Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
600 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 97 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 | C2×#4714 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #74254 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#4464 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #69552 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×#1295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10085 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#875 | C22 | 1 | 2 | yes | — |
| 8 | C2×#869 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×#97 | C22 | 1 | 2 | yes | — |
| 10 | C3×#382 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#821 | — | 2 | 2 | no | — |
| 12 | C2×#816 | — | 2 | 2 | no | — |
| 13 | #4991 | — | 2 | 2 | no | — |
| 14 | C2×#810 | — | 2 | 2 | no | — |
| 15 | C2×#803 | C22 | 1 | 1 | yes | characteristic |
| 16 | #4714 | — | 2 | 4 | no | — |
| 17 | #4714 | — | 2 | 4 | no | — |
| 18 | #4673 | — | 2 | 2 | no | — |
| 19 | #4671 | — | 2 | 2 | no | — |
| 20 | #4464 | — | 2 | 4 | no | — |
| 21 | C2×#489 | — | 3 | 2 | no | — |
| 22 | #2819 | — | 3 | 1 | no | — |
| 23 | #1960 | — | 3 | 1 | no | — |
| 24 | C2×#329 | — | 3 | 2 | no | — |
| 25 | #1295 | D3 | 1 | 1 | yes | characteristic, complemented |
| 26 | C22×D12 | — | 2 | 2 | no | — |
| 27 | C22×D12 | D4 | 1 | 2 | yes | — |
| 28 | C22×Dic6 | — | 2 | 1 | no | — |
| 29 | C22×Dic6 | D4 | 1 | 2 | yes | — |
| 30 | C22×C3×C8 | — | 2 | 2 | no | — |
| 31 | C2×C3×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 32 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×(C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 34 | C2×(C2×D3)⋊C4 | — | 2 | 2 | no | — |
| 35 | C2×(C2×D3)⋊C4 | — | 2 | 4 | no | — |
| 36 | C2×C4⋊Dic3 | — | 2 | 2 | no | — |
| 37 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 38 | #875 | — | 2 | 2 | no | — |
| 39 | #875 | — | 2 | 2 | no | — |
| 40 | #875 | — | 2 | 2 | no | — |
| 41 | #875 | — | 2 | 2 | no | — |
| 42 | #869 | — | 2 | 2 | no | — |
| 43 | #869 | — | 2 | 1 | no | — |
| 44 | #869 | D4 | 1 | 2 | yes | complemented |
| 45 | C3×#97 | — | 2 | 2 | no | — |
| 46 | C3×#97 | D4 | 1 | 4 | yes | complemented |
| 47 | C3×#95 | — | 2 | 2 | no | — |
| 48 | C3⋊#95 | — | 2 | 2 | no | — |
| 49 | #821 | — | 2 | 8 | no | — |
| 50 | #816 | — | 2 | 8 | no | — |
| 51 | #810 | — | 4 | 4 | no | — |
| 52 | #803 | — | 2 | 2 | no | — |
| 53 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 54 | C2×#124 | — | 3 | 2 | no | — |
| 55 | #489 | — | 6 | 4 | no | — |
| 56 | #489 | — | 6 | 4 | no | — |
| 57 | C2×#103 | — | 6 | 2 | no | — |
| 58 | C2×#103 | — | 6 | 2 | no | — |
| 59 | C2×#102 | — | 6 | 2 | no | — |
| 60 | C2×#97 | — | 3 | 1 | no | — |
| 61 | C2×#97 | C2×D3 | 1 | 2 | yes | — |
| 62 | #394 | — | 6 | 2 | no | — |
| 63 | #393 | — | 6 | 2 | no | — |
| 64 | #392 | — | 6 | 2 | no | — |
| 65 | #382 | C2×D3 | 1 | 1 | yes | characteristic |
| 66 | #329 | — | 6 | 4 | no | — |
| 67 | C23×D3 | — | 2 | 2 | no | — |
| 68 | C22×C3×C4 | — | 2 | 2 | no | — |
| 69 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 71 | C22×Dic3 | — | 2 | 2 | no | — |
| 72 | C22×Dic3 | — | 4 | 1 | no | — |
| 73 | C2×D12 | — | 2 | 2 | no | — |
| 74 | C2×D12 | — | 2 | 2 | no | — |
| 75 | C2×D12 | — | 4 | 4 | no | — |
| 76 | C2×D12 | — | 8 | 2 | no | — |
| 77 | C2×D12 | — | 8 | 4 | no | — |
| 78 | C2×Dic6 | — | 2 | 2 | no | — |
| 79 | C2×Dic6 | — | 2 | 2 | no | — |
| 80 | C2×Dic6 | — | 2 | 2 | no | — |
| 81 | C2×Dic6 | — | 4 | 4 | no | — |
| 82 | C2×Dic6 | — | 4 | 2 | no | — |
| 83 | C2×Dic6 | — | 4 | 2 | no | — |
| 84 | C2×Dic6 | — | 4 | 1 | no | — |
| 85 | C2×C3×C8 | — | 2 | 4 | no | — |
| 86 | C2×C3×C8 | — | 4 | 4 | no | — |
| 87 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 88 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 89 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 91 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 92 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 93 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 94 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 95 | (C2×D3)⋊C4 | — | 8 | 4 | no | — |
| 96 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 97 | C4⋊Dic3 | — | 4 | 2 | no | — |
| 98 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 99 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 100 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 101 | C22×Q8 | — | 3 | 2 | no | — |
| 102 | C22×Q8 | — | 6 | 1 | no | — |
| 103 | C22×D4 | — | 3 | 2 | no | — |
| 104 | C22×D4 | — | 6 | 2 | no | — |
| 105 | C22×C8 | — | 2 | 2 | no | — |
| 106 | C22×C8 | — | 6 | 1 | no | — |
| 107 | C4⋊Q8 | — | 3 | 2 | no | — |
| 108 | C4⋊Q8 | — | 6 | 2 | no | — |
| 109 | C4⋊Q8 | — | 6 | 1 | no | — |
| 110 | #124 | — | 6 | 2 | no | — |
| 111 | #124 | — | 6 | 2 | no | — |
| 112 | #124 | — | 6 | 2 | no | — |
| 113 | #124 | — | 6 | 2 | no | — |
| 114 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 115 | C2×C4⋊C4 | D12 | 1 | 2 | yes | — |
| 116 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 117 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 118 | C2×C22⋊C4 | — | 6 | 4 | no | — |
| 119 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 120 | #103 | — | 6 | 8 | no | — |
| 121 | #103 | — | 12 | 4 | no | — |
| 122 | #102 | — | 6 | 8 | no | — |
| 123 | #97 | — | 2 | 2 | no | — |
| 124 | #97 | — | 6 | 2 | no | — |
| 125 | #97 | D12 | 1 | 4 | yes | complemented |
| 126 | #95 | — | 2 | 2 | no | — |
| 127 | #95 | — | 6 | 2 | no | — |
| 128 | C23×C3 | #120 | 1 | 1 | yes | characteristic |
| 129 | C22×D3 | — | 4 | 2 | no | — |
| 130 | C22×D3 | — | 4 | 2 | no | — |
| 131 | C22×D3 | — | 4 | 2 | no | — |
| 132 | C22×D3 | — | 4 | 4 | no | — |
| 133 | C22×D3 | — | 4 | 4 | no | — |
| 134 | C2×C3×C4 | — | 2 | 1 | no | — |
| 135 | C2×C3×C4 | — | 2 | 1 | no | — |
| 136 | C2×C3×C4 | — | 2 | 4 | no | — |
| 137 | C2×C3×C4 | — | 4 | 2 | no | — |
| 138 | C2×C3×C4 | — | 4 | 4 | no | — |
| 139 | C2×C3×C4 | #120 | 1 | 2 | yes | — |
| 140 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 141 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 142 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 143 | C2×Dic3 | — | 4 | 2 | no | — |
| 144 | C2×Dic3 | — | 4 | 2 | no | — |
| 145 | C2×Dic3 | — | 4 | 4 | no | — |
| 146 | C2×Dic3 | — | 8 | 1 | no | — |
| 147 | C2×Dic3 | — | 8 | 1 | no | — |
| 148 | D12 | — | 4 | 4 | no | — |
| 149 | D12 | — | 8 | 2 | no | — |
| 150 | D12 | — | 8 | 8 | no | — |
| 151 | Dic6 | — | 4 | 4 | no | — |
| 152 | Dic6 | — | 4 | 4 | no | — |
| 153 | Dic6 | — | 8 | 2 | no | — |
| 154 | Dic6 | — | 8 | 2 | no | — |
| 155 | C3×C8 | — | 4 | 4 | no | — |
| 156 | C3⋊C8 | — | 4 | 2 | no | — |
| 157 | C24 | — | 6 | 2 | no | — |
| 158 | C2×Q8 | — | 6 | 2 | no | — |
| 159 | C2×Q8 | — | 6 | 2 | no | — |
| 160 | C2×Q8 | — | 6 | 2 | no | — |
| 161 | C2×Q8 | — | 12 | 4 | no | — |
| 162 | C2×Q8 | — | 12 | 2 | no | — |
| 163 | C2×Q8 | — | 12 | 2 | no | — |
| 164 | C2×Q8 | — | 12 | 1 | no | — |
| 165 | C2×D4 | — | 6 | 2 | no | — |
| 166 | C2×D4 | — | 6 | 2 | no | — |
| 167 | C2×D4 | — | 12 | 4 | no | — |
| 168 | C2×D4 | — | 24 | 2 | no | — |
| 169 | C2×D4 | — | 24 | 4 | no | — |
| 170 | C22×C4 | — | 2 | 2 | no | — |
| 171 | C22×C4 | — | 6 | 2 | no | — |
| 172 | C22×C4 | — | 12 | 1 | no | — |
| 173 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 174 | C22×C4 | D3×D4 | 1 | 2 | yes | — |
| 175 | C2×C8 | — | 2 | 4 | no | — |
| 176 | C2×C8 | — | 4 | 4 | no | — |
| 177 | C2×C8 | — | 6 | 2 | no | — |
| 178 | C2×C8 | — | 12 | 1 | no | — |
| 179 | C2×C8 | — | 12 | 1 | no | — |
| 180 | C4⋊C4 | — | 4 | 2 | no | — |
| 181 | C4⋊C4 | — | 6 | 4 | no | — |
| 182 | C4⋊C4 | — | 12 | 2 | no | — |
| 183 | C22⋊C4 | — | 12 | 2 | no | — |
| 184 | C22⋊C4 | — | 12 | 2 | no | — |
| 185 | C22⋊C4 | — | 12 | 2 | no | — |
| 186 | C22⋊C4 | — | 12 | 2 | no | — |
| 187 | C22⋊C4 | — | 24 | 4 | no | — |
| 188 | C42 | C2×D12 | 1 | 2 | yes | — |
| 189 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 190 | C42 | #239 | 1 | 1 | yes | characteristic |
| 191 | C22×C3 | #392 | 1 | 1 | yes | characteristic |
| 192 | C22×C3 | #453 | 1 | 1 | yes | characteristic |
| 193 | C22×C3 | #454 | 1 | 1 | yes | characteristic |
| 194 | C22×C3 | #455 | 1 | 4 | yes | — |
| 195 | C2×D3 | — | 8 | 4 | no | — |
| 196 | C2×D3 | — | 8 | 4 | no | — |
| 197 | C2×D3 | — | 8 | 2 | no | — |
| 198 | C2×D3 | — | 8 | 2 | no | — |
| 199 | C2×D3 | — | 8 | 2 | no | — |
| 200 | C3×C4 | — | 2 | 2 | no | — |
| 201 | C3×C4 | — | 2 | 2 | no | — |
| 202 | C3×C4 | — | 2 | 1 | no | — |
| 203 | C3×C4 | — | 2 | 1 | no | — |
| 204 | C3×C4 | — | 8 | 2 | no | — |
| 205 | Dic3 | — | 8 | 2 | no | — |
| 206 | Dic3 | — | 8 | 2 | no | — |
| 207 | C23 | — | 12 | 2 | no | — |
| 208 | C23 | — | 12 | 2 | no | — |
| 209 | C23 | — | 12 | 2 | no | — |
| 210 | C23 | — | 12 | 4 | no | — |
| 211 | C23 | — | 12 | 4 | no | — |
| 212 | C23 | #882 | 1 | 1 | yes | characteristic |
| 213 | Q8 | — | 12 | 4 | no | — |
| 214 | Q8 | — | 12 | 4 | no | — |
| 215 | Q8 | — | 24 | 2 | no | — |
| 216 | Q8 | — | 24 | 2 | no | — |
| 217 | D4 | — | 12 | 4 | no | — |
| 218 | D4 | — | 24 | 2 | no | — |
| 219 | D4 | — | 24 | 8 | no | — |
| 220 | C2×C4 | — | 2 | 1 | no | — |
| 221 | C2×C4 | — | 2 | 1 | no | — |
| 222 | C2×C4 | — | 2 | 4 | no | — |
| 223 | C2×C4 | — | 4 | 2 | no | — |
| 224 | C2×C4 | — | 4 | 4 | no | — |
| 225 | C2×C4 | — | 12 | 2 | no | — |
| 226 | C2×C4 | — | 12 | 2 | no | — |
| 227 | C2×C4 | — | 12 | 4 | no | — |
| 228 | C2×C4 | — | 24 | 1 | no | — |
| 229 | C2×C4 | — | 24 | 1 | no | — |
| 230 | C2×C4 | #874 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | #882 | 1 | 2 | yes | — |
| 232 | C2×C4 | #895 | 1 | 2 | yes | — |
| 233 | C2×C4 | #895 | 1 | 1 | yes | characteristic |
| 234 | C8 | — | 4 | 4 | no | — |
| 235 | C8 | — | 12 | 2 | no | — |
| 236 | C2×C3 | #1659 | 1 | 2 | yes | — |
| 237 | C2×C3 | #1662 | 1 | 2 | yes | — |
| 238 | C2×C3 | #2004 | 1 | 2 | yes | — |
| 239 | C2×C3 | #2006 | 1 | 1 | yes | characteristic |
| 240 | D3 | — | 16 | 2 | no | — |
| 241 | C22 | — | 24 | 4 | no | — |
| 242 | C22 | — | 24 | 4 | no | — |
| 243 | C22 | — | 24 | 2 | no | — |
| 244 | C22 | — | 24 | 2 | no | — |
| 245 | C22 | — | 24 | 2 | no | — |
| 246 | C22 | #4671 | 1 | 1 | yes | characteristic |
| 247 | C22 | #4747 | 1 | 1 | yes | characteristic |
| 248 | C22 | #4753 | 1 | 1 | yes | characteristic |
| 249 | C22 | #4825 | 1 | 4 | yes | — |
| 250 | C4 | — | 2 | 1 | no | — |
| 251 | C4 | — | 2 | 1 | no | — |
| 252 | C4 | — | 2 | 2 | no | — |
| 253 | C4 | — | 2 | 2 | no | — |
| 254 | C4 | — | 8 | 2 | no | — |
| 255 | C4 | — | 24 | 2 | no | — |
| 256 | C4 | — | 24 | 2 | no | — |
| 257 | C3 | #10085 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 258 | C2 | — | 48 | 2 | no | — |
| 259 | C2 | #68353 | 1 | 2 | yes | — |
| 260 | C2 | #68374 | 1 | 2 | yes | — |
| 261 | C2 | #68762 | 1 | 2 | yes | — |
| 262 | C2 | #69008 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4714 | C2 | semidirect |
| 3. C2×#4464 | C2 | semidirect |
| 4. #69552 | C2 | semidirect |
| 5. C3×#1295 | C2 | semidirect |
| 25. #1295 | D3 | semidirect |
| 44. #869 | D4 | semidirect |
| 46. C3×#97 | D4 | semidirect |
| 125. #97 | D12 | semidirect |
| 257. C3 | #10085 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4714 | 2 |
| #74254 | 2 |
| C2×#4464 | 2 |
| #69552 | 2 |
| C3×#1295 | 2 |
| #10085 | 3 |