Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1304 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 98 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×#6911 | C3 | 1 | 4 | yes | maximal, complemented |
| 2 | C3×#6909 | C3 | 1 | 4 | yes | maximal, complemented |
| 3 | C32×#732 | C3 | 1 | 2 | yes | maximal, complemented |
| 4 | C32×#731 | C3 | 1 | 2 | yes | maximal, complemented |
| 5 | #10624880 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C32×3−1+2 | C32 | 1 | 4 | yes | — |
| 7 | C32×H(3,𝔽3) | C32 | 1 | 2 | yes | — |
| 8 | C32×H(3,𝔽3) | C32 | 1 | 2 | yes | — |
| 9 | C32×H(3,𝔽3) | C32 | 1 | 4 | yes | — |
| 10 | C33×C9 | C32 | 1 | 1 | yes | characteristic |
| 11 | #6911 | — | 3 | 12 | no | — |
| 12 | #6911 | — | 3 | 24 | no | — |
| 13 | #6909 | — | 3 | 24 | no | — |
| 14 | #6909 | — | 3 | 12 | no | — |
| 15 | C3×#732 | — | 3 | 6 | no | — |
| 16 | C3×#732 | — | 3 | 12 | no | — |
| 17 | C3×#732 | — | 3 | 12 | no | — |
| 18 | C3×#732 | — | 3 | 6 | no | — |
| 19 | C3×#731 | — | 3 | 12 | no | — |
| 20 | C3×#731 | — | 3 | 6 | no | — |
| 21 | C3×#731 | — | 3 | 6 | no | — |
| 22 | C3×AΣL(1,𝔽27) | — | 3 | 12 | no | — |
| 23 | C32⋊H(3,𝔽3) | — | 3 | 6 | no | — |
| 24 | C32⋊H(3,𝔽3) | — | 3 | 6 | no | — |
| 25 | C34 | — | 3 | 2 | no | — |
| 26 | C34 | — | 3 | 2 | no | — |
| 27 | C34 | — | 3 | 4 | no | — |
| 28 | C34 | C33 | 1 | 1 | yes | characteristic |
| 29 | C3×3−1+2 | — | 3 | 8 | no | — |
| 30 | C3×3−1+2 | — | 3 | 8 | no | — |
| 31 | C3×3−1+2 | — | 3 | 4 | no | — |
| 32 | C3×3−1+2 | — | 3 | 4 | no | — |
| 33 | C3×3−1+2 | — | 3 | 4 | no | — |
| 34 | C3×3−1+2 | — | 3 | 8 | no | — |
| 35 | C3×3−1+2 | — | 3 | 8 | no | — |
| 36 | C3×3−1+2 | H(3,𝔽3) | 1 | 12 | yes | complemented |
| 37 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 38 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 39 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 40 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 41 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 42 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 43 | C3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 44 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 45 | C3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 46 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 47 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 48 | C3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 49 | C3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 50 | C3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 51 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 52 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 53 | C3×H(3,𝔽3) | — | 3 | 8 | no | — |
| 54 | C3×H(3,𝔽3) | — | 3 | 27 | no | — |
| 55 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 6 | yes | — |
| 56 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 6 | yes | — |
| 57 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 12 | yes | complemented |
| 58 | C32×C9 | — | 3 | 4 | no | — |
| 59 | C32×C9 | — | 3 | 2 | no | — |
| 60 | C32×C9 | — | 3 | 4 | no | — |
| 61 | C32×C9 | — | 3 | 2 | no | — |
| 62 | C32×C9 | — | 3 | 4 | no | — |
| 63 | C32×C9 | H(3,𝔽3) | 1 | 3 | yes | complemented |
| 64 | #732 | — | 3 | 54 | no | — |
| 65 | #732 | — | 3 | 108 | no | — |
| 66 | #731 | — | 3 | 54 | no | — |
| 67 | AΣL(1,𝔽27) | — | 3 | 108 | no | — |
| 68 | C33 | — | 3 | 6 | no | — |
| 69 | C33 | — | 3 | 6 | no | — |
| 70 | C33 | — | 3 | 12 | no | — |
| 71 | C33 | — | 9 | 3 | no | — |
| 72 | C33 | — | 9 | 4 | no | — |
| 73 | C33 | — | 9 | 2 | no | — |
| 74 | C33 | — | 9 | 2 | no | — |
| 75 | C33 | — | 9 | 4 | no | — |
| 76 | C33 | — | 9 | 4 | no | — |
| 77 | C33 | — | 9 | 4 | no | — |
| 78 | C33 | — | 9 | 4 | no | — |
| 79 | C33 | — | 27 | 6 | no | — |
| 80 | C33 | — | 27 | 6 | no | — |
| 81 | C33 | — | 27 | 12 | no | — |
| 82 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 83 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 84 | C33 | C3×H(3,𝔽3) | 1 | 4 | yes | — |
| 85 | C33 | C3×H(3,𝔽3) | 1 | 4 | yes | — |
| 86 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 87 | 3−1+2 | — | 3 | 24 | no | — |
| 88 | 3−1+2 | — | 3 | 12 | no | — |
| 89 | 3−1+2 | — | 9 | 8 | no | — |
| 90 | 3−1+2 | — | 9 | 8 | no | — |
| 91 | 3−1+2 | — | 9 | 8 | no | — |
| 92 | H(3,𝔽3) | — | 3 | 12 | no | — |
| 93 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 94 | H(3,𝔽3) | — | 3 | 12 | no | — |
| 95 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 96 | H(3,𝔽3) | — | 3 | 24 | no | — |
| 97 | H(3,𝔽3) | — | 3 | 12 | no | — |
| 98 | H(3,𝔽3) | — | 9 | 8 | no | — |
| 99 | H(3,𝔽3) | — | 9 | 8 | no | — |
| 100 | H(3,𝔽3) | — | 9 | 8 | no | — |
| 101 | H(3,𝔽3) | — | 9 | 4 | no | — |
| 102 | H(3,𝔽3) | — | 9 | 4 | no | — |
| 103 | H(3,𝔽3) | — | 9 | 8 | no | — |
| 104 | H(3,𝔽3) | — | 9 | 8 | no | — |
| 105 | H(3,𝔽3) | — | 27 | 27 | no | — |
| 106 | C3×C9 | — | 3 | 6 | no | — |
| 107 | C3×C9 | — | 3 | 6 | no | — |
| 108 | C3×C9 | — | 3 | 12 | no | — |
| 109 | C3×C9 | — | 9 | 4 | no | — |
| 110 | C3×C9 | — | 9 | 4 | no | — |
| 111 | C3×C9 | — | 9 | 4 | no | — |
| 112 | C3×C9 | — | 9 | 4 | no | — |
| 113 | C3×C9 | — | 9 | 1 | no | — |
| 114 | C3×C9 | — | 9 | 4 | no | — |
| 115 | C32 | — | 3 | 6 | no | — |
| 116 | C32 | — | 3 | 6 | no | — |
| 117 | C32 | — | 9 | 4 | no | — |
| 118 | C32 | — | 9 | 4 | no | — |
| 119 | C32 | — | 9 | 1 | no | — |
| 120 | C32 | — | 9 | 6 | no | — |
| 121 | C32 | — | 9 | 6 | no | — |
| 122 | C32 | — | 9 | 12 | no | — |
| 123 | C32 | — | 9 | 18 | no | — |
| 124 | C32 | — | 9 | 18 | no | — |
| 125 | C32 | — | 27 | 12 | no | — |
| 126 | C32 | — | 27 | 12 | no | — |
| 127 | C32 | — | 27 | 6 | no | — |
| 128 | C32 | — | 27 | 12 | no | — |
| 129 | C32 | — | 27 | 12 | no | — |
| 130 | C32 | — | 27 | 12 | no | — |
| 131 | C32 | — | 27 | 6 | no | — |
| 132 | C32 | — | 27 | 12 | no | — |
| 133 | C32 | C32⋊H(3,𝔽3) | 1 | 4 | yes | — |
| 134 | C32 | C32⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 135 | C32 | C32⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 136 | C32 | C32⋊H(3,𝔽3) | 1 | 4 | yes | — |
| 137 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 138 | C9 | — | 9 | 3 | no | — |
| 139 | C9 | — | 9 | 12 | no | — |
| 140 | C3 | — | 3 | 4 | no | — |
| 141 | C3 | — | 3 | 2 | no | — |
| 142 | C3 | — | 3 | 4 | no | — |
| 143 | C3 | — | 3 | 2 | no | — |
| 144 | C3 | — | 27 | 6 | no | — |
| 145 | C3 | — | 27 | 6 | no | — |
| 146 | C3 | — | 27 | 12 | no | — |
| 147 | C3 | #10624562 | 1 | 1 | yes | characteristic |
| 148 | C3 | #10624760 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C34 | C33 |
| LCS | G ▷ C34 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6911 | C3 | semidirect |
| 2. C3×#6909 | C3 | semidirect |
| 3. C32×#732 | C3 | semidirect |
| 4. C32×#731 | C3 | semidirect |
| 5. #10624880 | C3 | semidirect |
| 36. C3×3−1+2 | H(3,𝔽3) | semidirect |
| 57. C3×H(3,𝔽3) | H(3,𝔽3) | semidirect |
| 63. C32×C9 | H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6911 | 3 |
| C3×#6909 | 3 |
| C32×#732 | 3 |
| C32×#731 | 3 |
| #10624880 | 3 |