Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
586 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 586 conjugacy classes of subgroups — too many to draw. Only its 62 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 | H(3,𝔽3)2 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10624778 | C3 | 1 | 4 | yes | maximal, complemented |
| 3 | #10624731 | C3 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×H(3,𝔽3)⋊C9 | C3 | 1 | 2 | yes | maximal, complemented |
| 5 | #10624620 | C3 | 1 | 4 | yes | maximal, complemented |
| 6 | C32×H(3,𝔽3) | — | 3 | 2 | no | — |
| 7 | C32×H(3,𝔽3) | C32 | 1 | 2 | yes | — |
| 8 | C33×C9 | C32 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3×#731 | — | 3 | 6 | no | — |
| 10 | #6891 | C32 | 1 | 4 | yes | complemented |
| 11 | C32⋊H(3,𝔽3) | — | 3 | 6 | no | — |
| 12 | C32⋊H(3,𝔽3) | — | 3 | 4 | no | — |
| 13 | C32⋊H(3,𝔽3) | C32 | 1 | 2 | yes | — |
| 14 | C3×C32⋊C9 | C32 | 1 | 4 | yes | complemented |
| 15 | H(3,𝔽3)⋊C9 | — | 3 | 8 | no | — |
| 16 | H(3,𝔽3)⋊C9 | — | 3 | 4 | no | — |
| 17 | H(3,𝔽3)⋊C9 | — | 3 | 4 | no | — |
| 18 | H(3,𝔽3)⋊C9 | — | 3 | 2 | no | — |
| 19 | #6859 | — | 3 | 12 | no | — |
| 20 | C34 | — | 3 | 3 | no | — |
| 21 | C34 | — | 3 | 2 | no | — |
| 22 | C34 | C33 | 1 | 1 | yes | characteristic |
| 23 | C3×3−1+2 | — | 3 | 4 | no | — |
| 24 | C3×H(3,𝔽3) | — | 3 | 6 | no | — |
| 25 | C3×H(3,𝔽3) | — | 3 | 12 | no | — |
| 26 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 27 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 28 | C3×H(3,𝔽3) | — | 3 | 6 | no | — |
| 29 | C3×H(3,𝔽3) | — | 3 | 6 | no | — |
| 30 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 31 | C3×H(3,𝔽3) | — | 9 | 4 | no | — |
| 32 | C3×H(3,𝔽3) | — | 9 | 12 | no | — |
| 33 | C3×H(3,𝔽3) | — | 9 | 6 | no | — |
| 34 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 4 | yes | — |
| 35 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 2 | yes | — |
| 36 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 6 | yes | — |
| 37 | C32×C9 | — | 3 | 2 | no | — |
| 38 | C32×C9 | — | 3 | 2 | no | — |
| 39 | C32×C9 | — | 3 | 4 | no | — |
| 40 | C32×C9 | — | 9 | 3 | no | — |
| 41 | #731 | — | 9 | 18 | no | — |
| 42 | C32⋊C9 | — | 3 | 8 | no | — |
| 43 | C32⋊C9 | — | 3 | 4 | no | — |
| 44 | C32⋊C9 | — | 3 | 4 | no | — |
| 45 | C32⋊C9 | — | 3 | 8 | no | — |
| 46 | C33 | — | 3 | 2 | no | — |
| 47 | C33 | — | 3 | 2 | no | — |
| 48 | C33 | — | 3 | 4 | no | — |
| 49 | C33 | — | 3 | 4 | no | — |
| 50 | C33 | — | 3 | 6 | no | — |
| 51 | C33 | — | 9 | 1 | no | — |
| 52 | C33 | — | 9 | 1 | no | — |
| 53 | C33 | — | 9 | 6 | no | — |
| 54 | C33 | — | 9 | 18 | no | — |
| 55 | C33 | — | 9 | 2 | no | — |
| 56 | C33 | — | 9 | 4 | no | — |
| 57 | C33 | — | 27 | 2 | no | — |
| 58 | C33 | — | 27 | 2 | no | — |
| 59 | C33 | — | 27 | 3 | no | — |
| 60 | C33 | — | 27 | 3 | no | — |
| 61 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 62 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 63 | 3−1+2 | — | 9 | 12 | no | — |
| 64 | H(3,𝔽3) | — | 3 | 4 | no | — |
| 65 | H(3,𝔽3) | — | 3 | 4 | no | — |
| 66 | H(3,𝔽3) | — | 3 | 8 | no | — |
| 67 | H(3,𝔽3) | — | 9 | 12 | no | — |
| 68 | H(3,𝔽3) | — | 9 | 6 | no | — |
| 69 | H(3,𝔽3) | — | 9 | 36 | no | — |
| 70 | H(3,𝔽3) | — | 9 | 4 | no | — |
| 71 | H(3,𝔽3) | — | 9 | 8 | no | — |
| 72 | H(3,𝔽3) | — | 27 | 2 | no | — |
| 73 | H(3,𝔽3) | — | 27 | 6 | no | — |
| 74 | H(3,𝔽3) | — | 27 | 6 | no | — |
| 75 | H(3,𝔽3) | #731 | 1 | 6 | yes | complemented |
| 76 | C3×C9 | — | 3 | 6 | no | — |
| 77 | C3×C9 | — | 9 | 4 | no | — |
| 78 | C3×C9 | — | 9 | 1 | no | — |
| 79 | C3×C9 | — | 9 | 6 | no | — |
| 80 | C3×C9 | — | 9 | 4 | no | — |
| 81 | C3×C9 | — | 9 | 4 | no | — |
| 82 | C3×C9 | — | 27 | 4 | no | — |
| 83 | C32 | — | 3 | 2 | no | — |
| 84 | C32 | — | 3 | 2 | no | — |
| 85 | C32 | — | 3 | 2 | no | — |
| 86 | C32 | — | 3 | 6 | no | — |
| 87 | C32 | — | 3 | 4 | no | — |
| 88 | C32 | — | 3 | 2 | no | — |
| 89 | C32 | — | 3 | 2 | no | — |
| 90 | C32 | — | 9 | 1 | no | — |
| 91 | C32 | — | 9 | 4 | no | — |
| 92 | C32 | — | 9 | 4 | no | — |
| 93 | C32 | — | 9 | 2 | no | — |
| 94 | C32 | — | 9 | 2 | no | — |
| 95 | C32 | — | 9 | 2 | no | — |
| 96 | C32 | — | 9 | 4 | no | — |
| 97 | C32 | — | 9 | 4 | no | — |
| 98 | C32 | — | 9 | 12 | no | — |
| 99 | C32 | — | 9 | 6 | no | — |
| 100 | C32 | — | 9 | 4 | no | — |
| 101 | C32 | — | 9 | 6 | no | — |
| 102 | C32 | — | 9 | 4 | no | — |
| 103 | C32 | — | 27 | 6 | no | — |
| 104 | C32 | — | 27 | 12 | no | — |
| 105 | C32 | — | 27 | 27 | no | — |
| 106 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 107 | C32 | C3×#731 | 1 | 2 | yes | — |
| 108 | C32 | #6910 | 1 | 4 | yes | — |
| 109 | C9 | — | 9 | 3 | no | — |
| 110 | C9 | — | 27 | 4 | no | — |
| 111 | C9 | — | 27 | 4 | no | — |
| 112 | C3 | — | 3 | 2 | no | — |
| 113 | C3 | — | 3 | 4 | no | — |
| 114 | C3 | — | 9 | 2 | no | — |
| 115 | C3 | — | 9 | 6 | no | — |
| 116 | C3 | — | 27 | 4 | no | — |
| 117 | C3 | — | 27 | 6 | no | — |
| 118 | C3 | #10624741 | 1 | 2 | yes | — |
| 119 | C3 | UT(4,𝔽3) | 1 | 1 | yes | characteristic |
| 120 | C3 | #10624760 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C34 | C33 |
| prime radical | H(3,𝔽3)2 | C3 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624778 | C3 | semidirect |
| 3. #10624731 | C3 | semidirect |
| 4. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 5. #10624620 | C3 | semidirect |
| 8. C33×C9 | C32 | semidirect |
| 10. #6891 | C32 | semidirect |
| 14. C3×C32⋊C9 | C32 | semidirect |
| 75. H(3,𝔽3) | #731 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)2 | 3 |
| #10624778 | 3 |
| #10624731 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| #10624620 | 3 |