Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
100 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 13 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 | C32⋊(C42⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊#857 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#805 | — | 3 | 1 | no | maximal |
| 4 | (C3⋊S4)⋊C3 | — | 4 | 1 | no | maximal |
| 5 | C3×#235 | — | 3 | 1 | no | — |
| 6 | C32×C42 | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3×#208 | — | 3 | 1 | no | — |
| 8 | C3×C42⋊C3 | — | 2 | 1 | no | — |
| 9 | C3×C42⋊C3 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22⋊H(3,𝔽3) | — | 4 | 1 | no | — |
| 11 | #857 | — | 9 | 1 | no | — |
| 12 | C3×#104 | — | 9 | 1 | no | — |
| 13 | #805 | — | 3 | 1 | no | — |
| 14 | C3⋊S4 | — | 4 | 1 | no | — |
| 15 | C2×C32×C4 | — | 3 | 1 | no | — |
| 16 | C3×#67 | — | 6 | 1 | no | — |
| 17 | C3×D12 | — | 3 | 1 | no | — |
| 18 | C3×C4×D3 | — | 6 | 1 | no | — |
| 19 | C3×Dic6 | — | 3 | 1 | no | — |
| 20 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 21 | (C3⋊D3)⋊C3 | — | 16 | 1 | no | — |
| 22 | C3×#41 | — | 9 | 1 | no | — |
| 23 | #235 | — | 3 | 1 | no | — |
| 24 | C3×M16 | — | 9 | 1 | no | — |
| 25 | C3×C42 | — | 3 | 1 | no | — |
| 26 | C3×C42 | C3×D3 | 1 | 1 | yes | characteristic |
| 27 | #208 | — | 3 | 1 | no | — |
| 28 | C42⋊C3 | — | 3 | 1 | no | — |
| 29 | C42⋊C3 | — | 6 | 1 | no | — |
| 30 | C22×C32 | S4 | 1 | 1 | yes | characteristic |
| 31 | C2×C3×D3 | — | 6 | 1 | no | — |
| 32 | C3×A4 | — | 4 | 1 | no | — |
| 33 | C3×A4 | — | 8 | 1 | no | — |
| 34 | C32×C4 | — | 3 | 1 | no | — |
| 35 | C32×C4 | — | 3 | 1 | no | — |
| 36 | C3×Dic3 | — | 6 | 1 | no | — |
| 37 | #104 | — | 9 | 1 | no | Sylow |
| 38 | H(3,𝔽3) | — | 16 | 1 | no | Sylow |
| 39 | S4 | — | 36 | 1 | no | — |
| 40 | C3×Q8 | — | 9 | 1 | no | — |
| 41 | C3×D4 | — | 9 | 1 | no | — |
| 42 | C3×D4 | — | 18 | 1 | no | — |
| 43 | C2×C3×C4 | — | 3 | 1 | no | — |
| 44 | C2×C3×C4 | — | 3 | 1 | no | — |
| 45 | C2×C3×C4 | — | 6 | 1 | no | — |
| 46 | C2×C3×C4 | — | 18 | 1 | no | — |
| 47 | #67 | — | 6 | 1 | no | — |
| 48 | D12 | — | 3 | 1 | no | — |
| 49 | C4×D3 | — | 6 | 1 | no | — |
| 50 | Dic6 | — | 3 | 1 | no | — |
| 51 | C3×C8 | — | 18 | 1 | no | — |
| 52 | C3⋊C8 | — | 6 | 1 | no | — |
| 53 | C2×C32 | — | 3 | 1 | no | — |
| 54 | C3⋊D3 | — | 16 | 1 | no | — |
| 55 | C3×D3 | — | 12 | 1 | no | — |
| 56 | #41 | — | 9 | 1 | no | — |
| 57 | M16 | — | 9 | 1 | no | — |
| 58 | C42 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 59 | C22×C3 | — | 3 | 1 | no | — |
| 60 | C22×C3 | — | 18 | 1 | no | — |
| 61 | C22×C3 | C3×S4 | 1 | 1 | yes | characteristic |
| 62 | C2×D3 | — | 6 | 1 | no | — |
| 63 | A4 | — | 12 | 1 | no | — |
| 64 | A4 | — | 24 | 1 | no | — |
| 65 | C3×C4 | — | 3 | 1 | no | — |
| 66 | C3×C4 | — | 3 | 1 | no | — |
| 67 | C3×C4 | — | 3 | 1 | no | — |
| 68 | C3×C4 | — | 3 | 1 | no | — |
| 69 | C3×C4 | — | 6 | 1 | no | — |
| 70 | C3×C4 | — | 6 | 1 | no | — |
| 71 | C3×C4 | — | 18 | 1 | no | — |
| 72 | Dic3 | — | 6 | 1 | no | — |
| 73 | C32 | — | 16 | 1 | no | — |
| 74 | C32 | — | 32 | 1 | no | — |
| 75 | C32 | #857 | 1 | 1 | yes | characteristic, complemented |
| 76 | Q8 | — | 9 | 1 | no | — |
| 77 | D4 | — | 9 | 1 | no | — |
| 78 | D4 | — | 18 | 1 | no | — |
| 79 | C2×C4 | — | 3 | 1 | no | — |
| 80 | C2×C4 | — | 18 | 1 | no | — |
| 81 | C8 | — | 18 | 1 | no | — |
| 82 | C2×C3 | — | 3 | 1 | no | — |
| 83 | C2×C3 | — | 3 | 1 | no | — |
| 84 | C2×C3 | — | 6 | 1 | no | — |
| 85 | C2×C3 | — | 36 | 1 | no | — |
| 86 | D3 | — | 12 | 1 | no | — |
| 87 | D3 | — | 144 | 1 | no | — |
| 88 | C22 | — | 18 | 1 | no | — |
| 89 | C22 | (C3⋊S4)⋊C3 | 1 | 1 | yes | characteristic |
| 90 | C4 | — | 3 | 1 | no | — |
| 91 | C4 | — | 3 | 1 | no | — |
| 92 | C4 | — | 18 | 1 | no | — |
| 93 | C3 | — | 3 | 1 | no | — |
| 94 | C3 | — | 48 | 1 | no | — |
| 95 | C3 | — | 96 | 1 | no | — |
| 96 | C3 | C3×#857 | 1 | 1 | yes | characteristic |
| 97 | C2 | — | 3 | 1 | no | — |
| 98 | C2 | — | 36 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C42⋊C3 | C2×C3 |
| Frattini, socle | C22×C3 | C3×S4 |
| Fitting | C32×C42 | D3 |
| LCS | G ▷ C3×C42⋊C3 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32⋊(C42⋊C3) | C2 | semidirect |
| 2. C3⋊#857 | C3 | semidirect |
| 6. C32×C42 | D3 | semidirect |
| 9. C3×C42⋊C3 | C2×C3 | semidirect |
| 58. C42 | (C3⋊D3)⋊C3 | semidirect |
| 75. C32 | #857 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32⋊(C42⋊C3) | 2 |
| C3⋊#857 | 3 |
| C3×#805 | 3 |
| (C3⋊S4)⋊C3 | 4 |