Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
1013 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 1013 conjugacy classes of subgroups — too many to draw. Only its 128 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×AΣL(1,𝔽27) | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×C9×H(3,𝔽3) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×C32⋊C9 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×3−1+2⋊C9 | C3 | 1 | 4 | yes | maximal, complemented |
| 5 | C3×H(3,𝔽3)⋊C9 | C3 | 1 | 2 | yes | maximal, complemented |
| 6 | C3×#6866 | C3 | 1 | 2 | yes | maximal, complemented |
| 7 | #10624558 | C3 | 1 | 2 | yes | maximal, complemented |
| 8 | C35 | C32 | 1 | 1 | yes | characteristic |
| 9 | C32×3−1+2 | C32 | 1 | 2 | yes | — |
| 10 | C32×H(3,𝔽3) | C32 | 1 | 1 | yes | characteristic |
| 11 | C33×C9 | — | 3 | 1 | no | — |
| 12 | C33×C9 | C32 | 1 | 2 | yes | — |
| 13 | C33×C9 | C32 | 1 | 1 | yes | characteristic |
| 14 | C3×AΣL(1,𝔽27) | — | 3 | 3 | no | — |
| 15 | C3×AΣL(1,𝔽27) | — | 3 | 6 | no | — |
| 16 | C3×AΣL(1,𝔽27) | C9 | 1 | 9 | yes | complemented |
| 17 | C9×H(3,𝔽3) | — | 3 | 3 | no | — |
| 18 | C9×H(3,𝔽3) | — | 3 | 6 | no | — |
| 19 | C3×C32⋊C9 | — | 3 | 2 | no | — |
| 20 | C3×C32⋊C9 | — | 3 | 1 | no | — |
| 21 | C3×C32⋊C9 | — | 3 | 2 | no | — |
| 22 | C3×C32⋊C9 | — | 3 | 9 | no | — |
| 23 | C3×C32⋊C9 | C32 | 1 | 4 | yes | — |
| 24 | C3×C32⋊C9 | C32 | 1 | 2 | yes | — |
| 25 | 3−1+2⋊C9 | — | 3 | 12 | no | — |
| 26 | H(3,𝔽3)⋊C9 | — | 3 | 6 | no | — |
| 27 | #6866 | — | 3 | 6 | no | — |
| 28 | #6855 | — | 3 | 4 | no | — |
| 29 | #6855 | — | 3 | 2 | no | — |
| 30 | C34 | — | 3 | 6 | no | — |
| 31 | C34 | — | 3 | 3 | no | — |
| 32 | C34 | — | 3 | 1 | no | — |
| 33 | C34 | — | 3 | 1 | no | — |
| 34 | C34 | — | 3 | 2 | no | — |
| 35 | C34 | — | 3 | 2 | no | — |
| 36 | C34 | — | 3 | 1 | no | — |
| 37 | C34 | — | 3 | 2 | no | — |
| 38 | C34 | — | 3 | 4 | no | — |
| 39 | C34 | — | 9 | 4 | no | — |
| 40 | C34 | — | 9 | 2 | no | — |
| 41 | C34 | C3×C9 | 1 | 3 | yes | complemented |
| 42 | C34 | C33 | 1 | 1 | yes | characteristic |
| 43 | C3×3−1+2 | — | 3 | 4 | no | — |
| 44 | C3×3−1+2 | — | 3 | 4 | no | — |
| 45 | C3×3−1+2 | — | 3 | 2 | no | — |
| 46 | C3×3−1+2 | — | 3 | 4 | no | — |
| 47 | C3×3−1+2 | — | 3 | 8 | no | — |
| 48 | C3×3−1+2 | C3×C9 | 1 | 6 | yes | complemented |
| 49 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 50 | C3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 51 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 52 | C3×H(3,𝔽3) | C3×C9 | 1 | 3 | yes | — |
| 53 | C3×H(3,𝔽3) | H(3,𝔽3) | 1 | 3 | yes | — |
| 54 | C3×H(3,𝔽3) | 3−1+2 | 1 | 6 | yes | — |
| 55 | C32×C9 | — | 3 | 9 | no | — |
| 56 | C32×C9 | — | 3 | 1 | no | — |
| 57 | C32×C9 | — | 3 | 2 | no | — |
| 58 | C32×C9 | — | 3 | 2 | no | — |
| 59 | C32×C9 | — | 3 | 2 | no | — |
| 60 | C32×C9 | — | 3 | 2 | no | — |
| 61 | C32×C9 | — | 3 | 3 | no | — |
| 62 | C32×C9 | — | 3 | 6 | no | — |
| 63 | C32×C9 | — | 3 | 2 | no | — |
| 64 | C32×C9 | — | 3 | 4 | no | — |
| 65 | C32×C9 | — | 3 | 4 | no | — |
| 66 | C32×C9 | — | 9 | 2 | no | — |
| 67 | C32×C9 | — | 9 | 6 | no | — |
| 68 | C32×C9 | — | 9 | 3 | no | — |
| 69 | C32×C9 | — | 9 | 4 | no | — |
| 70 | C32×C9 | H(3,𝔽3) | 1 | 2 | yes | — |
| 71 | C32×C9 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 72 | AΣL(1,𝔽27) | — | 3 | 27 | no | — |
| 73 | C32⋊C9 | — | 3 | 18 | no | — |
| 74 | C32⋊C9 | — | 3 | 18 | no | — |
| 75 | C32⋊C9 | — | 3 | 2 | no | — |
| 76 | C32⋊C9 | — | 3 | 2 | no | — |
| 77 | C32⋊C9 | — | 3 | 9 | no | — |
| 78 | C32⋊C9 | — | 3 | 4 | no | — |
| 79 | C32⋊C9 | — | 3 | 4 | no | — |
| 80 | C32⋊C9 | — | 3 | 4 | no | — |
| 81 | C32⋊C9 | H(3,𝔽3) | 1 | 6 | yes | — |
| 82 | C33 | — | 3 | 4 | no | — |
| 83 | C33 | — | 3 | 1 | no | — |
| 84 | C33 | — | 3 | 2 | no | — |
| 85 | C33 | — | 3 | 2 | no | — |
| 86 | C33 | — | 3 | 18 | no | — |
| 87 | C33 | — | 3 | 9 | no | — |
| 88 | C33 | — | 3 | 3 | no | — |
| 89 | C33 | — | 3 | 3 | no | — |
| 90 | C33 | — | 3 | 3 | no | — |
| 91 | C33 | — | 3 | 6 | no | — |
| 92 | C33 | — | 3 | 3 | no | — |
| 93 | C33 | — | 3 | 6 | no | — |
| 94 | C33 | — | 3 | 3 | no | — |
| 95 | C33 | — | 3 | 6 | no | — |
| 96 | C33 | — | 9 | 6 | no | — |
| 97 | C33 | — | 9 | 6 | no | — |
| 98 | C33 | — | 9 | 12 | no | — |
| 99 | C33 | — | 9 | 12 | no | — |
| 100 | C33 | — | 9 | 24 | no | — |
| 101 | C33 | — | 9 | 6 | no | — |
| 102 | C33 | — | 9 | 3 | no | — |
| 103 | C33 | — | 9 | 12 | no | — |
| 104 | C33 | — | 9 | 6 | no | — |
| 105 | C33 | — | 9 | 12 | no | — |
| 106 | C33 | — | 9 | 4 | no | — |
| 107 | C33 | — | 9 | 4 | no | — |
| 108 | C33 | — | 9 | 1 | no | — |
| 109 | C33 | — | 9 | 4 | no | — |
| 110 | C33 | — | 9 | 2 | no | — |
| 111 | C33 | — | 9 | 2 | no | — |
| 112 | C33 | — | 9 | 1 | no | — |
| 113 | C33 | — | 9 | 2 | no | — |
| 114 | C33 | — | 9 | 2 | no | — |
| 115 | C33 | — | 9 | 2 | no | — |
| 116 | C33 | C32×C9 | 1 | 1 | yes | characteristic |
| 117 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 118 | C33 | C3×H(3,𝔽3) | 1 | 2 | yes | — |
| 119 | C33 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 120 | C33 | C3×3−1+2 | 1 | 2 | yes | — |
| 121 | C33 | #737 | 1 | 4 | yes | — |
| 122 | C33 | #737 | 1 | 2 | yes | — |
| 123 | 3−1+2 | — | 3 | 6 | no | — |
| 124 | 3−1+2 | — | 3 | 12 | no | — |
| 125 | 3−1+2 | — | 9 | 8 | no | — |
| 126 | 3−1+2 | — | 9 | 4 | no | — |
| 127 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 128 | H(3,𝔽3) | — | 3 | 12 | no | — |
| 129 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 130 | H(3,𝔽3) | C32⋊C9 | 1 | 9 | yes | complemented |
| 131 | C3×C9 | — | 3 | 1 | no | — |
| 132 | C3×C9 | — | 3 | 2 | no | — |
| 133 | C3×C9 | — | 3 | 6 | no | — |
| 134 | C3×C9 | — | 3 | 2 | no | — |
| 135 | C3×C9 | — | 3 | 6 | no | — |
| 136 | C3×C9 | — | 3 | 6 | no | — |
| 137 | C3×C9 | — | 3 | 2 | no | — |
| 138 | C3×C9 | — | 3 | 4 | no | — |
| 139 | C3×C9 | — | 9 | 3 | no | — |
| 140 | C3×C9 | — | 9 | 6 | no | — |
| 141 | C3×C9 | — | 9 | 27 | no | — |
| 142 | C3×C9 | — | 9 | 2 | no | — |
| 143 | C3×C9 | — | 9 | 2 | no | — |
| 144 | C3×C9 | — | 9 | 4 | no | — |
| 145 | C3×C9 | — | 9 | 2 | no | — |
| 146 | C3×C9 | — | 9 | 4 | no | — |
| 147 | C3×C9 | — | 9 | 4 | no | — |
| 148 | C3×C9 | — | 9 | 6 | no | — |
| 149 | C3×C9 | — | 9 | 12 | no | — |
| 150 | C3×C9 | — | 9 | 8 | no | — |
| 151 | C3×C9 | — | 9 | 4 | no | — |
| 152 | C3×C9 | — | 9 | 4 | no | — |
| 153 | C3×C9 | AΣL(1,𝔽27) | 1 | 3 | yes | complemented |
| 154 | C32 | — | 3 | 2 | no | — |
| 155 | C32 | — | 3 | 4 | no | — |
| 156 | C32 | — | 3 | 4 | no | — |
| 157 | C32 | — | 3 | 8 | no | — |
| 158 | C32 | — | 3 | 1 | no | — |
| 159 | C32 | — | 3 | 2 | no | — |
| 160 | C32 | — | 3 | 2 | no | — |
| 161 | C32 | — | 3 | 4 | no | — |
| 162 | C32 | — | 3 | 1 | no | — |
| 163 | C32 | — | 3 | 2 | no | — |
| 164 | C32 | — | 3 | 2 | no | — |
| 165 | C32 | — | 3 | 4 | no | — |
| 166 | C32 | — | 3 | 9 | no | — |
| 167 | C32 | — | 3 | 9 | no | — |
| 168 | C32 | — | 9 | 9 | no | — |
| 169 | C32 | — | 9 | 18 | no | — |
| 170 | C32 | — | 9 | 36 | no | — |
| 171 | C32 | — | 9 | 18 | no | — |
| 172 | C32 | — | 9 | 6 | no | — |
| 173 | C32 | — | 9 | 6 | no | — |
| 174 | C32 | — | 9 | 12 | no | — |
| 175 | C32 | — | 9 | 12 | no | — |
| 176 | C32 | — | 9 | 3 | no | — |
| 177 | C32 | — | 9 | 3 | no | — |
| 178 | C32 | — | 9 | 6 | no | — |
| 179 | C32 | — | 9 | 6 | no | — |
| 180 | C32 | — | 9 | 3 | no | — |
| 181 | C32 | — | 9 | 3 | no | — |
| 182 | C32 | — | 9 | 6 | no | — |
| 183 | C32 | — | 9 | 6 | no | — |
| 184 | C32 | C3×C32⋊C9 | 1 | 1 | yes | characteristic |
| 185 | C32 | C9×H(3,𝔽3) | 1 | 2 | yes | — |
| 186 | C32 | C9×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 187 | C32 | C32⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 188 | C32 | C9⋊H(3,𝔽3) | 1 | 4 | yes | — |
| 189 | C32 | C9⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 190 | C32 | #6893 | 1 | 2 | yes | — |
| 191 | C32 | C3×AΣL(1,𝔽27) | 1 | 1 | yes | characteristic |
| 192 | C32 | #6908 | 1 | 2 | yes | — |
| 193 | C32 | #6909 | 1 | 2 | yes | — |
| 194 | C32 | #6913 | 1 | 4 | yes | — |
| 195 | C9 | — | 3 | 3 | no | — |
| 196 | C9 | — | 9 | 6 | no | — |
| 197 | C9 | — | 9 | 2 | no | — |
| 198 | C9 | — | 9 | 6 | no | — |
| 199 | C9 | — | 9 | 6 | no | — |
| 200 | C9 | — | 27 | 4 | no | — |
| 201 | C3 | — | 3 | 1 | no | — |
| 202 | C3 | — | 3 | 2 | no | — |
| 203 | C3 | — | 3 | 4 | no | — |
| 204 | C3 | — | 3 | 2 | no | — |
| 205 | C3 | — | 9 | 9 | no | — |
| 206 | C3 | — | 9 | 9 | no | — |
| 207 | C3 | #10624555 | 1 | 1 | yes | characteristic |
| 208 | C3 | C9×AΣL(1,𝔽27) | 1 | 1 | yes | characteristic |
| 209 | C3 | AΣL(1,𝔽27)⋊C9 | 1 | 2 | yes | — |
| 210 | C3 | #10624742 | 1 | 1 | yes | characteristic |
| 211 | C3 | #10624746 | 1 | 2 | yes | — |
| 212 | C3 | #10624794 | 1 | 2 | yes | — |
| 213 | C3 | #10624797 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | C3×H(3,𝔽3) |
| commutator | C33 | C32×C9 |
| Frattini, cosocle | C34 | C33 |
| prime radical | C32×AΣL(1,𝔽27) | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C9×H(3,𝔽3) | C3 | semidirect |
| 3. C32×C32⋊C9 | C3 | semidirect |
| 4. C3×3−1+2⋊C9 | C3 | semidirect |
| 5. C3×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 6. C3×#6866 | C3 | semidirect |
| 7. #10624558 | C3 | semidirect |
| 16. C3×AΣL(1,𝔽27) | C9 | semidirect |
| 41. C34 | C3×C9 | semidirect |
| 48. C3×3−1+2 | C3×C9 | semidirect |
| 130. H(3,𝔽3) | C32⋊C9 | semidirect |
| 153. C3×C9 | AΣL(1,𝔽27) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×AΣL(1,𝔽27) | 3 |
| C3×C9×H(3,𝔽3) | 3 |
| C32×C32⋊C9 | 3 |
| C3×3−1+2⋊C9 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| C3×#6866 | 3 |
| #10624558 | 3 |