Subgroups — C₂×#121 #522

Order 64.

213 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#121	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂⁴	C₂²	1	2	yes	—
8	C₂⁴	C₂²	1	1	yes	characteristic
9	C₂×D₄	—	2	8	no	—
10	C₂×D₄	—	2	4	no	—
11	C₂×D₄	C₂²	1	8	yes	complemented
12	C₂×D₄	C₂²	1	4	yes	complemented
13	C₂²×C₄	—	2	4	no	—
14	C₂²×C₄	C₂²	1	1	yes	characteristic
15	C₂²×C₄	C₂²	1	2	yes	—
16	C₂²×C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₄	C₂²	1	4	yes	complemented
18	C₄⋊C₄	C₂²	1	4	yes	complemented
19	C₂²⋊C₄	C₂²	1	8	yes	complemented
20	C₂³	—	2	2	no	—
21	C₂³	—	2	2	no	—
22	C₂³	—	2	4	no	—
23	C₂³	—	2	4	no	—
24	C₂³	—	2	4	no	—
25	C₂³	D₄	1	4	yes	complemented
26	C₂³	C₂³	1	4	yes	—
27	C₂³	C₂³	1	2	yes	—
28	C₂³	C₂³	1	1	yes	characteristic
29	D₄	—	2	16	no	—
30	D₄	—	2	8	no	—
31	C₂×C₄	—	2	2	no	—
32	C₂×C₄	—	2	4	no	—
33	C₂×C₄	—	2	8	no	—
34	C₂×C₄	D₄	1	4	yes	complemented
35	C₂×C₄	C₂³	1	2	yes	—
36	C₂×C₄	C₂³	1	4	yes	—
37	C₂×C₄	C₂³	1	2	yes	—
38	C₂²	—	2	8	no	—
39	C₂²	—	2	2	no	—
40	C₂²	—	2	2	no	—
41	C₂²	—	2	4	no	—
42	C₂²	—	2	4	no	—
43	C₂²	—	2	4	no	—
44	C₂²	—	4	8	no	—
45	C₂²	C₂×D₄	1	2	yes	—
46	C₂²	C₂×D₄	1	2	yes	—
47	C₂²	C₂×D₄	1	4	yes	complemented
48	C₂²	#41	1	2	yes	—
49	C₂²	C₂⁴	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	4	no	—
52	C₄	C₂×D₄	1	4	yes	complemented
53	C₂	—	2	4	no	—
54	C₂	—	4	4	no	—
55	C₂	#121	1	4	yes	direct product factor
56	C₂	C₂²×D₄	1	1	yes	characteristic
57	C₂	C₂²×D₄	1	1	yes	characteristic
58	C₂	C₂×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴
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. C₂²×D₄	C₂	semidirect
2. C₂²×D₄	C₂	semidirect
3. C₂³×C₄	C₂	semidirect
4. #121	C₂	direct (N × H)
5. C₂×C₄⋊C₄	C₂	semidirect
6. C₂×C₂²⋊C₄	C₂	semidirect
11. C₂×D₄	C₂²	semidirect
12. C₂×D₄	C₂²	semidirect
17. C₂²×C₄	C₂²	semidirect
18. C₄⋊C₄	C₂²	semidirect
19. C₂²⋊C₄	C₂²	semidirect
25. C₂³	D₄	semidirect
34. C₂×C₄	D₄	semidirect
47. C₂²	C₂×D₄	semidirect
52. C₄	C₂×D₄	semidirect
55. C₂	#121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₄	2
C₂³×C₄	2
#121	2
C₂×C₄⋊C₄	2
C₂×C₂²⋊C₄	2

Subgroups — C2×#121 #522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#121 #522