Subgroups — C₂²×#543 #62670

Order 256.

4655 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#543	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₂²⋊Q₈	C₂	1	12	yes	maximal, complemented
3	C₂³×C₂²⋊C₄	C₂	1	3	yes	maximal
4	C₂⁶	C₂²	1	1	yes	characteristic
5	C₂³×Q₈	C₂²	1	4	yes	complemented
6	C₂⁴×C₄	C₂²	1	6	yes	—
7	#543	C₂²	1	256	yes	direct product factor
8	C₂×C₂²⋊Q₈	C₂²	1	288	yes	complemented
9	C₂²×C₄⋊C₄	C₂²	1	12	yes	complemented
10	C₂²×C₂²⋊C₄	C₂²	1	12	yes	—
11	C₂²×C₂²⋊C₄	C₂²	1	72	yes	—
12	C₂⁵	—	4	8	no	—
13	C₂⁵	Q₈	1	16	yes	complemented
14	C₂⁵	C₂³	1	3	yes	—
15	C₂⁵	C₂³	1	12	yes	—
16	C₂²×Q₈	—	4	16	no	—
17	C₂²×Q₈	C₂³	1	48	yes	complemented
18	C₂³×C₄	—	4	24	no	—
19	C₂³×C₄	C₂³	1	12	yes	—
20	C₂³×C₄	C₂³	1	72	yes	—
21	C₂²⋊Q₈	C₂³	1	768	yes	complemented
22	C₂×C₄⋊C₄	C₂³	1	144	yes	complemented
23	C₂×C₂²⋊C₄	C₂³	1	144	yes	—
24	C₂×C₂²⋊C₄	C₂³	1	192	yes	—
25	C₂⁴	—	2	24	no	—
26	C₂⁴	—	4	64	no	—
27	C₂⁴	—	4	48	no	—
28	C₂⁴	C₂×Q₈	1	24	yes	—
29	C₂⁴	C₂×Q₈	1	96	yes	complemented
30	C₂⁴	C₂⁴	1	18	yes	—
31	C₂⁴	C₂⁴	1	16	yes	—
32	C₂⁴	C₂⁴	1	1	yes	characteristic
33	C₂×Q₈	—	4	96	no	—
34	C₂×Q₈	C₂⁴	1	64	yes	complemented
35	C₂²×C₄	—	2	48	no	—
36	C₂²×C₄	—	4	144	no	—
37	C₂²×C₄	C₂⁴	1	72	yes	—
38	C₂²×C₄	C₂⁴	1	96	yes	—
39	C₄⋊C₄	C₂⁴	1	192	yes	complemented
40	C₂²⋊C₄	C₂⁴	1	192	yes	—
41	C₂³	—	2	96	no	—
42	C₂³	—	2	72	no	—
43	C₂³	—	4	192	no	—
44	C₂³	—	4	32	no	—
45	C₂³	C₂²×Q₈	1	72	yes	—
46	C₂³	C₂²×Q₈	1	4	yes	—
47	C₂³	C₂²×Q₈	1	64	yes	complemented
48	C₂³	H(5,𝔽₂)	1	8	yes	—
49	C₂³	C₂⁵	1	12	yes	—
50	C₂³	C₂⁵	1	3	yes	—
51	Q₈	—	4	64	no	—
52	C₂×C₄	—	2	144	no	—
53	C₂×C₄	—	4	96	no	—
54	C₂×C₄	C₂⁵	1	48	yes	—
55	C₂²	—	2	24	no	—
56	C₂²	—	2	144	no	—
57	C₂²	—	4	64	no	—
58	C₂²	#543	1	16	yes	direct product factor
59	C₂²	C₂³×Q₈	1	24	yes	—
60	C₂²	C₂³×Q₈	1	6	yes	—
61	C₂²	C₂×H(5,𝔽₂)	1	12	yes	—
62	C₂²	C₂⁶	1	1	yes	characteristic
63	C₄	—	2	48	no	—
64	C₂	—	2	24	no	—
65	C₂	C₂×#543	1	12	yes	direct product factor
66	C₂	C₂⁴×Q₈	1	1	yes	characteristic
67	C₂	C₂²×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#543	C₂	direct (N × H)
2. C₂²×C₂²⋊Q₈	C₂	semidirect
5. C₂³×Q₈	C₂²	semidirect
7. #543	C₂²	direct (N × H)
8. C₂×C₂²⋊Q₈	C₂²	semidirect
9. C₂²×C₄⋊C₄	C₂²	semidirect
13. C₂⁵	Q₈	semidirect
17. C₂²×Q₈	C₂³	semidirect
21. C₂²⋊Q₈	C₂³	semidirect
22. C₂×C₄⋊C₄	C₂³	semidirect
29. C₂⁴	C₂×Q₈	semidirect
34. C₂×Q₈	C₂⁴	semidirect
39. C₄⋊C₄	C₂⁴	semidirect
47. C₂³	C₂²×Q₈	semidirect
58. C₂²	#543	direct (N × H)
65. C₂	C₂×#543	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#543	2
C₂²×C₂²⋊Q₈	2
C₂³×C₂²⋊C₄	2

Subgroups — C22×#543 #62670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#543 #62670