Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
542 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 41 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×#63703 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10593109 | — | 3 | 3 | no | maximal |
| 3 | #10592943 | — | 3 | 1 | no | maximal |
| 4 | C2×C3⋊((C22⋊C9)⋊C2) | — | 4 | 3 | no | maximal |
| 5 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 6 | #64146 | — | 3 | 3 | no | — |
| 7 | #63703 | D3 | 1 | 3 | yes | complemented |
| 8 | #63661 | — | 3 | 3 | no | — |
| 9 | C2×C3×C22⋊C9 | — | 4 | 3 | no | — |
| 10 | C3⋊((C22⋊C9)⋊C2) | — | 4 | 6 | no | — |
| 11 | C3×Q8⋊C9 | — | 8 | 1 | no | — |
| 12 | #5244 | — | 9 | 3 | no | — |
| 13 | #5244 | — | 9 | 1 | no | — |
| 14 | C23×C3⋊D3 | — | 3 | 1 | no | — |
| 15 | C32×#41 | — | 6 | 1 | no | — |
| 16 | C2×C32×D4 | — | 3 | 3 | no | — |
| 17 | C2×#638 | — | 6 | 3 | no | — |
| 18 | C2×(C22⋊C9)⋊C2 | — | 12 | 9 | no | — |
| 19 | #3757 | — | 3 | 3 | no | — |
| 20 | #3752 | — | 6 | 3 | no | — |
| 21 | C2×C3⋊D9 | — | 16 | 1 | no | — |
| 22 | C3×C22⋊C9 | — | 4 | 3 | no | — |
| 23 | C3×H(5,𝔽2) | D9 | 1 | 3 | yes | complemented |
| 24 | C3×H(5,𝔽2) | C3⋊D3 | 1 | 1 | yes | characteristic |
| 25 | #937 | — | 9 | 9 | no | — |
| 26 | #937 | — | 9 | 3 | no | — |
| 27 | #834 | — | 9 | 9 | no | — |
| 28 | #834 | — | 9 | 3 | no | — |
| 29 | C23×C32 | — | 3 | 1 | no | — |
| 30 | C23×C32 | S4 | 1 | 3 | yes | — |
| 31 | C22×C3⋊D3 | — | 6 | 3 | no | — |
| 32 | C22×C3⋊D3 | — | 12 | 2 | no | — |
| 33 | C32×Q8 | — | 2 | 1 | no | — |
| 34 | C32×D4 | — | 6 | 3 | no | — |
| 35 | C2×C32×C4 | — | 3 | 3 | no | — |
| 36 | #638 | — | 12 | 6 | no | — |
| 37 | C2×C3⋊Dic3 | — | 6 | 3 | no | — |
| 38 | C2×C22⋊C9 | — | 4 | 9 | no | — |
| 39 | (C22⋊C9)⋊C2 | — | 12 | 18 | no | — |
| 40 | Q8⋊C9 | — | 8 | 3 | no | — |
| 41 | B(4,𝔽2) | — | 27 | 1 | no | Sylow |
| 42 | C2×C3×C9 | — | 16 | 1 | no | — |
| 43 | C3⋊D9 | — | 16 | 2 | no | — |
| 44 | C23×D3 | — | 9 | 1 | no | — |
| 45 | C23×D3 | — | 9 | 3 | no | — |
| 46 | C3×#41 | — | 6 | 3 | no | — |
| 47 | C3×#41 | — | 6 | 1 | no | — |
| 48 | C2×C3×D4 | — | 3 | 9 | no | — |
| 49 | C2×C3×D4 | — | 3 | 3 | no | — |
| 50 | C2×#67 | — | 18 | 9 | no | — |
| 51 | C2×#67 | — | 18 | 3 | no | — |
| 52 | #217 | — | 9 | 3 | no | — |
| 53 | #217 | — | 9 | 9 | no | — |
| 54 | (C2×D3)⋊C4 | — | 18 | 3 | no | — |
| 55 | (C2×D3)⋊C4 | — | 18 | 9 | no | — |
| 56 | C22×C32 | — | 3 | 3 | no | — |
| 57 | C22×C32 | — | 4 | 3 | no | — |
| 58 | C22×C32 | — | 12 | 1 | no | — |
| 59 | C2×C3⋊D3 | — | 6 | 6 | no | — |
| 60 | C2×C3⋊D3 | — | 12 | 3 | no | — |
| 61 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 62 | C32×C4 | — | 6 | 1 | no | — |
| 63 | C3⋊Dic3 | — | 12 | 3 | no | — |
| 64 | C2×D9 | — | 48 | 3 | no | — |
| 65 | C22⋊C9 | — | 4 | 9 | no | — |
| 66 | H(5,𝔽2) | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 67 | #120 | — | 27 | 3 | no | — |
| 68 | #99 | — | 27 | 3 | no | — |
| 69 | C3×C9 | — | 16 | 1 | no | Sylow |
| 70 | C23×C3 | — | 3 | 1 | no | — |
| 71 | C23×C3 | — | 3 | 3 | no | — |
| 72 | C23×C3 | (C22⋊C9)⋊C2 | 1 | 9 | yes | complemented |
| 73 | C23×C3 | C3⋊S4 | 1 | 3 | yes | — |
| 74 | C22×D3 | — | 18 | 3 | no | — |
| 75 | C22×D3 | — | 18 | 9 | no | — |
| 76 | C22×D3 | — | 36 | 6 | no | — |
| 77 | C22×D3 | — | 36 | 2 | no | — |
| 78 | C3×Q8 | — | 2 | 1 | no | — |
| 79 | C3×Q8 | — | 2 | 3 | no | — |
| 80 | C3×D4 | — | 6 | 3 | no | — |
| 81 | C3×D4 | — | 6 | 9 | no | — |
| 82 | C2×C3×C4 | — | 3 | 3 | no | — |
| 83 | C2×C3×C4 | — | 3 | 9 | no | — |
| 84 | #67 | — | 36 | 6 | no | — |
| 85 | #67 | — | 36 | 18 | no | — |
| 86 | C2×Dic3 | — | 18 | 3 | no | — |
| 87 | C2×Dic3 | — | 18 | 9 | no | — |
| 88 | C2×C32 | — | 6 | 3 | no | — |
| 89 | C2×C32 | #1020 | 1 | 1 | yes | characteristic |
| 90 | C3⋊D3 | — | 12 | 2 | no | — |
| 91 | C2×C9 | — | 16 | 3 | no | — |
| 92 | D9 | — | 48 | 6 | no | — |
| 93 | C24 | — | 27 | 1 | no | — |
| 94 | #41 | — | 6 | 1 | no | — |
| 95 | C2×D4 | — | 3 | 3 | no | — |
| 96 | C2×D4 | — | 54 | 3 | no | — |
| 97 | C22⋊C4 | — | 27 | 3 | no | — |
| 98 | C22⋊C4 | — | 54 | 3 | no | — |
| 99 | C22×C3 | — | 3 | 3 | no | — |
| 100 | C22×C3 | — | 3 | 9 | no | — |
| 101 | C22×C3 | — | 4 | 9 | no | — |
| 102 | C22×C3 | — | 4 | 3 | no | — |
| 103 | C22×C3 | — | 12 | 3 | no | — |
| 104 | C22×C3 | — | 12 | 1 | no | — |
| 105 | C2×D3 | — | 18 | 18 | no | — |
| 106 | C2×D3 | — | 18 | 6 | no | — |
| 107 | C2×D3 | — | 36 | 9 | no | — |
| 108 | C2×D3 | — | 36 | 3 | no | — |
| 109 | C2×D3 | — | 36 | 1 | no | — |
| 110 | C2×D3 | — | 36 | 3 | no | — |
| 111 | C3×C4 | — | 6 | 1 | no | — |
| 112 | C3×C4 | — | 6 | 3 | no | — |
| 113 | Dic3 | — | 36 | 9 | no | — |
| 114 | Dic3 | — | 36 | 3 | no | — |
| 115 | C32 | #5934 | 1 | 1 | yes | characteristic |
| 116 | C9 | — | 16 | 3 | no | — |
| 117 | C23 | — | 3 | 1 | no | — |
| 118 | C23 | — | 54 | 3 | no | — |
| 119 | C23 | — | 108 | 2 | no | — |
| 120 | C23 | C3⋊((C22⋊C9)⋊C2) | 1 | 3 | yes | complemented |
| 121 | Q8 | — | 2 | 1 | no | — |
| 122 | D4 | — | 6 | 3 | no | — |
| 123 | D4 | — | 108 | 6 | no | — |
| 124 | C2×C4 | — | 3 | 3 | no | — |
| 125 | C2×C4 | — | 54 | 3 | no | — |
| 126 | C2×C3 | — | 6 | 9 | no | — |
| 127 | C2×C3 | — | 6 | 3 | no | — |
| 128 | C2×C3 | #64188 | 1 | 3 | yes | — |
| 129 | C2×C3 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 130 | D3 | — | 36 | 6 | no | — |
| 131 | D3 | — | 36 | 2 | no | — |
| 132 | C22 | — | 3 | 3 | no | — |
| 133 | C22 | — | 4 | 3 | no | — |
| 134 | C22 | — | 12 | 1 | no | — |
| 135 | C22 | — | 54 | 6 | no | — |
| 136 | C22 | — | 108 | 3 | no | — |
| 137 | C22 | — | 108 | 1 | no | — |
| 138 | C4 | — | 6 | 1 | no | — |
| 139 | C4 | — | 108 | 3 | no | — |
| 140 | C3 | #10593109 | 1 | 3 | yes | complemented |
| 141 | C3 | C3⋊#5934 | 1 | 1 | yes | characteristic |
| 142 | C2 | — | 6 | 3 | no | — |
| 143 | C2 | — | 108 | 2 | no | — |
| 144 | C2 | C3⋊#64188 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#64188 |
| commutator, cosocle, hypocenter | C3×#63703 | C2 |
| Frattini | C2×C3 | C3⋊#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | #1020 |
| LCS | G ▷ C3×#63703 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63703 | C2 | semidirect |
| 7. #63703 | D3 | semidirect |
| 23. C3×H(5,𝔽2) | D9 | semidirect |
| 66. H(5,𝔽2) | C3⋊D9 | semidirect |
| 72. C23×C3 | (C22⋊C9)⋊C2 | semidirect |
| 120. C23 | C3⋊((C22⋊C9)⋊C2) | semidirect |
| 140. C3 | #10593109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63703 | 2 |
| #10593109 | 3 |
| #10592943 | 3 |
| C2×C3⋊((C22⋊C9)⋊C2) | 4 |