Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
184 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 17 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 | #72986 | C2 | 1 | 2 | yes | maximal |
| 2 | #1726⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #9763 | — | 3 | 1 | no | maximal, Sylow |
| 4 | #5398 | — | 4 | 1 | no | maximal |
| 5 | #376⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | #4621 | — | 2 | 2 | no | — |
| 7 | #3170 | — | 3 | 1 | no | — |
| 8 | #1726 | D3 | 1 | 1 | yes | characteristic |
| 9 | #1568 | — | 3 | 1 | no | — |
| 10 | #1531 | — | 3 | 2 | no | — |
| 11 | #1506 | — | 3 | 2 | no | — |
| 12 | A4⋊Q8 | — | 4 | 2 | no | — |
| 13 | C8×A4 | — | 4 | 1 | no | — |
| 14 | #95⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 15 | #533 | — | 3 | 2 | no | — |
| 16 | #500 | — | 3 | 1 | no | — |
| 17 | #494 | — | 3 | 1 | no | — |
| 18 | #487 | — | 3 | 2 | no | — |
| 19 | #467 | — | 6 | 1 | no | — |
| 20 | #467 | — | 6 | 1 | no | — |
| 21 | C4×Q16 | — | 6 | 1 | no | — |
| 22 | C4×Q16 | — | 6 | 1 | no | — |
| 23 | #432 | — | 3 | 2 | no | — |
| 24 | C2×#97 | — | 3 | 1 | no | — |
| 25 | C2×C4×C8 | — | 3 | 1 | no | — |
| 26 | #376 | C2×D3 | 1 | 1 | yes | characteristic |
| 27 | #335 | — | 6 | 1 | no | — |
| 28 | #334 | — | 6 | 1 | no | — |
| 29 | #330 | — | 6 | 4 | no | — |
| 30 | #328 | — | 6 | 2 | no | — |
| 31 | #328 | — | 6 | 2 | no | — |
| 32 | C4×A4 | — | 4 | 1 | no | — |
| 33 | SL(2,ℤ/4) | — | 8 | 2 | no | — |
| 34 | Dic12 | — | 16 | 1 | no | — |
| 35 | C2×Q16 | — | 6 | 1 | no | — |
| 36 | C2×Q16 | — | 6 | 1 | no | — |
| 37 | C22×C8 | — | 3 | 1 | no | — |
| 38 | C22×C8 | S4 | 1 | 1 | yes | characteristic |
| 39 | C4⋊Q8 | — | 3 | 2 | no | — |
| 40 | C4⋊Q8 | — | 3 | 2 | no | — |
| 41 | #125 | — | 6 | 2 | no | — |
| 42 | C22⋊Q8 | — | 6 | 2 | no | — |
| 43 | C22⋊Q8 | — | 6 | 2 | no | — |
| 44 | C4×Q8 | — | 6 | 2 | no | — |
| 45 | C4×Q8 | — | 6 | 2 | no | — |
| 46 | #117 | — | 6 | 2 | no | — |
| 47 | C2×C42 | — | 3 | 1 | no | — |
| 48 | #107 | — | 6 | 1 | no | — |
| 49 | #107 | — | 6 | 1 | no | — |
| 50 | C4⋊C8 | — | 6 | 2 | no | — |
| 51 | #103 | — | 6 | 2 | no | — |
| 52 | #103 | — | 6 | 2 | no | — |
| 53 | C22⋊C8 | — | 6 | 2 | no | — |
| 54 | #97 | — | 3 | 2 | no | — |
| 55 | #97 | — | 6 | 1 | no | — |
| 56 | #97 | — | 6 | 2 | no | — |
| 57 | C4×C8 | — | 3 | 1 | no | — |
| 58 | C4×C8 | — | 3 | 1 | no | — |
| 59 | C4×C8 | — | 3 | 1 | no | — |
| 60 | C4×C8 | — | 3 | 1 | no | — |
| 61 | #95 | — | 3 | 1 | no | — |
| 62 | #95 | D12 | 1 | 1 | yes | characteristic |
| 63 | C2×A4 | — | 4 | 1 | no | — |
| 64 | Dic6 | — | 16 | 2 | no | — |
| 65 | C3×C8 | — | 16 | 1 | no | — |
| 66 | C2×Q8 | — | 6 | 2 | no | — |
| 67 | C2×Q8 | — | 6 | 2 | no | — |
| 68 | C22×C4 | — | 3 | 1 | no | — |
| 69 | C22×C4 | — | 3 | 1 | no | — |
| 70 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 71 | Q16 | — | 12 | 2 | no | — |
| 72 | Q16 | — | 12 | 1 | no | — |
| 73 | Q16 | — | 12 | 1 | no | — |
| 74 | C2×C8 | — | 3 | 1 | no | — |
| 75 | C2×C8 | — | 3 | 1 | no | — |
| 76 | C2×C8 | — | 3 | 1 | no | — |
| 77 | C2×C8 | — | 3 | 1 | no | — |
| 78 | C2×C8 | — | 6 | 2 | no | — |
| 79 | C2×C8 | — | 12 | 2 | no | — |
| 80 | C4⋊C4 | — | 6 | 2 | no | — |
| 81 | C4⋊C4 | — | 6 | 2 | no | — |
| 82 | C4⋊C4 | — | 6 | 2 | no | — |
| 83 | C4⋊C4 | — | 6 | 2 | no | — |
| 84 | C4⋊C4 | — | 12 | 2 | no | — |
| 85 | C4⋊C4 | — | 12 | 2 | no | — |
| 86 | C22⋊C4 | — | 12 | 2 | no | — |
| 87 | C42 | — | 3 | 1 | no | — |
| 88 | C42 | — | 3 | 1 | no | — |
| 89 | C42 | — | 3 | 1 | no | — |
| 90 | C42 | — | 3 | 1 | no | — |
| 91 | C42 | — | 12 | 2 | no | — |
| 92 | A4 | — | 4 | 1 | no | — |
| 93 | C3×C4 | — | 16 | 1 | no | — |
| 94 | Dic3 | — | 32 | 2 | no | — |
| 95 | C23 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 96 | Q8 | — | 12 | 2 | no | — |
| 97 | Q8 | — | 12 | 2 | no | — |
| 98 | Q8 | — | 24 | 2 | no | — |
| 99 | C2×C4 | — | 3 | 1 | no | — |
| 100 | C2×C4 | — | 3 | 1 | no | — |
| 101 | C2×C4 | — | 3 | 1 | no | — |
| 102 | C2×C4 | — | 3 | 1 | no | — |
| 103 | C2×C4 | — | 3 | 1 | no | — |
| 104 | C2×C4 | — | 3 | 1 | no | — |
| 105 | C2×C4 | — | 6 | 1 | no | — |
| 106 | C2×C4 | — | 6 | 1 | no | — |
| 107 | C2×C4 | — | 12 | 1 | no | — |
| 108 | C2×C4 | — | 12 | 2 | no | — |
| 109 | C2×C4 | — | 12 | 2 | no | — |
| 110 | C8 | — | 3 | 2 | no | — |
| 111 | C8 | — | 3 | 1 | no | — |
| 112 | C8 | — | 6 | 1 | no | — |
| 113 | C8 | — | 24 | 2 | no | — |
| 114 | C8 | #857 | 1 | 1 | yes | characteristic |
| 115 | C2×C3 | — | 16 | 1 | no | — |
| 116 | C22 | — | 3 | 1 | no | — |
| 117 | C22 | — | 3 | 1 | no | — |
| 118 | C22 | #5403 | 1 | 1 | yes | characteristic |
| 119 | C4 | — | 3 | 1 | no | — |
| 120 | C4 | — | 6 | 1 | no | — |
| 121 | C4 | — | 6 | 1 | no | — |
| 122 | C4 | — | 6 | 1 | no | — |
| 123 | C4 | — | 6 | 1 | no | — |
| 124 | C4 | — | 24 | 2 | no | — |
| 125 | C4 | — | 24 | 2 | no | — |
| 126 | C4 | C2×#857 | 1 | 1 | yes | characteristic |
| 127 | C3 | — | 16 | 1 | no | Sylow |
| 128 | C2 | — | 3 | 1 | no | — |
| 129 | C2 | — | 3 | 1 | no | — |
| 130 | C2 | C4⋊#857 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊#857 |
| commutator, cosocle | #376⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #1726 | D3 |
| socle | C23 | C4⋊S4 |
| prime radical, hypocenter | #95⋊C3 | D4 |
| hypercenter | C8 | #857 |
| LCS | G ▷ #376⋊C3 ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72986 | 2 |
| #1726⋊C3 | 2 |
| #9763 | 3 |
| #5398 | 4 |