Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
287 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 29 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 | C9×C22⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊#5931 | — | 3 | 1 | no | maximal |
| 3 | #10593106 | — | 3 | 2 | no | maximal |
| 4 | #10588908 | — | 3 | 1 | no | maximal |
| 5 | C2×C9⋊S4 | — | 4 | 1 | no | maximal |
| 6 | C9⋊GL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 7 | C3×C22⋊SL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 8 | C22×Q8×C9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C22⋊(Q8⋊C9) | D3 | 1 | 2 | yes | complemented |
| 10 | C2×#3675 | — | 3 | 2 | no | — |
| 11 | #63492 | — | 3 | 1 | no | — |
| 12 | #63395 | — | 3 | 2 | no | — |
| 13 | #63389 | — | 3 | 1 | no | — |
| 14 | C2×C9×A4 | — | 4 | 1 | no | — |
| 15 | C9⋊S4 | — | 4 | 2 | no | — |
| 16 | C9×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 17 | C9×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 18 | #5931 | — | 9 | 3 | no | — |
| 19 | #5227 | — | 9 | 1 | no | — |
| 20 | C2×C3⋊S4 | — | 12 | 1 | no | — |
| 21 | C3⋊GL(2,𝔽3) | — | 12 | 2 | no | — |
| 22 | C2×(C22⋊C9)⋊C2 | — | 12 | 2 | no | — |
| 23 | C2×Q8×C9 | — | 3 | 2 | no | — |
| 24 | C2×Q8×C9 | — | 6 | 1 | no | — |
| 25 | C22×C4×C9 | — | 3 | 1 | no | — |
| 26 | C2×#611 | — | 6 | 1 | no | — |
| 27 | C2×D36 | — | 3 | 1 | no | — |
| 28 | #3689 | — | 12 | 4 | no | — |
| 29 | #3675 | — | 6 | 2 | no | — |
| 30 | #3675 | — | 6 | 2 | no | — |
| 31 | (C2×D9)⋊C4 | — | 6 | 1 | no | — |
| 32 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 33 | C2×C9⋊C8 | — | 3 | 2 | no | — |
| 34 | C2×C3⋊D9 | — | 16 | 1 | no | — |
| 35 | C9×A4 | — | 4 | 1 | no | — |
| 36 | C22×C3×Q8 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 37 | C22⋊SL(2,𝔽3) | D9 | 1 | 3 | yes | complemented |
| 38 | C2×#215 | — | 9 | 2 | no | — |
| 39 | #930 | — | 9 | 1 | no | — |
| 40 | #835 | — | 9 | 2 | no | — |
| 41 | #830 | — | 9 | 1 | no | — |
| 42 | C2×C3×A4 | — | 4 | 1 | no | — |
| 43 | C3⋊S4 | — | 12 | 2 | no | — |
| 44 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 45 | C3×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 46 | C23×C9 | S4 | 1 | 1 | yes | characteristic |
| 47 | C22×D9 | — | 6 | 1 | no | — |
| 48 | C2×C22⋊C9 | — | 4 | 2 | no | — |
| 49 | (C22⋊C9)⋊C2 | — | 12 | 4 | no | — |
| 50 | Q8×C9 | — | 2 | 1 | no | — |
| 51 | Q8×C9 | — | 3 | 2 | no | — |
| 52 | Q8×C9 | — | 6 | 1 | no | — |
| 53 | Q8×C9 | S4 | 1 | 2 | yes | complemented |
| 54 | C2×C4×C9 | — | 3 | 1 | no | — |
| 55 | C2×C4×C9 | — | 3 | 1 | no | — |
| 56 | C2×C4×C9 | — | 6 | 2 | no | — |
| 57 | #611 | — | 12 | 1 | no | — |
| 58 | #611 | — | 12 | 1 | no | — |
| 59 | C2×Dic9 | — | 6 | 1 | no | — |
| 60 | D36 | — | 6 | 2 | no | — |
| 61 | Q8⋊C9 | — | 4 | 4 | no | — |
| 62 | Q8⋊C9 | — | 8 | 2 | no | — |
| 63 | C9⋊C8 | — | 6 | 2 | no | — |
| 64 | #448 | — | 27 | 1 | no | Sylow |
| 65 | C2×C3×C9 | — | 16 | 1 | no | — |
| 66 | C3⋊D9 | — | 16 | 2 | no | — |
| 67 | C2×S4 | — | 36 | 3 | no | — |
| 68 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 69 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 70 | C22×C3×C4 | — | 3 | 1 | no | — |
| 71 | C2×#67 | — | 18 | 1 | no | — |
| 72 | C2×D12 | — | 9 | 1 | no | — |
| 73 | GL(2,𝔽3) | — | 36 | 6 | no | — |
| 74 | #215 | — | 18 | 2 | no | — |
| 75 | #215 | — | 18 | 2 | no | — |
| 76 | (C2×D3)⋊C4 | — | 18 | 1 | no | — |
| 77 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 78 | C2×C3⋊C8 | — | 9 | 2 | no | — |
| 79 | C2×C3⋊D3 | — | 48 | 1 | no | — |
| 80 | C3×A4 | — | 4 | 1 | no | — |
| 81 | C22×C9 | — | 3 | 1 | no | — |
| 82 | C22×C9 | — | 3 | 1 | no | — |
| 83 | C22×C9 | GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 84 | C2×D9 | — | 12 | 1 | no | — |
| 85 | C2×D9 | — | 12 | 1 | no | — |
| 86 | C2×D9 | — | 12 | 1 | no | — |
| 87 | C2×D9 | — | 48 | 2 | no | — |
| 88 | C22⋊C9 | — | 4 | 2 | no | — |
| 89 | C4×C9 | — | 3 | 2 | no | — |
| 90 | C4×C9 | — | 6 | 1 | no | — |
| 91 | Dic9 | — | 12 | 1 | no | — |
| 92 | C22×Q8 | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 93 | C2×SD16 | — | 27 | 2 | no | — |
| 94 | #121 | — | 27 | 1 | no | — |
| 95 | #103 | — | 27 | 2 | no | — |
| 96 | C22⋊C8 | — | 27 | 1 | no | — |
| 97 | C3×C9 | — | 16 | 1 | no | Sylow |
| 98 | C23×C3 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 99 | C22×D3 | — | 18 | 1 | no | — |
| 100 | C2×A4 | — | 4 | 3 | no | — |
| 101 | S4 | — | 36 | 6 | no | — |
| 102 | C3×Q8 | — | 2 | 1 | no | — |
| 103 | C3×Q8 | — | 3 | 2 | no | — |
| 104 | C3×Q8 | — | 6 | 1 | no | — |
| 105 | C3×Q8 | C3⋊S4 | 1 | 2 | yes | — |
| 106 | C2×C3×C4 | — | 3 | 1 | no | — |
| 107 | C2×C3×C4 | — | 3 | 1 | no | — |
| 108 | C2×C3×C4 | — | 6 | 2 | no | — |
| 109 | #67 | — | 36 | 1 | no | — |
| 110 | #67 | — | 36 | 1 | no | — |
| 111 | C2×Dic3 | — | 18 | 1 | no | — |
| 112 | D12 | — | 18 | 2 | no | — |
| 113 | SL(2,𝔽3) | — | 4 | 6 | no | — |
| 114 | SL(2,𝔽3) | — | 8 | 3 | no | — |
| 115 | C3⋊C8 | — | 18 | 2 | no | — |
| 116 | C2×C32 | — | 16 | 1 | no | — |
| 117 | C3⋊D3 | — | 48 | 2 | no | — |
| 118 | C2×C9 | — | 3 | 1 | no | — |
| 119 | C2×C9 | — | 3 | 1 | no | — |
| 120 | C2×C9 | — | 16 | 2 | no | — |
| 121 | C2×C9 | #1020 | 1 | 1 | yes | characteristic |
| 122 | D9 | — | 24 | 1 | no | — |
| 123 | D9 | — | 48 | 4 | no | — |
| 124 | C2×Q8 | — | 3 | 2 | no | — |
| 125 | C2×Q8 | — | 6 | 1 | no | — |
| 126 | C2×D4 | — | 27 | 1 | no | — |
| 127 | C2×D4 | — | 54 | 1 | no | — |
| 128 | C22×C4 | — | 3 | 1 | no | — |
| 129 | SD16 | — | 54 | 2 | no | — |
| 130 | SD16 | — | 54 | 2 | no | — |
| 131 | C2×C8 | — | 27 | 2 | no | — |
| 132 | C4⋊C4 | — | 27 | 1 | no | — |
| 133 | C22⋊C4 | — | 54 | 1 | no | — |
| 134 | C22×C3 | — | 3 | 1 | no | — |
| 135 | C22×C3 | — | 3 | 1 | no | — |
| 136 | C22×C3 | C3⋊GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 137 | C2×D3 | — | 36 | 1 | no | — |
| 138 | C2×D3 | — | 36 | 1 | no | — |
| 139 | C2×D3 | — | 36 | 1 | no | — |
| 140 | C2×D3 | — | 144 | 3 | no | — |
| 141 | A4 | — | 4 | 3 | no | — |
| 142 | C3×C4 | — | 3 | 2 | no | — |
| 143 | C3×C4 | — | 6 | 1 | no | — |
| 144 | Dic3 | — | 36 | 1 | no | — |
| 145 | C32 | — | 16 | 1 | no | — |
| 146 | C9 | — | 16 | 2 | no | — |
| 147 | C9 | #5931 | 1 | 1 | yes | characteristic, complemented |
| 148 | C23 | — | 54 | 1 | no | — |
| 149 | C23 | C9⋊S4 | 1 | 1 | yes | characteristic |
| 150 | Q8 | — | 2 | 1 | no | — |
| 151 | Q8 | — | 3 | 2 | no | — |
| 152 | Q8 | — | 6 | 1 | no | — |
| 153 | Q8 | C9⋊S4 | 1 | 2 | yes | complemented |
| 154 | D4 | — | 54 | 2 | no | — |
| 155 | D4 | — | 108 | 1 | no | — |
| 156 | D4 | — | 108 | 1 | no | — |
| 157 | C2×C4 | — | 3 | 1 | no | — |
| 158 | C2×C4 | — | 3 | 1 | no | — |
| 159 | C2×C4 | — | 6 | 2 | no | — |
| 160 | C2×C4 | — | 54 | 1 | no | — |
| 161 | C8 | — | 54 | 2 | no | — |
| 162 | C2×C3 | — | 3 | 1 | no | — |
| 163 | C2×C3 | — | 3 | 1 | no | — |
| 164 | C2×C3 | — | 16 | 3 | no | — |
| 165 | C2×C3 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 166 | D3 | — | 72 | 1 | no | — |
| 167 | D3 | — | 144 | 6 | no | — |
| 168 | C22 | — | 3 | 1 | no | — |
| 169 | C22 | — | 3 | 1 | no | — |
| 170 | C22 | — | 108 | 1 | no | — |
| 171 | C22 | — | 108 | 1 | no | — |
| 172 | C22 | — | 108 | 1 | no | — |
| 173 | C22 | C9⋊GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 174 | C4 | — | 3 | 2 | no | — |
| 175 | C4 | — | 6 | 1 | no | — |
| 176 | C4 | — | 108 | 1 | no | — |
| 177 | C3 | — | 16 | 3 | no | — |
| 178 | C3 | C3⋊#5931 | 1 | 1 | yes | characteristic |
| 179 | C2 | — | 3 | 1 | no | — |
| 180 | C2 | — | 3 | 1 | no | — |
| 181 | C2 | — | 216 | 1 | no | — |
| 182 | C2 | C9⋊#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊#1020 |
| commutator, cosocle, hypocenter | C9×C22⋊SL(2,𝔽3) | C2 |
| Frattini | C2×C3 | C3⋊#1020 |
| Fitting | C22×Q8×C9 | D3 |
| socle | C23×C3 | C3⋊S4 |
| LCS | G ▷ C9×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×C22⋊SL(2,𝔽3) | C2 | semidirect |
| 8. C22×Q8×C9 | D3 | semidirect |
| 9. C22⋊(Q8⋊C9) | D3 | semidirect |
| 37. C22⋊SL(2,𝔽3) | D9 | semidirect |
| 53. Q8×C9 | S4 | semidirect |
| 83. C22×C9 | GL(2,𝔽3) | semidirect |
| 92. C22×Q8 | C3⋊D9 | semidirect |
| 147. C9 | #5931 | semidirect |
| 153. Q8 | C9⋊S4 | semidirect |
| 173. C22 | C9⋊GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C22⋊SL(2,𝔽3) | 2 |
| C3⋊#5931 | 3 |
| #10593106 | 3 |
| #10588908 | 3 |
| C2×C9⋊S4 | 4 |
| C9⋊GL(2,𝔽3) | 4 |