Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
200 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 26 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 | C3⋊D3×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×C3⋊#226 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | D3×#226 | — | 3 | 2 | no | maximal |
| 4 | #63944 | — | 3 | 1 | no | maximal |
| 5 | C32×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 6 | #6294 | — | 4 | 1 | no | maximal |
| 7 | Q8×C3⋊D3 | D3 | 1 | 1 | yes | characteristic |
| 8 | #3797 | — | 3 | 1 | no | — |
| 9 | D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 10 | D3×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 11 | C3⋊#226 | D3 | 1 | 2 | yes | complemented |
| 12 | C3×#226 | — | 3 | 2 | no | — |
| 13 | C3×#226 | — | 6 | 2 | no | — |
| 14 | C3×#216 | — | 3 | 2 | no | — |
| 15 | #3719 | — | 3 | 2 | no | — |
| 16 | #3710 | — | 3 | 1 | no | — |
| 17 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 18 | C3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 19 | C2×#226 | — | 9 | 1 | no | — |
| 20 | D3×Q16 | — | 9 | 2 | no | — |
| 21 | C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 22 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 23 | #634 | — | 3 | 1 | no | — |
| 24 | C3×Dic6 | — | 3 | 2 | no | — |
| 25 | C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 26 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 27 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 28 | C3×SL(2,𝔽3) | C2×D3 | 1 | 2 | yes | — |
| 29 | #627 | — | 6 | 1 | no | — |
| 30 | #624 | — | 6 | 1 | no | — |
| 31 | D3×Dic3 | — | 12 | 2 | no | — |
| 32 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 33 | C2×C33 | — | 4 | 1 | no | — |
| 34 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 35 | D3×Q8 | — | 3 | 2 | no | — |
| 36 | D3×Q8 | — | 6 | 1 | no | — |
| 37 | D3×Q8 | — | 9 | 2 | no | — |
| 38 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 39 | #226 | — | 3 | 2 | no | — |
| 40 | #226 | — | 6 | 2 | no | — |
| 41 | C3×Q16 | — | 9 | 2 | no | — |
| 42 | #216 | — | 3 | 2 | no | — |
| 43 | #216 | — | 9 | 2 | no | — |
| 44 | Dic12 | — | 9 | 2 | no | — |
| 45 | D3×C8 | — | 9 | 2 | no | — |
| 46 | C2×C3⋊D3 | S4 | 1 | 1 | yes | characteristic |
| 47 | C2×C3×D3 | — | 12 | 2 | no | — |
| 48 | C2×C3×D3 | — | 24 | 1 | no | — |
| 49 | C32×C4 | — | 3 | 1 | no | — |
| 50 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 51 | C3⋊Dic3 | — | 4 | 2 | no | — |
| 52 | C3×Dic3 | — | 6 | 2 | no | — |
| 53 | C3×Dic3 | — | 12 | 2 | no | — |
| 54 | C3×Dic3 | — | 24 | 2 | no | — |
| 55 | C2×Q16 | — | 27 | 1 | no | Sylow |
| 56 | C33 | — | 4 | 1 | no | Sylow |
| 57 | C3×Q8 | — | 2 | 1 | no | — |
| 58 | C3×Q8 | — | 9 | 2 | no | — |
| 59 | C3×Q8 | D32 | 1 | 2 | yes | — |
| 60 | C2×Dic3 | — | 36 | 1 | no | — |
| 61 | C4×D3 | — | 9 | 2 | no | — |
| 62 | C4×D3 | — | 18 | 1 | no | — |
| 63 | C4×D3 | — | 18 | 2 | no | — |
| 64 | Dic6 | — | 3 | 2 | no | — |
| 65 | Dic6 | — | 9 | 2 | no | — |
| 66 | Dic6 | — | 18 | 1 | no | — |
| 67 | Dic6 | — | 18 | 2 | no | — |
| 68 | SL(2,𝔽3) | — | 2 | 2 | no | — |
| 69 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 70 | SL(2,𝔽3) | D32 | 1 | 1 | yes | characteristic |
| 71 | C3×C8 | — | 9 | 2 | no | — |
| 72 | C3⋊C8 | — | 3 | 2 | no | — |
| 73 | C2×C32 | — | 4 | 2 | no | — |
| 74 | C2×C32 | — | 8 | 1 | no | — |
| 75 | C2×C32 | — | 8 | 2 | no | — |
| 76 | C2×C32 | — | 16 | 1 | no | — |
| 77 | C2×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 78 | C3⋊D3 | #226 | 1 | 2 | yes | complemented |
| 79 | C3×D3 | — | 12 | 4 | no | — |
| 80 | C3×D3 | — | 24 | 2 | no | — |
| 81 | C2×Q8 | — | 9 | 1 | no | — |
| 82 | C2×Q8 | — | 27 | 1 | no | — |
| 83 | Q16 | — | 9 | 2 | no | — |
| 84 | Q16 | — | 27 | 2 | no | — |
| 85 | C2×C8 | — | 27 | 1 | no | — |
| 86 | C22×C3 | — | 36 | 1 | no | — |
| 87 | C2×D3 | — | 3 | 2 | no | — |
| 88 | C2×D3 | — | 6 | 1 | no | — |
| 89 | C3×C4 | — | 3 | 2 | no | — |
| 90 | C3×C4 | — | 6 | 1 | no | — |
| 91 | C3×C4 | — | 18 | 2 | no | — |
| 92 | Dic3 | — | 6 | 2 | no | — |
| 93 | Dic3 | — | 9 | 2 | no | — |
| 94 | Dic3 | — | 12 | 2 | no | — |
| 95 | Dic3 | — | 18 | 1 | no | — |
| 96 | Dic3 | — | 24 | 2 | no | — |
| 97 | C32 | — | 4 | 2 | no | — |
| 98 | C32 | — | 8 | 1 | no | — |
| 99 | C32 | — | 8 | 2 | no | — |
| 100 | C32 | — | 16 | 1 | no | — |
| 101 | C32 | C2×#226 | 1 | 1 | yes | characteristic, complemented |
| 102 | Q8 | — | 9 | 2 | no | — |
| 103 | Q8 | — | 27 | 1 | no | — |
| 104 | Q8 | — | 54 | 1 | no | — |
| 105 | Q8 | #1113 | 1 | 1 | yes | characteristic |
| 106 | C2×C4 | — | 27 | 1 | no | — |
| 107 | C2×C4 | — | 54 | 1 | no | — |
| 108 | C8 | — | 9 | 2 | no | — |
| 109 | C2×C3 | — | 2 | 1 | no | — |
| 110 | C2×C3 | — | 4 | 1 | no | — |
| 111 | C2×C3 | — | 8 | 2 | no | — |
| 112 | C2×C3 | — | 16 | 1 | no | — |
| 113 | C2×C3 | — | 36 | 2 | no | — |
| 114 | C2×C3 | D3×S4 | 1 | 2 | yes | — |
| 115 | D3 | — | 3 | 4 | no | — |
| 116 | D3 | — | 6 | 2 | no | — |
| 117 | C22 | — | 9 | 1 | no | — |
| 118 | C4 | — | 3 | 1 | no | — |
| 119 | C4 | — | 18 | 2 | no | — |
| 120 | C4 | — | 27 | 1 | no | — |
| 121 | C3 | — | 2 | 1 | no | — |
| 122 | C3 | — | 4 | 1 | no | — |
| 123 | C3 | — | 8 | 2 | no | — |
| 124 | C3 | — | 16 | 1 | no | — |
| 125 | C3 | D3×#226 | 1 | 2 | yes | complemented |
| 126 | C2 | — | 9 | 2 | no | — |
| 127 | C2 | #89101 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #89101 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C22 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| prime radical | C3⋊D3×SL(2,𝔽3) | C2 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C3⋊#226 | C2 | semidirect |
| 11. C3⋊#226 | D3 | semidirect |
| 78. C3⋊D3 | #226 | semidirect |
| 101. C32 | C2×#226 | semidirect |
| 125. C3 | D3×#226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×SL(2,𝔽3) | 2 |
| C3×C3⋊#226 | 2 |
| D3×#226 | 3 |
| #63944 | 3 |
| #6294 | 4 |