Subgroups — C₂×D₄² #3462

Order 128.

1268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄	C₂	1	4	yes	maximal, complemented
2	D₄²	C₂	1	16	yes	maximal, direct product factor
3	C₂×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#121	C₂	1	4	yes	maximal, complemented
5	C₂×#120	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×D₄	C₂	1	2	yes	maximal, complemented
7	C₂⁵	C₂²	1	4	yes	complemented
8	C₂²×D₄	—	2	32	no	—
9	C₂²×D₄	C₂²	1	2	yes	—
10	C₂²×D₄	C₂²	1	8	yes	complemented
11	C₂²×D₄	C₂²	1	4	yes	complemented
12	C₂²×D₄	C₂²	1	2	yes	complemented
13	C₂²×D₄	C₂²	1	32	yes	complemented
14	C₂³×C₄	C₂²	1	4	yes	complemented
15	#127	C₂²	1	8	yes	complemented
16	#121	C₂²	1	32	yes	complemented
17	#120	C₂²	1	32	yes	complemented
18	C₄×D₄	C₂²	1	16	yes	complemented
19	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
21	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
22	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
23	C₂⁴	—	2	32	no	—
24	C₂⁴	—	4	8	no	—
25	C₂⁴	C₂³	1	4	yes	—
26	C₂⁴	C₂³	1	4	yes	—
27	C₂⁴	C₂³	1	16	yes	complemented
28	C₂×D₄	—	2	32	no	—
29	C₂×D₄	—	2	128	no	—
30	C₂×D₄	—	2	16	no	—
31	C₂×D₄	—	4	4	no	—
32	C₂×D₄	D₄	1	16	yes	direct product factor
33	C₂×D₄	C₂³	1	8	yes	—
34	C₂×D₄	C₂³	1	32	yes	complemented
35	C₂×D₄	C₂³	1	16	yes	complemented
36	C₂×D₄	C₂³	1	8	yes	complemented
37	C₂²×C₄	—	2	16	no	—
38	C₂²×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₄	C₂³	1	4	yes	—
40	C₂²×C₄	C₂³	1	2	yes	—
41	C₂²×C₄	C₂³	1	16	yes	complemented
42	C₄⋊C₄	C₂³	1	8	yes	complemented
43	C₂²⋊C₄	C₂³	1	16	yes	complemented
44	C₂²⋊C₄	C₂³	1	16	yes	complemented
45	C₄²	C₂³	1	4	yes	complemented
46	C₂³	—	2	8	no	—
47	C₂³	—	2	8	no	—
48	C₂³	—	2	16	no	—
49	C₂³	—	2	64	no	—
50	C₂³	—	2	8	no	—
51	C₂³	—	4	16	no	—
52	C₂³	—	4	64	no	—
53	C₂³	C₂×D₄	1	16	yes	complemented
54	C₂³	C₂⁴	1	8	yes	—
55	C₂³	C₂⁴	1	8	yes	—
56	C₂³	C₂⁴	1	1	yes	characteristic
57	D₄	—	2	64	no	—
58	D₄	—	2	32	no	—
59	D₄	—	4	8	no	—
60	D₄	C₂×D₄	1	32	yes	direct product factor
61	C₂×C₄	—	2	8	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	32	no	—
64	C₂×C₄	C₂×D₄	1	8	yes	complemented
65	C₂×C₄	C₂⁴	1	2	yes	—
66	C₂×C₄	C₂⁴	1	8	yes	—
67	C₂×C₄	C₂⁴	1	4	yes	—
68	C₂²	—	2	32	no	—
69	C₂²	—	2	8	no	—
70	C₂²	—	2	8	no	—
71	C₂²	—	2	16	no	—
72	C₂²	—	2	8	no	—
73	C₂²	—	4	64	no	—
74	C₂²	—	4	16	no	—
75	C₂²	C₂²×D₄	1	4	yes	—
76	C₂²	C₂²×D₄	1	16	yes	complemented
77	C₂²	H(5,𝔽₂)	1	2	yes	—
78	C₂²	C₂⁵	1	1	yes	characteristic
79	C₄	—	2	8	no	—
80	C₄	—	2	2	no	—
81	C₄	C₂²×D₄	1	8	yes	complemented
82	C₂	—	2	16	no	—
83	C₂	—	4	8	no	—
84	C₂	D₄²	1	4	yes	direct product factor
85	C₂	C₂³×D₄	1	2	yes	—
86	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×D₄	C₂	semidirect
2. D₄²	C₂	direct (N × H)
3. C₂×#127	C₂	semidirect
4. C₂×#121	C₂	semidirect
5. C₂×#120	C₂	semidirect
6. C₂×C₄×D₄	C₂	semidirect
7. C₂⁵	C₂²	semidirect
10. C₂²×D₄	C₂²	semidirect
11. C₂²×D₄	C₂²	semidirect
12. C₂²×D₄	C₂²	semidirect
13. C₂²×D₄	C₂²	semidirect
14. C₂³×C₄	C₂²	semidirect
15. #127	C₂²	semidirect
16. #121	C₂²	semidirect
17. #120	C₂²	semidirect
18. C₄×D₄	C₂²	semidirect
19. C₂×C₄⋊C₄	C₂²	semidirect
20. C₂×C₂²⋊C₄	C₂²	semidirect
21. C₂×C₂²⋊C₄	C₂²	semidirect
22. C₂×C₄²	C₂²	semidirect
27. C₂⁴	C₂³	semidirect
32. C₂×D₄	D₄	direct (N × H)
34. C₂×D₄	C₂³	semidirect
35. C₂×D₄	C₂³	semidirect
36. C₂×D₄	C₂³	semidirect
41. C₂²×C₄	C₂³	semidirect
42. C₄⋊C₄	C₂³	semidirect
43. C₂²⋊C₄	C₂³	semidirect
44. C₂²⋊C₄	C₂³	semidirect
45. C₄²	C₂³	semidirect
53. C₂³	C₂×D₄	semidirect
60. D₄	C₂×D₄	direct (N × H)
64. C₂×C₄	C₂×D₄	semidirect
76. C₂²	C₂²×D₄	semidirect
81. C₄	C₂²×D₄	semidirect
84. C₂	D₄²	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×D₄	2
D₄²	2
C₂×#127	2
C₂×#121	2
C₂×#120	2
C₂×C₄×D₄	2

Subgroups — C2×D42 #3462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄² #3462