Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #83336 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #71834 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #70774 | C2 | 1 | 2 | yes | maximal |
| 5 | #67743 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #9438 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C3×#533 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#874 | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×#446 | — | 2 | 2 | no | — |
| 10 | C3×C8×Q8 | — | 2 | 2 | no | — |
| 11 | C3×#433 | — | 2 | 2 | no | — |
| 12 | C3×#431 | C22 | 1 | 2 | yes | — |
| 13 | #5088 | — | 2 | 2 | no | — |
| 14 | #5067 | C4 | 1 | 4 | yes | — |
| 15 | #5001 | — | 2 | 2 | no | — |
| 16 | #4924 | C22 | 1 | 2 | yes | — |
| 17 | C4×#208 | C22 | 1 | 1 | yes | characteristic |
| 18 | #4547 | — | 2 | 2 | no | — |
| 19 | #4485 | — | 2 | 2 | no | — |
| 20 | #4485 | — | 2 | 2 | no | — |
| 21 | #3138 | — | 3 | 1 | no | — |
| 22 | #2986 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | #1561 | — | 3 | 1 | no | — |
| 24 | #1545 | — | 3 | 2 | no | — |
| 25 | #1299 | — | 3 | 2 | no | — |
| 26 | C22×D12 | — | 2 | 1 | no | — |
| 27 | C22×D12 | — | 2 | 2 | no | — |
| 28 | C3×C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 29 | C3×#125 | C2×C4 | 1 | 2 | yes | — |
| 30 | C3×C22⋊Q8 | — | 2 | 2 | no | — |
| 31 | C3×C4×Q8 | — | 2 | 2 | no | — |
| 32 | C3×#117 | D4 | 1 | 2 | yes | — |
| 33 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×#208 | — | 2 | 1 | no | — |
| 35 | C2×#208 | — | 2 | 2 | no | — |
| 36 | #874 | — | 2 | 2 | no | — |
| 37 | #874 | — | 2 | 1 | no | — |
| 38 | #874 | C2×C4 | 1 | 2 | yes | — |
| 39 | C3×C4⋊C8 | — | 2 | 4 | no | — |
| 40 | C3×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 41 | C3×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 42 | C3×C22⋊C8 | — | 2 | 2 | no | — |
| 43 | C3×#97 | — | 2 | 2 | no | — |
| 44 | C3×C4×C8 | — | 2 | 2 | no | — |
| 45 | C3×C4×C8 | — | 2 | 2 | no | — |
| 46 | #809 | — | 2 | 8 | no | — |
| 47 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 48 | C4⋊(C3⋊C8) | — | 2 | 2 | no | — |
| 49 | #803 | C2×C4 | 1 | 2 | yes | — |
| 50 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 51 | #533 | C2×D3 | 1 | 1 | yes | characteristic |
| 52 | C2×#127 | — | 3 | 1 | no | — |
| 53 | #490 | — | 3 | 4 | no | — |
| 54 | #486 | — | 6 | 2 | no | — |
| 55 | #446 | — | 2 | 2 | no | — |
| 56 | C8×Q8 | — | 2 | 2 | no | — |
| 57 | #433 | — | 2 | 2 | no | — |
| 58 | #431 | C2×D3 | 1 | 2 | yes | — |
| 59 | #423 | — | 3 | 2 | no | — |
| 60 | #418 | — | 6 | 2 | no | — |
| 61 | C4×M16 | — | 3 | 1 | no | — |
| 62 | #332 | — | 6 | 2 | no | — |
| 63 | #331 | — | 6 | 2 | no | — |
| 64 | #331 | — | 6 | 2 | no | — |
| 65 | C23×D3 | — | 4 | 1 | no | — |
| 66 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 67 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C22×C3×C4 | #41 | 1 | 2 | yes | — |
| 69 | C2×D12 | — | 2 | 2 | no | — |
| 70 | C2×D12 | — | 2 | 4 | no | — |
| 71 | C2×D12 | — | 4 | 4 | no | — |
| 72 | C2×D12 | — | 4 | 1 | no | — |
| 73 | C2×D12 | — | 4 | 2 | no | — |
| 74 | C2×D12 | — | 8 | 2 | no | — |
| 75 | C2×D12 | — | 8 | 2 | no | — |
| 76 | C2×C3×C8 | — | 2 | 2 | no | — |
| 77 | C2×C3×C8 | — | 2 | 2 | no | — |
| 78 | C3×C4⋊C4 | — | 2 | 4 | no | — |
| 79 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 80 | C3×C4⋊C4 | C22⋊C4 | 1 | 4 | yes | — |
| 81 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 82 | C3×C42 | — | 2 | 2 | no | — |
| 83 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C3×C42 | C2×D4 | 1 | 2 | yes | — |
| 86 | #208 | — | 4 | 2 | no | — |
| 87 | #208 | — | 8 | 2 | no | — |
| 88 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 89 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 90 | C22×D4 | — | 6 | 2 | no | — |
| 91 | C22×D4 | — | 6 | 1 | no | — |
| 92 | C2×M16 | — | 6 | 1 | no | — |
| 93 | C2×M16 | — | 6 | 2 | no | — |
| 94 | C4⋊Q8 | D12 | 1 | 2 | yes | — |
| 95 | #127 | — | 3 | 2 | no | — |
| 96 | #127 | — | 6 | 1 | no | — |
| 97 | #127 | — | 6 | 2 | no | — |
| 98 | #125 | C4×D3 | 1 | 2 | yes | — |
| 99 | C22⋊Q8 | — | 2 | 2 | no | — |
| 100 | C4×Q8 | — | 2 | 2 | no | — |
| 101 | #117 | #67 | 1 | 2 | yes | — |
| 102 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 103 | C4⋊C8 | — | 2 | 4 | no | — |
| 104 | C4⋊C8 | — | 6 | 2 | no | — |
| 105 | C4⋊C8 | — | 6 | 2 | no | — |
| 106 | C4⋊C8 | #67 | 1 | 2 | yes | — |
| 107 | C4⋊C8 | #67 | 1 | 2 | yes | — |
| 108 | #102 | — | 6 | 8 | no | — |
| 109 | C22⋊C8 | — | 2 | 2 | no | — |
| 110 | #97 | — | 2 | 2 | no | — |
| 111 | #97 | — | 3 | 2 | no | — |
| 112 | C4×C8 | — | 2 | 2 | no | — |
| 113 | C4×C8 | — | 2 | 2 | no | — |
| 114 | C4×C8 | — | 6 | 1 | no | — |
| 115 | C23×C3 | #123 | 1 | 1 | yes | characteristic |
| 116 | C22×D3 | — | 4 | 2 | no | — |
| 117 | C22×D3 | — | 8 | 2 | no | — |
| 118 | C22×D3 | — | 8 | 4 | no | — |
| 119 | C3×Q8 | — | 4 | 2 | no | — |
| 120 | C3×Q8 | — | 4 | 2 | no | — |
| 121 | C2×C3×C4 | — | 2 | 2 | no | — |
| 122 | C2×C3×C4 | — | 2 | 2 | no | — |
| 123 | C2×C3×C4 | — | 2 | 2 | no | — |
| 124 | C2×C3×C4 | — | 2 | 4 | no | — |
| 125 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C2×C3×C4 | C4×D4 | 1 | 2 | yes | — |
| 127 | C2×C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 128 | C2×C3×C4 | #121 | 1 | 2 | yes | — |
| 129 | D12 | — | 4 | 4 | no | — |
| 130 | D12 | — | 8 | 4 | no | — |
| 131 | D12 | — | 8 | 2 | no | — |
| 132 | D12 | — | 8 | 2 | no | — |
| 133 | D12 | — | 8 | 2 | no | — |
| 134 | C3×C8 | — | 4 | 2 | no | — |
| 135 | C3×C8 | — | 4 | 2 | no | — |
| 136 | C3⋊C8 | — | 4 | 2 | no | — |
| 137 | C3⋊C8 | — | 8 | 2 | no | — |
| 138 | C24 | — | 12 | 1 | no | — |
| 139 | C2×Q8 | — | 2 | 2 | no | — |
| 140 | C2×D4 | — | 6 | 2 | no | — |
| 141 | C2×D4 | — | 6 | 4 | no | — |
| 142 | C2×D4 | — | 12 | 4 | no | — |
| 143 | C2×D4 | — | 12 | 2 | no | — |
| 144 | C2×D4 | — | 12 | 1 | no | — |
| 145 | C2×D4 | — | 24 | 2 | no | — |
| 146 | C2×D4 | — | 24 | 2 | no | — |
| 147 | C22×C4 | #235 | 1 | 2 | yes | — |
| 148 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 149 | M16 | — | 12 | 2 | no | — |
| 150 | M16 | — | 24 | 2 | no | — |
| 151 | C2×C8 | — | 2 | 2 | no | — |
| 152 | C2×C8 | — | 2 | 2 | no | — |
| 153 | C2×C8 | — | 6 | 2 | no | — |
| 154 | C2×C8 | — | 12 | 2 | no | — |
| 155 | C4⋊C4 | — | 2 | 4 | no | — |
| 156 | C4⋊C4 | — | 2 | 2 | no | — |
| 157 | C4⋊C4 | (C2×D3)⋊C4 | 1 | 4 | yes | — |
| 158 | C22⋊C4 | — | 2 | 2 | no | — |
| 159 | C42 | — | 2 | 2 | no | — |
| 160 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 161 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 162 | C42 | C2×#67 | 1 | 2 | yes | — |
| 163 | C22×C3 | — | 2 | 1 | no | — |
| 164 | C22×C3 | — | 2 | 2 | no | — |
| 165 | C22×C3 | #386 | 1 | 1 | yes | characteristic |
| 166 | C2×D3 | — | 8 | 2 | no | — |
| 167 | C2×D3 | — | 8 | 4 | no | — |
| 168 | C2×D3 | — | 16 | 4 | no | — |
| 169 | C3×C4 | — | 2 | 1 | no | — |
| 170 | C3×C4 | — | 2 | 2 | no | — |
| 171 | C3×C4 | — | 4 | 4 | no | — |
| 172 | C3×C4 | #454 | 1 | 2 | yes | — |
| 173 | C3×C4 | #471 | 1 | 2 | yes | — |
| 174 | C3×C4 | #472 | 1 | 2 | yes | — |
| 175 | C23 | — | 12 | 2 | no | — |
| 176 | C23 | — | 24 | 2 | no | — |
| 177 | C23 | — | 24 | 4 | no | — |
| 178 | C23 | #929 | 1 | 1 | yes | characteristic |
| 179 | Q8 | — | 4 | 2 | no | — |
| 180 | Q8 | — | 4 | 2 | no | — |
| 181 | D4 | — | 12 | 4 | no | — |
| 182 | D4 | — | 24 | 4 | no | — |
| 183 | D4 | — | 24 | 2 | no | — |
| 184 | D4 | — | 24 | 2 | no | — |
| 185 | D4 | — | 24 | 2 | no | — |
| 186 | C2×C4 | — | 2 | 2 | no | — |
| 187 | C2×C4 | — | 2 | 2 | no | — |
| 188 | C2×C4 | — | 2 | 2 | no | — |
| 189 | C2×C4 | — | 2 | 4 | no | — |
| 190 | C2×C4 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 191 | C2×C4 | C4×#67 | 1 | 2 | yes | — |
| 192 | C2×C4 | #930 | 1 | 2 | yes | — |
| 193 | C2×C4 | #953 | 1 | 1 | yes | characteristic |
| 194 | C8 | — | 4 | 2 | no | — |
| 195 | C8 | — | 4 | 2 | no | — |
| 196 | C8 | — | 12 | 2 | no | — |
| 197 | C8 | — | 24 | 2 | no | — |
| 198 | C2×C3 | — | 4 | 1 | no | — |
| 199 | C2×C3 | #1888 | 1 | 1 | yes | characteristic |
| 200 | C2×C3 | #1907 | 1 | 2 | yes | — |
| 201 | D3 | — | 16 | 2 | no | — |
| 202 | C22 | — | 2 | 2 | no | — |
| 203 | C22 | — | 2 | 1 | no | — |
| 204 | C22 | — | 24 | 4 | no | — |
| 205 | C22 | — | 24 | 2 | no | — |
| 206 | C22 | — | 48 | 4 | no | — |
| 207 | C22 | #5213 | 1 | 1 | yes | characteristic |
| 208 | C4 | — | 2 | 2 | no | — |
| 209 | C4 | — | 2 | 1 | no | — |
| 210 | C4 | — | 4 | 4 | no | — |
| 211 | C4 | #5141 | 1 | 2 | yes | — |
| 212 | C4 | #5142 | 1 | 2 | yes | — |
| 213 | C4 | #5245 | 1 | 2 | yes | — |
| 214 | C3 | #9438 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 215 | C2 | — | 4 | 1 | no | — |
| 216 | C2 | — | 48 | 2 | no | — |
| 217 | C2 | #71661 | 1 | 1 | yes | characteristic |
| 218 | C2 | #72018 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#2986 | C2 | semidirect |
| 3. #71834 | C2 | semidirect |
| 5. #67743 | C2 | semidirect |
| 22. #2986 | D3 | semidirect |
| 214. C3 | #9438 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2986 | 2 |
| #83336 | 2 |
| #71834 | 2 |
| #70774 | 2 |
| #67743 | 2 |
| #9438 | 3 |