Finite Group DatabasebetaFinite Group DatabasebetaOrder 192.
152 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 27 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 | 2−1+4⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #985 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #983 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #577 | — | 3 | 1 | no | maximal, Sylow |
| 7 | D3×D4 | — | 4 | 1 | no | maximal |
| 8 | #41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 10 | GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | #226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | 2−1+4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 15 | #137 | — | 3 | 1 | no | — |
| 16 | #137 | — | 3 | 2 | no | — |
| 17 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 18 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 19 | #135 | — | 3 | 2 | no | — |
| 20 | #135 | — | 3 | 1 | no | — |
| 21 | C2×SD16 | — | 3 | 1 | no | — |
| 22 | C2×SD16 | — | 3 | 2 | no | — |
| 23 | #131 | — | 3 | 1 | no | — |
| 24 | C22×D3 | — | 4 | 2 | no | — |
| 25 | C3×D4 | — | 4 | 1 | no | — |
| 26 | #67 | — | 4 | 2 | no | — |
| 27 | D12 | — | 4 | 1 | no | — |
| 28 | C4×D3 | — | 4 | 1 | no | — |
| 29 | SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 30 | #41 | — | 3 | 1 | no | — |
| 31 | #41 | — | 3 | 2 | no | — |
| 32 | #41 | — | 3 | 2 | no | — |
| 33 | #41 | — | 3 | 1 | no | — |
| 34 | #41 | — | 6 | 1 | no | — |
| 35 | #41 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 36 | C2×Q8 | — | 3 | 1 | no | — |
| 37 | C2×Q8 | C2×D3 | 1 | 2 | yes | complemented |
| 38 | C2×D4 | — | 3 | 2 | no | — |
| 39 | C2×D4 | — | 3 | 1 | no | — |
| 40 | C2×D4 | — | 6 | 2 | no | — |
| 41 | C2×D4 | — | 6 | 1 | no | — |
| 42 | Q16 | — | 3 | 2 | no | — |
| 43 | Q16 | — | 3 | 1 | no | — |
| 44 | SD16 | — | 3 | 2 | no | — |
| 45 | SD16 | — | 3 | 2 | no | — |
| 46 | SD16 | — | 3 | 2 | no | — |
| 47 | SD16 | — | 3 | 1 | no | — |
| 48 | SD16 | — | 3 | 1 | no | — |
| 49 | SD16 | — | 3 | 2 | no | — |
| 50 | D8 | — | 3 | 2 | no | — |
| 51 | D8 | — | 3 | 1 | no | — |
| 52 | M16 | — | 3 | 2 | no | — |
| 53 | M16 | — | 3 | 1 | no | — |
| 54 | C2×C8 | — | 3 | 2 | no | — |
| 55 | C2×C8 | — | 3 | 1 | no | — |
| 56 | C22×C3 | — | 4 | 2 | no | — |
| 57 | C2×D3 | — | 4 | 1 | no | — |
| 58 | C2×D3 | — | 4 | 2 | no | — |
| 59 | C2×D3 | — | 8 | 4 | no | — |
| 60 | C3×C4 | — | 4 | 1 | no | — |
| 61 | Dic3 | — | 4 | 1 | no | — |
| 62 | C23 | — | 6 | 1 | no | — |
| 63 | C23 | — | 6 | 2 | no | — |
| 64 | Q8 | — | 3 | 2 | no | — |
| 65 | Q8 | — | 3 | 1 | no | — |
| 66 | Q8 | — | 3 | 1 | no | — |
| 67 | Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 68 | D4 | — | 3 | 1 | no | — |
| 69 | D4 | — | 3 | 2 | no | — |
| 70 | D4 | — | 3 | 2 | no | — |
| 71 | D4 | — | 3 | 1 | no | — |
| 72 | D4 | — | 6 | 2 | no | — |
| 73 | D4 | — | 6 | 1 | no | — |
| 74 | D4 | — | 6 | 2 | no | — |
| 75 | D4 | — | 6 | 1 | no | — |
| 76 | D4 | S4 | 1 | 1 | yes | characteristic |
| 77 | C2×C4 | — | 3 | 2 | no | — |
| 78 | C2×C4 | — | 3 | 1 | no | — |
| 79 | C2×C4 | — | 6 | 1 | no | — |
| 80 | C2×C4 | — | 6 | 2 | no | — |
| 81 | C2×C4 | — | 6 | 1 | no | — |
| 82 | C8 | — | 3 | 1 | no | — |
| 83 | C8 | — | 3 | 1 | no | — |
| 84 | C8 | — | 3 | 2 | no | — |
| 85 | C2×C3 | — | 4 | 1 | no | — |
| 86 | C2×C3 | — | 8 | 2 | no | — |
| 87 | D3 | — | 4 | 2 | no | — |
| 88 | D3 | — | 8 | 2 | no | — |
| 89 | C22 | — | 3 | 1 | no | — |
| 90 | C22 | — | 6 | 1 | no | — |
| 91 | C22 | — | 6 | 2 | no | — |
| 92 | C22 | — | 24 | 2 | no | — |
| 93 | C22 | — | 24 | 1 | no | — |
| 94 | C22 | C2×S4 | 1 | 2 | yes | — |
| 95 | C4 | — | 3 | 1 | no | — |
| 96 | C4 | — | 3 | 2 | no | — |
| 97 | C4 | — | 6 | 1 | no | — |
| 98 | C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 99 | C3 | — | 4 | 1 | no | Sylow |
| 100 | C2 | — | 2 | 2 | no | — |
| 101 | C2 | — | 6 | 1 | no | — |
| 102 | C2 | — | 12 | 1 | no | — |
| 103 | C2 | — | 12 | 2 | no | — |
| 104 | C2 | C22×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23 |
| Fitting | 2−1+4 | D3 |
| hypercenter | D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+4⋊C3 | C2 | semidirect |
| 2. #986 | C2 | semidirect |
| 3. #985 | C2 | semidirect |
| 4. #983 | C2 | semidirect |
| 5. C2×GL(2,𝔽3) | C2 | semidirect |
| 8. #41⋊C3 | C22 | semidirect |
| 9. C2×SL(2,𝔽3) | C22 | semidirect |
| 10. GL(2,𝔽3) | C22 | semidirect |
| 11. GL(2,𝔽3) | C22 | semidirect |
| 12. #226 | C22 | semidirect |
| 13. 2−1+4 | D3 | semidirect |
| 35. #41 | C2×D3 | semidirect |
| 37. C2×Q8 | C2×D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊C3 | 2 |
| #986 | 2 |
| #985 | 2 |
| #983 | 2 |
| C2×GL(2,𝔽3) | 2 |
| #577 | 3 |
| D3×D4 | 4 |