Subgroups — #525

Order 64.

165 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	C₂³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#123	C₂	1	2	yes	maximal, complemented
4	C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	#121	C₂	1	2	yes	maximal, complemented
6	#120	C₂	1	2	yes	maximal, complemented
7	C₄×D₄	C₂	1	4	yes	maximal, complemented
8	#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴	C₂²	1	1	yes	characteristic
10	#41	—	2	4	no	—
11	C₂×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₄	C₂²	1	4	yes	—
13	C₂×D₄	C₂²	1	2	yes	—
14	C₂×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₄	—	2	4	no	—
16	C₂²×C₄	C₂²	1	2	yes	—
17	C₂²×C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₄	C₂²	1	2	yes	—
19	C₂²×C₄	C₂²	1	1	yes	characteristic
20	C₂²×C₄	C₂²	1	2	yes	complemented
21	C₄⋊C₄	C₂²	1	2	yes	complemented
22	C₄⋊C₄	C₂²	1	4	yes	complemented
23	C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂²⋊C₄	C₂²	1	4	yes	—
25	C₂²⋊C₄	C₂²	1	4	yes	complemented
26	C₄²	C₂²	1	2	yes	complemented
27	C₂³	—	2	2	no	—
28	C₂³	—	2	4	no	—
29	C₂³	C₂³	1	2	yes	—
30	C₂³	C₂³	1	1	yes	characteristic
31	C₂³	C₂³	1	2	yes	—
32	Q₈	—	2	2	no	—
33	D₄	—	2	4	no	—
34	D₄	—	2	8	no	—
35	D₄	—	2	2	no	—
36	C₂×C₄	—	2	2	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	—	2	4	no	—
40	C₂×C₄	D₄	1	4	yes	complemented
41	C₂×C₄	C₂³	1	4	yes	—
42	C₂×C₄	C₂³	1	2	yes	—
43	C₂×C₄	C₂³	1	2	yes	—
44	C₂×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₄	C₂³	1	1	yes	characteristic
46	C₂²	—	2	8	no	—
47	C₂²	—	2	2	no	—
48	C₂²	—	2	2	no	—
49	C₂²	—	2	2	no	—
50	C₂²	—	2	2	no	—
51	C₂²	—	2	4	no	—
52	C₂²	C₂×D₄	1	2	yes	—
53	C₂²	#41	1	4	yes	complemented
54	C₂²	C₂⁴	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	2	2	no	—
57	C₄	—	2	4	no	—
58	C₄	C₂×D₄	1	2	yes	—
59	C₄	C₂×D₄	1	2	yes	—
60	C₂	—	2	2	no	—
61	C₂	—	2	4	no	—
62	C₂	—	4	2	no	—
63	C₂	C₂²×D₄	1	1	yes	characteristic
64	C₂	C₂×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³
commutator, Frattini, socle, cosocle	C₂²	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. C₂³×C₄	C₂	semidirect
3. #123	C₂	semidirect
4. C₂²⋊Q₈	C₂	semidirect
5. #121	C₂	semidirect
6. #120	C₂	semidirect
7. C₄×D₄	C₂	semidirect
8. #117	C₂	semidirect
11. C₂×Q₈	C₂²	semidirect
14. C₂×D₄	C₂²	semidirect
20. C₂²×C₄	C₂²	semidirect
21. C₄⋊C₄	C₂²	semidirect
22. C₄⋊C₄	C₂²	semidirect
25. C₂²⋊C₄	C₂²	semidirect
26. C₄²	C₂²	semidirect
40. C₂×C₄	D₄	semidirect
53. C₂²	#41	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
C₂³×C₄	2
#123	2
C₂²⋊Q₈	2
#121	2
#120	2
C₄×D₄	2
#117	2