Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
123 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 123 conjugacy classes of subgroups — too many to draw. Only its 15 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 | #11757434 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10619740 | — | 3 | 2 | no | maximal |
| 3 | C9×(C22⋊C9)⋊C2 | — | 3 | 1 | no | maximal |
| 4 | #10619649 | — | 3 | 1 | no | maximal |
| 5 | #92216 | — | 4 | 1 | no | maximal |
| 6 | C22×#732 | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×#4178 | — | 3 | 1 | no | — |
| 8 | #66358 | D3 | 1 | 2 | yes | complemented |
| 9 | C9×C22⋊C9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #66325 | — | 3 | 1 | no | — |
| 11 | #6879 | — | 4 | 1 | no | Sylow |
| 12 | #6258 | — | 3 | 2 | no | — |
| 13 | C3×(C22⋊C9)⋊C2 | — | 3 | 1 | no | — |
| 14 | #6235 | — | 3 | 1 | no | — |
| 15 | C9×#67 | — | 9 | 1 | no | — |
| 16 | C2×#732 | — | 3 | 1 | no | — |
| 17 | #4178 | — | 6 | 1 | no | — |
| 18 | #4178 | — | 12 | 2 | no | — |
| 19 | C9×D9 | — | 12 | 1 | no | — |
| 20 | C22×H(3,𝔽3) | — | 3 | 1 | no | — |
| 21 | C22×C3×C9 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 22 | C2×#272 | — | 3 | 1 | no | — |
| 23 | C2×D3×C9 | — | 9 | 1 | no | — |
| 24 | C22⋊H(3,𝔽3) | — | 3 | 2 | no | — |
| 25 | C3×C22⋊C9 | — | 3 | 1 | no | — |
| 26 | #1084 | — | 3 | 1 | no | — |
| 27 | C9×Dic3 | — | 9 | 1 | no | — |
| 28 | #732 | — | 4 | 2 | no | — |
| 29 | #732 | S4 | 1 | 1 | yes | characteristic, complemented |
| 30 | C92 | — | 4 | 1 | no | — |
| 31 | C3×S4 | — | 27 | 2 | no | — |
| 32 | C3×#67 | — | 9 | 1 | no | — |
| 33 | C3×#67 | — | 27 | 1 | no | — |
| 34 | (C22⋊C9)⋊C2 | — | 3 | 1 | no | — |
| 35 | D4×C9 | — | 27 | 1 | no | — |
| 36 | C2×H(3,𝔽3) | — | 3 | 1 | no | — |
| 37 | C2×H(3,𝔽3) | — | 6 | 1 | no | — |
| 38 | C2×C3×C9 | — | 3 | 1 | no | — |
| 39 | #272 | — | 6 | 1 | no | — |
| 40 | #272 | — | 12 | 2 | no | — |
| 41 | D3×C9 | — | 18 | 1 | no | — |
| 42 | C3×D9 | — | 12 | 1 | no | — |
| 43 | C22×C32 | — | 9 | 1 | no | — |
| 44 | C22×C32 | #272 | 1 | 1 | yes | characteristic |
| 45 | C2×C3×D3 | — | 9 | 1 | no | — |
| 46 | C2×C3×D3 | — | 27 | 1 | no | — |
| 47 | C3×A4 | — | 9 | 2 | no | — |
| 48 | C3×Dic3 | — | 9 | 1 | no | — |
| 49 | C3×Dic3 | — | 27 | 1 | no | — |
| 50 | C22×C9 | — | 3 | 1 | no | — |
| 51 | C22×C9 | — | 27 | 1 | no | — |
| 52 | C22⋊C9 | — | 3 | 1 | no | — |
| 53 | C22⋊C9 | — | 6 | 1 | no | — |
| 54 | C4×C9 | — | 27 | 1 | no | — |
| 55 | H(3,𝔽3) | — | 3 | 1 | no | — |
| 56 | H(3,𝔽3) | — | 12 | 2 | no | — |
| 57 | C3×C9 | — | 12 | 1 | no | — |
| 58 | C3×C9 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 59 | S4 | — | 27 | 2 | no | — |
| 60 | C3×D4 | — | 27 | 1 | no | — |
| 61 | #67 | — | 9 | 1 | no | — |
| 62 | #67 | — | 27 | 1 | no | — |
| 63 | C2×C32 | — | 3 | 1 | no | — |
| 64 | C2×C32 | — | 9 | 1 | no | — |
| 65 | C2×C32 | — | 18 | 1 | no | — |
| 66 | C3×D3 | — | 18 | 1 | no | — |
| 67 | C3×D3 | — | 54 | 1 | no | — |
| 68 | C3×D3 | — | 108 | 2 | no | — |
| 69 | C2×C9 | — | 9 | 1 | no | — |
| 70 | C2×C9 | — | 54 | 1 | no | — |
| 71 | D9 | — | 12 | 1 | no | — |
| 72 | C22×C3 | — | 3 | 1 | no | — |
| 73 | C22×C3 | — | 27 | 1 | no | — |
| 74 | C22×C3 | — | 27 | 1 | no | — |
| 75 | C22×C3 | #4185 | 1 | 1 | yes | characteristic |
| 76 | C2×D3 | — | 9 | 1 | no | — |
| 77 | C2×D3 | — | 27 | 1 | no | — |
| 78 | A4 | — | 27 | 2 | no | — |
| 79 | C3×C4 | — | 27 | 1 | no | — |
| 80 | Dic3 | — | 9 | 1 | no | — |
| 81 | Dic3 | — | 27 | 1 | no | — |
| 82 | C32 | — | 9 | 1 | no | — |
| 83 | C32 | — | 36 | 2 | no | — |
| 84 | C32 | #6258 | 1 | 1 | yes | characteristic |
| 85 | C9 | — | 3 | 1 | no | — |
| 86 | C9 | — | 12 | 1 | no | — |
| 87 | C9 | — | 24 | 1 | no | — |
| 88 | D4 | — | 27 | 1 | no | Sylow |
| 89 | C2×C3 | — | 3 | 1 | no | — |
| 90 | C2×C3 | — | 9 | 1 | no | — |
| 91 | C2×C3 | — | 27 | 1 | no | — |
| 92 | C2×C3 | — | 54 | 1 | no | — |
| 93 | C2×C3 | — | 54 | 1 | no | — |
| 94 | D3 | — | 18 | 1 | no | — |
| 95 | D3 | — | 54 | 1 | no | — |
| 96 | D3 | — | 108 | 2 | no | — |
| 97 | C22 | — | 27 | 1 | no | — |
| 98 | C22 | #92216 | 1 | 1 | yes | characteristic, complemented |
| 99 | C4 | — | 27 | 1 | no | — |
| 100 | C3 | — | 3 | 1 | no | — |
| 101 | C3 | — | 27 | 1 | no | — |
| 102 | C3 | — | 108 | 2 | no | — |
| 103 | C3 | #10619755 | 1 | 1 | yes | characteristic |
| 104 | C2 | — | 3 | 1 | no | — |
| 105 | C2 | — | 54 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #10619755 |
| commutator, cosocle, hypocenter | #11757434 | C2 |
| Frattini | C3×C9 | C3⋊S4 |
| Fitting | C22×#732 | D3 |
| socle | C22×C3 | #4185 |
| LCS | G ▷ #11757434 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11757434 | C2 | semidirect |
| 6. C22×#732 | D3 | semidirect |
| 8. #66358 | D3 | semidirect |
| 9. C9×C22⋊C9 | D3 | semidirect |
| 29. #732 | S4 | semidirect |
| 98. C22 | #92216 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11757434 | 2 |
| #10619740 | 3 |
| C9×(C22⋊C9)⋊C2 | 3 |
| #10619649 | 3 |
| #92216 | 4 |