Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
912 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 912 conjugacy classes of subgroups — too many to draw. Only its 113 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 | #79305 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #73318 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #70774 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #69396 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×#1276 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10290 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#874 | — | 2 | 4 | no | — |
| 8 | C2×#874 | — | 2 | 2 | no | — |
| 9 | C2×#874 | C22 | 1 | 2 | yes | — |
| 10 | #5470 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#431 | C22 | 1 | 2 | yes | — |
| 12 | C3×C43 | C22 | 1 | 1 | yes | characteristic |
| 13 | #4924 | C22 | 1 | 1 | yes | characteristic |
| 14 | #4848 | — | 2 | 4 | no | — |
| 15 | #4731 | — | 2 | 4 | no | — |
| 16 | #4694 | — | 2 | 2 | no | — |
| 17 | #4693 | — | 2 | 2 | no | — |
| 18 | #4523 | — | 2 | 2 | no | — |
| 19 | #4485 | — | 2 | 4 | no | — |
| 20 | #3208 | — | 3 | 2 | no | — |
| 21 | #2867 | — | 3 | 1 | no | — |
| 22 | #1994 | — | 3 | 1 | no | — |
| 23 | #1545 | — | 3 | 2 | no | — |
| 24 | #1276 | D3 | 1 | 1 | yes | characteristic, complemented |
| 25 | C22×D12 | — | 2 | 8 | no | — |
| 26 | C22×D12 | — | 2 | 2 | no | — |
| 27 | C22×D12 | — | 2 | 2 | no | — |
| 28 | C22×D12 | D4 | 1 | 4 | yes | — |
| 29 | C2×C3×C42 | — | 2 | 2 | no | — |
| 30 | C2×C3×C42 | D4 | 1 | 2 | yes | — |
| 31 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 32 | #926 | — | 2 | 1 | no | — |
| 33 | #924 | — | 2 | 2 | no | — |
| 34 | C2×#208 | — | 2 | 1 | no | — |
| 35 | C2×#204 | — | 2 | 8 | no | — |
| 36 | #874 | — | 2 | 2 | no | — |
| 37 | #874 | — | 2 | 2 | no | — |
| 38 | #874 | — | 2 | 2 | no | — |
| 39 | #874 | — | 2 | 2 | no | — |
| 40 | #874 | — | 4 | 16 | no | — |
| 41 | #874 | — | 4 | 4 | no | — |
| 42 | #874 | — | 4 | 4 | no | — |
| 43 | #870 | — | 2 | 1 | no | — |
| 44 | #869 | D4 | 1 | 2 | yes | complemented |
| 45 | C4×Dic6 | — | 2 | 2 | no | — |
| 46 | C3×C4⋊C8 | D4 | 1 | 4 | yes | complemented |
| 47 | C3×C22⋊C8 | — | 2 | 2 | no | — |
| 48 | C3×C4×C8 | — | 2 | 2 | no | — |
| 49 | #821 | — | 2 | 8 | no | — |
| 50 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 51 | #533 | — | 3 | 1 | no | — |
| 52 | C2×#127 | — | 3 | 2 | no | — |
| 53 | C2×#127 | — | 6 | 4 | no | — |
| 54 | C2×#127 | — | 6 | 2 | no | — |
| 55 | #492 | — | 6 | 2 | no | — |
| 56 | #486 | — | 6 | 2 | no | — |
| 57 | #460 | — | 6 | 4 | no | — |
| 58 | #450 | — | 6 | 4 | no | — |
| 59 | #431 | C2×D3 | 1 | 2 | yes | — |
| 60 | #423 | — | 3 | 1 | no | — |
| 61 | C43 | C2×D3 | 1 | 1 | yes | characteristic |
| 62 | #339 | — | 6 | 2 | no | — |
| 63 | #331 | — | 6 | 4 | no | — |
| 64 | C23×D3 | — | 2 | 4 | no | — |
| 65 | C22×C3×C4 | — | 2 | 2 | no | — |
| 66 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C22×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 68 | C2×D12 | — | 2 | 4 | no | — |
| 69 | C2×D12 | — | 2 | 4 | no | — |
| 70 | C2×D12 | — | 4 | 8 | no | — |
| 71 | C2×D12 | — | 4 | 32 | no | — |
| 72 | C2×D12 | — | 4 | 16 | no | — |
| 73 | C2×D12 | — | 4 | 8 | no | — |
| 74 | C2×D12 | — | 4 | 2 | no | — |
| 75 | C2×D12 | — | 4 | 2 | no | — |
| 76 | C2×D12 | — | 4 | 2 | no | — |
| 77 | C2×D12 | — | 4 | 2 | no | — |
| 78 | C2×D12 | — | 4 | 4 | no | — |
| 79 | C2×D12 | — | 4 | 4 | no | — |
| 80 | C2×Dic6 | — | 2 | 2 | no | — |
| 81 | C2×C3×C8 | — | 2 | 4 | no | — |
| 82 | C3×C42 | — | 2 | 8 | no | — |
| 83 | C3×C42 | — | 2 | 2 | no | — |
| 84 | C3×C42 | SD16 | 1 | 4 | yes | complemented |
| 85 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 86 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 88 | #217 | — | 4 | 1 | no | — |
| 89 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 90 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 91 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 92 | C4×Dic3 | — | 4 | 1 | no | — |
| 93 | #208 | — | 4 | 2 | no | — |
| 94 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 95 | #204 | — | 8 | 8 | no | — |
| 96 | C22×D4 | — | 3 | 4 | no | — |
| 97 | C22×D4 | — | 6 | 8 | no | — |
| 98 | C22×D4 | — | 6 | 2 | no | — |
| 99 | C22×D4 | — | 6 | 2 | no | — |
| 100 | C2×SD16 | — | 6 | 8 | no | — |
| 101 | C2×M16 | — | 6 | 1 | no | — |
| 102 | C4⋊Q8 | — | 3 | 2 | no | — |
| 103 | #127 | — | 6 | 2 | no | — |
| 104 | #127 | — | 6 | 2 | no | — |
| 105 | #127 | — | 6 | 2 | no | — |
| 106 | #127 | — | 6 | 2 | no | — |
| 107 | #127 | — | 12 | 16 | no | — |
| 108 | #127 | — | 12 | 4 | no | — |
| 109 | #127 | — | 12 | 4 | no | — |
| 110 | #125 | — | 6 | 1 | no | — |
| 111 | C22⋊Q8 | — | 6 | 2 | no | — |
| 112 | C4×Q8 | — | 6 | 2 | no | — |
| 113 | #117 | — | 6 | 1 | no | — |
| 114 | C2×C42 | — | 2 | 2 | no | — |
| 115 | C2×C42 | D12 | 1 | 2 | yes | — |
| 116 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 117 | C4⋊C8 | — | 6 | 2 | no | — |
| 118 | C4⋊C8 | D12 | 1 | 4 | yes | complemented |
| 119 | #102 | — | 6 | 8 | no | — |
| 120 | C22⋊C8 | — | 2 | 2 | no | — |
| 121 | C4×C8 | — | 2 | 2 | no | — |
| 122 | C23×C3 | #120 | 1 | 1 | yes | characteristic |
| 123 | C22×D3 | — | 4 | 8 | no | — |
| 124 | C22×D3 | — | 4 | 8 | no | — |
| 125 | C22×D3 | — | 4 | 4 | no | — |
| 126 | C22×D3 | — | 4 | 8 | no | — |
| 127 | C2×C3×C4 | — | 2 | 8 | no | — |
| 128 | C2×C3×C4 | — | 2 | 4 | no | — |
| 129 | C2×C3×C4 | — | 2 | 4 | no | — |
| 130 | C2×C3×C4 | #120 | 1 | 2 | yes | — |
| 131 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 132 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 133 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 134 | C2×C3×C4 | C2×SD16 | 1 | 2 | yes | — |
| 135 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 136 | C2×Dic3 | — | 4 | 2 | no | — |
| 137 | D12 | — | 4 | 8 | no | — |
| 138 | D12 | — | 8 | 4 | no | — |
| 139 | D12 | — | 8 | 32 | no | — |
| 140 | D12 | — | 8 | 4 | no | — |
| 141 | D12 | — | 8 | 4 | no | — |
| 142 | D12 | — | 8 | 4 | no | — |
| 143 | D12 | — | 8 | 2 | no | — |
| 144 | D12 | — | 8 | 2 | no | — |
| 145 | Dic6 | — | 4 | 4 | no | — |
| 146 | C3×C8 | — | 4 | 4 | no | — |
| 147 | C3⋊C8 | — | 8 | 1 | no | — |
| 148 | C24 | — | 6 | 4 | no | — |
| 149 | C2×Q8 | — | 6 | 2 | no | — |
| 150 | C2×D4 | — | 6 | 4 | no | — |
| 151 | C2×D4 | — | 6 | 4 | no | — |
| 152 | C2×D4 | — | 12 | 8 | no | — |
| 153 | C2×D4 | — | 12 | 32 | no | — |
| 154 | C2×D4 | — | 12 | 16 | no | — |
| 155 | C2×D4 | — | 12 | 8 | no | — |
| 156 | C2×D4 | — | 12 | 2 | no | — |
| 157 | C2×D4 | — | 12 | 2 | no | — |
| 158 | C2×D4 | — | 12 | 2 | no | — |
| 159 | C2×D4 | — | 12 | 2 | no | — |
| 160 | C2×D4 | — | 12 | 4 | no | — |
| 161 | C2×D4 | — | 12 | 4 | no | — |
| 162 | C22×C4 | — | 2 | 2 | no | — |
| 163 | C22×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 164 | C22×C4 | D3×D4 | 1 | 2 | yes | — |
| 165 | SD16 | — | 24 | 8 | no | — |
| 166 | M16 | — | 12 | 2 | no | — |
| 167 | C2×C8 | — | 2 | 4 | no | — |
| 168 | C2×C8 | — | 12 | 1 | no | — |
| 169 | C4⋊C4 | — | 6 | 2 | no | — |
| 170 | C4⋊C4 | — | 6 | 2 | no | — |
| 171 | C4⋊C4 | — | 12 | 2 | no | — |
| 172 | C22⋊C4 | — | 12 | 1 | no | — |
| 173 | C42 | — | 2 | 8 | no | — |
| 174 | C42 | — | 2 | 2 | no | — |
| 175 | C42 | — | 12 | 1 | no | — |
| 176 | C42 | #204 | 1 | 4 | yes | complemented |
| 177 | C42 | C2×D12 | 1 | 2 | yes | — |
| 178 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 179 | C42 | #239 | 1 | 1 | yes | characteristic |
| 180 | C22×C3 | — | 2 | 2 | no | — |
| 181 | C22×C3 | #392 | 1 | 1 | yes | characteristic |
| 182 | C22×C3 | #450 | 1 | 2 | yes | — |
| 183 | C2×D3 | — | 8 | 16 | no | — |
| 184 | C2×D3 | — | 8 | 4 | no | — |
| 185 | C2×D3 | — | 8 | 4 | no | — |
| 186 | C2×D3 | — | 8 | 4 | no | — |
| 187 | C3×C4 | — | 2 | 8 | no | — |
| 188 | C3×C4 | — | 2 | 2 | no | — |
| 189 | C3×C4 | #453 | 1 | 4 | yes | — |
| 190 | C3×C4 | #460 | 1 | 2 | yes | — |
| 191 | C3×C4 | #492 | 1 | 1 | yes | characteristic |
| 192 | C3×C4 | #496 | 1 | 1 | yes | characteristic |
| 193 | Dic3 | — | 8 | 2 | no | — |
| 194 | C23 | — | 12 | 8 | no | — |
| 195 | C23 | — | 12 | 8 | no | — |
| 196 | C23 | — | 12 | 4 | no | — |
| 197 | C23 | — | 12 | 8 | no | — |
| 198 | C23 | #882 | 1 | 1 | yes | characteristic |
| 199 | Q8 | — | 12 | 4 | no | — |
| 200 | D4 | — | 12 | 8 | no | — |
| 201 | D4 | — | 24 | 4 | no | — |
| 202 | D4 | — | 24 | 32 | no | — |
| 203 | D4 | — | 24 | 4 | no | — |
| 204 | D4 | — | 24 | 2 | no | — |
| 205 | D4 | — | 24 | 2 | no | — |
| 206 | D4 | — | 24 | 4 | no | — |
| 207 | D4 | — | 24 | 4 | no | — |
| 208 | C2×C4 | — | 2 | 8 | no | — |
| 209 | C2×C4 | — | 2 | 4 | no | — |
| 210 | C2×C4 | — | 2 | 4 | no | — |
| 211 | C2×C4 | — | 12 | 2 | no | — |
| 212 | C2×C4 | #874 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #882 | 1 | 2 | yes | — |
| 214 | C2×C4 | #895 | 1 | 2 | yes | — |
| 215 | C2×C4 | #895 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | C2×#204 | 1 | 2 | yes | — |
| 217 | C2×C4 | #908 | 1 | 2 | yes | — |
| 218 | C8 | — | 4 | 4 | no | — |
| 219 | C8 | — | 24 | 1 | no | — |
| 220 | C2×C3 | — | 2 | 2 | no | — |
| 221 | C2×C3 | #2002 | 1 | 2 | yes | — |
| 222 | C2×C3 | #2014 | 1 | 1 | yes | characteristic |
| 223 | D3 | — | 16 | 4 | no | — |
| 224 | C22 | — | 2 | 2 | no | — |
| 225 | C22 | — | 24 | 16 | no | — |
| 226 | C22 | — | 24 | 4 | no | — |
| 227 | C22 | — | 24 | 4 | no | — |
| 228 | C22 | — | 24 | 4 | no | — |
| 229 | C22 | #4671 | 1 | 1 | yes | characteristic |
| 230 | C22 | #4731 | 1 | 2 | yes | — |
| 231 | C4 | — | 2 | 8 | no | — |
| 232 | C4 | — | 2 | 2 | no | — |
| 233 | C4 | — | 24 | 2 | no | — |
| 234 | C4 | #4693 | 1 | 1 | yes | characteristic |
| 235 | C4 | #4712 | 1 | 1 | yes | characteristic |
| 236 | C4 | #4822 | 1 | 4 | yes | — |
| 237 | C4 | #4848 | 1 | 2 | yes | — |
| 238 | C3 | #10290 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 239 | C2 | — | 2 | 2 | no | — |
| 240 | C2 | — | 48 | 4 | no | — |
| 241 | C2 | #68676 | 1 | 1 | yes | characteristic |
| 242 | C2 | #68761 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #79305 | C2 | semidirect |
| 3. #70774 | C2 | semidirect |
| 4. #69396 | C2 | semidirect |
| 5. C3×#1276 | C2 | semidirect |
| 24. #1276 | D3 | semidirect |
| 44. #869 | D4 | semidirect |
| 46. C3×C4⋊C8 | D4 | semidirect |
| 84. C3×C42 | SD16 | semidirect |
| 118. C4⋊C8 | D12 | semidirect |
| 176. C42 | #204 | semidirect |
| 238. C3 | #10290 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #79305 | 2 |
| #73318 | 2 |
| #70774 | 2 |
| #69396 | 2 |
| C3×#1276 | 2 |
| #10290 | 3 |