Subgroups — #535

Order 64.

167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#127	C₂	1	2	yes	maximal, complemented
4	#124	C₂	1	2	yes	maximal, complemented
5	#121	C₂	1	4	yes	maximal, complemented
6	#120	C₂	1	4	yes	maximal, complemented
7	#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴	C₂²	1	2	yes	—
9	#41	—	2	4	no	—
10	C₂×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄	—	2	4	no	—
12	C₂×D₄	C₂²	1	2	yes	—
13	C₂×D₄	C₂²	1	2	yes	—
14	C₂×D₄	C₂²	1	2	yes	—
15	C₂×D₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₄	C₂²	1	8	yes	complemented
17	C₂²×C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₄	C₂²	1	2	yes	complemented
19	C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂²⋊C₄	C₂²	1	8	yes	complemented
22	C₄²	C₂²	1	2	yes	complemented
23	C₂³	—	2	4	no	—
24	C₂³	—	4	4	no	—
25	C₂³	C₂³	1	4	yes	—
26	C₂³	C₂³	1	2	yes	—
27	C₂³	C₂³	1	1	yes	characteristic
28	Q₈	—	2	2	no	—
29	D₄	—	2	4	no	—
30	D₄	—	2	4	no	—
31	D₄	—	2	4	no	—
32	D₄	—	2	2	no	—
33	D₄	—	4	8	no	—
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	D₄	1	4	yes	complemented
36	C₂×C₄	C₂³	1	4	yes	—
37	C₂×C₄	C₂³	1	2	yes	—
38	C₂×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₄	C₂³	1	1	yes	characteristic
40	C₂²	—	2	2	no	—
41	C₂²	—	2	2	no	—
42	C₂²	—	2	4	no	—
43	C₂²	—	2	4	no	—
44	C₂²	—	2	8	no	—
45	C₂²	—	4	8	no	—
46	C₂²	C₂×D₄	1	2	yes	—
47	C₂²	C₂⁴	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	2	4	no	—
50	C₄	C₂×D₄	1	2	yes	—
51	C₄	C₂×D₄	1	2	yes	—
52	C₂	—	2	2	no	—
53	C₂	—	4	2	no	—
54	C₂	—	4	4	no	—
55	C₂	C₂²×D₄	1	1	yes	characteristic
56	C₂	H(5,𝔽₂)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#41	C₂	semidirect
2. C₂²×D₄	C₂	semidirect
3. #127	C₂	semidirect
4. #124	C₂	semidirect
5. #121	C₂	semidirect
6. #120	C₂	semidirect
7. #117	C₂	semidirect
10. C₂×Q₈	C₂²	semidirect
15. C₂×D₄	C₂²	semidirect
16. C₂×D₄	C₂²	semidirect
18. C₂²×C₄	C₂²	semidirect
19. C₄⋊C₄	C₂²	semidirect
21. C₂²⋊C₄	C₂²	semidirect
22. C₄²	C₂²	semidirect
35. C₂×C₄	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41	2
C₂²×D₄	2
#127	2
#124	2
#121	2
#120	2
#117	2