Finite Group DatabasebetaFinite Group DatabasebetaOrder 2880.
167 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 8 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 | C2×#10624296 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×S6 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13088275 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×A6 | C22 | 1 | 1 | yes | characteristic |
| 5 | #10624296 | — | 2 | 1 | no | — |
| 6 | S6 | — | 2 | 1 | no | — |
| 7 | A6 | D4 | 1 | 1 | yes | characteristic |
| 8 | C2×AΓL(1,𝔽9) | — | 10 | 1 | no | maximal |
| 9 | C2×S5 | — | 12 | 1 | no | — |
| 10 | C2×PSU(3,𝔽2) | — | 10 | 1 | no | — |
| 11 | C2×GO+(4,𝔽2) | — | 10 | 1 | no | — |
| 12 | C2×AGL(1,𝔽9) | — | 10 | 1 | no | — |
| 13 | AΓL(1,𝔽9) | — | 10 | 4 | no | — |
| 14 | C2×A5 | — | 12 | 1 | no | — |
| 15 | S5 | — | 12 | 2 | no | — |
| 16 | C22×S4 | — | 30 | 1 | no | — |
| 17 | C4⋊AGL(1,𝔽5) | — | 36 | 1 | no | maximal |
| 18 | C2×D32 | — | 20 | 1 | no | — |
| 19 | C2×#157 | — | 10 | 1 | no | — |
| 20 | C2×#157 | — | 20 | 1 | no | — |
| 21 | PSU(3,𝔽2) | — | 10 | 2 | no | — |
| 22 | PSU(3,𝔽2) | — | 20 | 1 | no | — |
| 23 | GO+(4,𝔽2) | — | 10 | 2 | no | — |
| 24 | GO+(4,𝔽2) | — | 20 | 1 | no | — |
| 25 | AGL(1,𝔽9) | — | 10 | 2 | no | — |
| 26 | #450 | — | 45 | 1 | no | maximal, Sylow |
| 27 | A5 | — | 12 | 1 | no | — |
| 28 | C22×A4 | — | 30 | 1 | no | — |
| 29 | C2×S4 | — | 30 | 1 | no | — |
| 30 | C2×S4 | — | 30 | 1 | no | — |
| 31 | C2×S4 | — | 30 | 2 | no | — |
| 32 | C2×S4 | — | 30 | 2 | no | — |
| 33 | C2×AGL(1,𝔽5) | — | 36 | 1 | no | — |
| 34 | C2×AGL(1,𝔽5) | — | 36 | 1 | no | — |
| 35 | C4×D5 | — | 36 | 1 | no | — |
| 36 | C2×C3⋊D3 | — | 10 | 1 | no | — |
| 37 | C2×C3×D3 | — | 40 | 1 | no | — |
| 38 | D32 | — | 20 | 2 | no | — |
| 39 | D32 | — | 40 | 1 | no | — |
| 40 | #157 | — | 10 | 1 | no | — |
| 41 | #157 | — | 10 | 1 | no | — |
| 42 | #157 | — | 20 | 2 | no | — |
| 43 | C22×D4 | — | 45 | 1 | no | — |
| 44 | C2×SD16 | — | 45 | 1 | no | — |
| 45 | C2×SD16 | — | 45 | 1 | no | — |
| 46 | C22⋊Q8 | — | 45 | 1 | no | — |
| 47 | #102 | — | 45 | 1 | no | — |
| 48 | #102 | — | 45 | 1 | no | — |
| 49 | C22⋊C8 | — | 45 | 1 | no | — |
| 50 | C22×D3 | — | 120 | 1 | no | — |
| 51 | C2×A4 | — | 30 | 1 | no | — |
| 52 | C2×A4 | — | 30 | 2 | no | — |
| 53 | S4 | — | 30 | 2 | no | — |
| 54 | S4 | — | 30 | 1 | no | — |
| 55 | S4 | — | 30 | 1 | no | — |
| 56 | C2×D5 | — | 36 | 1 | no | — |
| 57 | AGL(1,𝔽5) | — | 72 | 1 | no | — |
| 58 | AGL(1,𝔽5) | — | 72 | 1 | no | — |
| 59 | C4×C5 | — | 36 | 1 | no | — |
| 60 | Dic5 | — | 36 | 1 | no | — |
| 61 | C2×C32 | — | 10 | 1 | no | — |
| 62 | C3⋊D3 | — | 10 | 1 | no | — |
| 63 | C3⋊D3 | — | 10 | 1 | no | — |
| 64 | C3×D3 | — | 40 | 2 | no | — |
| 65 | C24 | — | 30 | 1 | no | — |
| 66 | C2×Q8 | — | 45 | 1 | no | — |
| 67 | C2×D4 | — | 45 | 1 | no | — |
| 68 | C2×D4 | — | 45 | 1 | no | — |
| 69 | C2×D4 | — | 90 | 1 | no | — |
| 70 | C2×D4 | — | 90 | 1 | no | — |
| 71 | C2×D4 | — | 90 | 2 | no | — |
| 72 | C2×D4 | — | 90 | 1 | no | — |
| 73 | C22×C4 | — | 45 | 1 | no | — |
| 74 | SD16 | — | 90 | 1 | no | — |
| 75 | SD16 | — | 90 | 1 | no | — |
| 76 | SD16 | — | 90 | 2 | no | — |
| 77 | C2×C8 | — | 45 | 1 | no | — |
| 78 | C2×C8 | — | 45 | 1 | no | — |
| 79 | C4⋊C4 | — | 45 | 1 | no | — |
| 80 | C4⋊C4 | — | 90 | 1 | no | — |
| 81 | C22⋊C4 | — | 90 | 1 | no | — |
| 82 | C22×C3 | — | 120 | 1 | no | — |
| 83 | C2×D3 | — | 40 | 1 | no | — |
| 84 | C2×D3 | — | 120 | 1 | no | — |
| 85 | C2×D3 | — | 120 | 2 | no | — |
| 86 | C2×D3 | — | 120 | 2 | no | — |
| 87 | A4 | — | 30 | 1 | no | — |
| 88 | C2×C5 | — | 36 | 1 | no | — |
| 89 | D5 | — | 36 | 1 | no | — |
| 90 | D5 | — | 36 | 1 | no | — |
| 91 | C32 | — | 10 | 1 | no | Sylow |
| 92 | C23 | — | 30 | 1 | no | — |
| 93 | C23 | — | 30 | 2 | no | — |
| 94 | C23 | — | 45 | 1 | no | — |
| 95 | C23 | — | 90 | 1 | no | — |
| 96 | C23 | — | 90 | 2 | no | — |
| 97 | Q8 | — | 90 | 1 | no | — |
| 98 | Q8 | — | 90 | 1 | no | — |
| 99 | D4 | — | 45 | 1 | no | — |
| 100 | D4 | — | 45 | 1 | no | — |
| 101 | D4 | — | 45 | 2 | no | — |
| 102 | D4 | — | 90 | 1 | no | — |
| 103 | D4 | — | 90 | 1 | no | — |
| 104 | D4 | — | 90 | 2 | no | — |
| 105 | D4 | — | 90 | 2 | no | — |
| 106 | C2×C4 | — | 45 | 1 | no | — |
| 107 | C2×C4 | — | 45 | 1 | no | — |
| 108 | C2×C4 | — | 90 | 1 | no | — |
| 109 | C2×C4 | — | 90 | 1 | no | — |
| 110 | C2×C4 | — | 90 | 1 | no | — |
| 111 | C2×C4 | — | 90 | 1 | no | — |
| 112 | C8 | — | 90 | 1 | no | — |
| 113 | C8 | — | 90 | 1 | no | — |
| 114 | C2×C3 | — | 40 | 1 | no | — |
| 115 | C2×C3 | — | 120 | 2 | no | — |
| 116 | D3 | — | 40 | 2 | no | — |
| 117 | D3 | — | 120 | 1 | no | — |
| 118 | D3 | — | 120 | 1 | no | — |
| 119 | C5 | — | 36 | 1 | no | Sylow |
| 120 | C22 | — | 30 | 1 | no | — |
| 121 | C22 | — | 30 | 1 | no | — |
| 122 | C22 | — | 45 | 2 | no | — |
| 123 | C22 | — | 45 | 1 | no | — |
| 124 | C22 | — | 90 | 1 | no | — |
| 125 | C22 | — | 90 | 2 | no | — |
| 126 | C22 | — | 90 | 1 | no | — |
| 127 | C22 | — | 180 | 1 | no | — |
| 128 | C4 | — | 36 | 1 | no | — |
| 129 | C4 | — | 45 | 1 | no | — |
| 130 | C4 | — | 45 | 1 | no | — |
| 131 | C4 | — | 90 | 1 | no | — |
| 132 | C4 | — | 180 | 1 | no | — |
| 133 | C3 | — | 40 | 1 | no | — |
| 134 | C2 | — | 30 | 2 | no | — |
| 135 | C2 | — | 45 | 1 | no | — |
| 136 | C2 | — | 45 | 1 | no | — |
| 137 | C2 | PΓL(2,𝔽9) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, hypercenter | C2 | PΓL(2,𝔽9) |
| commutator, socle, cosocle | C2×A6 | C22 |
| prime radical | C2×S6 | C2 |
| hypocenter | A6 | D4 |
| LCS | G ▷ C2×A6 ▷ A6 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#10624296 | C2 | semidirect |
| 3. #13088275 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10624296 | 2 |
| C2×S6 | 2 |
| #13088275 | 2 |
| C2×AΓL(1,𝔽9) | 10 |
| C4⋊AGL(1,𝔽5) | 36 |
| #450 | 45 |