Subgroups — #62652

Order 256.

5837 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₄×H(5,𝔽₂)	C₂	1	32	yes	maximal, complemented
3	#3428	C₂	1	12	yes	maximal, complemented
4	C₂×#518	C₂	1	18	yes	maximal, complemented
5	C₂×H(5,𝔽₂)	C₄	1	32	yes	complemented
6	C₂×H(5,𝔽₂)	C₂²	1	16	yes	—
7	C₂²×#41	C₂²	1	6	yes	—
8	C₂³×D₄	C₂²	1	9	yes	—
9	#520	C₂²	1	96	yes	complemented
10	#519	C₂²	1	8	yes	complemented
11	#518	C₂²	1	72	yes	complemented
12	#518	C₂²	1	144	yes	complemented
13	C₄×#41	C₂²	1	96	yes	complemented
14	C₂×C₄×D₄	C₂²	1	144	yes	complemented
15	C₂×#117	C₂²	1	36	yes	complemented
16	C₂²×C₂²⋊C₄	C₂²	1	24	yes	complemented
17	C₂⁵	C₂³	1	6	yes	—
18	H(5,𝔽₂)	C₂×C₄	1	256	yes	complemented
19	C₂×#41	C₂×C₄	1	96	yes	complemented
20	C₂×#41	C₂³	1	48	yes	—
21	C₂²×Q₈	C₂³	1	2	yes	—
22	C₂²×D₄	C₂×C₄	1	144	yes	complemented
23	C₂²×D₄	C₂³	1	72	yes	—
24	C₂²×D₄	C₂³	1	18	yes	—
25	C₂³×C₄	C₂³	1	9	yes	—
26	C₂³×C₄	C₂³	1	48	yes	complemented
27	C₄×Q₈	C₂³	1	64	yes	complemented
28	C₄×D₄	C₂³	1	576	yes	complemented
29	#117	C₂³	1	48	yes	—
30	#117	C₂³	1	144	yes	complemented
31	C₂×C₄⋊C₄	C₂³	1	72	yes	complemented
32	C₂×C₂²⋊C₄	C₂³	1	72	yes	—
33	C₂×C₂²⋊C₄	C₂³	1	144	yes	complemented
34	C₂×C₄²	C₂³	1	72	yes	complemented
35	C₂⁴	—	8	12	no	—
36	C₂⁴	C₂²×C₄	1	48	yes	complemented
37	C₂⁴	C₂⁴	1	9	yes	—
38	C₂⁴	C₂⁴	1	24	yes	—
39	#41	C₂²×C₄	1	384	yes	complemented
40	C₂×Q₈	C₂²×C₄	1	16	yes	complemented
41	C₂×Q₈	C₂⁴	1	8	yes	—
42	C₂×D₄	C₂²×C₄	1	144	yes	complemented
43	C₂×D₄	C₂²×C₄	1	576	yes	complemented
44	C₂×D₄	C₂⁴	1	72	yes	—
45	C₂²×C₄	—	8	48	no	—
46	C₂²×C₄	C₂²×C₄	1	72	yes	complemented
47	C₂²×C₄	C₂⁴	1	144	yes	—
48	C₂²×C₄	C₂⁴	1	36	yes	—
49	C₂²×C₄	C₂⁴	1	6	yes	—
50	C₄⋊C₄	C₂⁴	1	96	yes	—
51	C₂²⋊C₄	C₂⁴	1	16	yes	—
52	C₂²⋊C₄	C₂⁴	1	144	yes	—
53	C₄²	C₂⁴	1	96	yes	—
54	C₂³	—	4	48	no	—
55	C₂³	—	4	18	no	—
56	C₂³	—	8	48	no	—
57	C₂³	C₂³×C₄	1	36	yes	—
58	C₂³	C₂³×C₄	1	96	yes	—
59	C₂³	C₂⁵	1	1	yes	characteristic
60	C₂³	C₂⁵	1	18	yes	—
61	Q₈	C₂³×C₄	1	32	yes	—
62	D₄	C₂³×C₄	1	288	yes	—
63	C₂×C₄	—	4	144	no	—
64	C₂×C₄	C₂³×C₄	1	24	yes	—
65	C₂×C₄	C₂³×C₄	1	144	yes	—
66	C₂×C₄	C₂⁵	1	32	yes	—
67	C₂×C₄	C₂⁵	1	12	yes	—
68	C₂²	—	2	36	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	36	no	—
71	C₂²	—	4	96	no	—
72	C₂²	C₂⁴×C₄	1	2	yes	—
73	C₂²	C₂⁴×C₄	1	36	yes	—
74	C₂²	C₂⁶	1	1	yes	characteristic
75	C₄	—	2	32	no	—
76	C₄	C₂⁴×C₄	1	24	yes	—
77	C₂	—	2	2	no	—
78	C₂	—	2	36	no	—
79	C₂	C₂⁵×C₄	1	1	yes	characteristic
80	C₂	H(7,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²	C₂⁶
commutator	C₂	C₂⁵×C₄
prime radical	C₂²×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×H(5,𝔽₂)	C₂	semidirect
3. #3428	C₂	semidirect
4. C₂×#518	C₂	semidirect
5. C₂×H(5,𝔽₂)	C₄	semidirect
9. #520	C₂²	semidirect
10. #519	C₂²	semidirect
11. #518	C₂²	semidirect
12. #518	C₂²	semidirect
13. C₄×#41	C₂²	semidirect
14. C₂×C₄×D₄	C₂²	semidirect
15. C₂×#117	C₂²	semidirect
16. C₂²×C₂²⋊C₄	C₂²	semidirect
18. H(5,𝔽₂)	C₂×C₄	semidirect
19. C₂×#41	C₂×C₄	semidirect
22. C₂²×D₄	C₂×C₄	semidirect
26. C₂³×C₄	C₂³	semidirect
27. C₄×Q₈	C₂³	semidirect
28. C₄×D₄	C₂³	semidirect
30. #117	C₂³	semidirect
31. C₂×C₄⋊C₄	C₂³	semidirect
33. C₂×C₂²⋊C₄	C₂³	semidirect
34. C₂×C₄²	C₂³	semidirect
36. C₂⁴	C₂²×C₄	semidirect
39. #41	C₂²×C₄	semidirect
40. C₂×Q₈	C₂²×C₄	semidirect
42. C₂×D₄	C₂²×C₄	semidirect
43. C₂×D₄	C₂²×C₄	semidirect
46. C₂²×C₄	C₂²×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)	2
C₄×H(5,𝔽₂)	2
#3428	2
C₂×#518	2