Subgroups — #547

Order 64.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	2	yes	maximal, complemented
2	C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#123	C₂	1	4	yes	maximal, complemented
4	C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	#121	C₂	1	2	yes	maximal, complemented
6	C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	#41	—	2	8	no	—
10	C₂×Q₈	C₂²	1	2	yes	—
11	C₂×D₄	C₂²	1	1	yes	characteristic
12	C₂×D₄	C₂²	1	4	yes	—
13	C₂×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄	C₂²	1	2	yes	—
15	C₂²×C₄	C₂²	1	4	yes	—
16	C₂²×C₄	C₂²	1	2	yes	—
17	C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄⋊C₄	C₂²	1	2	yes	—
19	C₄⋊C₄	C₂²	1	4	yes	—
20	C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂²⋊C₄	C₂²	1	4	yes	—
25	C₄²	C₂²	1	1	yes	characteristic
26	C₂³	C₂³	1	2	yes	—
27	C₂³	C₂³	1	2	yes	—
28	Q₈	—	2	4	no	—
29	D₄	—	2	8	no	—
30	D₄	—	2	2	no	—
31	D₄	D₄	1	4	yes	complemented
32	C₂×C₄	—	2	8	no	—
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	C₂³	1	1	yes	characteristic
36	C₂×C₄	C₂³	1	2	yes	—
37	C₂×C₄	C₂³	1	2	yes	—
38	C₂×C₄	C₂³	1	4	yes	—
39	C₂×C₄	C₂³	1	1	yes	characteristic
40	C₂×C₄	C₂³	1	1	yes	characteristic
41	C₂²	—	2	2	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	2	2	no	—
44	C₂²	—	2	2	no	—
45	C₂²	—	2	2	no	—
46	C₂²	C₂×D₄	1	4	yes	—
47	C₂²	C₂⁴	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	2	4	no	—
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	C₂×D₄	1	2	yes	—
53	C₄	#41	1	2	yes	complemented
54	C₂	—	2	4	no	—
55	C₂	—	4	2	no	—
56	C₂	C₂²×D₄	1	1	yes	characteristic
57	C₂	C₂×#41	1	1	yes	characteristic
58	C₂	2₋¹⁺⁴	1	1	yes	characteristic

	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. C₄⋊Q₈	C₂	semidirect
3. #123	C₂	semidirect
4. C₂²⋊Q₈	C₂	semidirect
5. #121	C₂	semidirect
6. C₄×D₄	C₂	semidirect
7. C₄×D₄	C₂	semidirect
8. C₂×C₄⋊C₄	C₂	semidirect
31. D₄	D₄	semidirect
53. C₄	#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
C₄⋊Q₈	2
#123	2
C₂²⋊Q₈	2
#121	2
C₄×D₄	2
C₂×C₄⋊C₄	2