Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
185 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 17 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⋊#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(B(4,𝔽2)⋊C3) | — | 3 | 1 | no | maximal |
| 3 | #10588924 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×D9⋊A4 | — | 4 | 1 | no | maximal |
| 5 | C3×#997 | D3 | 1 | 1 | yes | characteristic |
| 6 | C9×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | #63703 | — | 2 | 1 | no | — |
| 8 | #63499 | — | 3 | 1 | no | — |
| 9 | #63393 | — | 3 | 1 | no | — |
| 10 | C2×C9⋊A4 | — | 4 | 1 | no | — |
| 11 | C2×C9⋊A4 | — | 8 | 1 | no | — |
| 12 | D9⋊A4 | — | 4 | 2 | no | — |
| 13 | C9⋊SL(2,𝔽3) | — | 8 | 1 | no | — |
| 14 | #5244 | — | 3 | 1 | no | — |
| 15 | B(4,𝔽2)⋊C3 | — | 9 | 1 | no | — |
| 16 | C2×D3×A4 | — | 12 | 1 | no | — |
| 17 | C23×D9 | A4 | 1 | 1 | yes | characteristic, complemented |
| 18 | C9×#41 | — | 6 | 1 | no | — |
| 19 | C2×D4×C9 | — | 3 | 1 | no | — |
| 20 | C2×D4×C9 | — | 6 | 1 | no | — |
| 21 | C2×#611 | — | 6 | 1 | no | — |
| 22 | #3676 | — | 3 | 1 | no | — |
| 23 | (C2×D9)⋊C4 | — | 6 | 1 | no | — |
| 24 | C2×AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 25 | C9⋊A4 | — | 4 | 1 | no | — |
| 26 | C9⋊A4 | — | 8 | 1 | no | — |
| 27 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 28 | #997 | — | 3 | 1 | no | — |
| 29 | #937 | — | 9 | 1 | no | — |
| 30 | #834 | — | 9 | 1 | no | — |
| 31 | C2×C3×A4 | — | 4 | 1 | no | — |
| 32 | C2×C3×A4 | — | 8 | 1 | no | — |
| 33 | D3×A4 | — | 12 | 2 | no | — |
| 34 | C3×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 35 | C23×C9 | — | 2 | 1 | no | — |
| 36 | C23×C9 | — | 3 | 1 | no | — |
| 37 | C23×C9 | C2×A4 | 1 | 1 | yes | characteristic |
| 38 | C22×D9 | — | 4 | 2 | no | — |
| 39 | C22×D9 | — | 6 | 1 | no | — |
| 40 | C2×C22⋊C9 | — | 8 | 2 | no | — |
| 41 | C2×C22⋊C9 | — | 8 | 1 | no | — |
| 42 | Q8×C9 | — | 2 | 1 | no | — |
| 43 | D4×C9 | — | 6 | 2 | no | — |
| 44 | D4×C9 | — | 6 | 1 | no | — |
| 45 | C2×C4×C9 | — | 3 | 1 | no | — |
| 46 | C2×C4×C9 | — | 6 | 1 | no | — |
| 47 | #611 | — | 12 | 2 | no | — |
| 48 | C2×Dic9 | — | 6 | 1 | no | — |
| 49 | Q8⋊C9 | — | 8 | 2 | no | — |
| 50 | B(4,𝔽2) | — | 9 | 1 | no | Sylow |
| 51 | C2×3−1+2 | — | 16 | 1 | no | — |
| 52 | AGL(1,ℤ/9) | — | 16 | 2 | no | — |
| 53 | C23×D3 | — | 3 | 1 | no | — |
| 54 | C22×A4 | — | 36 | 1 | no | — |
| 55 | C3×#41 | — | 6 | 1 | no | — |
| 56 | C2×C3×D4 | — | 3 | 1 | no | — |
| 57 | C2×C3×D4 | — | 6 | 1 | no | — |
| 58 | C2×#67 | — | 18 | 1 | no | — |
| 59 | #217 | — | 9 | 1 | no | — |
| 60 | (C2×D3)⋊C4 | — | 18 | 1 | no | — |
| 61 | C2×C3×D3 | — | 48 | 1 | no | — |
| 62 | C3×A4 | — | 4 | 1 | no | — |
| 63 | C3×A4 | — | 8 | 1 | no | — |
| 64 | C22×C9 | — | 3 | 1 | no | — |
| 65 | C22×C9 | — | 4 | 1 | no | — |
| 66 | C22×C9 | — | 6 | 1 | no | — |
| 67 | C22×C9 | — | 8 | 1 | no | — |
| 68 | C22×C9 | — | 12 | 1 | no | — |
| 69 | C2×D9 | — | 4 | 1 | no | — |
| 70 | C2×D9 | — | 6 | 2 | no | — |
| 71 | C2×D9 | — | 12 | 1 | no | — |
| 72 | C22⋊C9 | — | 8 | 1 | no | — |
| 73 | C22⋊C9 | — | 8 | 2 | no | — |
| 74 | C4×C9 | — | 6 | 1 | no | — |
| 75 | Dic9 | — | 12 | 1 | no | — |
| 76 | H(5,𝔽2) | AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 77 | #120 | — | 27 | 1 | no | — |
| 78 | #99 | — | 27 | 1 | no | — |
| 79 | 3−1+2 | — | 16 | 1 | no | Sylow |
| 80 | C23×C3 | — | 2 | 1 | no | — |
| 81 | C23×C3 | — | 3 | 1 | no | — |
| 82 | C23×C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 83 | C22×D3 | — | 12 | 2 | no | — |
| 84 | C22×D3 | — | 18 | 1 | no | — |
| 85 | C2×A4 | — | 12 | 1 | no | — |
| 86 | C2×A4 | — | 24 | 1 | no | — |
| 87 | C2×A4 | — | 36 | 2 | no | — |
| 88 | C3×Q8 | — | 2 | 1 | no | — |
| 89 | C3×D4 | — | 6 | 2 | no | — |
| 90 | C3×D4 | — | 6 | 1 | no | — |
| 91 | C2×C3×C4 | — | 3 | 1 | no | — |
| 92 | C2×C3×C4 | — | 6 | 1 | no | — |
| 93 | #67 | — | 36 | 2 | no | — |
| 94 | C2×Dic3 | — | 18 | 1 | no | — |
| 95 | SL(2,𝔽3) | — | 24 | 1 | no | — |
| 96 | C2×C32 | — | 16 | 1 | no | — |
| 97 | C3×D3 | — | 48 | 2 | no | — |
| 98 | C2×C9 | — | 6 | 1 | no | — |
| 99 | C2×C9 | — | 12 | 1 | no | — |
| 100 | C2×C9 | — | 32 | 1 | no | — |
| 101 | C2×C9 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 102 | D9 | — | 4 | 2 | no | — |
| 103 | C24 | — | 9 | 1 | no | — |
| 104 | #41 | — | 6 | 1 | no | — |
| 105 | C2×D4 | — | 3 | 1 | no | — |
| 106 | C2×D4 | — | 6 | 1 | no | — |
| 107 | C2×D4 | — | 54 | 1 | no | — |
| 108 | C22⋊C4 | — | 27 | 1 | no | — |
| 109 | C22⋊C4 | — | 54 | 1 | no | — |
| 110 | C22×C3 | — | 3 | 1 | no | — |
| 111 | C22×C3 | — | 4 | 1 | no | — |
| 112 | C22×C3 | — | 6 | 1 | no | — |
| 113 | C22×C3 | — | 8 | 1 | no | — |
| 114 | C22×C3 | — | 12 | 1 | no | — |
| 115 | C22×C3 | — | 144 | 1 | no | — |
| 116 | C2×D3 | — | 12 | 1 | no | — |
| 117 | C2×D3 | — | 18 | 2 | no | — |
| 118 | C2×D3 | — | 36 | 1 | no | — |
| 119 | A4 | — | 12 | 1 | no | — |
| 120 | A4 | — | 24 | 1 | no | — |
| 121 | C3×C4 | — | 6 | 1 | no | — |
| 122 | Dic3 | — | 36 | 1 | no | — |
| 123 | C32 | — | 16 | 1 | no | — |
| 124 | C9 | — | 32 | 1 | no | — |
| 125 | C9 | B(4,𝔽2)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 126 | C23 | — | 2 | 1 | no | — |
| 127 | C23 | — | 3 | 1 | no | — |
| 128 | C23 | — | 36 | 2 | no | — |
| 129 | C23 | — | 54 | 1 | no | — |
| 130 | C23 | D9⋊A4 | 1 | 1 | yes | characteristic |
| 131 | Q8 | — | 2 | 1 | no | — |
| 132 | D4 | — | 6 | 2 | no | — |
| 133 | D4 | — | 6 | 1 | no | — |
| 134 | D4 | — | 108 | 2 | no | — |
| 135 | C2×C4 | — | 3 | 1 | no | — |
| 136 | C2×C4 | — | 6 | 1 | no | — |
| 137 | C2×C4 | — | 54 | 1 | no | — |
| 138 | C2×C3 | — | 6 | 1 | no | — |
| 139 | C2×C3 | — | 12 | 1 | no | — |
| 140 | C2×C3 | — | 48 | 1 | no | — |
| 141 | C2×C3 | — | 144 | 2 | no | — |
| 142 | C2×C3 | C3⋊(#120⋊C3) | 1 | 1 | yes | characteristic |
| 143 | D3 | — | 12 | 2 | no | — |
| 144 | C22 | — | 3 | 1 | no | — |
| 145 | C22 | — | 4 | 1 | no | — |
| 146 | C22 | — | 6 | 1 | no | — |
| 147 | C22 | — | 8 | 1 | no | — |
| 148 | C22 | — | 12 | 1 | no | — |
| 149 | C22 | — | 36 | 1 | no | — |
| 150 | C22 | — | 54 | 2 | no | — |
| 151 | C22 | — | 108 | 1 | no | — |
| 152 | C4 | — | 6 | 1 | no | — |
| 153 | C4 | — | 108 | 1 | no | — |
| 154 | C3 | — | 48 | 1 | no | — |
| 155 | C3 | C3⋊(B(4,𝔽2)⋊C3) | 1 | 1 | yes | characteristic |
| 156 | C2 | — | 6 | 1 | no | — |
| 157 | C2 | — | 12 | 1 | no | — |
| 158 | C2 | — | 36 | 2 | no | — |
| 159 | C2 | #11720760 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11720760 |
| commutator, Fitting, cosocle, hypocenter | C9×H(5,𝔽2) | C2×C3 |
| Frattini | C23×C3 | D3×A4 |
| socle | C2×C3 | C3⋊(#120⋊C3) |
| LCS | G ▷ C9×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9⋊#997 | C2 | semidirect |
| 3. #10588924 | C3 | semidirect |
| 6. C9×H(5,𝔽2) | C2×C3 | semidirect |
| 17. C23×D9 | A4 | semidirect |
| 76. H(5,𝔽2) | AGL(1,ℤ/9) | semidirect |
| 125. C9 | B(4,𝔽2)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9⋊#997 | 2 |
| C3⋊(B(4,𝔽2)⋊C3) | 3 |
| #10588924 | 3 |
| C2×D9⋊A4 | 4 |