Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
135 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | B(4,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #455 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #354 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | AΣL(1,𝔽16) | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 7 | #137 | — | 2 | 1 | no | — |
| 8 | #124 | — | 2 | 1 | no | — |
| 9 | #124 | — | 2 | 2 | no | — |
| 10 | #124 | — | 2 | 1 | no | — |
| 11 | #124 | C22 | 1 | 1 | yes | characteristic |
| 12 | #120 | — | 2 | 2 | no | — |
| 13 | #120 | — | 2 | 2 | no | — |
| 14 | #120 | C22 | 1 | 2 | yes | — |
| 15 | #104 | — | 2 | 1 | no | — |
| 16 | #100 | C22 | 1 | 1 | yes | characteristic |
| 17 | #99 | — | 2 | 2 | no | — |
| 18 | #99 | C22 | 1 | 2 | yes | — |
| 19 | C24 | D4 | 1 | 2 | yes | complemented |
| 20 | #41 | — | 2 | 1 | no | — |
| 21 | #41 | — | 4 | 1 | no | — |
| 22 | C2×Q8 | — | 2 | 1 | no | — |
| 23 | C2×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×D4 | — | 2 | 2 | no | — |
| 25 | C2×D4 | — | 2 | 1 | no | — |
| 26 | C2×D4 | — | 2 | 1 | no | — |
| 27 | C2×D4 | — | 2 | 2 | no | — |
| 28 | C2×D4 | — | 4 | 1 | no | — |
| 29 | C2×D4 | — | 4 | 2 | no | — |
| 30 | C2×D4 | C23 | 1 | 1 | yes | characteristic |
| 31 | Q16 | — | 4 | 1 | no | — |
| 32 | SD16 | — | 4 | 1 | no | — |
| 33 | M16 | — | 2 | 1 | no | — |
| 34 | C22⋊C4 | — | 2 | 2 | no | — |
| 35 | C22⋊C4 | — | 2 | 2 | no | — |
| 36 | C22⋊C4 | — | 2 | 2 | no | — |
| 37 | C22⋊C4 | — | 2 | 2 | no | — |
| 38 | C22⋊C4 | — | 2 | 2 | no | — |
| 39 | C22⋊C4 | — | 4 | 2 | no | — |
| 40 | C22⋊C4 | D4 | 1 | 2 | yes | complemented |
| 41 | C42 | — | 2 | 1 | no | — |
| 42 | C42 | D4 | 1 | 1 | yes | characteristic, complemented |
| 43 | C23 | — | 2 | 2 | no | — |
| 44 | C23 | — | 2 | 2 | no | — |
| 45 | C23 | — | 4 | 2 | no | — |
| 46 | C23 | — | 8 | 2 | no | — |
| 47 | C23 | C2×D4 | 1 | 2 | yes | — |
| 48 | Q8 | — | 2 | 1 | no | — |
| 49 | Q8 | — | 4 | 1 | no | — |
| 50 | Q8 | — | 8 | 1 | no | — |
| 51 | D4 | — | 2 | 1 | no | — |
| 52 | D4 | — | 4 | 1 | no | — |
| 53 | D4 | — | 4 | 2 | no | — |
| 54 | D4 | — | 4 | 1 | no | — |
| 55 | D4 | — | 8 | 2 | no | — |
| 56 | D4 | — | 8 | 1 | no | — |
| 57 | D4 | — | 8 | 1 | no | — |
| 58 | D4 | — | 8 | 4 | no | — |
| 59 | C2×C4 | — | 2 | 2 | no | — |
| 60 | C2×C4 | — | 2 | 1 | no | — |
| 61 | C2×C4 | — | 2 | 2 | no | — |
| 62 | C2×C4 | — | 2 | 1 | no | — |
| 63 | C2×C4 | — | 4 | 1 | no | — |
| 64 | C2×C4 | — | 4 | 2 | no | — |
| 65 | C2×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C8 | — | 4 | 1 | no | — |
| 67 | C22 | — | 2 | 2 | no | — |
| 68 | C22 | — | 4 | 2 | no | — |
| 69 | C22 | — | 4 | 2 | no | — |
| 70 | C22 | — | 4 | 2 | no | — |
| 71 | C22 | — | 4 | 4 | no | — |
| 72 | C22 | — | 4 | 1 | no | — |
| 73 | C22 | — | 4 | 2 | no | — |
| 74 | C22 | — | 8 | 2 | no | — |
| 75 | C22 | — | 8 | 2 | no | — |
| 76 | C22 | #120 | 1 | 1 | yes | characteristic |
| 77 | C4 | — | 2 | 1 | no | — |
| 78 | C4 | — | 4 | 2 | no | — |
| 79 | C4 | — | 4 | 1 | no | — |
| 80 | C4 | — | 4 | 1 | no | — |
| 81 | C4 | — | 4 | 1 | no | — |
| 82 | C4 | — | 8 | 2 | no | — |
| 83 | C2 | — | 2 | 1 | no | — |
| 84 | C2 | — | 4 | 2 | no | — |
| 85 | C2 | — | 8 | 2 | no | — |
| 86 | C2 | — | 8 | 1 | no | — |
| 87 | C2 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽4) | C2 | semidirect |
| 2. B(4,𝔽2) | C2 | semidirect |
| 3. #455 | C2 | semidirect |
| 4. #354 | C2 | semidirect |
| 5. AΣL(1,𝔽16) | C2 | semidirect |
| 19. C24 | D4 | semidirect |
| 23. C2×Q8 | D4 | semidirect |
| 40. C22⋊C4 | D4 | semidirect |
| 42. C42 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽4) | 2 |
| B(4,𝔽2) | 2 |
| #455 | 2 |
| #354 | 2 |
| AΣL(1,𝔽16) | 2 |