Subgroups — #62653

Order 256.

5465 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₄×2₋¹⁺⁴	C₂	1	32	yes	maximal, complemented
3	#3428	C₂	1	20	yes	maximal, complemented
4	C₂×#519	C₂	1	10	yes	maximal, complemented
5	C₂×2₋¹⁺⁴	C₄	1	32	yes	complemented
6	C₂×2₋¹⁺⁴	C₂²	1	16	yes	—
7	C₂²×#41	C₂²	1	10	yes	—
8	C₂³×Q₈	C₂²	1	5	yes	—
9	#520	C₂²	1	160	yes	complemented
10	#519	C₂²	1	40	yes	—
11	#519	C₂²	1	80	yes	complemented
12	#518	C₂²	1	40	yes	—
13	C₄×#41	C₂²	1	160	yes	complemented
14	C₂×C₄×Q₈	C₂²	1	80	yes	complemented
15	C₂×#117	C₂²	1	60	yes	—
16	2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
17	C₂×#41	C₂×C₄	1	160	yes	complemented
18	C₂×#41	C₂³	1	80	yes	—
19	C₂²×Q₈	C₂×C₄	1	80	yes	complemented
20	C₂²×Q₈	C₂³	1	40	yes	—
21	C₂²×Q₈	C₂³	1	10	yes	—
22	C₂²×D₄	C₂³	1	10	yes	—
23	C₂³×C₄	C₂³	1	15	yes	—
24	C₄×Q₈	C₂³	1	320	yes	—
25	C₄×D₄	C₂³	1	320	yes	—
26	#117	C₂³	1	80	yes	—
27	#117	C₂³	1	240	yes	—
28	C₂×C₄⋊C₄	C₂³	1	120	yes	—
29	C₂×C₂²⋊C₄	C₂³	1	40	yes	—
30	C₂×C₄²	C₂³	1	120	yes	—
31	C₂⁴	C₂⁴	1	5	yes	—
32	#41	C₂²×C₄	1	640	yes	—
33	C₂×Q₈	C₂²×C₄	1	80	yes	—
34	C₂×Q₈	C₂²×C₄	1	320	yes	—
35	C₂×Q₈	C₂⁴	1	40	yes	—
36	C₂×D₄	C₂²×C₄	1	80	yes	—
37	C₂×D₄	C₂⁴	1	40	yes	—
38	C₂²×C₄	C₂²×C₄	1	120	yes	—
39	C₂²×C₄	C₂⁴	1	80	yes	—
40	C₂²×C₄	C₂⁴	1	60	yes	—
41	C₂²×C₄	C₂⁴	1	10	yes	—
42	C₄⋊C₄	C₂⁴	1	160	yes	—
43	C₂²⋊C₄	C₂⁴	1	16	yes	—
44	C₂²⋊C₄	C₂⁴	1	80	yes	—
45	C₄²	C₂⁴	1	160	yes	—
46	C₂³	—	4	10	no	—
47	C₂³	C₂³×C₄	1	20	yes	—
48	C₂³	C₂⁵	1	1	yes	characteristic
49	C₂³	C₂⁵	1	10	yes	—
50	Q₈	C₂³×C₄	1	160	yes	—
51	D₄	C₂³×C₄	1	160	yes	—
52	C₂×C₄	—	4	80	no	—
53	C₂×C₄	C₂³×C₄	1	40	yes	—
54	C₂×C₄	C₂³×C₄	1	240	yes	—
55	C₂×C₄	C₂⁵	1	32	yes	—
56	C₂×C₄	C₂⁵	1	20	yes	—
57	C₂²	—	2	20	no	—
58	C₂²	—	2	2	no	—
59	C₂²	—	4	20	no	—
60	C₂²	C₂⁴×C₄	1	2	yes	—
61	C₂²	C₂⁴×C₄	1	20	yes	—
62	C₂²	C₂⁶	1	1	yes	characteristic
63	C₄	—	2	32	no	—
64	C₄	C₂⁴×C₄	1	40	yes	—
65	C₂	—	2	2	no	—
66	C₂	—	2	20	no	—
67	C₂	C₂⁵×C₄	1	1	yes	characteristic
68	C₂	2₋¹⁺⁶	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²	C₂⁶
commutator	C₂	C₂⁵×C₄
prime radical	C₂²×2₋¹⁺⁴	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. C₄×2₋¹⁺⁴	C₂	semidirect
3. #3428	C₂	semidirect
4. C₂×#519	C₂	semidirect
5. C₂×2₋¹⁺⁴	C₄	semidirect
9. #520	C₂²	semidirect
11. #519	C₂²	semidirect
13. C₄×#41	C₂²	semidirect
14. C₂×C₄×Q₈	C₂²	semidirect
16. 2₋¹⁺⁴	C₂×C₄	semidirect
17. C₂×#41	C₂×C₄	semidirect
19. C₂²×Q₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴	2
C₄×2₋¹⁺⁴	2
#3428	2
C₂×#519	2