Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
240 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 240 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×#5464 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #85558 | — | 3 | 3 | no | maximal |
| 3 | #71951 | — | 3 | 1 | no | maximal |
| 4 | C3⋊#857 | — | 4 | 3 | no | maximal |
| 5 | C3×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #5464 | D3 | 1 | 3 | yes | complemented |
| 7 | #5062 | — | 3 | 3 | no | — |
| 8 | #4542 | — | 3 | 3 | no | — |
| 9 | C3×C22⋊A4 | — | 8 | 1 | no | — |
| 10 | C3×C42⋊C3 | — | 4 | 3 | no | — |
| 11 | #2202 | — | 9 | 1 | no | Sylow |
| 12 | #1007 | — | 3 | 1 | no | — |
| 13 | C3×#124 | — | 3 | 3 | no | — |
| 14 | C3×#120 | — | 6 | 1 | no | — |
| 15 | #932 | — | 6 | 3 | no | — |
| 16 | #857 | — | 12 | 9 | no | — |
| 17 | #836 | — | 3 | 3 | no | — |
| 18 | #805 | — | 6 | 3 | no | — |
| 19 | C3⋊S4 | — | 16 | 1 | no | — |
| 20 | H(3,𝔽4) | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | #454 | — | 9 | 3 | no | — |
| 22 | #355 | — | 9 | 3 | no | — |
| 23 | C24×C3 | — | 2 | 1 | no | — |
| 24 | C22⋊A4 | — | 8 | 3 | no | — |
| 25 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 26 | C2×C3×D4 | — | 3 | 3 | no | — |
| 27 | #239 | — | 12 | 1 | no | — |
| 28 | D3×Q8 | — | 12 | 1 | no | — |
| 29 | #235 | — | 6 | 3 | no | — |
| 30 | C3×C22⋊C4 | — | 6 | 3 | no | — |
| 31 | C3×C42 | S4 | 1 | 3 | yes | — |
| 32 | #214 | — | 12 | 3 | no | — |
| 33 | #213 | — | 12 | 3 | no | — |
| 34 | #208 | — | 6 | 3 | no | — |
| 35 | C42⋊C3 | — | 4 | 9 | no | — |
| 36 | C3×A4 | — | 16 | 1 | no | — |
| 37 | C3×A4 | — | 32 | 4 | no | — |
| 38 | 2−1+4 | — | 9 | 1 | no | — |
| 39 | AGL(1,ℤ/8) | — | 18 | 3 | no | — |
| 40 | #124 | — | 3 | 3 | no | — |
| 41 | #120 | — | 6 | 1 | no | — |
| 42 | #104 | — | 18 | 3 | no | — |
| 43 | #101 | — | 9 | 3 | no | — |
| 44 | C23×C3 | — | 6 | 1 | no | — |
| 45 | C23×C3 | — | 24 | 1 | no | — |
| 46 | S4 | — | 48 | 3 | no | — |
| 47 | C3×Q8 | — | 12 | 1 | no | — |
| 48 | C3×D4 | — | 12 | 3 | no | — |
| 49 | C2×C3×C4 | — | 3 | 3 | no | — |
| 50 | #67 | — | 12 | 1 | no | — |
| 51 | D12 | — | 6 | 3 | no | — |
| 52 | C4×D3 | — | 12 | 3 | no | — |
| 53 | Dic6 | — | 6 | 3 | no | — |
| 54 | C3⋊C8 | — | 12 | 3 | no | — |
| 55 | C3⋊D3 | — | 64 | 1 | no | — |
| 56 | C24 | — | 2 | 1 | no | — |
| 57 | #41 | — | 18 | 3 | no | — |
| 58 | #41 | — | 36 | 1 | no | — |
| 59 | C2×Q8 | — | 3 | 1 | no | — |
| 60 | C2×Q8 | — | 36 | 1 | no | — |
| 61 | C2×D4 | — | 3 | 3 | no | — |
| 62 | SD16 | — | 36 | 3 | no | — |
| 63 | D8 | — | 36 | 3 | no | — |
| 64 | M16 | — | 18 | 3 | no | — |
| 65 | C22⋊C4 | — | 6 | 3 | no | — |
| 66 | C42 | C3⋊S4 | 1 | 3 | yes | — |
| 67 | C22×C3 | — | 8 | 4 | no | — |
| 68 | C22×C3 | — | 12 | 3 | no | — |
| 69 | C22×C3 | #1020 | 1 | 1 | yes | characteristic |
| 70 | C2×D3 | — | 12 | 1 | no | — |
| 71 | A4 | — | 16 | 3 | no | — |
| 72 | A4 | — | 32 | 12 | no | — |
| 73 | C3×C4 | — | 6 | 3 | no | — |
| 74 | Dic3 | — | 12 | 1 | no | — |
| 75 | C32 | — | 64 | 1 | no | Sylow |
| 76 | C23 | — | 6 | 1 | no | — |
| 77 | C23 | — | 24 | 1 | no | — |
| 78 | Q8 | — | 12 | 1 | no | — |
| 79 | Q8 | — | 18 | 3 | no | — |
| 80 | D4 | — | 12 | 3 | no | — |
| 81 | D4 | — | 18 | 3 | no | — |
| 82 | D4 | — | 36 | 1 | no | — |
| 83 | C2×C4 | — | 3 | 3 | no | — |
| 84 | C2×C4 | — | 36 | 3 | no | — |
| 85 | C8 | — | 36 | 3 | no | — |
| 86 | C2×C3 | — | 3 | 1 | no | — |
| 87 | C2×C3 | — | 24 | 1 | no | — |
| 88 | D3 | — | 24 | 1 | no | — |
| 89 | D3 | — | 192 | 3 | no | — |
| 90 | C22 | — | 8 | 4 | no | — |
| 91 | C22 | — | 12 | 3 | no | — |
| 92 | C22 | — | 36 | 1 | no | — |
| 93 | C22 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 94 | C4 | — | 6 | 3 | no | — |
| 95 | C4 | — | 36 | 1 | no | — |
| 96 | C3 | — | 64 | 3 | no | — |
| 97 | C3 | #85558 | 1 | 1 | yes | characteristic, complemented |
| 98 | C2 | — | 3 | 1 | no | — |
| 99 | C2 | — | 24 | 1 | no | — |
| 100 | C2 | — | 72 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#5464 | C2 |
| Frattini | C22 | C3⋊#1020 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C22×C3 | #1020 |
| LCS | G ▷ C3×#5464 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#5464 | C2 | semidirect |
| 5. C3×H(3,𝔽4) | D3 | semidirect |
| 6. #5464 | D3 | semidirect |
| 20. H(3,𝔽4) | C3⋊D3 | semidirect |
| 97. C3 | #85558 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#5464 | 2 |
| #85558 | 3 |
| #71951 | 3 |
| C3⋊#857 | 4 |