Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1232 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #5945 | C2 | 1 | 10 | yes | maximal, complemented |
| 2 | C2×#994 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | H(7,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | C3×2−1+4 | — | 4 | 1 | no | maximal |
| 5 | 2−1+4⋊C3 | C22 | 1 | 10 | yes | complemented |
| 6 | #994 | C22 | 1 | 10 | yes | complemented |
| 7 | C2×#41⋊C3 | C22 | 1 | 15 | yes | complemented |
| 8 | #585 | — | 3 | 5 | no | — |
| 9 | #585 | — | 3 | 1 | no | — |
| 10 | #585 | C2×C3 | 1 | 10 | yes | complemented |
| 11 | C2×H(5,𝔽2) | — | 3 | 10 | no | — |
| 12 | C2×H(5,𝔽2) | C2×C3 | 1 | 5 | yes | complemented |
| 13 | C3×#41 | — | 4 | 10 | no | — |
| 14 | C2×C3×Q8 | — | 4 | 5 | no | — |
| 15 | #41⋊C3 | C23 | 1 | 10 | yes | complemented |
| 16 | C2×SL(2,𝔽3) | C23 | 1 | 5 | yes | complemented |
| 17 | 2−1+4 | — | 3 | 5 | no | — |
| 18 | 2−1+4 | — | 3 | 10 | no | — |
| 19 | 2−1+4 | A4 | 1 | 1 | yes | characteristic |
| 20 | 2−1+4 | C22×C3 | 1 | 10 | yes | — |
| 21 | H(5,𝔽2) | — | 3 | 20 | no | — |
| 22 | H(5,𝔽2) | — | 3 | 30 | no | — |
| 23 | H(5,𝔽2) | — | 3 | 10 | no | — |
| 24 | H(5,𝔽2) | — | 3 | 30 | no | — |
| 25 | H(5,𝔽2) | C22×C3 | 1 | 10 | yes | — |
| 26 | C2×#41 | — | 3 | 30 | no | — |
| 27 | C2×#41 | — | 3 | 20 | no | — |
| 28 | C2×#41 | — | 3 | 10 | no | — |
| 29 | C2×#41 | — | 3 | 5 | no | — |
| 30 | C2×#41 | C22×C3 | 1 | 15 | yes | — |
| 31 | C22×D4 | — | 3 | 20 | no | — |
| 32 | C22×D4 | — | 3 | 15 | no | — |
| 33 | C3×Q8 | — | 4 | 10 | no | — |
| 34 | C3×D4 | — | 4 | 10 | no | — |
| 35 | C2×C3×C4 | — | 4 | 15 | no | — |
| 36 | SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 37 | C24 | — | 3 | 10 | no | — |
| 38 | #41 | — | 3 | 60 | no | — |
| 39 | #41 | — | 3 | 10 | no | — |
| 40 | #41 | — | 3 | 20 | no | — |
| 41 | #41 | — | 3 | 10 | no | — |
| 42 | #41 | — | 3 | 20 | no | — |
| 43 | #41 | — | 3 | 30 | no | — |
| 44 | #41 | — | 3 | 30 | no | — |
| 45 | #41 | C2×A4 | 1 | 10 | yes | — |
| 46 | #41 | C23×C3 | 1 | 10 | yes | — |
| 47 | C2×Q8 | — | 3 | 10 | no | — |
| 48 | C2×Q8 | — | 3 | 10 | no | — |
| 49 | C2×Q8 | C2×A4 | 1 | 5 | yes | — |
| 50 | C2×Q8 | C23×C3 | 1 | 5 | yes | — |
| 51 | C2×D4 | — | 3 | 15 | no | — |
| 52 | C2×D4 | — | 3 | 60 | no | — |
| 53 | C2×D4 | — | 3 | 60 | no | — |
| 54 | C2×D4 | — | 3 | 30 | no | — |
| 55 | C2×D4 | — | 3 | 30 | no | — |
| 56 | C2×D4 | — | 3 | 15 | no | — |
| 57 | C22×C4 | — | 3 | 20 | no | — |
| 58 | C22×C4 | — | 3 | 15 | no | — |
| 59 | C22×C3 | — | 4 | 5 | no | — |
| 60 | C3×C4 | — | 4 | 10 | no | — |
| 61 | C23 | — | 3 | 20 | no | — |
| 62 | C23 | — | 3 | 15 | no | — |
| 63 | C23 | — | 24 | 10 | no | — |
| 64 | Q8 | — | 3 | 5 | no | — |
| 65 | Q8 | — | 3 | 10 | no | — |
| 66 | Q8 | C22×A4 | 1 | 10 | yes | — |
| 67 | Q8 | C24×C3 | 1 | 1 | yes | characteristic |
| 68 | D4 | — | 3 | 10 | no | — |
| 69 | D4 | — | 3 | 30 | no | — |
| 70 | D4 | — | 3 | 20 | no | — |
| 71 | D4 | — | 3 | 30 | no | — |
| 72 | D4 | C22×A4 | 1 | 10 | yes | — |
| 73 | C2×C4 | — | 3 | 20 | no | — |
| 74 | C2×C4 | — | 3 | 5 | no | — |
| 75 | C2×C4 | — | 3 | 30 | no | — |
| 76 | C2×C4 | — | 3 | 10 | no | — |
| 77 | C2×C4 | C22×A4 | 1 | 15 | yes | — |
| 78 | C2×C3 | — | 4 | 1 | no | — |
| 79 | C2×C3 | — | 8 | 5 | no | — |
| 80 | C22 | — | 3 | 10 | no | — |
| 81 | C22 | — | 12 | 15 | no | — |
| 82 | C22 | — | 12 | 20 | no | — |
| 83 | C22 | C23×A4 | 1 | 5 | yes | — |
| 84 | C4 | — | 3 | 5 | no | — |
| 85 | C4 | — | 3 | 1 | no | — |
| 86 | C4 | C23×A4 | 1 | 10 | yes | — |
| 87 | C3 | — | 4 | 1 | no | Sylow |
| 88 | C2 | — | 2 | 5 | no | — |
| 89 | C2 | — | 6 | 10 | no | — |
| 90 | C2 | C24×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C24×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C3 |
| Fitting | H(7,𝔽2) | C3 |
| hypercenter | 2−1+4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5945 | C2 | semidirect |
| 2. C2×#994 | C2 | semidirect |
| 3. H(7,𝔽2) | C3 | semidirect |
| 5. 2−1+4⋊C3 | C22 | semidirect |
| 6. #994 | C22 | semidirect |
| 7. C2×#41⋊C3 | C22 | semidirect |
| 10. #585 | C2×C3 | semidirect |
| 12. C2×H(5,𝔽2) | C2×C3 | semidirect |
| 15. #41⋊C3 | C23 | semidirect |
| 16. C2×SL(2,𝔽3) | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5945 | 2 |
| C2×#994 | 2 |
| H(7,𝔽2) | 3 |
| C3×2−1+4 | 4 |