Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
709 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 709 conjugacy classes of subgroups — too many to draw. Only its 67 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×#5395 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×C4⋊GL(2,𝔽3) | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #73167 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #33167 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C22×GL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 6 | C22×#226 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C4×SL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 8 | #5395 | — | 2 | 6 | no | — |
| 9 | C4⋊GL(2,𝔽3) | — | 2 | 6 | no | — |
| 10 | #4671 | — | 4 | 1 | no | maximal |
| 11 | C2×#464 | — | 3 | 3 | no | — |
| 12 | C2×#460 | — | 3 | 3 | no | — |
| 13 | #2657 | — | 3 | 1 | no | — |
| 14 | #2276 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | #1983 | — | 3 | 3 | no | — |
| 16 | #1982 | — | 3 | 3 | no | — |
| 17 | #1843 | — | 3 | 1 | no | — |
| 18 | C22×D12 | — | 4 | 3 | no | — |
| 19 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 20 | C2×GL(2,𝔽3) | — | 2 | 6 | no | — |
| 21 | C2×GL(2,𝔽3) | — | 2 | 3 | no | — |
| 22 | C2×#226 | — | 2 | 3 | no | — |
| 23 | C2×(C2×D3)⋊C4 | — | 4 | 3 | no | — |
| 24 | C4×SL(2,𝔽3) | D4 | 1 | 6 | yes | complemented |
| 25 | C3×#95 | — | 4 | 1 | no | — |
| 26 | C22×Q16 | — | 3 | 1 | no | — |
| 27 | C22×SD16 | — | 3 | 3 | no | — |
| 28 | C2×#127 | — | 3 | 3 | no | — |
| 29 | C2×#124 | — | 3 | 3 | no | — |
| 30 | C2×C4×Q8 | C2×D3 | 1 | 3 | yes | — |
| 31 | #464 | — | 6 | 6 | no | — |
| 32 | #464 | — | 6 | 6 | no | — |
| 33 | #460 | — | 6 | 6 | no | — |
| 34 | #460 | — | 6 | 6 | no | — |
| 35 | C2×C4⋊C8 | — | 3 | 3 | no | — |
| 36 | C2×C4⋊C8 | — | 3 | 3 | no | — |
| 37 | C2×#103 | — | 3 | 3 | no | — |
| 38 | C2×#102 | — | 3 | 3 | no | — |
| 39 | C2×#102 | — | 3 | 6 | no | — |
| 40 | #392 | — | 6 | 3 | no | — |
| 41 | #392 | — | 6 | 1 | no | — |
| 42 | C4×C4⋊C4 | — | 3 | 3 | no | — |
| 43 | #376 | — | 3 | 1 | no | — |
| 44 | C23×D3 | — | 4 | 3 | no | — |
| 45 | C22×C3×C4 | — | 4 | 3 | no | — |
| 46 | C22×Dic3 | — | 4 | 1 | no | — |
| 47 | C2×D12 | — | 8 | 6 | no | — |
| 48 | C2×D12 | — | 8 | 12 | no | — |
| 49 | C2×SL(2,𝔽3) | C2×D4 | 1 | 3 | yes | — |
| 50 | GL(2,𝔽3) | — | 4 | 3 | no | — |
| 51 | #226 | — | 4 | 1 | no | — |
| 52 | (C2×D3)⋊C4 | — | 8 | 6 | no | — |
| 53 | C22×Q8 | — | 3 | 1 | no | — |
| 54 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 55 | C22×D4 | — | 3 | 3 | no | — |
| 56 | C22×D4 | — | 6 | 3 | no | — |
| 57 | C22×D4 | — | 6 | 3 | no | — |
| 58 | C2×Q16 | — | 6 | 3 | no | — |
| 59 | C2×Q16 | — | 6 | 3 | no | — |
| 60 | C2×SD16 | — | 6 | 6 | no | — |
| 61 | C2×SD16 | — | 6 | 3 | no | — |
| 62 | C2×SD16 | — | 6 | 6 | no | — |
| 63 | C2×SD16 | — | 6 | 3 | no | — |
| 64 | C22×C8 | — | 3 | 3 | no | — |
| 65 | C22×C8 | — | 3 | 1 | no | — |
| 66 | #127 | — | 6 | 6 | no | — |
| 67 | #127 | — | 6 | 6 | no | — |
| 68 | #124 | — | 6 | 6 | no | — |
| 69 | #124 | — | 6 | 6 | no | — |
| 70 | C4×Q8 | — | 3 | 6 | no | — |
| 71 | C4×Q8 | D12 | 1 | 6 | yes | complemented |
| 72 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 73 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 74 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 75 | C2×C42 | — | 3 | 3 | no | — |
| 76 | C2×C42 | — | 6 | 1 | no | — |
| 77 | C4⋊C8 | — | 6 | 6 | no | — |
| 78 | C4⋊C8 | — | 6 | 6 | no | — |
| 79 | #103 | — | 6 | 6 | no | — |
| 80 | #102 | — | 6 | 12 | no | — |
| 81 | #102 | — | 6 | 6 | no | — |
| 82 | #95 | — | 3 | 3 | no | — |
| 83 | #95 | S4 | 1 | 1 | yes | characteristic |
| 84 | C23×C3 | — | 4 | 1 | no | — |
| 85 | C22×D3 | — | 4 | 6 | no | — |
| 86 | C22×D3 | — | 8 | 3 | no | — |
| 87 | C22×D3 | — | 8 | 6 | no | — |
| 88 | C22×D3 | — | 8 | 6 | no | — |
| 89 | C22×D3 | — | 8 | 3 | no | — |
| 90 | C2×C3×C4 | — | 4 | 6 | no | — |
| 91 | C2×C3×C4 | — | 8 | 6 | no | — |
| 92 | C2×Dic3 | — | 8 | 3 | no | — |
| 93 | D12 | — | 8 | 24 | no | — |
| 94 | SL(2,𝔽3) | #127 | 1 | 1 | yes | characteristic |
| 95 | C24 | — | 6 | 3 | no | — |
| 96 | C2×Q8 | — | 3 | 3 | no | — |
| 97 | C2×Q8 | — | 6 | 3 | no | — |
| 98 | C2×Q8 | — | 6 | 3 | no | — |
| 99 | C2×Q8 | C2×D12 | 1 | 3 | yes | — |
| 100 | C2×D4 | — | 6 | 3 | no | — |
| 101 | C2×D4 | — | 6 | 6 | no | — |
| 102 | C2×D4 | — | 6 | 3 | no | — |
| 103 | C2×D4 | — | 6 | 6 | no | — |
| 104 | C2×D4 | — | 12 | 6 | no | — |
| 105 | C2×D4 | — | 12 | 6 | no | — |
| 106 | C2×D4 | — | 24 | 6 | no | — |
| 107 | C2×D4 | — | 24 | 3 | no | — |
| 108 | C2×D4 | — | 24 | 3 | no | — |
| 109 | C22×C4 | — | 3 | 1 | no | — |
| 110 | C22×C4 | — | 3 | 3 | no | — |
| 111 | C22×C4 | — | 6 | 1 | no | — |
| 112 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 113 | Q16 | — | 12 | 3 | no | — |
| 114 | Q16 | — | 12 | 1 | no | — |
| 115 | SD16 | — | 12 | 3 | no | — |
| 116 | SD16 | — | 12 | 6 | no | — |
| 117 | SD16 | — | 12 | 3 | no | — |
| 118 | C2×C8 | — | 6 | 6 | no | — |
| 119 | C2×C8 | — | 6 | 3 | no | — |
| 120 | C2×C8 | — | 6 | 3 | no | — |
| 121 | C4⋊C4 | — | 3 | 6 | no | — |
| 122 | C4⋊C4 | — | 6 | 3 | no | — |
| 123 | C22⋊C4 | — | 12 | 6 | no | — |
| 124 | C22⋊C4 | — | 12 | 6 | no | — |
| 125 | C42 | — | 3 | 6 | no | — |
| 126 | C42 | — | 3 | 6 | no | — |
| 127 | C42 | — | 24 | 1 | no | — |
| 128 | C22×C3 | — | 4 | 3 | no | — |
| 129 | C22×C3 | — | 4 | 1 | no | — |
| 130 | C22×C3 | — | 4 | 3 | no | — |
| 131 | C2×D3 | — | 8 | 12 | no | — |
| 132 | C2×D3 | — | 8 | 6 | no | — |
| 133 | C2×D3 | — | 16 | 3 | no | — |
| 134 | C2×D3 | — | 16 | 6 | no | — |
| 135 | C2×D3 | — | 16 | 3 | no | — |
| 136 | C3×C4 | — | 8 | 6 | no | — |
| 137 | Dic3 | — | 16 | 1 | no | — |
| 138 | C23 | — | 12 | 3 | no | — |
| 139 | C23 | — | 12 | 6 | no | — |
| 140 | C23 | — | 12 | 3 | no | — |
| 141 | C23 | — | 12 | 3 | no | — |
| 142 | C23 | — | 12 | 6 | no | — |
| 143 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 144 | Q8 | — | 3 | 3 | no | — |
| 145 | Q8 | — | 6 | 1 | no | — |
| 146 | Q8 | — | 12 | 1 | no | — |
| 147 | Q8 | — | 12 | 3 | no | — |
| 148 | Q8 | #874 | 1 | 1 | yes | characteristic |
| 149 | D4 | — | 12 | 3 | no | — |
| 150 | D4 | — | 12 | 3 | no | — |
| 151 | D4 | — | 12 | 6 | no | — |
| 152 | D4 | — | 24 | 12 | no | — |
| 153 | D4 | — | 24 | 12 | no | — |
| 154 | C2×C4 | — | 2 | 6 | no | — |
| 155 | C2×C4 | — | 3 | 3 | no | — |
| 156 | C2×C4 | — | 3 | 3 | no | — |
| 157 | C2×C4 | — | 3 | 6 | no | — |
| 158 | C2×C4 | — | 6 | 3 | no | — |
| 159 | C2×C4 | — | 6 | 3 | no | — |
| 160 | C2×C4 | — | 12 | 3 | no | — |
| 161 | C2×C4 | C4⋊S4 | 1 | 6 | yes | — |
| 162 | C8 | — | 12 | 3 | no | — |
| 163 | C8 | — | 12 | 1 | no | — |
| 164 | C2×C3 | — | 4 | 1 | no | — |
| 165 | C2×C3 | — | 4 | 3 | no | — |
| 166 | C2×C3 | — | 4 | 3 | no | — |
| 167 | D3 | — | 16 | 6 | no | — |
| 168 | C22 | — | 24 | 3 | no | — |
| 169 | C22 | — | 24 | 6 | no | — |
| 170 | C22 | — | 24 | 3 | no | — |
| 171 | C22 | — | 24 | 6 | no | — |
| 172 | C22 | — | 24 | 3 | no | — |
| 173 | C22 | C2×C4⋊S4 | 1 | 3 | yes | — |
| 174 | C22 | #5925 | 1 | 1 | yes | characteristic |
| 175 | C22 | #5926 | 1 | 3 | yes | — |
| 176 | C4 | — | 2 | 6 | no | — |
| 177 | C4 | — | 3 | 1 | no | — |
| 178 | C4 | — | 3 | 3 | no | — |
| 179 | C4 | — | 6 | 6 | no | — |
| 180 | C4 | — | 24 | 1 | no | — |
| 181 | C3 | — | 4 | 1 | no | Sylow |
| 182 | C2 | — | 48 | 3 | no | — |
| 183 | C2 | #85289 | 1 | 1 | yes | characteristic |
| 184 | C2 | #85355 | 1 | 3 | yes | — |
| 185 | C2 | #85368 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C22×S4 |
| commutator, cosocle | C22×SL(2,𝔽3) | C23 |
| Fitting | #2276 | D3 |
| hypercenter | #95 | S4 |
| hypocenter | SL(2,𝔽3) | #127 |
| LCS | G ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ #95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#5395 | C2 | semidirect |
| 2. C2×C4⋊GL(2,𝔽3) | C2 | semidirect |
| 3. #73167 | C2 | semidirect |
| 14. #2276 | D3 | semidirect |
| 24. C4×SL(2,𝔽3) | D4 | semidirect |
| 71. C4×Q8 | D12 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5395 | 2 |
| C2×C4⋊GL(2,𝔽3) | 2 |
| #73167 | 2 |
| #33167 | 3 |
| #4671 | 4 |