Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
543 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 38 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⋊#5948 | — | 3 | 1 | no | maximal |
| 3 | C22×#3701 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | A4×Dic9 | — | 4 | 1 | no | maximal |
| 5 | #88615 | — | 4 | 4 | no | maximal |
| 6 | C3×C22⋊SL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 7 | C22×Q8×C9 | C2×C3 | 1 | 1 | yes | characteristic |
| 8 | C2×#3701 | — | 3 | 4 | no | — |
| 9 | C2×#3701 | — | 3 | 4 | no | — |
| 10 | C22×D36 | — | 3 | 1 | no | — |
| 11 | C22×C4×D9 | — | 3 | 1 | no | — |
| 12 | C22⋊(Q8⋊C9) | — | 2 | 1 | no | — |
| 13 | C2×C9×A4 | — | 4 | 1 | no | — |
| 14 | C9×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 15 | C22×#239 | — | 3 | 1 | no | — |
| 16 | #5948 | — | 9 | 1 | no | — |
| 17 | Dic3×A4 | — | 12 | 1 | no | — |
| 18 | C3⋊(#41⋊C3) | — | 12 | 4 | no | — |
| 19 | C23×D9 | — | 3 | 1 | no | — |
| 20 | C2×Q8×C9 | — | 3 | 4 | no | — |
| 21 | C22×C4×C9 | — | 3 | 1 | no | — |
| 22 | C22×Dic9 | A4 | 1 | 1 | yes | characteristic |
| 23 | #3701 | — | 3 | 12 | no | — |
| 24 | #3701 | — | 3 | 4 | no | — |
| 25 | #3701 | — | 3 | 4 | no | — |
| 26 | #3701 | A4 | 1 | 4 | yes | complemented |
| 27 | C2×D36 | — | 3 | 12 | no | — |
| 28 | C2×C4×D9 | — | 3 | 6 | no | — |
| 29 | C2×C4×D9 | — | 3 | 6 | no | — |
| 30 | C9×A4 | — | 4 | 1 | no | — |
| 31 | C3×Dic9 | — | 16 | 1 | no | — |
| 32 | C22×C3×Q8 | C3×D3 | 1 | 1 | yes | characteristic |
| 33 | C2×#239 | — | 9 | 4 | no | — |
| 34 | C2×#239 | — | 9 | 4 | no | — |
| 35 | C22×D12 | — | 9 | 1 | no | — |
| 36 | C22×C4×D3 | — | 9 | 1 | no | — |
| 37 | C22⋊SL(2,𝔽3) | — | 2 | 1 | no | — |
| 38 | C22⋊SL(2,𝔽3) | D9 | 1 | 1 | yes | characteristic, complemented |
| 39 | C2×C3×A4 | — | 4 | 1 | no | — |
| 40 | C3×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 41 | C23×C9 | C2×A4 | 1 | 1 | yes | characteristic |
| 42 | C22×D9 | — | 3 | 6 | no | — |
| 43 | C22×D9 | — | 6 | 3 | no | — |
| 44 | C22×D9 | — | 6 | 1 | no | — |
| 45 | C2×C22⋊C9 | — | 8 | 1 | no | — |
| 46 | Q8×C9 | — | 3 | 4 | no | — |
| 47 | Q8×C9 | C2×A4 | 1 | 4 | yes | — |
| 48 | C2×C4×C9 | — | 3 | 6 | no | — |
| 49 | C2×Dic9 | — | 3 | 1 | no | — |
| 50 | C2×Dic9 | — | 3 | 1 | no | — |
| 51 | D36 | — | 3 | 4 | no | — |
| 52 | D36 | — | 3 | 12 | no | — |
| 53 | C4×D9 | — | 3 | 12 | no | — |
| 54 | C4×D9 | — | 3 | 4 | no | — |
| 55 | Q8⋊C9 | — | 8 | 4 | no | — |
| 56 | C22×#41 | — | 9 | 1 | no | Sylow |
| 57 | C2×C3×C9 | — | 16 | 1 | no | — |
| 58 | C23×D3 | — | 9 | 1 | no | — |
| 59 | C2×C3×Q8 | — | 3 | 4 | no | — |
| 60 | C22×C3×C4 | — | 3 | 1 | no | — |
| 61 | C22×Dic3 | — | 3 | 1 | no | — |
| 62 | #239 | — | 3 | 4 | no | — |
| 63 | #239 | — | 9 | 4 | no | — |
| 64 | #239 | — | 9 | 12 | no | — |
| 65 | #239 | — | 9 | 4 | no | — |
| 66 | C2×D12 | — | 9 | 12 | no | — |
| 67 | C2×C4×D3 | — | 9 | 6 | no | — |
| 68 | C2×C4×D3 | — | 9 | 6 | no | — |
| 69 | #41⋊C3 | — | 36 | 4 | no | — |
| 70 | C4×A4 | — | 36 | 1 | no | — |
| 71 | C3×A4 | — | 4 | 1 | no | — |
| 72 | C3×Dic3 | — | 48 | 1 | no | — |
| 73 | C22×C9 | — | 3 | 1 | no | — |
| 74 | C22×C9 | — | 3 | 1 | no | — |
| 75 | C22×C9 | #41⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 76 | C2×D9 | — | 3 | 4 | no | — |
| 77 | C2×D9 | — | 6 | 6 | no | — |
| 78 | C2×D9 | — | 6 | 6 | no | — |
| 79 | C22⋊C9 | — | 8 | 1 | no | — |
| 80 | C4×C9 | — | 3 | 4 | no | — |
| 81 | Dic9 | — | 3 | 1 | no | — |
| 82 | Dic9 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 83 | C2×#41 | — | 27 | 4 | no | — |
| 84 | C2×#41 | — | 27 | 4 | no | — |
| 85 | C22×Q8 | C3×D9 | 1 | 1 | yes | characteristic |
| 86 | C22×D4 | — | 27 | 1 | no | — |
| 87 | C23×C4 | — | 27 | 1 | no | — |
| 88 | C3×C9 | — | 16 | 1 | no | Sylow |
| 89 | C23×C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 90 | C22×D3 | — | 9 | 6 | no | — |
| 91 | C22×D3 | — | 18 | 3 | no | — |
| 92 | C22×D3 | — | 18 | 1 | no | — |
| 93 | C2×A4 | — | 4 | 1 | no | — |
| 94 | C2×A4 | — | 8 | 1 | no | — |
| 95 | C3×Q8 | — | 3 | 4 | no | — |
| 96 | C3×Q8 | D3×A4 | 1 | 4 | yes | — |
| 97 | C2×C3×C4 | — | 3 | 6 | no | — |
| 98 | C2×Dic3 | — | 9 | 1 | no | — |
| 99 | C2×Dic3 | — | 9 | 1 | no | — |
| 100 | D12 | — | 9 | 4 | no | — |
| 101 | D12 | — | 9 | 12 | no | — |
| 102 | C4×D3 | — | 9 | 4 | no | — |
| 103 | C4×D3 | — | 9 | 12 | no | — |
| 104 | SL(2,𝔽3) | — | 4 | 4 | no | — |
| 105 | SL(2,𝔽3) | — | 8 | 4 | no | — |
| 106 | C2×C32 | — | 16 | 1 | no | — |
| 107 | C2×C9 | — | 3 | 1 | no | — |
| 108 | C2×C9 | — | 3 | 1 | no | — |
| 109 | C2×C9 | — | 32 | 1 | no | — |
| 110 | C2×C9 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 111 | D9 | — | 6 | 4 | no | — |
| 112 | C24 | — | 27 | 1 | no | — |
| 113 | #41 | — | 9 | 4 | no | — |
| 114 | #41 | — | 27 | 4 | no | — |
| 115 | #41 | — | 27 | 12 | no | — |
| 116 | #41 | — | 27 | 4 | no | — |
| 117 | C2×Q8 | — | 3 | 4 | no | — |
| 118 | C2×D4 | — | 27 | 12 | no | — |
| 119 | C22×C4 | — | 3 | 1 | no | — |
| 120 | C22×C4 | — | 9 | 1 | no | — |
| 121 | C22×C4 | — | 27 | 6 | no | — |
| 122 | C22×C4 | — | 27 | 6 | no | — |
| 123 | C22×C3 | — | 3 | 1 | no | — |
| 124 | C22×C3 | — | 3 | 1 | no | — |
| 125 | C22×C3 | C3⋊(#41⋊C3) | 1 | 1 | yes | characteristic |
| 126 | C2×D3 | — | 9 | 4 | no | — |
| 127 | C2×D3 | — | 18 | 6 | no | — |
| 128 | C2×D3 | — | 18 | 6 | no | — |
| 129 | A4 | — | 4 | 1 | no | — |
| 130 | A4 | — | 8 | 1 | no | — |
| 131 | C3×C4 | — | 3 | 4 | no | — |
| 132 | C3×C4 | — | 144 | 1 | no | — |
| 133 | Dic3 | — | 3 | 1 | no | — |
| 134 | Dic3 | — | 9 | 1 | no | — |
| 135 | C32 | — | 16 | 1 | no | — |
| 136 | C9 | — | 32 | 1 | no | — |
| 137 | C9 | #5948 | 1 | 1 | yes | characteristic, complemented |
| 138 | C23 | — | 27 | 6 | no | — |
| 139 | C23 | — | 54 | 3 | no | — |
| 140 | C23 | — | 54 | 1 | no | — |
| 141 | C23 | A4×D9 | 1 | 1 | yes | characteristic |
| 142 | Q8 | — | 3 | 4 | no | — |
| 143 | Q8 | A4×D9 | 1 | 4 | yes | — |
| 144 | D4 | — | 27 | 4 | no | — |
| 145 | D4 | — | 27 | 12 | no | — |
| 146 | C2×C4 | — | 3 | 6 | no | — |
| 147 | C2×C4 | — | 27 | 4 | no | — |
| 148 | C2×C4 | — | 27 | 12 | no | — |
| 149 | C2×C4 | — | 27 | 1 | no | — |
| 150 | C2×C4 | — | 27 | 1 | no | — |
| 151 | C2×C3 | — | 3 | 1 | no | — |
| 152 | C2×C3 | — | 3 | 1 | no | — |
| 153 | C2×C3 | — | 16 | 1 | no | — |
| 154 | C2×C3 | — | 32 | 1 | no | — |
| 155 | C2×C3 | D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 156 | D3 | — | 18 | 4 | no | — |
| 157 | C22 | — | 3 | 1 | no | — |
| 158 | C22 | — | 3 | 1 | no | — |
| 159 | C22 | — | 27 | 4 | no | — |
| 160 | C22 | — | 54 | 6 | no | — |
| 161 | C22 | — | 54 | 6 | no | — |
| 162 | C22 | #88615 | 1 | 1 | yes | characteristic, complemented |
| 163 | C4 | — | 3 | 4 | no | — |
| 164 | C4 | — | 9 | 1 | no | — |
| 165 | C4 | — | 27 | 1 | no | — |
| 166 | C3 | — | 16 | 1 | no | — |
| 167 | C3 | — | 32 | 1 | no | — |
| 168 | C3 | C3⋊#5948 | 1 | 1 | yes | characteristic |
| 169 | C2 | — | 3 | 1 | no | — |
| 170 | C2 | — | 3 | 1 | no | — |
| 171 | C2 | — | 54 | 4 | no | — |
| 172 | C2 | D9×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C22×Q8×C9 | C2×C3 |
| Frattini | C2×C3 | D3×C22⋊A4 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ C22×Q8×C9 | |
| 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 |
| 3. C22×#3701 | C3 | semidirect |
| 26. #3701 | A4 | semidirect |
| 38. C22⋊SL(2,𝔽3) | D9 | semidirect |
| 75. C22×C9 | #41⋊C3 | semidirect |
| 137. C9 | #5948 | semidirect |
| 162. C22 | #88615 | 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⋊#5948 | 3 |
| C22×#3701 | 3 |
| A4×Dic9 | 4 |
| #88615 | 4 |