Subgroups — C₂²×#504 #62622

Order 256.

1059 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×#504	C₂	1	24	yes	maximal, direct product factor
3	C₂²×M₃₂	C₂	1	3	yes	maximal, complemented
4	C₂³×C₁₆	C₂	1	3	yes	maximal, complemented
5	C₂²×#41	C₄	1	1	yes	characteristic
6	C₂×#131	C₄	1	12	yes	—
7	C₂×#131	C₂²	1	12	yes	—
8	C₂²×M₁₆	C₄	1	3	yes	—
9	C₂³×C₈	C₂²	1	3	yes	—
10	#504	C₂²	1	64	yes	direct product factor
11	C₂×M₃₂	C₂²	1	36	yes	complemented
12	C₂²×C₁₆	C₂²	1	3	yes	—
13	C₂²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆	C₂²	1	36	yes	complemented
15	C₂×#41	C₈	1	12	yes	—
16	C₂×#41	C₂×C₄	1	12	yes	—
17	C₂²×Q₈	C₈	1	1	yes	characteristic
18	C₂²×D₄	C₈	1	3	yes	—
19	C₂³×C₄	C₂×C₄	1	3	yes	—
20	#131	C₂×C₄	1	48	yes	—
21	#131	C₂³	1	16	yes	—
22	C₂×M₁₆	C₂×C₄	1	36	yes	—
23	C₂²×C₈	C₂×C₄	1	3	yes	—
24	C₂²×C₈	C₂×C₄	1	18	yes	—
25	C₂²×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	18	yes	—
27	M₃₂	C₂³	1	48	yes	complemented
28	C₂×C₁₆	C₂³	1	18	yes	—
29	C₂×C₁₆	C₂³	1	6	yes	—
30	C₂×C₁₆	C₂³	1	48	yes	complemented
31	C₂⁴	C₂×C₈	1	3	yes	—
32	#41	C₂×C₈	1	48	yes	—
33	#41	C₂²×C₄	1	16	yes	—
34	C₂×Q₈	C₂×C₈	1	12	yes	—
35	C₂×D₄	C₂×C₈	1	36	yes	—
36	C₂²×C₄	C₂×C₈	1	3	yes	—
37	C₂²×C₄	C₂×C₈	1	18	yes	—
38	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄	C₂²×C₄	1	18	yes	—
40	M₁₆	C₂²×C₄	1	48	yes	—
41	C₂×C₈	C₂²×C₄	1	3	yes	—
42	C₂×C₈	C₂²×C₄	1	18	yes	—
43	C₂×C₈	C₂²×C₄	1	36	yes	—
44	C₂×C₈	C₂⁴	1	3	yes	—
45	C₂×C₈	C₂⁴	1	12	yes	—
46	C₁₆	C₂⁴	1	4	yes	—
47	C₁₆	C₂⁴	1	12	yes	—
48	C₂³	—	2	12	no	—
49	C₂³	C₂²×C₈	1	18	yes	—
50	C₂³	C₂²×C₈	1	1	yes	characteristic
51	Q₈	C₂²×C₈	1	16	yes	—
52	D₄	C₂²×C₈	1	48	yes	—
53	C₂×C₄	C₂²×C₈	1	3	yes	—
54	C₂×C₄	C₂²×C₈	1	18	yes	—
55	C₂×C₄	C₂²×C₈	1	36	yes	—
56	C₂×C₄	C₂³×C₄	1	3	yes	—
57	C₂×C₄	C₂³×C₄	1	12	yes	—
58	C₈	C₂³×C₄	1	3	yes	—
59	C₈	C₂³×C₄	1	12	yes	—
60	C₈	C₂⁵	1	1	yes	characteristic
61	C₂²	—	2	36	no	—
62	C₂²	#504	1	4	yes	direct product factor
63	C₂²	C₂³×C₈	1	12	yes	—
64	C₂²	C₂³×C₈	1	3	yes	—
65	C₄	C₂³×C₈	1	3	yes	—
66	C₄	C₂³×C₈	1	12	yes	—
67	C₄	C₂⁴×C₄	1	1	yes	characteristic
68	C₂	—	2	12	no	—
69	C₂	C₂×#504	1	6	yes	direct product factor
70	C₂	C₂⁴×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂²
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₈	C₂⁵
socle	C₂³	C₂²×C₈
prime radical	C₂²×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#504	C₂	direct (N × H)
3. C₂²×M₃₂	C₂	semidirect
4. C₂³×C₁₆	C₂	semidirect
10. #504	C₂²	direct (N × H)
11. C₂×M₃₂	C₂²	semidirect
14. C₂²×C₁₆	C₂²	semidirect
27. M₃₂	C₂³	semidirect
30. C₂×C₁₆	C₂³	semidirect
62. C₂²	#504	direct (N × H)
69. C₂	C₂×#504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131	2
C₂×#504	2
C₂²×M₃₂	2
C₂³×C₁₆	2

Subgroups — C22×#504 #62622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#504 #62622