Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
369 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 63 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 | #85337 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×#131⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #73126 | C2 | 1 | 4 | yes | maximal |
| 4 | #33222 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C2×#131⋊C3 | C22 | 1 | 2 | yes | — |
| 6 | C4×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | #5426 | C22 | 1 | 4 | yes | — |
| 8 | #5391 | — | 2 | 2 | no | — |
| 9 | #5386 | — | 2 | 2 | no | — |
| 10 | #4682 | — | 4 | 1 | no | maximal |
| 11 | C2×#500 | — | 3 | 1 | no | — |
| 12 | #3079 | — | 3 | 2 | no | — |
| 13 | C4×#131 | D3 | 1 | 1 | yes | characteristic |
| 14 | #1997 | — | 3 | 2 | no | — |
| 15 | #1952 | — | 3 | 4 | no | — |
| 16 | #1848 | — | 3 | 1 | no | — |
| 17 | #1816 | — | 3 | 4 | no | — |
| 18 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 19 | #869 | — | 4 | 2 | no | — |
| 20 | #131⋊C3 | Q8 | 1 | 4 | yes | complemented |
| 21 | C4×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 22 | C4×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 23 | #859 | — | 2 | 4 | no | — |
| 24 | C3×C4×C8 | — | 4 | 1 | no | — |
| 25 | #818 | — | 4 | 4 | no | — |
| 26 | C2×#131 | C2×D3 | 1 | 2 | yes | — |
| 27 | C2×C4⋊Q8 | — | 3 | 2 | no | — |
| 28 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | #500 | — | 6 | 2 | no | — |
| 30 | #500 | — | 6 | 1 | no | — |
| 31 | #500 | — | 6 | 1 | no | — |
| 32 | #477 | — | 6 | 2 | no | — |
| 33 | #476 | — | 6 | 2 | no | — |
| 34 | #475 | — | 6 | 2 | no | — |
| 35 | #474 | — | 6 | 2 | no | — |
| 36 | C2×#108 | — | 3 | 4 | no | — |
| 37 | #428 | — | 3 | 4 | no | — |
| 38 | C2×#107 | — | 3 | 4 | no | — |
| 39 | #423 | — | 3 | 2 | no | — |
| 40 | #417 | — | 3 | 4 | no | — |
| 41 | #405 | — | 3 | 2 | no | — |
| 42 | C4×M16 | — | 3 | 1 | no | — |
| 43 | C2×C4×C8 | — | 3 | 1 | no | — |
| 44 | #343 | — | 3 | 8 | no | — |
| 45 | C2×Dic6 | — | 8 | 2 | no | — |
| 46 | #41⋊C3 | C2×Q8 | 1 | 2 | yes | — |
| 47 | C2×SL(2,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | C2×C3×C8 | — | 4 | 2 | no | — |
| 49 | C3×C42 | — | 4 | 1 | no | — |
| 50 | C4⋊Dic3 | — | 4 | 4 | no | — |
| 51 | C4⋊Dic3 | — | 8 | 2 | no | — |
| 52 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 53 | C22×Q8 | — | 6 | 2 | no | — |
| 54 | #131 | — | 3 | 4 | no | — |
| 55 | #131 | Dic6 | 1 | 4 | yes | complemented |
| 56 | C2×M16 | — | 3 | 2 | no | — |
| 57 | C2×M16 | — | 3 | 4 | no | — |
| 58 | C22×C8 | — | 3 | 2 | no | — |
| 59 | C4⋊Q8 | — | 6 | 2 | no | — |
| 60 | C4⋊Q8 | — | 6 | 2 | no | — |
| 61 | C4⋊Q8 | — | 6 | 2 | no | — |
| 62 | C4⋊Q8 | — | 6 | 2 | no | — |
| 63 | C4×Q8 | D12 | 1 | 1 | yes | characteristic |
| 64 | C4×Q8 | D12 | 1 | 1 | yes | characteristic |
| 65 | C4×D4 | — | 3 | 1 | no | — |
| 66 | C4×D4 | — | 3 | 1 | no | — |
| 67 | #117 | — | 3 | 1 | no | — |
| 68 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 69 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 70 | C2×C42 | — | 3 | 1 | no | — |
| 71 | #108 | — | 6 | 8 | no | — |
| 72 | #107 | — | 6 | 4 | no | — |
| 73 | #107 | — | 6 | 4 | no | — |
| 74 | #107 | — | 6 | 4 | no | — |
| 75 | #106 | — | 6 | 4 | no | — |
| 76 | C4⋊C8 | — | 6 | 2 | no | — |
| 77 | C4⋊C8 | — | 6 | 2 | no | — |
| 78 | #103 | — | 6 | 4 | no | — |
| 79 | #102 | — | 6 | 4 | no | — |
| 80 | #97 | — | 6 | 1 | no | — |
| 81 | C4×C8 | — | 3 | 2 | no | — |
| 82 | C4×C8 | — | 3 | 1 | no | — |
| 83 | C4×C8 | S4 | 1 | 1 | yes | characteristic |
| 84 | C2×C3×C4 | — | 4 | 1 | no | — |
| 85 | C2×C3×C4 | — | 4 | 1 | no | — |
| 86 | C2×C3×C4 | — | 4 | 1 | no | — |
| 87 | C2×Dic3 | — | 8 | 4 | no | — |
| 88 | Dic6 | — | 16 | 4 | no | — |
| 89 | SL(2,𝔽3) | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 90 | C3×C8 | — | 4 | 4 | no | — |
| 91 | #41 | — | 3 | 2 | no | — |
| 92 | #41 | C2×Dic6 | 1 | 2 | yes | — |
| 93 | C2×Q8 | — | 12 | 2 | no | — |
| 94 | C2×Q8 | — | 12 | 2 | no | — |
| 95 | C2×Q8 | — | 24 | 4 | no | — |
| 96 | C2×Q8 | C2×D12 | 1 | 1 | yes | characteristic |
| 97 | C2×D4 | — | 3 | 1 | no | — |
| 98 | C22×C4 | — | 3 | 1 | no | — |
| 99 | C22×C4 | — | 3 | 1 | no | — |
| 100 | C22×C4 | — | 3 | 1 | no | — |
| 101 | C22×C4 | — | 6 | 4 | no | — |
| 102 | M16 | — | 3 | 4 | no | — |
| 103 | M16 | — | 6 | 2 | no | — |
| 104 | M16 | — | 6 | 8 | no | — |
| 105 | C2×C8 | — | 3 | 2 | no | — |
| 106 | C2×C8 | — | 3 | 4 | no | — |
| 107 | C2×C8 | — | 3 | 4 | no | — |
| 108 | C2×C8 | — | 6 | 4 | no | — |
| 109 | C2×C8 | C2×S4 | 1 | 2 | yes | — |
| 110 | C4⋊C4 | — | 3 | 1 | no | — |
| 111 | C4⋊C4 | — | 3 | 1 | no | — |
| 112 | C4⋊C4 | — | 6 | 4 | no | — |
| 113 | C4⋊C4 | — | 6 | 4 | no | — |
| 114 | C4⋊C4 | — | 12 | 2 | no | — |
| 115 | C4⋊C4 | — | 12 | 2 | no | — |
| 116 | C22⋊C4 | — | 6 | 1 | no | — |
| 117 | C42 | — | 3 | 1 | no | — |
| 118 | C42 | — | 3 | 1 | no | — |
| 119 | C42 | — | 3 | 1 | no | — |
| 120 | C42 | C2×S4 | 1 | 1 | yes | characteristic |
| 121 | C22×C3 | — | 4 | 1 | no | — |
| 122 | C3×C4 | — | 4 | 2 | no | — |
| 123 | C3×C4 | — | 4 | 2 | no | — |
| 124 | C3×C4 | — | 4 | 2 | no | — |
| 125 | Dic3 | — | 16 | 4 | no | — |
| 126 | C23 | — | 3 | 1 | no | — |
| 127 | Q8 | — | 3 | 1 | no | — |
| 128 | Q8 | — | 24 | 4 | no | — |
| 129 | Q8 | — | 24 | 4 | no | — |
| 130 | Q8 | #869 | 1 | 1 | yes | characteristic |
| 131 | D4 | — | 3 | 2 | no | — |
| 132 | D4 | — | 6 | 1 | no | — |
| 133 | C2×C4 | — | 3 | 1 | no | — |
| 134 | C2×C4 | — | 3 | 2 | no | — |
| 135 | C2×C4 | — | 3 | 2 | no | — |
| 136 | C2×C4 | — | 3 | 1 | no | — |
| 137 | C2×C4 | — | 3 | 1 | no | — |
| 138 | C2×C4 | — | 6 | 2 | no | — |
| 139 | C2×C4 | — | 6 | 2 | no | — |
| 140 | C2×C4 | — | 12 | 4 | no | — |
| 141 | C2×C4 | — | 24 | 4 | no | — |
| 142 | C2×C4 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 143 | C2×C4 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 144 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 145 | C8 | — | 3 | 4 | no | — |
| 146 | C8 | — | 12 | 4 | no | — |
| 147 | C8 | A4⋊Q8 | 1 | 4 | yes | — |
| 148 | C2×C3 | — | 4 | 1 | no | — |
| 149 | C2×C3 | — | 4 | 1 | no | — |
| 150 | C2×C3 | — | 4 | 1 | no | — |
| 151 | C22 | — | 3 | 2 | no | — |
| 152 | C22 | — | 6 | 1 | no | — |
| 153 | C22 | — | 6 | 1 | no | — |
| 154 | C22 | C2×C4⋊S4 | 1 | 1 | yes | characteristic |
| 155 | C4 | — | 3 | 1 | no | — |
| 156 | C4 | — | 3 | 1 | no | — |
| 157 | C4 | — | 6 | 1 | no | — |
| 158 | C4 | — | 6 | 1 | no | — |
| 159 | C4 | — | 24 | 4 | no | — |
| 160 | C4 | #5403 | 1 | 2 | yes | — |
| 161 | C4 | #5407 | 1 | 2 | yes | — |
| 162 | C4 | C2×A4⋊Q8 | 1 | 2 | yes | — |
| 163 | C3 | — | 4 | 1 | no | Sylow |
| 164 | C2 | — | 6 | 2 | no | — |
| 165 | C2 | #85283 | 1 | 1 | yes | characteristic |
| 166 | C2 | C2×#5403 | 1 | 1 | yes | characteristic |
| 167 | C2 | C2×#5407 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 20. #131⋊C3 | Q8 | semidirect |
| 55. #131 | Dic6 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #85337 | 2 |
| C4×#131⋊C3 | 2 |
| #73126 | 2 |
| #33222 | 3 |
| #4682 | 4 |