Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
386 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 50 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×#732 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10624768 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10624733 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C9×C32⋊C9 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10624481 | C3 | 1 | 9 | yes | maximal, complemented |
| 6 | C32×H(3,𝔽3) | — | 3 | 1 | no | — |
| 7 | C33×C9 | C32 | 1 | 1 | yes | characteristic |
| 8 | C3×#733 | — | 3 | 1 | no | — |
| 9 | C3×#733 | — | 3 | 1 | no | — |
| 10 | C3×#733 | C32 | 1 | 3 | yes | complemented |
| 11 | C3×#732 | — | 3 | 1 | no | — |
| 12 | C3×#732 | — | 3 | 1 | no | — |
| 13 | C3×#732 | — | 3 | 9 | no | — |
| 14 | C3×#732 | C32 | 1 | 3 | yes | — |
| 15 | #6891 | — | 3 | 1 | no | — |
| 16 | #6891 | — | 3 | 1 | no | — |
| 17 | C3×C32⋊C9 | — | 3 | 1 | no | — |
| 18 | C3×C92 | — | 3 | 1 | no | — |
| 19 | #6868 | — | 3 | 1 | no | — |
| 20 | #6868 | — | 3 | 1 | no | — |
| 21 | #6868 | C32 | 1 | 3 | yes | complemented |
| 22 | #6855 | — | 3 | 1 | no | — |
| 23 | #6855 | C32 | 1 | 3 | yes | complemented |
| 24 | C34 | — | 9 | 1 | no | — |
| 25 | C34 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 26 | C3×3−1+2 | — | 3 | 3 | no | — |
| 27 | C3×3−1+2 | — | 9 | 1 | no | — |
| 28 | C3×H(3,𝔽3) | — | 3 | 9 | no | — |
| 29 | C3×H(3,𝔽3) | — | 3 | 3 | no | — |
| 30 | C3×H(3,𝔽3) | — | 9 | 3 | no | — |
| 31 | C3×H(3,𝔽3) | — | 9 | 3 | no | — |
| 32 | C3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 33 | C3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 34 | C32×C9 | — | 3 | 1 | no | — |
| 35 | C32×C9 | — | 3 | 1 | no | — |
| 36 | C32×C9 | — | 3 | 1 | no | — |
| 37 | C32×C9 | — | 3 | 3 | no | — |
| 38 | C32×C9 | — | 3 | 3 | no | — |
| 39 | C32×C9 | — | 9 | 1 | no | — |
| 40 | C32×C9 | — | 9 | 1 | no | — |
| 41 | C32×C9 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 42 | C32×C9 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 43 | C32×C9 | C33 | 1 | 1 | yes | characteristic |
| 44 | #733 | — | 9 | 3 | no | — |
| 45 | #733 | — | 9 | 3 | no | — |
| 46 | #733 | — | 9 | 3 | no | — |
| 47 | #732 | — | 3 | 9 | no | — |
| 48 | #732 | — | 3 | 9 | no | — |
| 49 | #732 | — | 3 | 9 | no | — |
| 50 | C32⋊C9 | — | 3 | 9 | no | — |
| 51 | C32⋊C9 | — | 3 | 3 | no | — |
| 52 | C32⋊C9 | — | 9 | 1 | no | — |
| 53 | C32⋊C9 | — | 9 | 1 | no | — |
| 54 | C32⋊C9 | — | 9 | 1 | no | — |
| 55 | C32⋊C9 | — | 9 | 1 | no | — |
| 56 | C32⋊C9 | — | 9 | 1 | no | — |
| 57 | C92 | — | 9 | 3 | no | — |
| 58 | C33 | — | 3 | 3 | no | — |
| 59 | C33 | — | 3 | 1 | no | — |
| 60 | C33 | — | 3 | 1 | no | — |
| 61 | C33 | — | 3 | 1 | no | — |
| 62 | C33 | — | 9 | 1 | no | — |
| 63 | C33 | — | 9 | 1 | no | — |
| 64 | C33 | — | 9 | 9 | no | — |
| 65 | C33 | — | 9 | 3 | no | — |
| 66 | C33 | — | 27 | 3 | no | — |
| 67 | C33 | — | 27 | 3 | no | — |
| 68 | C33 | — | 27 | 3 | no | — |
| 69 | C33 | #732 | 1 | 1 | yes | characteristic |
| 70 | C33 | #732 | 1 | 1 | yes | characteristic |
| 71 | C33 | #732 | 1 | 1 | yes | characteristic |
| 72 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 73 | 3−1+2 | — | 9 | 3 | no | — |
| 74 | 3−1+2 | — | 9 | 3 | no | — |
| 75 | 3−1+2 | — | 9 | 3 | no | — |
| 76 | 3−1+2 | — | 27 | 3 | no | — |
| 77 | H(3,𝔽3) | — | 3 | 9 | no | — |
| 78 | H(3,𝔽3) | — | 9 | 9 | no | — |
| 79 | H(3,𝔽3) | — | 9 | 9 | no | — |
| 80 | H(3,𝔽3) | — | 9 | 3 | no | — |
| 81 | H(3,𝔽3) | — | 9 | 3 | no | — |
| 82 | C3×C9 | — | 3 | 1 | no | — |
| 83 | C3×C9 | — | 3 | 1 | no | — |
| 84 | C3×C9 | — | 3 | 1 | no | — |
| 85 | C3×C9 | — | 3 | 9 | no | — |
| 86 | C3×C9 | — | 3 | 1 | no | — |
| 87 | C3×C9 | — | 3 | 1 | no | — |
| 88 | C3×C9 | — | 3 | 1 | no | — |
| 89 | C3×C9 | — | 3 | 1 | no | — |
| 90 | C3×C9 | — | 3 | 1 | no | — |
| 91 | C3×C9 | — | 3 | 1 | no | — |
| 92 | C3×C9 | — | 3 | 1 | no | — |
| 93 | C3×C9 | — | 3 | 1 | no | — |
| 94 | C3×C9 | — | 9 | 3 | no | — |
| 95 | C3×C9 | — | 9 | 3 | no | — |
| 96 | C3×C9 | — | 9 | 3 | no | — |
| 97 | C3×C9 | — | 9 | 3 | no | — |
| 98 | C3×C9 | — | 9 | 3 | no | — |
| 99 | C3×C9 | — | 9 | 3 | no | — |
| 100 | C3×C9 | — | 9 | 3 | no | — |
| 101 | C3×C9 | — | 9 | 3 | no | — |
| 102 | C3×C9 | AΣL(1,𝔽27) | 1 | 3 | yes | — |
| 103 | C32 | — | 3 | 1 | no | — |
| 104 | C32 | — | 3 | 1 | no | — |
| 105 | C32 | — | 3 | 1 | no | — |
| 106 | C32 | — | 3 | 1 | no | — |
| 107 | C32 | — | 3 | 1 | no | — |
| 108 | C32 | — | 3 | 1 | no | — |
| 109 | C32 | — | 3 | 1 | no | — |
| 110 | C32 | — | 3 | 1 | no | — |
| 111 | C32 | — | 3 | 1 | no | — |
| 112 | C32 | — | 3 | 1 | no | — |
| 113 | C32 | — | 3 | 1 | no | — |
| 114 | C32 | — | 3 | 1 | no | — |
| 115 | C32 | — | 3 | 1 | no | — |
| 116 | C32 | — | 3 | 1 | no | — |
| 117 | C32 | — | 9 | 3 | no | — |
| 118 | C32 | — | 9 | 3 | no | — |
| 119 | C32 | — | 9 | 3 | no | — |
| 120 | C32 | — | 9 | 9 | no | — |
| 121 | C32 | — | 27 | 27 | no | — |
| 122 | C32 | — | 27 | 3 | no | — |
| 123 | C32 | — | 27 | 3 | no | — |
| 124 | C32 | — | 27 | 3 | no | — |
| 125 | C32 | #6878 | 1 | 3 | yes | — |
| 126 | C32 | C3×AΣL(1,𝔽27) | 1 | 1 | yes | characteristic |
| 127 | C32 | C3×#732 | 1 | 1 | yes | characteristic |
| 128 | C32 | #6909 | 1 | 1 | yes | characteristic |
| 129 | C32 | #6911 | 1 | 1 | yes | characteristic |
| 130 | C9 | — | 3 | 3 | no | — |
| 131 | C9 | — | 9 | 9 | no | — |
| 132 | C9 | — | 9 | 1 | no | — |
| 133 | C9 | — | 9 | 1 | no | — |
| 134 | C9 | — | 27 | 3 | no | — |
| 135 | C9 | — | 27 | 3 | no | — |
| 136 | C3 | — | 3 | 1 | no | — |
| 137 | C3 | — | 3 | 3 | no | — |
| 138 | C3 | — | 3 | 1 | no | — |
| 139 | C3 | — | 3 | 1 | no | — |
| 140 | C3 | — | 9 | 1 | no | — |
| 141 | C3 | — | 9 | 1 | no | — |
| 142 | C3 | — | 27 | 9 | no | — |
| 143 | C3 | #10624596 | 1 | 1 | yes | characteristic |
| 144 | C3 | C3×#6878 | 1 | 1 | yes | characteristic |
| 145 | C3 | #10624846 | 1 | 1 | yes | characteristic |
| 146 | C3 | #10624848 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C3×#732 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| prime radical | C32×#732 | C3 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10624768 | C3 | semidirect |
| 3. #10624733 | C3 | semidirect |
| 4. C9×C32⋊C9 | C3 | semidirect |
| 5. #10624481 | C3 | semidirect |
| 10. C3×#733 | C32 | semidirect |
| 21. #6868 | C32 | semidirect |
| 23. #6855 | C32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#732 | 3 |
| #10624768 | 3 |
| #10624733 | 3 |
| C9×C32⋊C9 | 3 |
| #10624481 | 3 |