Subgroups — #397

Order 64.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#95	C₂	1	2	yes	maximal, complemented
5	#95	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴	C₂²	1	1	yes	characteristic
7	C₂²×C₄	C₂²	1	2	yes	—
8	C₂²×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₄	C₂²	1	2	yes	—
10	C₂²×C₄	C₂²	1	1	yes	characteristic
11	C₄⋊C₄	—	2	2	no	—
12	C₂²⋊C₄	—	2	4	no	—
13	C₂²⋊C₄	—	2	2	no	—
14	C₂³	—	2	1	no	—
15	C₂³	—	2	1	no	—
16	C₂³	—	2	2	no	—
17	C₂³	—	2	2	no	—
18	C₂³	D₄	1	2	yes	—
19	C₂³	C₂³	1	1	yes	characteristic
20	C₂×C₄	—	2	2	no	—
21	C₂×C₄	—	2	2	no	—
22	C₂×C₄	—	2	2	no	—
23	C₂×C₄	—	2	1	no	—
24	C₂×C₄	—	2	2	no	—
25	C₂×C₄	—	2	2	no	—
26	C₂×C₄	—	2	2	no	—
27	C₂×C₄	—	2	2	no	—
28	C₂×C₄	—	2	2	no	—
29	C₂×C₄	D₄	1	2	yes	—
30	C₂²	—	2	2	no	—
31	C₂²	—	2	4	no	—
32	C₂²	—	4	1	no	—
33	C₂²	—	4	1	no	—
34	C₂²	—	4	2	no	—
35	C₂²	C₂×D₄	1	1	yes	characteristic
36	C₂²	C₂×D₄	1	1	yes	characteristic
37	C₂²	#41	1	1	yes	characteristic
38	C₂²	#41	1	2	yes	—
39	C₂²	#41	1	2	yes	—
40	C₄	—	2	2	no	—
41	C₄	—	4	2	no	—
42	C₄	—	4	2	no	—
43	C₄	—	4	1	no	—
44	C₂	—	4	2	no	—
45	C₂	#121	1	1	yes	characteristic
46	C₂	#123	1	1	yes	characteristic
47	C₂	#123	1	2	yes	—
48	C₂	#124	1	1	yes	characteristic
49	C₂	#126	1	2	yes	—

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

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

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