Subgroups — #380

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	2	yes	maximal
4	#95	C₂	1	2	yes	maximal, complemented
5	C₂⁴	C₂²	1	1	yes	characteristic
6	C₂²×C₄	—	2	4	no	—
7	C₂²×C₄	C₄	1	4	yes	complemented
8	C₂²×C₄	C₂²	1	4	yes	—
9	C₂²×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄	C₂²	1	1	yes	characteristic
11	C₄⋊C₄	—	2	4	no	—
12	C₂²⋊C₄	—	2	4	no	—
13	C₂³	—	2	2	no	—
14	C₂³	—	2	2	no	—
15	C₂³	C₂×C₄	1	2	yes	—
16	C₂³	D₄	1	2	yes	—
17	C₂³	Q₈	1	2	yes	—
18	C₂³	C₂³	1	1	yes	characteristic
19	C₂×C₄	—	2	2	no	—
20	C₂×C₄	—	2	8	no	—
21	C₂×C₄	—	2	4	no	—
22	C₂×C₄	—	2	4	no	—
23	C₂×C₄	—	2	4	no	—
24	C₂×C₄	C₂×C₄	1	2	yes	—
25	C₂×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₄	D₄	1	4	yes	—
27	C₂²	—	2	2	no	—
28	C₂²	—	2	2	no	—
29	C₂²	—	2	4	no	—
30	C₂²	—	2	4	no	—
31	C₂²	C₄⋊C₄	1	4	yes	complemented
32	C₂²	C₂²×C₄	1	1	yes	characteristic
33	C₂²	C₂×D₄	1	2	yes	—
34	C₂²	C₂×D₄	1	1	yes	characteristic
35	C₂²	C₂×Q₈	1	1	yes	characteristic
36	C₂²	#41	1	2	yes	—
37	C₄	—	2	2	no	—
38	C₄	—	4	4	no	—
39	C₄	C₂²⋊C₄	1	4	yes	complemented
40	C₂	—	2	4	no	—
41	C₂	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂	#117	1	1	yes	characteristic
44	C₂	#121	1	2	yes	—
45	C₂	C₂²⋊Q₈	1	2	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.

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