Subgroups — #63076

Order 256.

5398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585	C₂	1	1	yes	maximal, characteristic
2	#3573	C₂	1	32	yes	maximal, complemented
3	C₂×#568	C₂	1	30	yes	maximal, complemented
4	#585	C₄	1	16	yes	—
5	#585	C₂²	1	16	yes	—
6	C₂×2₋¹⁺⁴	C₄	1	6	yes	—
7	C₂×H(5,𝔽₂)	C₄	1	10	yes	—
8	C₂²×#41	C₂²	1	15	yes	—
9	#568	C₂²	1	80	yes	complemented
10	#568	C₂²	1	240	yes	complemented
11	C₂×#131	C₂²	1	240	yes	complemented
12	C₂²×M₁₆	C₂²	1	60	yes	complemented
13	2₋¹⁺⁴	C₂×C₄	1	96	yes	—
14	H(5,𝔽₂)	C₂×C₄	1	160	yes	—
15	C₂×#41	C₂×C₄	1	60	yes	—
16	C₂×#41	C₂×C₄	1	120	yes	—
17	C₂×#41	C₂³	1	120	yes	—
18	C₂×#41	C₂³	1	20	yes	—
19	C₂²×Q₈	C₂×C₄	1	15	yes	—
20	C₂²×D₄	C₂×C₄	1	45	yes	—
21	C₂³×C₄	C₂³	1	15	yes	—
22	#131	C₂³	1	640	yes	complemented
23	C₂×M₁₆	C₂³	1	120	yes	—
24	C₂×M₁₆	C₂³	1	360	yes	complemented
25	C₂²×C₈	C₂³	1	120	yes	complemented
26	C₂⁴	C₂²×C₄	1	15	yes	—
27	#41	C₂²×C₄	1	80	yes	—
28	#41	C₂²×C₄	1	480	yes	—
29	#41	C₂⁴	1	80	yes	—
30	C₂×Q₈	C₂²×C₄	1	20	yes	—
31	C₂×Q₈	C₂²×C₄	1	120	yes	—
32	C₂×D₄	C₂²×C₄	1	60	yes	—
33	C₂×D₄	C₂²×C₄	1	360	yes	—
34	C₂²×C₄	C₂²×C₄	1	45	yes	—
35	C₂²×C₄	C₂²×C₄	1	60	yes	—
36	C₂²×C₄	C₂⁴	1	60	yes	—
37	C₂²×C₄	C₂⁴	1	15	yes	—
38	M₁₆	C₂⁴	1	16	yes	—
39	M₁₆	C₂⁴	1	240	yes	—
40	C₂×C₈	C₂⁴	1	240	yes	—
41	C₂³	—	8	15	no	—
42	C₂³	C₂³×C₄	1	15	yes	—
43	C₂³	C₂³×C₄	1	60	yes	—
44	Q₈	C₂³×C₄	1	80	yes	—
45	D₄	C₂³×C₄	1	240	yes	—
46	C₂×C₄	C₂³×C₄	1	15	yes	—
47	C₂×C₄	C₂³×C₄	1	30	yes	—
48	C₂×C₄	C₂³×C₄	1	180	yes	—
49	C₂×C₄	C₂⁵	1	30	yes	—
50	C₂×C₄	C₂⁵	1	1	yes	characteristic
51	C₈	C₂⁵	1	32	yes	—
52	C₂²	—	4	15	no	—
53	C₂²	—	4	60	no	—
54	C₂²	C₂⁴×C₄	1	1	yes	characteristic
55	C₂²	C₂⁴×C₄	1	30	yes	—
56	C₄	C₂⁴×C₄	1	1	yes	characteristic
57	C₄	C₂⁴×C₄	1	30	yes	—
58	C₄	C₂⁶	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	—	2	30	no	—
61	C₂	C₂⁵×C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₄	C₂⁶
commutator, socle	C₂	C₂⁵×C₄
prime radical	C₂×#585	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3573	C₂	semidirect
3. C₂×#568	C₂	semidirect
9. #568	C₂²	semidirect
10. #568	C₂²	semidirect
11. C₂×#131	C₂²	semidirect
12. C₂²×M₁₆	C₂²	semidirect
22. #131	C₂³	semidirect
24. C₂×M₁₆	C₂³	semidirect
25. C₂²×C₈	C₂³	semidirect

Maximal subgroups

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

subgroup	index
C₂×#585	2
#3573	2
C₂×#568	2