | — |
| 141 | C3×D4 | — | 4 | 2 | no | — |
| 142 | C3×D4 | — | 8 | 2 | no | — |
| 143 | C3×D4 | — | 8 | 2 | no | — |
| 144 | C2×C3×C4 | — | 2 | 2 | no | — |
| 145 | C2×C3×C4 | — | 2 | 2 | no | — |
| 146 | C2×C3×C4 | — | 4 | 2 | no | — |
| 147 | C2×C3×C4 | — | 4 | 2 | no | — |
| 148 | C2×C3×C4 | — | 4 | 2 | no | — |
| 149 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 150 | C2×C3×C4 | C4×D4 | 1 | 2 | yes | — |
| 151 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 152 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 153 | #67 | — | 8 | 1 | no | — |
| 154 | #67 | — | 8 | 1 | no | — |
| 155 | C2×Dic3 | — | 4 | 1 | no | — |
| 156 | D12 | — | 4 | 2 | no | — |
| 157 | D12 | — | 4 | 4 | no | — |
| 158 | C4×D3 | — | 8 | 2 | no | — |
| 159 | Dic6 | — | 4 | 2 | no | — |
| 160 | C3×C8 | — | 4 | 4 | no | — |
| 161 | C3⋊C8 | — | 2 | 4 | no | — |
| 162 | C3⋊C8 | — | 8 | 2 | no | — |
| 163 | #41 | — | 4 | 2 | no | — |
| 164 | #41 | — | 4 | 2 | no | — |
| 165 | #41 | — | 12 | 2 | no | — |
| 166 | #41 | — | 12 | 2 | no | — |
| 167 | C2×Q8 | — | 6 | 1 | no | — |
| 168 | C2×Q8 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 169 | C2×D4 | — | 2 | 2 | no | — |
| 170 | C2×D4 | — | 2 | 2 | no | — |
| 171 | C2×D4 | — | 6 | 1 | no | — |
| 172 | C2×D4 | — | 6 | 2 | no | — |
| 173 | C2×D4 | — | 12 | 1 | no | — |
| 174 | C2×D4 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 175 | C22×C4 | — | 2 | 2 | no | — |
| 176 | C22×C4 | — | 12 | 1 | no | — |
| 177 | C22×C4 | #235 | 1 | 2 | yes | — |
| 178 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 179 | Q16 | — | 12 | 4 | no | — |
| 180 | SD16 | — | 12 | 4 | no | — |
| 181 | SD16 | — | 12 | 4 | no | — |
| 182 | D8 | — | 12 | 4 | no | — |
| 183 | M16 | — | 8 | 2 | no | — |
| 184 | C2×C8 | — | 2 | 4 | no | — |
| 185 | C2×C8 | — | 6 | 4 | no | — |
| 186 | C2×C8 | — | 6 | 2 | no | — |
| 187 | C2×C8 | — | 8 | 2 | no | — |
| 188 | C2×C8 | — | 12 | 2 | no | — |
| 189 | C4⋊C4 | — | 2 | 2 | no | — |
| 190 | C4⋊C4 | — | 6 | 2 | no | — |
| 191 | C22⋊C4 | — | 2 | 4 | no | — |
| 192 | C22⋊C4 | — | 12 | 2 | no | — |
| 193 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 194 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 195 | C42 | C2×#67 | 1 | 2 | yes | — |
| 196 | C22×C3 | — | 2 | 2 | no | — |
| 197 | C22×C3 | — | 2 | 1 | no | — |
| 198 | C22×C3 | — | 4 | 4 | no | — |
| 199 | C22×C3 | — | 4 | 2 | no | — |
| 200 | C22×C3 | #386 | 1 | 1 | yes | characteristic |
| 201 | C2×D3 | — | 8 | 2 | no | — |
| 202 | C2×D3 | — | 8 | 1 | no | — |
| 203 | C3×C4 | — | 2 | 2 | no | — |
| 204 | C3×C4 | — | 2 | 2 | no | — |
| 205 | C3×C4 | — | 2 | 1 | no | — |
| 206 | C3×C4 | — | 4 | 2 | no | — |
| 207 | C3×C4 | #453 | 1 | 2 | yes | — |
| 208 | Dic3 | — | 8 | 1 | no | — |
| 209 | C23 | — | 2 | 2 | no | — |
| 210 | C23 | — | 12 | 1 | no | — |
| 211 | C23 | #929 | 1 | 1 | yes | characteristic |
| 212 | Q8 | — | 4 | 2 | no | — |
| 213 | Q8 | — | 12 | 2 | no | — |
| 214 | D4 | — | 4 | 2 | no | — |
| 215 | D4 | — | 8 | 2 | no | — |
| 216 | D4 | — | 8 | 2 | no | — |
| 217 | D4 | — | 12 | 2 | no | — |
| 218 | D4 | — | 12 | 4 | no | — |
| 219 | D4 | — | 24 | 1 | no | — |
| 220 | D4 | — | 24 | 1 | no | — |
| 221 | C2×C4 | — | 2 | 2 | no | — |
| 222 | C2×C4 | — | 2 | 2 | no | — |
| 223 | C2×C4 | — | 4 | 2 | no | — |
| 224 | C2×C4 | — | 4 | 2 | no | — |
| 225 | C2×C4 | — | 4 | 2 | no | — |
| 226 | C2×C4 | — | 12 | 1 | no | — |
| 227 | C2×C4 | — | 24 | 2 | no | — |
| 228 | C2×C4 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | C4×#67 | 1 | 2 | yes | — |
| 230 | C2×C4 | #930 | 1 | 2 | yes | — |
| 231 | C2×C4 | #953 | 1 | 1 | yes | characteristic |
| 232 | C8 | — | 4 | 4 | no | — |
| 233 | C8 | — | 6 | 4 | no | — |
| 234 | C8 | — | 24 | 2 | no | — |
| 235 | C2×C3 | — | 4 | 1 | no | — |
| 236 | C2×C3 | — | 8 | 2 | no | — |
| 237 | C2×C3 | #1881 | 1 | 2 | yes | — |
| 238 | C2×C3 | #1887 | 1 | 1 | yes | characteristic |
| 239 | D3 | — | 16 | 1 | no | — |
| 240 | C22 | — | 2 | 2 | no | — |
| 241 | C22 | — | 2 | 1 | no | — |
| 242 | C22 | — | 4 | 4 | no | — |
| 243 | C22 | — | 4 | 2 | no | — |
| 244 | C22 | — | 24 | 2 | no | — |
| 245 | C22 | — | 24 | 1 | no | — |
| 246 | C22 | #5213 | 1 | 1 | yes | characteristic |
| 247 | C4 | — | 2 | 2 | no | — |
| 248 | C4 | — | 2 | 2 | no | — |
| 249 | C4 | — | 2 | 1 | no | — |
| 250 | C4 | — | 4 | 2 | no | — |
| 251 | C4 | — | 24 | 1 | no | — |
| 252 | C4 | #5241 | 1 | 2 | yes | — |
| 253 | C3 | #9007 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 254 | C2 | — | 4 | 1 | no | — |
| 255 | C2 | — | 8 | 2 | no | — |
| 256 | C2 | — | 48 | 1 | no | — |
| 257 | C2 | #71650 | 1 | 1 | yes | characteristic |
| 258 | C2 | #72020 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#2980 | C2 | semidirect |
| 3. #71805 | C2 | semidirect |
| 4. #70801 | C2 | semidirect |
| 5. #67751 | C2 | semidirect |
| 23. #2980 | D3 | semidirect |
| 253. C3 | #9007 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2980 | 2 |
| #83286 | 2 |
| #71805 | 2 |
| #70801 | 2 |
| #67751 | 2 |
| #9007 | 3 |