Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
166 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 14 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 | D3×SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊#6729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#6729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#6729 | — | 3 | 1 | no | maximal |
| 5 | C3×SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | D3×GL(2,𝔽3) | — | 5 | 1 | no | maximal, Hall |
| 7 | D3×#170 | — | 6 | 1 | no | maximal |
| 8 | C2×SL(2,𝔽5) | — | 3 | 1 | no | — |
| 9 | #6729 | — | 3 | 1 | no | — |
| 10 | #6729 | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 11 | D3×#67 | — | 10 | 1 | no | maximal |
| 12 | D3×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 13 | C3⋊GL(2,𝔽3) | — | 5 | 1 | no | — |
| 14 | C3×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 15 | D3×Dic5 | — | 6 | 1 | no | — |
| 16 | C3⋊#170 | — | 6 | 1 | no | — |
| 17 | C3×#170 | — | 6 | 1 | no | — |
| 18 | SL(2,𝔽5) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×GL(2,𝔽3) | — | 15 | 1 | no | — |
| 20 | D3×SD16 | — | 15 | 1 | no | — |
| 21 | C2×#170 | — | 18 | 1 | no | — |
| 22 | C22×C3×D3 | — | 10 | 1 | no | — |
| 23 | C2×D32 | — | 10 | 1 | no | — |
| 24 | #638 | — | 10 | 1 | no | — |
| 25 | C3×#67 | — | 10 | 1 | no | — |
| 26 | C3×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 27 | #626 | — | 10 | 1 | no | — |
| 28 | #625 | — | 10 | 1 | no | — |
| 29 | D3×Dic3 | — | 10 | 1 | no | — |
| 30 | C2×C5×D3 | — | 6 | 1 | no | — |
| 31 | Dic15 | — | 6 | 1 | no | — |
| 32 | C3×Dic5 | — | 6 | 1 | no | — |
| 33 | C2×#67 | — | 30 | 1 | no | — |
| 34 | D3×Q8 | — | 5 | 1 | no | — |
| 35 | D3×D4 | — | 15 | 1 | no | — |
| 36 | C2×SL(2,𝔽3) | — | 15 | 1 | no | — |
| 37 | GL(2,𝔽3) | — | 5 | 1 | no | — |
| 38 | GL(2,𝔽3) | — | 15 | 1 | no | — |
| 39 | GL(2,𝔽3) | — | 30 | 1 | no | — |
| 40 | C3×SD16 | — | 15 | 1 | no | — |
| 41 | #215 | — | 15 | 1 | no | — |
| 42 | #214 | — | 15 | 1 | no | — |
| 43 | #204 | — | 15 | 1 | no | — |
| 44 | D3×C8 | — | 15 | 1 | no | — |
| 45 | C2×Dic5 | — | 18 | 1 | no | — |
| 46 | #170 | — | 6 | 1 | no | — |
| 47 | #170 | — | 18 | 1 | no | — |
| 48 | C22×C32 | — | 10 | 1 | no | — |
| 49 | C2×C3⋊D3 | — | 10 | 1 | no | — |
| 50 | C2×C3×D3 | — | 10 | 1 | no | — |
| 51 | C2×C3×D3 | — | 10 | 1 | no | — |
| 52 | C2×C3×D3 | — | 10 | 1 | no | — |
| 53 | C2×C3×D3 | — | 20 | 2 | no | — |
| 54 | D32 | — | 20 | 2 | no | — |
| 55 | C3⋊Dic3 | — | 10 | 1 | no | — |
| 56 | C3×Dic3 | — | 10 | 1 | no | — |
| 57 | C2×SD16 | — | 45 | 1 | no | Sylow |
| 58 | C2×C3×C5 | — | 6 | 1 | no | — |
| 59 | C5×D3 | — | 6 | 2 | no | — |
| 60 | C23×C3 | — | 30 | 1 | no | — |
| 61 | C22×D3 | — | 10 | 1 | no | — |
| 62 | C22×D3 | — | 30 | 1 | no | — |
| 63 | C3×Q8 | — | 5 | 1 | no | — |
| 64 | C3×D4 | — | 15 | 1 | no | — |
| 65 | #67 | — | 10 | 1 | no | — |
| 66 | #67 | — | 30 | 1 | no | — |
| 67 | #67 | — | 30 | 1 | no | — |
| 68 | #67 | — | 30 | 1 | no | — |
| 69 | #67 | — | 30 | 1 | no | — |
| 70 | #67 | — | 60 | 1 | no | — |
| 71 | C2×Dic3 | — | 30 | 1 | no | — |
| 72 | D12 | — | 15 | 1 | no | — |
| 73 | C4×D3 | — | 15 | 1 | no | — |
| 74 | Dic6 | — | 15 | 1 | no | — |
| 75 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 76 | SL(2,𝔽3) | — | 10 | 1 | no | — |
| 77 | C3×C8 | — | 15 | 1 | no | — |
| 78 | C3⋊C8 | — | 15 | 1 | no | — |
| 79 | C22×C5 | — | 18 | 1 | no | — |
| 80 | Dic5 | — | 6 | 1 | no | — |
| 81 | Dic5 | — | 18 | 1 | no | — |
| 82 | C2×C32 | — | 10 | 1 | no | — |
| 83 | C2×C32 | — | 20 | 1 | no | — |
| 84 | C3⋊D3 | — | 20 | 1 | no | — |
| 85 | C3×D3 | — | 10 | 2 | no | — |
| 86 | C3×D3 | — | 20 | 1 | no | — |
| 87 | C3×D3 | — | 20 | 1 | no | — |
| 88 | C2×Q8 | — | 15 | 1 | no | — |
| 89 | C2×D4 | — | 45 | 1 | no | — |
| 90 | SD16 | — | 15 | 1 | no | — |
| 91 | SD16 | — | 45 | 1 | no | — |
| 92 | SD16 | — | 45 | 1 | no | — |
| 93 | SD16 | — | 45 | 1 | no | — |
| 94 | C2×C8 | — | 45 | 1 | no | — |
| 95 | C3×C5 | — | 6 | 1 | no | — |
| 96 | C22×C3 | — | 10 | 1 | no | — |
| 97 | C22×C3 | — | 10 | 1 | no | — |
| 98 | C22×C3 | — | 20 | 1 | no | — |
| 99 | C22×C3 | — | 30 | 1 | no | — |
| 100 | C22×C3 | — | 30 | 1 | no | — |
| 101 | C22×C3 | — | 60 | 2 | no | — |
| 102 | C2×D3 | — | 10 | 1 | no | — |
| 103 | C2×D3 | — | 10 | 1 | no | — |
| 104 | C2×D3 | — | 20 | 2 | no | — |
| 105 | C2×D3 | — | 30 | 1 | no | — |
| 106 | C2×D3 | — | 60 | 1 | no | — |
| 107 | C2×D3 | — | 60 | 2 | no | — |
| 108 | C2×D3 | S5 | 1 | 1 | yes | characteristic |
| 109 | C3×C4 | — | 15 | 1 | no | — |
| 110 | Dic3 | — | 10 | 1 | no | — |
| 111 | Dic3 | — | 15 | 1 | no | — |
| 112 | Dic3 | — | 30 | 1 | no | — |
| 113 | Dic3 | — | 60 | 1 | no | — |
| 114 | C2×C5 | — | 6 | 1 | no | — |
| 115 | C2×C5 | — | 18 | 2 | no | — |
| 116 | C32 | — | 10 | 1 | no | Sylow |
| 117 | C23 | — | 30 | 1 | no | — |
| 118 | Q8 | — | 5 | 1 | no | — |
| 119 | Q8 | — | 45 | 1 | no | — |
| 120 | D4 | — | 15 | 1 | no | — |
| 121 | D4 | — | 45 | 1 | no | — |
| 122 | D4 | — | 90 | 1 | no | — |
| 123 | C2×C4 | — | 45 | 1 | no | — |
| 124 | C8 | — | 15 | 1 | no | — |
| 125 | C8 | — | 45 | 1 | no | — |
| 126 | C2×C3 | — | 10 | 1 | no | — |
| 127 | C2×C3 | — | 20 | 1 | no | — |
| 128 | C2×C3 | — | 20 | 1 | no | — |
| 129 | C2×C3 | — | 20 | 1 | no | — |
| 130 | C2×C3 | — | 30 | 2 | no | — |
| 131 | C2×C3 | — | 40 | 1 | no | — |
| 132 | C2×C3 | — | 60 | 1 | no | — |
| 133 | C2×C3 | C2×S5 | 1 | 1 | yes | characteristic |
| 134 | D3 | — | 20 | 1 | no | — |
| 135 | D3 | — | 20 | 1 | no | — |
| 136 | D3 | — | 60 | 1 | no | — |
| 137 | D3 | — | 120 | 1 | no | — |
| 138 | D3 | #6729 | 1 | 2 | yes | direct product factor |
| 139 | C5 | — | 6 | 1 | no | Sylow |
| 140 | C22 | — | 3 | 1 | no | — |
| 141 | C22 | — | 10 | 1 | no | — |
| 142 | C22 | — | 30 | 1 | no | — |
| 143 | C22 | — | 60 | 2 | no | — |
| 144 | C4 | — | 15 | 1 | no | — |
| 145 | C4 | — | 45 | 1 | no | — |
| 146 | C3 | — | 10 | 1 | no | — |
| 147 | C3 | — | 20 | 1 | no | — |
| 148 | C3 | C2×#6729 | 1 | 1 | yes | characteristic, complemented |
| 149 | C2 | — | 3 | 2 | no | — |
| 150 | C2 | — | 20 | 1 | no | — |
| 151 | C2 | — | 60 | 1 | no | — |
| 152 | C2 | D3×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S5 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽5) | C22 |
| Fitting, socle | C2×C3 | C2×S5 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D3×SL(2,𝔽5) | C2 | semidirect |
| 2. C3⋊#6729 | C2 | semidirect |
| 3. C3×#6729 | C2 | semidirect |
| 5. C3×SL(2,𝔽5) | C22 | semidirect |
| 10. #6729 | D3 | direct (N × H) |
| 18. SL(2,𝔽5) | C2×D3 | semidirect |
| 138. D3 | #6729 | direct (N × H) |
| 148. C3 | C2×#6729 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×SL(2,𝔽5) | 2 |
| C3⋊#6729 | 2 |
| C3×#6729 | 2 |
| C2×#6729 | 3 |
| D3×GL(2,𝔽3) | 5 |
| D3×#170 | 6 |
| D3×#67 | 10 |