Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
157 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 157 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 | C3×#338⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #68727 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #2017⋊C3 | — | 3 | 1 | no | maximal |
| 4 | A4×C3⋊C8 | — | 4 | 1 | no | maximal |
| 5 | #5748 | — | 3 | 1 | no | — |
| 6 | #5129 | — | 3 | 1 | no | — |
| 7 | C3×#338 | C2×C3 | 1 | 1 | yes | characteristic |
| 8 | #338⋊C3 | — | 2 | 1 | no | — |
| 9 | #338⋊C3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C3×C4×A4 | — | 4 | 1 | no | — |
| 11 | #2017 | — | 3 | 1 | no | Sylow |
| 12 | C2×#235 | — | 3 | 1 | no | — |
| 13 | C2×C3×M16 | — | 3 | 1 | no | — |
| 14 | C22×C3⋊C8 | A4 | 1 | 1 | yes | characteristic |
| 15 | #909 | — | 6 | 2 | no | — |
| 16 | #908 | — | 6 | 2 | no | — |
| 17 | #904 | — | 6 | 2 | no | — |
| 18 | C8×A4 | — | 12 | 1 | no | — |
| 19 | (C2×D3)⋊C8 | — | 6 | 2 | no | — |
| 20 | C2×C3×A4 | — | 4 | 1 | no | — |
| 21 | C3×C3⋊C8 | — | 16 | 1 | no | — |
| 22 | #575 | — | 9 | 1 | no | — |
| 23 | #408 | — | 9 | 1 | no | — |
| 24 | #338 | C3×D3 | 1 | 1 | yes | characteristic |
| 25 | C22×C3×C4 | C2×A4 | 1 | 1 | yes | characteristic |
| 26 | C2×#67 | — | 6 | 1 | no | — |
| 27 | #235 | — | 6 | 2 | no | — |
| 28 | #235 | — | 6 | 2 | no | — |
| 29 | C2×D12 | — | 3 | 1 | no | — |
| 30 | C2×C4×D3 | — | 6 | 1 | no | — |
| 31 | C2×Dic6 | — | 3 | 1 | no | — |
| 32 | C4×A4 | — | 4 | 1 | no | — |
| 33 | C4×A4 | — | 8 | 1 | no | — |
| 34 | C3×M16 | — | 6 | 2 | no | — |
| 35 | C2×C3×C8 | — | 3 | 2 | no | — |
| 36 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 37 | Dic12 | — | 6 | 2 | no | — |
| 38 | D24 | — | 6 | 2 | no | — |
| 39 | #204 | — | 6 | 4 | no | — |
| 40 | C3×A4 | — | 4 | 1 | no | — |
| 41 | C32×C4 | — | 16 | 1 | no | — |
| 42 | C2×#41 | — | 9 | 1 | no | — |
| 43 | #137 | — | 18 | 2 | no | — |
| 44 | AGL(1,ℤ/8) | — | 18 | 2 | no | — |
| 45 | #135 | — | 18 | 2 | no | — |
| 46 | C2×M16 | — | 3 | 1 | no | — |
| 47 | C22×C8 | — | 3 | 1 | no | — |
| 48 | C22⋊C8 | — | 18 | 2 | no | — |
| 49 | C23×C3 | #41⋊C3 | 1 | 1 | yes | characteristic |
| 50 | C22×D3 | — | 6 | 1 | no | — |
| 51 | C2×A4 | — | 4 | 1 | no | — |
| 52 | C2×A4 | — | 8 | 1 | no | — |
| 53 | C2×C3×C4 | — | 3 | 1 | no | — |
| 54 | C2×C3×C4 | — | 3 | 1 | no | — |
| 55 | #67 | — | 12 | 2 | no | — |
| 56 | C2×Dic3 | — | 6 | 1 | no | — |
| 57 | D12 | — | 6 | 2 | no | — |
| 58 | C4×D3 | — | 12 | 1 | no | — |
| 59 | C4×D3 | — | 12 | 1 | no | — |
| 60 | Dic6 | — | 6 | 2 | no | — |
| 61 | C3×C8 | — | 6 | 2 | no | — |
| 62 | C3×C8 | — | 48 | 1 | no | — |
| 63 | C3⋊C8 | — | 4 | 1 | no | — |
| 64 | C2×C32 | — | 16 | 1 | no | — |
| 65 | #41 | — | 18 | 2 | no | — |
| 66 | #41 | — | 18 | 2 | no | — |
| 67 | C2×Q8 | — | 9 | 1 | no | — |
| 68 | C2×D4 | — | 9 | 1 | no | — |
| 69 | C2×D4 | — | 18 | 1 | no | — |
| 70 | C22×C4 | — | 18 | 1 | no | — |
| 71 | C22×C4 | D3×A4 | 1 | 1 | yes | characteristic |
| 72 | Q16 | — | 18 | 2 | no | — |
| 73 | SD16 | — | 18 | 4 | no | — |
| 74 | D8 | — | 18 | 2 | no | — |
| 75 | M16 | — | 6 | 2 | no | — |
| 76 | C2×C8 | — | 3 | 2 | no | — |
| 77 | C2×C8 | — | 18 | 1 | no | — |
| 78 | C22×C3 | — | 3 | 1 | no | — |
| 79 | C22×C3 | — | 4 | 1 | no | — |
| 80 | C2×D3 | — | 12 | 2 | no | — |
| 81 | C2×D3 | — | 12 | 1 | no | — |
| 82 | A4 | — | 4 | 1 | no | — |
| 83 | A4 | — | 8 | 1 | no | — |
| 84 | C3×C4 | — | 3 | 1 | no | — |
| 85 | C3×C4 | — | 16 | 1 | no | — |
| 86 | C3×C4 | — | 32 | 1 | no | — |
| 87 | C3×C4 | #127⋊C3 | 1 | 1 | yes | characteristic |
| 88 | Dic3 | — | 12 | 1 | no | — |
| 89 | C32 | — | 16 | 1 | no | Sylow |
| 90 | C23 | — | 18 | 1 | no | — |
| 91 | C23 | C3⋊(#41⋊C3) | 1 | 1 | yes | characteristic |
| 92 | Q8 | — | 18 | 2 | no | — |
| 93 | D4 | — | 18 | 2 | no | — |
| 94 | D4 | — | 36 | 2 | no | — |
| 95 | C2×C4 | — | 3 | 1 | no | — |
| 96 | C2×C4 | — | 3 | 1 | no | — |
| 97 | C2×C4 | — | 18 | 1 | no | — |
| 98 | C2×C4 | — | 36 | 1 | no | — |
| 99 | C2×C4 | — | 36 | 1 | no | — |
| 100 | C8 | — | 6 | 2 | no | — |
| 101 | C8 | — | 12 | 1 | no | — |
| 102 | C2×C3 | — | 6 | 1 | no | — |
| 103 | C2×C3 | — | 16 | 1 | no | — |
| 104 | C2×C3 | — | 32 | 1 | no | — |
| 105 | C2×C3 | #392⋊C3 | 1 | 1 | yes | characteristic |
| 106 | D3 | — | 24 | 1 | no | — |
| 107 | C22 | — | 3 | 1 | no | — |
| 108 | C22 | — | 4 | 1 | no | — |
| 109 | C22 | — | 36 | 2 | no | — |
| 110 | C22 | — | 36 | 1 | no | — |
| 111 | C4 | — | 3 | 1 | no | — |
| 112 | C4 | — | 36 | 1 | no | — |
| 113 | C4 | C3⋊(#127⋊C3) | 1 | 1 | yes | characteristic |
| 114 | C3 | — | 16 | 1 | no | — |
| 115 | C3 | — | 32 | 1 | no | — |
| 116 | C3 | #2017⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 117 | C2 | — | 6 | 1 | no | — |
| 118 | C2 | — | 72 | 1 | no | — |
| 119 | C2 | C3⋊(#392⋊C3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(#127⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C3×#338 | C2×C3 |
| Frattini | C22×C4 | D3×A4 |
| socle | C2×C3 | #392⋊C3 |
| LCS | G ▷ C3×#338 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#338⋊C3 | C2 | semidirect |
| 2. #68727 | C3 | semidirect |
| 9. #338⋊C3 | D3 | semidirect |
| 116. C3 | #2017⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#338⋊C3 | 2 |
| #68727 | 3 |
| #2017⋊C3 | 3 |
| A4×C3⋊C8 | 4 |