Subgroups — #435

Order 64.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄	C₄	1	1	yes	characteristic
9	C₂²×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄	C₂²	1	1	yes	characteristic
11	M₁₆	—	2	2	no	—
12	C₂×C₈	—	2	2	no	—
13	C₂×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₄	C₄	1	1	yes	characteristic
18	C₂²⋊C₄	C₄	1	1	yes	characteristic
19	C₂²⋊C₄	C₄	1	1	yes	characteristic
20	C₄²	C₂²	1	1	yes	characteristic
21	C₂³	C₂×C₄	1	1	yes	characteristic
22	C₂³	C₂×C₄	1	1	yes	characteristic
23	D₄	—	2	2	no	—
24	C₂×C₄	—	2	1	no	—
25	C₂×C₄	—	2	1	no	—
26	C₂×C₄	—	2	1	no	—
27	C₂×C₄	—	2	1	no	—
28	C₂×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄	C₂³	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	2	1	no	—
35	C₈	—	2	1	no	—
36	C₈	D₄	1	2	yes	complemented
37	C₂²	—	2	1	no	—
38	C₂²	—	2	1	no	—
39	C₂²	—	2	1	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	2	1	no	—
42	C₂²	M₁₆	1	2	yes	complemented
43	C₂²	C₂²×C₄	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	2	1	no	—
46	C₄	—	2	1	no	—
47	C₄	—	2	1	no	—
48	C₄	C₂×D₄	1	1	yes	characteristic
49	C₄	#41	1	1	yes	characteristic
50	C₂	—	2	2	no	—
51	C₂	—	4	1	no	—
52	C₂	C₄×D₄	1	1	yes	characteristic
53	C₂	C₂×M₁₆	1	1	yes	characteristic
54	C₂	#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂³
commutator, socle	C₂²	C₂²×C₄
prime radical	C₂×D₄	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆	C₂	semidirect
2. C₂²×C₈	C₂	semidirect
4. C₄⋊C₈	C₂	semidirect
5. C₂²⋊C₈	C₂	semidirect
6. C₂²⋊C₈	C₂	semidirect
7. #97	C₂	semidirect
36. C₈	D₄	semidirect
42. C₂²	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆	2
C₂²×C₈	2
C₄×D₄	2
C₄⋊C₈	2
C₂²⋊C₈	2
#97	2