Subgroups — #385

Order 64.

117 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×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#95	C₂	1	2	yes	maximal, complemented
6	C₂⁴	C₂²	1	1	yes	characteristic
7	C₂²×C₄	—	2	2	no	—
8	C₂²×C₄	—	2	4	no	—
9	C₂²×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₄	C₂²	1	2	yes	—
12	C₂²×C₄	C₂²	1	2	yes	—
13	C₄⋊C₄	—	2	4	no	—
14	C₂²⋊C₄	—	2	2	no	—
15	C₂²⋊C₄	C₄	1	4	yes	complemented
16	C₂³	—	2	2	no	—
17	C₂³	—	2	2	no	—
18	C₂³	C₂×C₄	1	2	yes	—
19	C₂³	D₄	1	2	yes	—
20	C₂³	Q₈	1	2	yes	—
21	C₂³	C₂³	1	1	yes	characteristic
22	C₂×C₄	—	2	8	no	—
23	C₂×C₄	—	2	2	no	—
24	C₂×C₄	—	2	2	no	—
25	C₂×C₄	—	2	2	no	—
26	C₂×C₄	—	2	2	no	—
27	C₂×C₄	—	2	1	no	—
28	C₂×C₄	—	2	1	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	—	2	2	no	—
31	C₂×C₄	C₂×C₄	1	4	yes	complemented
32	C₂×C₄	D₄	1	4	yes	—
33	C₂²	—	2	4	no	—
34	C₂²	—	2	2	no	—
35	C₂²	—	2	4	no	—
36	C₂²	—	2	2	no	—
37	C₂²	C₄⋊C₄	1	4	yes	complemented
38	C₂²	C₂²×C₄	1	1	yes	characteristic
39	C₂²	C₂×D₄	1	1	yes	characteristic
40	C₂²	C₂×D₄	1	2	yes	—
41	C₂²	C₂×Q₈	1	1	yes	characteristic
42	C₂²	#41	1	2	yes	—
43	C₄	—	2	4	no	—
44	C₄	—	2	4	no	—
45	C₄	—	4	1	no	—
46	C₄	—	4	1	no	—
47	C₂	—	2	4	no	—
48	C₂	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂	C₄×D₄	1	2	yes	—
50	C₂	#120	1	1	yes	characteristic
51	C₂	C₂²⋊Q₈	1	2	yes	—
52	C₂	#123	1	1	yes	characteristic

	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.

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