Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
243 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 40 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 | #10592770 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#4545 | C2 | 1 | 2 | yes | maximal |
| 3 | #67539 | — | 3 | 2 | no | maximal |
| 4 | C32×#127 | C22 | 1 | 1 | yes | characteristic |
| 5 | #64079 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #63659 | — | 2 | 2 | no | — |
| 7 | #63630 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3×#834 | — | 2 | 2 | no | — |
| 9 | #5071 | — | 3 | 2 | no | — |
| 10 | #5071 | — | 6 | 1 | no | — |
| 11 | C3×#353 | — | 3 | 2 | no | — |
| 12 | #4545 | D3 | 1 | 2 | yes | — |
| 13 | C2×C32×D4 | — | 4 | 1 | no | — |
| 14 | C2×C32×D4 | D4 | 1 | 2 | yes | — |
| 15 | C32×C42 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C4⋊(C3⋊Dic3) | — | 2 | 1 | no | — |
| 17 | #3750 | — | 4 | 1 | no | — |
| 18 | C2×C3⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 19 | C3×#217 | — | 4 | 2 | no | — |
| 20 | #1412 | — | 9 | 1 | no | Sylow |
| 21 | C3×#127 | — | 2 | 1 | no | — |
| 22 | C3×#127 | C2×D3 | 1 | 2 | yes | — |
| 23 | #870 | — | 3 | 2 | no | — |
| 24 | #870 | — | 6 | 1 | no | — |
| 25 | C3×#99 | — | 6 | 2 | no | — |
| 26 | #834 | — | 2 | 2 | no | — |
| 27 | #832 | — | 6 | 4 | no | — |
| 28 | #832 | — | 6 | 2 | no | — |
| 29 | #832 | — | 6 | 2 | no | — |
| 30 | #803 | — | 3 | 2 | no | — |
| 31 | #803 | — | 6 | 1 | no | — |
| 32 | C23×C32 | — | 2 | 2 | no | — |
| 33 | C32×D4 | — | 4 | 2 | no | — |
| 34 | C32×D4 | — | 8 | 2 | no | — |
| 35 | C2×C32×C4 | — | 2 | 1 | no | — |
| 36 | C2×C32×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 37 | C2×C3⋊Dic3 | — | 4 | 1 | no | — |
| 38 | C2×C3×Dic3 | — | 8 | 2 | no | — |
| 39 | C3⋊(C3⋊C8) | — | 4 | 1 | no | — |
| 40 | #490 | — | 9 | 1 | no | — |
| 41 | #353 | — | 3 | 2 | no | — |
| 42 | C2×C3×D4 | — | 2 | 2 | no | — |
| 43 | C2×C3×D4 | — | 4 | 2 | no | — |
| 44 | C2×C3×D4 | — | 4 | 2 | no | — |
| 45 | C2×C3×D4 | D12 | 1 | 2 | yes | — |
| 46 | C2×C3×D4 | #67 | 1 | 2 | yes | — |
| 47 | C3×C22⋊C4 | — | 12 | 2 | no | — |
| 48 | C3×C42 | — | 2 | 1 | no | — |
| 49 | C3×C42 | C4×D3 | 1 | 2 | yes | — |
| 50 | #217 | — | 4 | 2 | no | — |
| 51 | C4⋊Dic3 | — | 6 | 2 | no | — |
| 52 | C4⋊Dic3 | — | 6 | 2 | no | — |
| 53 | Dic3⋊C4 | — | 12 | 2 | no | — |
| 54 | Dic3⋊C4 | — | 12 | 2 | no | — |
| 55 | C2×C3⋊C8 | — | 6 | 2 | no | — |
| 56 | C2×C3⋊C8 | — | 6 | 2 | no | — |
| 57 | C22×C32 | — | 4 | 2 | no | — |
| 58 | C22×C32 | — | 8 | 2 | no | — |
| 59 | C22×C32 | #99 | 1 | 1 | yes | characteristic |
| 60 | C32×C4 | — | 2 | 1 | no | — |
| 61 | C32×C4 | — | 4 | 1 | no | — |
| 62 | C3⋊Dic3 | — | 8 | 1 | no | — |
| 63 | C3×Dic3 | — | 16 | 2 | no | — |
| 64 | #127 | D32 | 1 | 1 | yes | characteristic |
| 65 | #125 | — | 9 | 1 | no | — |
| 66 | #102 | — | 18 | 2 | no | — |
| 67 | #99 | — | 6 | 2 | no | — |
| 68 | #97 | — | 9 | 1 | no | — |
| 69 | C23×C3 | — | 2 | 2 | no | — |
| 70 | C23×C3 | — | 2 | 2 | no | — |
| 71 | C23×C3 | — | 4 | 2 | no | — |
| 72 | C3×D4 | — | 4 | 2 | no | — |
| 73 | C3×D4 | — | 4 | 2 | no | — |
| 74 | C3×D4 | — | 4 | 4 | no | — |
| 75 | C3×D4 | — | 8 | 4 | no | — |
| 76 | C3×D4 | — | 8 | 4 | no | — |
| 77 | C2×C3×C4 | — | 2 | 2 | no | — |
| 78 | C2×C3×C4 | — | 2 | 2 | no | — |
| 79 | C2×C3×C4 | — | 2 | 1 | no | — |
| 80 | C2×C3×C4 | — | 24 | 2 | no | — |
| 81 | C2×C3×C4 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 82 | C2×Dic3 | — | 8 | 2 | no | — |
| 83 | C2×Dic3 | — | 12 | 2 | no | — |
| 84 | C2×Dic3 | — | 12 | 2 | no | — |
| 85 | C3⋊C8 | — | 12 | 2 | no | — |
| 86 | C3⋊C8 | — | 12 | 2 | no | — |
| 87 | C2×C32 | — | 2 | 1 | no | — |
| 88 | C2×C32 | — | 8 | 2 | no | — |
| 89 | C2×C32 | #354 | 1 | 1 | yes | characteristic |
| 90 | C2×D4 | — | 4 | 1 | no | — |
| 91 | C2×D4 | #626 | 1 | 2 | yes | — |
| 92 | C2×C8 | — | 18 | 1 | no | — |
| 93 | C4⋊C4 | — | 18 | 1 | no | — |
| 94 | C4⋊C4 | — | 36 | 1 | no | — |
| 95 | C22⋊C4 | — | 12 | 2 | no | — |
| 96 | C42 | #624 | 1 | 1 | yes | characteristic |
| 97 | C22×C3 | — | 2 | 1 | no | — |
| 98 | C22×C3 | — | 4 | 2 | no | — |
| 99 | C22×C3 | — | 4 | 2 | no | — |
| 100 | C22×C3 | — | 8 | 2 | no | — |
| 101 | C22×C3 | — | 8 | 2 | no | — |
| 102 | C22×C3 | — | 8 | 2 | no | — |
| 103 | C22×C3 | — | 8 | 4 | no | — |
| 104 | C22×C3 | #806 | 1 | 2 | yes | — |
| 105 | C3×C4 | — | 2 | 2 | no | — |
| 106 | C3×C4 | — | 2 | 2 | no | — |
| 107 | C3×C4 | — | 4 | 2 | no | — |
| 108 | C3×C4 | — | 4 | 2 | no | — |
| 109 | C3×C4 | — | 48 | 2 | no | — |
| 110 | Dic3 | — | 16 | 2 | no | — |
| 111 | Dic3 | — | 24 | 2 | no | — |
| 112 | Dic3 | — | 24 | 2 | no | — |
| 113 | C32 | #1412 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 114 | C23 | — | 2 | 2 | no | — |
| 115 | D4 | — | 4 | 2 | no | — |
| 116 | D4 | — | 8 | 2 | no | — |
| 117 | C2×C4 | — | 2 | 1 | no | — |
| 118 | C2×C4 | — | 24 | 2 | no | — |
| 119 | C2×C4 | — | 36 | 1 | no | — |
| 120 | C2×C4 | #3722 | 1 | 1 | yes | characteristic |
| 121 | C8 | — | 36 | 1 | no | — |
| 122 | C2×C3 | — | 2 | 2 | no | — |
| 123 | C2×C3 | — | 2 | 2 | no | — |
| 124 | C2×C3 | — | 2 | 1 | no | — |
| 125 | C2×C3 | — | 8 | 2 | no | — |
| 126 | C2×C3 | — | 8 | 2 | no | — |
| 127 | C2×C3 | — | 16 | 2 | no | — |
| 128 | C2×C3 | #4473 | 1 | 2 | yes | — |
| 129 | C22 | — | 4 | 2 | no | — |
| 130 | C22 | — | 8 | 2 | no | — |
| 131 | C22 | #63581 | 1 | 1 | yes | characteristic |
| 132 | C4 | — | 2 | 1 | no | — |
| 133 | C4 | — | 4 | 1 | no | — |
| 134 | C4 | — | 48 | 2 | no | — |
| 135 | C4 | — | 72 | 1 | no | — |
| 136 | C3 | — | 2 | 1 | no | — |
| 137 | C3 | #67539 | 1 | 2 | yes | complemented |
| 138 | C2 | — | 2 | 1 | no | — |
| 139 | C2 | — | 8 | 2 | no | — |
| 140 | C2 | #10589181 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10589181 |
| commutator | C32×C42 | C2×C4 |
| Frattini, hypercenter | #127 | D32 |
| Fitting, prime radical, cosocle | C32×#127 | C22 |
| socle | C2×C32 | #354 |
| hypocenter | C32 | #1412 |
| LCS | G ▷ C32×C42 ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #64079 | C4 | semidirect |
| 7. #63630 | C4 | semidirect |
| 113. C32 | #1412 | semidirect |
| 137. C3 | #67539 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10592770 | 2 |
| C3×#4545 | 2 |
| #67539 | 3 |