Subgroups — #381

Order 64.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₄	C₂	1	2	yes	maximal
3	#95	C₂	1	4	yes	maximal, complemented
4	C₂⁴	C₂²	1	1	yes	characteristic
5	C₂²×C₄	—	2	4	no	—
6	C₂²×C₄	C₄	1	4	yes	complemented
7	C₂²×C₄	C₂²	1	4	yes	—
8	C₂²×C₄	C₂²	1	2	yes	—
9	C₂²⋊C₄	—	2	4	no	—
10	C₂³	—	2	4	no	—
11	C₂³	C₂×C₄	1	2	yes	—
12	C₂³	D₄	1	4	yes	—
13	C₂³	C₂³	1	1	yes	characteristic
14	C₂×C₄	—	2	8	no	—
15	C₂×C₄	—	2	4	no	—
16	C₂×C₄	—	2	4	no	—
17	C₂×C₄	—	2	8	no	—
18	C₂×C₄	C₂×C₄	1	4	yes	—
19	C₂²	—	2	4	no	—
20	C₂²	—	2	8	no	—
21	C₂²	C₂²⋊C₄	1	4	yes	—
22	C₂²	C₂²×C₄	1	1	yes	characteristic
23	C₂²	C₂×D₄	1	2	yes	—
24	C₂²	#41	1	4	yes	—
25	C₄	—	2	4	no	—
26	C₄	—	4	4	no	—
27	C₂	—	2	4	no	—
28	C₂	C₂×C₂²⋊C₄	1	1	yes	characteristic
29	C₂	#117	1	2	yes	—
30	C₂	#123	1	4	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂³
commutator	C₂²	C₂²×C₄
prime radical	C₂⁴	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

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

N	H ≅ G/N	type
3. #95	C₂	semidirect
6. C₂²×C₄	C₄	semidirect

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