Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
246 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 19 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 | C4⋊#857 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C8×C42⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #9844 | — | 3 | 1 | no | maximal, Sylow |
| 4 | #5402 | — | 4 | 1 | no | maximal |
| 5 | C2×#857 | — | 2 | 2 | no | — |
| 6 | C4×C42⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | #3169 | — | 3 | 1 | no | — |
| 8 | #1962 | — | 3 | 2 | no | — |
| 9 | C42×C8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #1567 | — | 3 | 1 | no | — |
| 11 | #1492 | — | 3 | 2 | no | — |
| 12 | C4⋊S4 | — | 4 | 2 | no | — |
| 13 | C8×A4 | — | 4 | 1 | no | — |
| 14 | C2×C42⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 15 | #857 | — | 4 | 2 | no | — |
| 16 | #532 | — | 3 | 2 | no | — |
| 17 | #500 | — | 3 | 1 | no | — |
| 18 | #493 | — | 3 | 1 | no | — |
| 19 | #486 | — | 3 | 2 | no | — |
| 20 | #466 | — | 6 | 1 | no | — |
| 21 | #466 | — | 6 | 1 | no | — |
| 22 | C4×D8 | — | 6 | 1 | no | — |
| 23 | C4×D8 | — | 6 | 1 | no | — |
| 24 | #423 | — | 3 | 2 | no | — |
| 25 | C2×#104 | — | 6 | 2 | no | — |
| 26 | C2×#104 | — | 6 | 2 | no | — |
| 27 | C2×C4×C8 | — | 3 | 1 | no | — |
| 28 | C2×C4×C8 | — | 3 | 1 | no | — |
| 29 | C43 | C2×D3 | 1 | 1 | yes | characteristic |
| 30 | #335 | — | 6 | 1 | no | — |
| 31 | #335 | — | 6 | 1 | no | — |
| 32 | D4⋊C8 | — | 6 | 4 | no | — |
| 33 | C2×S4 | — | 8 | 2 | no | — |
| 34 | C4×A4 | — | 4 | 1 | no | — |
| 35 | D24 | — | 16 | 1 | no | — |
| 36 | C42⋊C3 | D8 | 1 | 1 | yes | characteristic, complemented |
| 37 | C2×#41 | — | 6 | 2 | no | — |
| 38 | C2×D8 | — | 6 | 1 | no | — |
| 39 | C2×D8 | — | 6 | 1 | no | — |
| 40 | C2×M16 | — | 6 | 2 | no | — |
| 41 | C22×C8 | — | 3 | 1 | no | — |
| 42 | C22×C8 | S4 | 1 | 1 | yes | characteristic |
| 43 | C4⋊Q8 | — | 3 | 2 | no | — |
| 44 | #127 | — | 3 | 2 | no | — |
| 45 | #124 | — | 6 | 2 | no | — |
| 46 | #121 | — | 6 | 2 | no | — |
| 47 | #121 | — | 6 | 2 | no | — |
| 48 | C4×D4 | — | 6 | 2 | no | — |
| 49 | C4×D4 | — | 6 | 2 | no | — |
| 50 | C2×C42 | — | 3 | 1 | no | — |
| 51 | C2×C42 | — | 3 | 1 | no | — |
| 52 | C2×C42 | D12 | 1 | 1 | yes | characteristic |
| 53 | #107 | — | 6 | 1 | no | — |
| 54 | #107 | — | 6 | 1 | no | — |
| 55 | C4⋊C8 | — | 6 | 2 | no | — |
| 56 | C4⋊C8 | — | 6 | 2 | no | — |
| 57 | #104 | — | 12 | 2 | no | — |
| 58 | #104 | — | 12 | 4 | no | — |
| 59 | #104 | — | 12 | 2 | no | — |
| 60 | #102 | — | 6 | 2 | no | — |
| 61 | #102 | — | 6 | 2 | no | — |
| 62 | C4×C8 | — | 3 | 1 | no | — |
| 63 | C4×C8 | — | 3 | 1 | no | — |
| 64 | C4×C8 | — | 3 | 1 | no | — |
| 65 | C4×C8 | — | 3 | 1 | no | — |
| 66 | C4×C8 | — | 3 | 2 | no | — |
| 67 | C4×C8 | — | 6 | 1 | no | — |
| 68 | C2×A4 | — | 4 | 1 | no | — |
| 69 | S4 | — | 16 | 2 | no | — |
| 70 | D12 | — | 16 | 2 | no | — |
| 71 | C3×C8 | — | 16 | 1 | no | — |
| 72 | #41 | — | 12 | 2 | no | — |
| 73 | #41 | — | 12 | 2 | no | — |
| 74 | #41 | — | 12 | 2 | no | — |
| 75 | #41 | — | 12 | 2 | no | — |
| 76 | C2×Q8 | — | 6 | 2 | no | — |
| 77 | C2×D4 | — | 6 | 2 | no | — |
| 78 | C2×D4 | — | 6 | 2 | no | — |
| 79 | C2×D4 | — | 6 | 2 | no | — |
| 80 | C2×D4 | — | 12 | 2 | no | — |
| 81 | C22×C4 | — | 3 | 1 | no | — |
| 82 | C22×C4 | — | 3 | 1 | no | — |
| 83 | C22×C4 | — | 12 | 2 | no | — |
| 84 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 85 | D8 | — | 12 | 2 | no | — |
| 86 | D8 | — | 12 | 1 | no | — |
| 87 | D8 | — | 12 | 1 | no | — |
| 88 | M16 | — | 12 | 2 | no | — |
| 89 | M16 | — | 12 | 2 | no | — |
| 90 | C2×C8 | — | 3 | 1 | no | — |
| 91 | C2×C8 | — | 3 | 1 | no | — |
| 92 | C2×C8 | — | 3 | 1 | no | — |
| 93 | C2×C8 | — | 3 | 1 | no | — |
| 94 | C2×C8 | — | 6 | 2 | no | — |
| 95 | C2×C8 | — | 12 | 2 | no | — |
| 96 | C4⋊C4 | — | 6 | 2 | no | — |
| 97 | C4⋊C4 | — | 6 | 2 | no | — |
| 98 | C22⋊C4 | — | 12 | 2 | no | — |
| 99 | C22⋊C4 | — | 12 | 2 | no | — |
| 100 | C42 | — | 3 | 1 | no | — |
| 101 | C42 | — | 3 | 1 | no | — |
| 102 | C42 | — | 3 | 1 | no | — |
| 103 | C42 | — | 3 | 1 | no | — |
| 104 | C42 | — | 3 | 1 | no | — |
| 105 | C42 | — | 3 | 2 | no | — |
| 106 | C42 | — | 6 | 1 | no | — |
| 107 | C42 | D24 | 1 | 1 | yes | characteristic, complemented |
| 108 | C2×D3 | — | 32 | 2 | no | — |
| 109 | A4 | — | 4 | 1 | no | — |
| 110 | C3×C4 | — | 16 | 1 | no | — |
| 111 | C23 | — | 12 | 2 | no | — |
| 112 | C23 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 113 | Q8 | — | 12 | 2 | no | — |
| 114 | Q8 | — | 12 | 2 | no | — |
| 115 | D4 | — | 12 | 2 | no | — |
| 116 | D4 | — | 12 | 4 | no | — |
| 117 | D4 | — | 12 | 2 | no | — |
| 118 | D4 | — | 12 | 2 | no | — |
| 119 | D4 | — | 12 | 2 | no | — |
| 120 | D4 | — | 24 | 2 | no | — |
| 121 | D4 | — | 24 | 2 | no | — |
| 122 | C2×C4 | — | 3 | 1 | no | — |
| 123 | C2×C4 | — | 3 | 1 | no | — |
| 124 | C2×C4 | — | 3 | 1 | no | — |
| 125 | C2×C4 | — | 3 | 1 | no | — |
| 126 | C2×C4 | — | 3 | 1 | no | — |
| 127 | C2×C4 | — | 3 | 1 | no | — |
| 128 | C2×C4 | — | 3 | 1 | no | — |
| 129 | C2×C4 | — | 3 | 1 | no | — |
| 130 | C2×C4 | — | 3 | 1 | no | — |
| 131 | C2×C4 | — | 3 | 1 | no | — |
| 132 | C2×C4 | — | 6 | 2 | no | — |
| 133 | C2×C4 | — | 12 | 2 | no | — |
| 134 | C2×C4 | — | 24 | 2 | no | — |
| 135 | C2×C4 | — | 24 | 2 | no | — |
| 136 | C8 | — | 3 | 2 | no | — |
| 137 | C8 | — | 3 | 1 | no | — |
| 138 | C8 | — | 6 | 1 | no | — |
| 139 | C8 | — | 24 | 2 | no | — |
| 140 | C8 | #857 | 1 | 1 | yes | characteristic, complemented |
| 141 | C2×C3 | — | 16 | 1 | no | — |
| 142 | D3 | — | 64 | 2 | no | — |
| 143 | C22 | — | 3 | 1 | no | — |
| 144 | C22 | — | 3 | 1 | no | — |
| 145 | C22 | — | 24 | 2 | no | — |
| 146 | C22 | — | 24 | 2 | no | — |
| 147 | C22 | — | 24 | 2 | no | — |
| 148 | C22 | #5402 | 1 | 1 | yes | characteristic |
| 149 | C4 | — | 3 | 1 | no | — |
| 150 | C4 | — | 3 | 1 | no | — |
| 151 | C4 | — | 3 | 1 | no | — |
| 152 | C4 | — | 3 | 1 | no | — |
| 153 | C4 | — | 3 | 2 | no | — |
| 154 | C4 | — | 3 | 1 | no | — |
| 155 | C4 | — | 6 | 1 | no | — |
| 156 | C4 | — | 24 | 2 | no | — |
| 157 | C4 | C2×#857 | 1 | 1 | yes | characteristic |
| 158 | C3 | — | 16 | 1 | no | Sylow |
| 159 | C2 | — | 3 | 1 | no | — |
| 160 | C2 | — | 3 | 1 | no | — |
| 161 | C2 | — | 48 | 2 | no | — |
| 162 | C2 | C4⋊#857 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊#857 |
| commutator, cosocle | C4×C42⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | C42×C8 | D3 |
| socle | C23 | C4⋊S4 |
| hypercenter | C8 | #857 |
| hypocenter | C42⋊C3 | D8 |
| LCS | G ▷ C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4⋊#857 | C2 | semidirect |
| 2. C8×C42⋊C3 | C2 | semidirect |
| 9. C42×C8 | D3 | semidirect |
| 36. C42⋊C3 | D8 | semidirect |
| 107. C42 | D24 | semidirect |
| 140. C8 | #857 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊#857 | 2 |
| C8×C42⋊C3 | 2 |
| #9844 | 3 |
| #5402 | 4 |