Subgroups — #536

Order 64.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
4	#124	C₂	1	2	yes	maximal, complemented
5	#123	C₂	1	4	yes	maximal, complemented
6	C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	#117	C₂	1	1	yes	maximal, characteristic, complemented
8	#41	—	2	4	no	—
9	C₂×Q₈	—	2	4	no	—
10	C₂×Q₈	C₂²	1	2	yes	—
11	C₂×Q₈	C₂²	1	2	yes	—
12	C₂×Q₈	C₂²	1	1	yes	characteristic
13	C₂×D₄	C₂²	1	2	yes	—
14	C₂×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₄	C₂²	1	2	yes	—
16	C₂²×C₄	C₂²	1	2	yes	—
17	C₂²×C₄	C₂²	1	1	yes	characteristic
18	C₄⋊C₄	C₂²	1	2	yes	—
19	C₄⋊C₄	C₂²	1	8	yes	—
20	C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂²⋊C₄	C₂²	1	8	yes	—
22	C₄²	C₂²	1	2	yes	—
23	C₂³	C₂³	1	2	yes	—
24	C₂³	C₂³	1	1	yes	characteristic
25	Q₈	—	2	4	no	—
26	Q₈	—	2	4	no	—
27	Q₈	—	2	2	no	—
28	D₄	—	2	4	no	—
29	D₄	—	2	2	no	—
30	C₂×C₄	—	2	4	no	—
31	C₂×C₄	—	2	4	no	—
32	C₂×C₄	D₄	1	4	yes	—
33	C₂×C₄	C₂³	1	4	yes	—
34	C₂×C₄	C₂³	1	4	yes	—
35	C₂×C₄	C₂³	1	2	yes	—
36	C₂×C₄	C₂³	1	1	yes	characteristic
37	C₂×C₄	C₂³	1	1	yes	characteristic
38	C₂²	—	2	2	no	—
39	C₂²	—	2	2	no	—
40	C₂²	—	2	4	no	—
41	C₂²	C₂×D₄	1	2	yes	—
42	C₂²	C₂⁴	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	2	4	no	—
45	C₄	—	2	4	no	—
46	C₄	C₂×D₄	1	2	yes	—
47	C₄	C₂×D₄	1	2	yes	—
48	C₂	—	2	2	no	—
49	C₂	—	4	2	no	—
50	C₂	C₂²×D₄	1	1	yes	characteristic
51	C₂	2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
prime radical	C₂×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ 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.