Subgroups — #342

Order 64.

71 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂⁴	C₂²	1	1	yes	characteristic
4	C₂²×C₄	—	2	2	no	—
5	C₂²×C₄	C₄	1	2	yes	complemented
6	C₂²⋊C₄	—	2	2	no	—
7	C₂²⋊C₄	—	2	2	no	—
8	C₂²⋊C₄	C₄	1	4	yes	complemented
9	C₂³	—	4	2	no	—
10	C₂³	C₂×C₄	1	1	yes	characteristic
11	C₂³	C₂×C₄	1	2	yes	—
12	C₂³	D₄	1	2	yes	—
13	C₂³	D₄	1	1	yes	characteristic
14	C₂³	Q₈	1	1	yes	characteristic
15	C₂×C₄	—	2	4	no	—
16	C₂×C₄	—	2	2	no	—
17	C₂×C₄	—	4	4	no	—
18	C₂×C₄	—	4	2	no	—
19	C₂²	—	2	2	no	—
20	C₂²	—	2	4	no	—
21	C₂²	—	4	2	no	—
22	C₂²	—	4	1	no	—
23	C₂²	—	4	1	no	—
24	C₂²	C₄²	1	1	yes	characteristic
25	C₂²	C₂²⋊C₄	1	2	yes	—
26	C₂²	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²	C₄⋊C₄	1	1	yes	characteristic
28	C₂²	C₄⋊C₄	1	2	yes	—
29	C₄	—	4	4	no	—
30	C₄	—	4	2	no	—
31	C₂	—	2	2	no	—
32	C₂	—	2	1	no	—
33	C₂	—	2	1	no	—
34	C₂	—	2	2	no	—
35	C₂	#95	1	1	yes	characteristic
36	C₂	#99	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²⋊C₄
commutator	C₂²	C₄²
Frattini, prime radical, cosocle	C₂⁴	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
5. C₂²×C₄	C₄	semidirect
8. C₂²⋊C₄	C₄	semidirect

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

subgroup	index
C₂×C₂²⋊C₄	2