Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
456 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 456 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 | #11757605 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11757604 | C2 | 1 | 2 | yes | maximal |
| 3 | #10620153 | — | 3 | 1 | no | maximal |
| 4 | #10619881 | — | 3 | 2 | no | maximal |
| 5 | C2×C32⋊H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 6 | C3⋊(C3⋊(C3⋊Dic3)) | D3 | 1 | 1 | yes | characteristic |
| 7 | C32×C3⋊Dic3 | — | 3 | 2 | no | — |
| 8 | C3⋊#1084 | D3 | 1 | 2 | yes | — |
| 9 | C3⋊((C3⋊Dic3)⋊C3) | — | 3 | 2 | no | — |
| 10 | C3⋊((C3⋊Dic3)⋊C3) | — | 6 | 1 | no | — |
| 11 | C3×(C3⋊Dic3)⋊C3 | — | 3 | 2 | no | — |
| 12 | C3×(C3⋊Dic3)⋊C3 | — | 6 | 2 | no | — |
| 13 | C32⋊H(3,𝔽3) | Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | #6296 | — | 9 | 1 | no | — |
| 15 | #6293 | — | 9 | 2 | no | — |
| 16 | #6293 | — | 9 | 6 | no | — |
| 17 | #6196 | — | 9 | 6 | no | — |
| 18 | C2×C34 | C2×D3 | 1 | 1 | yes | characteristic |
| 19 | C2×C3×H(3,𝔽3) | — | 2 | 2 | no | — |
| 20 | C2×C3×H(3,𝔽3) | — | 2 | 1 | no | — |
| 21 | C2×C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 22 | C2×C3×H(3,𝔽3) | C2×D3 | 1 | 2 | yes | — |
| 23 | C3⋊(C3⋊Dic3) | — | 3 | 2 | no | — |
| 24 | C3⋊(C3⋊Dic3) | — | 6 | 1 | no | — |
| 25 | C3⋊(C3⋊Dic3) | — | 9 | 6 | no | — |
| 26 | C3⋊(C3⋊Dic3) | — | 18 | 3 | no | — |
| 27 | C3×C3⋊Dic3 | — | 3 | 6 | no | — |
| 28 | C3×C3⋊Dic3 | — | 3 | 2 | no | — |
| 29 | C3×C3⋊Dic3 | — | 3 | 2 | no | — |
| 30 | C3×C3⋊Dic3 | — | 6 | 2 | no | — |
| 31 | C3×C3⋊Dic3 | — | 9 | 1 | no | — |
| 32 | C32×Dic3 | — | 9 | 6 | no | — |
| 33 | C32×Dic3 | — | 9 | 2 | no | — |
| 34 | #1084 | — | 9 | 6 | no | — |
| 35 | (C3⋊Dic3)⋊C3 | — | 3 | 6 | no | — |
| 36 | (C3⋊Dic3)⋊C3 | — | 6 | 6 | no | — |
| 37 | (C3⋊Dic3)⋊C3 | — | 9 | 6 | no | — |
| 38 | (C3⋊Dic3)⋊C3 | — | 18 | 3 | no | — |
| 39 | C34 | Dic6 | 1 | 1 | yes | characteristic, complemented |
| 40 | C3×H(3,𝔽3) | — | 2 | 2 | no | — |
| 41 | C3×H(3,𝔽3) | — | 2 | 1 | no | — |
| 42 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 43 | C3×H(3,𝔽3) | Dic6 | 1 | 2 | yes | complemented |
| 44 | #634 | — | 27 | 2 | no | — |
| 45 | #627 | — | 27 | 6 | no | — |
| 46 | #627 | — | 27 | 1 | no | — |
| 47 | #627 | — | 27 | 9 | no | — |
| 48 | #627 | — | 27 | 2 | no | — |
| 49 | C2×C33 | — | 2 | 1 | no | — |
| 50 | C2×C33 | — | 2 | 2 | no | — |
| 51 | C2×C33 | — | 3 | 6 | no | — |
| 52 | C2×C33 | — | 4 | 1 | no | — |
| 53 | C2×C33 | — | 6 | 3 | no | — |
| 54 | C2×C33 | D32 | 1 | 1 | yes | characteristic |
| 55 | C2×C33 | D32 | 1 | 2 | yes | — |
| 56 | C2×H(3,𝔽3) | — | 3 | 6 | no | — |
| 57 | C2×H(3,𝔽3) | — | 6 | 3 | no | — |
| 58 | C2×H(3,𝔽3) | — | 6 | 6 | no | — |
| 59 | C2×H(3,𝔽3) | — | 12 | 3 | no | — |
| 60 | C32×C4 | — | 27 | 2 | no | — |
| 61 | C3⋊Dic3 | — | 3 | 2 | no | — |
| 62 | C3⋊Dic3 | — | 3 | 2 | no | — |
| 63 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 64 | C3⋊Dic3 | — | 9 | 1 | no | — |
| 65 | C3⋊Dic3 | — | 9 | 6 | no | — |
| 66 | C3⋊Dic3 | — | 18 | 3 | no | — |
| 67 | C3⋊Dic3 | — | 27 | 2 | no | — |
| 68 | C3⋊Dic3 | — | 27 | 9 | no | — |
| 69 | C3⋊Dic3 | — | 54 | 2 | no | — |
| 70 | C3⋊Dic3 | — | 54 | 2 | no | — |
| 71 | C3⋊Dic3 | — | 54 | 1 | no | — |
| 72 | C3⋊Dic3 | — | 54 | 9 | no | — |
| 73 | C3×Dic3 | — | 9 | 18 | no | — |
| 74 | C3×Dic3 | — | 9 | 6 | no | — |
| 75 | C3×Dic3 | — | 9 | 6 | no | — |
| 76 | C3×Dic3 | — | 9 | 2 | no | — |
| 77 | C3×Dic3 | — | 27 | 2 | no | — |
| 78 | C3×Dic3 | — | 27 | 2 | no | — |
| 79 | C3×Dic3 | — | 54 | 2 | no | — |
| 80 | C3×Dic3 | — | 54 | 1 | no | — |
| 81 | C33 | — | 2 | 1 | no | — |
| 82 | C33 | — | 2 | 2 | no | — |
| 83 | C33 | — | 3 | 6 | no | — |
| 84 | C33 | — | 4 | 1 | no | — |
| 85 | C33 | — | 6 | 3 | no | — |
| 86 | C33 | #627 | 1 | 2 | yes | — |
| 87 | C33 | #627 | 1 | 1 | yes | characteristic, complemented |
| 88 | H(3,𝔽3) | — | 3 | 6 | no | — |
| 89 | H(3,𝔽3) | — | 6 | 3 | no | — |
| 90 | H(3,𝔽3) | — | 6 | 6 | no | — |
| 91 | H(3,𝔽3) | — | 12 | 3 | no | — |
| 92 | Dic6 | — | 81 | 2 | no | — |
| 93 | Dic6 | — | 81 | 6 | no | — |
| 94 | Dic6 | — | 81 | 1 | no | — |
| 95 | C2×C32 | — | 2 | 3 | no | — |
| 96 | C2×C32 | — | 3 | 2 | no | — |
| 97 | C2×C32 | — | 3 | 9 | no | — |
| 98 | C2×C32 | — | 3 | 2 | no | — |
| 99 | C2×C32 | — | 6 | 2 | no | — |
| 100 | C2×C32 | — | 6 | 1 | no | — |
| 101 | C2×C32 | — | 6 | 1 | no | — |
| 102 | C2×C32 | — | 6 | 9 | no | — |
| 103 | C2×C32 | — | 6 | 2 | no | — |
| 104 | C2×C32 | — | 6 | 2 | no | — |
| 105 | C2×C32 | — | 6 | 2 | no | — |
| 106 | C2×C32 | — | 12 | 2 | no | — |
| 107 | C2×C32 | — | 12 | 1 | no | — |
| 108 | C2×C32 | — | 12 | 1 | no | — |
| 109 | C2×C32 | — | 12 | 2 | no | — |
| 110 | C2×C32 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 6 | yes | — |
| 111 | C2×C32 | #1113 | 1 | 1 | yes | characteristic |
| 112 | C3×C4 | — | 27 | 6 | no | — |
| 113 | C3×C4 | — | 27 | 2 | no | — |
| 114 | C3×C4 | — | 81 | 1 | no | — |
| 115 | Dic3 | — | 9 | 6 | no | — |
| 116 | Dic3 | — | 9 | 2 | no | — |
| 117 | Dic3 | — | 27 | 2 | no | — |
| 118 | Dic3 | — | 27 | 2 | no | — |
| 119 | Dic3 | — | 54 | 2 | no | — |
| 120 | Dic3 | — | 54 | 1 | no | — |
| 121 | Dic3 | — | 81 | 6 | no | — |
| 122 | Dic3 | — | 162 | 3 | no | — |
| 123 | C32 | — | 2 | 3 | no | — |
| 124 | C32 | — | 3 | 2 | no | — |
| 125 | C32 | — | 3 | 9 | no | — |
| 126 | C32 | — | 3 | 2 | no | — |
| 127 | C32 | — | 6 | 2 | no | — |
| 128 | C32 | — | 6 | 1 | no | — |
| 129 | C32 | — | 6 | 1 | no | — |
| 130 | C32 | — | 6 | 9 | no | — |
| 131 | C32 | — | 6 | 2 | no | — |
| 132 | C32 | — | 6 | 2 | no | — |
| 133 | C32 | — | 6 | 2 | no | — |
| 134 | C32 | — | 12 | 2 | no | — |
| 135 | C32 | — | 12 | 1 | no | — |
| 136 | C32 | — | 12 | 1 | no | — |
| 137 | C32 | — | 12 | 2 | no | — |
| 138 | C32 | #6196 | 1 | 6 | yes | complemented |
| 139 | C32 | #6296 | 1 | 1 | yes | characteristic |
| 140 | Q8 | — | 243 | 1 | no | Sylow |
| 141 | C2×C3 | — | 2 | 1 | no | — |
| 142 | C2×C3 | — | 3 | 6 | no | — |
| 143 | C2×C3 | — | 6 | 3 | no | — |
| 144 | C2×C3 | — | 9 | 1 | no | — |
| 145 | C2×C3 | — | 18 | 2 | no | — |
| 146 | C2×C3 | — | 36 | 1 | no | — |
| 147 | C2×C3 | #66432 | 1 | 2 | yes | — |
| 148 | C4 | — | 27 | 2 | no | — |
| 149 | C4 | — | 81 | 1 | no | — |
| 150 | C3 | — | 2 | 1 | no | — |
| 151 | C3 | — | 3 | 6 | no | — |
| 152 | C3 | — | 6 | 3 | no | — |
| 153 | C3 | — | 9 | 1 | no | — |
| 154 | C3 | — | 18 | 2 | no | — |
| 155 | C3 | — | 36 | 1 | no | — |
| 156 | C3 | #10619900 | 1 | 2 | yes | — |
| 157 | C2 | #11757749 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11757749 |
| commutator, Fitting, prime radical, cosocle | C2×C32⋊H(3,𝔽3) | C22 |
| Frattini, socle | C2×C32 | #1113 |
| hypocenter | C32⋊H(3,𝔽3) | Q8 |
| LCS | G ▷ C2×C32⋊H(3,𝔽3) ▷ C32⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 13. C32⋊H(3,𝔽3) | Q8 | semidirect |
| 39. C34 | Dic6 | semidirect |
| 43. C3×H(3,𝔽3) | Dic6 | semidirect |
| 87. C33 | #627 | semidirect |
| 138. C32 | #6196 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11757605 | 2 |
| #11757604 | 2 |
| #10620153 | 3 |
| #10619881 | 3 |