Subgroups — #554

Order 64.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#125	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
4	C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄	C₂	1	2	yes	maximal
7	C₄×D₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₂×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Q₈	C₂²	1	2	yes	—
11	C₂×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄	C₂²	1	2	yes	—
13	C₂²×C₄	C₂²	1	4	yes	—
14	C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₄⋊C₄	C₂²	1	4	yes	—
16	C₄⋊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	2	yes	—
20	C₄⋊C₄	C₂²	1	4	yes	—
21	C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂²⋊C₄	C₂²	1	2	yes	—
23	C₄²	C₂²	1	2	yes	—
24	C₄²	C₂²	1	1	yes	characteristic
25	C₂³	C₂³	1	2	yes	—
26	Q₈	—	2	2	no	—
27	Q₈	—	4	2	no	—
28	D₄	Q₈	1	4	yes	complemented
29	C₂×C₄	—	2	4	no	—
30	C₂×C₄	—	4	4	no	—
31	C₂×C₄	C₂³	1	2	yes	—
32	C₂×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₄	C₂³	1	2	yes	—
34	C₂×C₄	C₂³	1	2	yes	—
35	C₂×C₄	C₂³	1	4	yes	—
36	C₂×C₄	C₂³	1	1	yes	characteristic
37	C₂×C₄	C₂³	1	1	yes	characteristic
38	C₂²	—	2	2	no	—
39	C₂²	—	2	2	no	—
40	C₂²	C₂×Q₈	1	4	yes	—
41	C₂²	C₂⁴	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	2	2	no	—
44	C₄	—	2	2	no	—
45	C₄	—	2	4	no	—
46	C₄	—	2	1	no	—
47	C₄	C₂×Q₈	1	2	yes	—
48	C₄	#41	1	2	yes	—
49	C₂	—	2	4	no	—
50	C₂	C₂²×Q₈	1	1	yes	characteristic
51	C₂	C₂×#41	1	1	yes	characteristic
52	C₂	H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
prime radical	C₂×D₄	C₂²
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. C₄⋊Q₈	C₂	semidirect
2. #125	C₂	semidirect
3. C₂²⋊Q₈	C₂	semidirect
4. C₂²⋊Q₈	C₂	semidirect
5. C₄×Q₈	C₂	semidirect
8. C₂×C₄⋊C₄	C₂	semidirect
28. D₄	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈	2
#125	2
C₂²⋊Q₈	2
C₄×Q₈	2
C₄×D₄	2
C₂×C₄⋊C₄	2