Finite Group DatabasebetaFinite Group DatabasebetaOrder 2916.
1597 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 1597 conjugacy classes of subgroups — too many to draw. Only its 101 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 | #13091718 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #13091717 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11758087 | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | D3×#4208 | — | 3 | 8 | no | maximal |
| 5 | #10624943 | C22 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | #92404 | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | C3⋊#4208 | — | 3 | 8 | no | — |
| 8 | C3⋊#4208 | — | 6 | 16 | no | — |
| 9 | #92394 | — | 2 | 8 | no | — |
| 10 | #92392 | C2×C3 | 1 | 2 | yes | — |
| 11 | C3×#4208 | D3 | 1 | 8 | yes | complemented |
| 12 | D3×#737 | — | 3 | 8 | no | — |
| 13 | C2×#4208 | — | 9 | 2 | no | — |
| 14 | D3×#272 | — | 3 | 8 | no | — |
| 15 | D32×C9 | — | 9 | 16 | no | — |
| 16 | 3−1+4 | — | 2 | 8 | no | — |
| 17 | 3−1+4 | — | 4 | 16 | no | — |
| 18 | H(5,𝔽3) | C22×C3 | 1 | 1 | yes | characteristic |
| 19 | C3×#737 | — | 2 | 16 | no | — |
| 20 | C3×#737 | C2×D3 | 1 | 8 | yes | complemented |
| 21 | C2×#737 | — | 9 | 2 | no | — |
| 22 | C3⋊#272 | — | 3 | 8 | no | — |
| 23 | C3⋊#272 | — | 6 | 16 | no | — |
| 24 | #4208 | — | 2 | 8 | no | — |
| 25 | #4208 | — | 9 | 2 | no | — |
| 26 | #4208 | — | 18 | 32 | no | — |
| 27 | #4208 | — | 18 | 12 | no | — |
| 28 | #4208 | C3⋊D3 | 1 | 2 | yes | complemented |
| 29 | C3×#272 | C3×D3 | 1 | 8 | yes | — |
| 30 | C9×C3⋊D3 | — | 9 | 16 | no | — |
| 31 | C9×C3⋊D3 | — | 18 | 12 | no | — |
| 32 | D3×3−1+2 | — | 6 | 32 | no | — |
| 33 | D3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 34 | C3×D3×C9 | — | 3 | 32 | no | — |
| 35 | C3×D32 | — | 9 | 16 | no | — |
| 36 | C2×#272 | — | 9 | 2 | no | — |
| 37 | C2×D3×C9 | — | 27 | 8 | no | — |
| 38 | #737 | — | 2 | 12 | no | — |
| 39 | #737 | — | 2 | 8 | no | — |
| 40 | #737 | — | 2 | 32 | no | — |
| 41 | #737 | — | 4 | 32 | no | — |
| 42 | #737 | — | 4 | 96 | no | — |
| 43 | #737 | — | 4 | 48 | no | — |
| 44 | #737 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 45 | C3×3−1+2 | — | 2 | 32 | no | — |
| 46 | C3×3−1+2 | — | 4 | 48 | no | — |
| 47 | C3×3−1+2 | — | 4 | 16 | no | — |
| 48 | C3×H(3,𝔽3) | — | 2 | 16 | no | — |
| 49 | C3×H(3,𝔽3) | C2×C3×D3 | 1 | 8 | yes | — |
| 50 | C32×C9 | — | 2 | 12 | no | — |
| 51 | C32×C9 | D32 | 1 | 16 | yes | complemented |
| 52 | C3×C3⋊D3 | — | 9 | 16 | no | — |
| 53 | C3×C3⋊D3 | — | 18 | 12 | no | — |
| 54 | C32×D3 | — | 3 | 32 | no | — |
| 55 | C2×3−1+2 | — | 18 | 8 | no | — |
| 56 | C2×H(3,𝔽3) | — | 9 | 2 | no | — |
| 57 | C2×C3×C9 | — | 9 | 8 | no | — |
| 58 | #272 | — | 9 | 2 | no | — |
| 59 | #272 | — | 18 | 32 | no | — |
| 60 | #272 | — | 18 | 12 | no | — |
| 61 | #272 | C3×C3⋊D3 | 1 | 2 | yes | — |
| 62 | D3×C9 | — | 3 | 8 | no | — |
| 63 | D3×C9 | — | 6 | 32 | no | — |
| 64 | D3×C9 | — | 27 | 8 | no | — |
| 65 | D3×C9 | — | 54 | 16 | no | — |
| 66 | C2×C3×D3 | — | 27 | 8 | no | — |
| 67 | D32 | — | 9 | 16 | no | — |
| 68 | C22×C9 | — | 81 | 1 | no | — |
| 69 | C33 | — | 2 | 12 | no | — |
| 70 | C33 | C3×D32 | 1 | 16 | yes | — |
| 71 | 3−1+2 | — | 2 | 8 | no | — |
| 72 | 3−1+2 | — | 4 | 96 | no | — |
| 73 | 3−1+2 | — | 4 | 48 | no | — |
| 74 | 3−1+2 | — | 4 | 32 | no | — |
| 75 | H(3,𝔽3) | — | 2 | 12 | no | — |
| 76 | H(3,𝔽3) | — | 2 | 32 | no | — |
| 77 | H(3,𝔽3) | C2×C3×C3⋊D3 | 1 | 2 | yes | — |
| 78 | C3×C9 | — | 2 | 32 | no | — |
| 79 | C3×C9 | — | 2 | 16 | no | — |
| 80 | C3×C9 | — | 4 | 16 | no | — |
| 81 | C3×C9 | — | 4 | 48 | no | — |
| 82 | C3×C9 | D3×C3⋊D3 | 1 | 8 | yes | — |
| 83 | C2×C32 | — | 9 | 8 | no | — |
| 84 | C3⋊D3 | — | 9 | 16 | no | — |
| 85 | C3⋊D3 | — | 18 | 12 | no | — |
| 86 | C3×D3 | — | 3 | 8 | no | — |
| 87 | C3×D3 | — | 9 | 32 | no | — |
| 88 | C3×D3 | — | 27 | 8 | no | — |
| 89 | C3×D3 | — | 54 | 16 | no | — |
| 90 | C2×C9 | — | 9 | 2 | no | — |
| 91 | C2×C9 | — | 18 | 8 | no | — |
| 92 | C2×C9 | — | 81 | 1 | no | — |
| 93 | C22×C3 | — | 81 | 1 | no | — |
| 94 | C2×D3 | — | 27 | 8 | no | — |
| 95 | C32 | — | 2 | 16 | no | — |
| 96 | C32 | — | 9 | 16 | no | — |
| 97 | C32 | — | 18 | 12 | no | — |
| 98 | C32 | C3×D3×C3⋊D3 | 1 | 8 | yes | — |
| 99 | C9 | — | 2 | 8 | no | — |
| 100 | C9 | — | 4 | 16 | no | — |
| 101 | C9 | C3⋊D32 | 1 | 1 | yes | characteristic |
| 102 | C2×C3 | — | 9 | 2 | no | — |
| 103 | C2×C3 | — | 27 | 8 | no | — |
| 104 | C2×C3 | — | 81 | 1 | no | — |
| 105 | D3 | — | 3 | 8 | no | — |
| 106 | D3 | — | 27 | 8 | no | — |
| 107 | D3 | — | 54 | 16 | no | — |
| 108 | C22 | — | 81 | 1 | no | Sylow |
| 109 | C3 | — | 3 | 8 | no | — |
| 110 | C3 | — | 6 | 16 | no | — |
| 111 | C3 | C3×C3⋊D32 | 1 | 1 | yes | characteristic |
| 112 | C2 | — | 9 | 2 | no | — |
| 113 | C2 | — | 81 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | C3⋊D32 |
| commutator, cosocle, hypocenter | H(5,𝔽3) | C22×C3 |
| Frattini, socle | C3 | C3×C3⋊D32 |
| Fitting | #10624943 | C22 |
| prime radical | #11758087 | C3 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13091718 | C2 | semidirect |
| 2. #13091717 | C2 | semidirect |
| 5. #10624943 | C22 | semidirect |
| 11. C3×#4208 | D3 | semidirect |
| 20. C3×#737 | C2×D3 | semidirect |
| 28. #4208 | C3⋊D3 | semidirect |
| 51. C32×C9 | D32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13091718 | 2 |
| #13091717 | 2 |
| #11758087 | 3 |
| D3×#4208 | 3 |