Subgroups — #62843

Order 256.

3565 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3501	C₂	1	48	yes	maximal, complemented
2	#3458	C₂	1	4	yes	maximal, complemented
3	C₂×#543	C₂	1	8	yes	maximal, complemented
4	C₂×#518	C₂	1	3	yes	maximal
5	C₂³×D₄	C₂²	1	1	yes	characteristic
6	#554	C₂²	1	192	yes	complemented
7	D₄×Q₈	C₂²	1	64	yes	complemented
8	#544	C₂²	1	48	yes	complemented
9	#543	C₂²	1	16	yes	—
10	#543	C₂²	1	96	yes	complemented
11	#533	C₂²	1	48	yes	complemented
12	C₂×C₂²⋊Q₈	C₂²	1	96	yes	complemented
13	#518	C₂²	1	48	yes	—
14	C₂×C₄×D₄	C₂²	1	24	yes	—
15	C₂×#117	C₂²	1	6	yes	—
16	C₂²×C₂²⋊C₄	C₂²	1	12	yes	—
17	C₂⁵	C₂³	1	2	yes	—
18	C₂²×Q₈	C₂³	1	32	yes	complemented
19	C₂²×D₄	Q₈	1	16	yes	complemented
20	C₂²×D₄	C₂³	1	12	yes	—
21	C₂³×C₄	C₂³	1	24	yes	—
22	C₂³×C₄	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈	C₂³	1	96	yes	complemented
24	#125	C₂³	1	96	yes	complemented
25	C₂²⋊Q₈	C₂³	1	192	yes	—
26	C₂²⋊Q₈	C₂³	1	384	yes	complemented
27	C₄×Q₈	C₂³	1	64	yes	complemented
28	C₄×D₄	C₂³	1	192	yes	—
29	#117	C₂³	1	48	yes	—
30	C₂×C₄⋊C₄	C₂³	1	12	yes	—
31	C₂×C₄⋊C₄	C₂³	1	96	yes	complemented
32	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
33	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
34	C₂×C₂²⋊C₄	C₂³	1	96	yes	—
35	C₂×C₄²	C₂³	1	12	yes	—
36	C₂⁴	—	8	4	no	—
37	C₂⁴	C₂×Q₈	1	16	yes	complemented
38	C₂⁴	C₂⁴	1	1	yes	characteristic
39	C₂⁴	C₂⁴	1	12	yes	—
40	C₂×Q₈	—	8	32	no	—
41	C₂×Q₈	C₂⁴	1	64	yes	—
42	C₂×D₄	C₂×Q₈	1	96	yes	complemented
43	C₂×D₄	C₂⁴	1	16	yes	—
44	C₂²×C₄	—	8	24	no	—
45	C₂²×C₄	C₂×Q₈	1	8	yes	complemented
46	C₂²×C₄	C₂⁴	1	24	yes	—
47	C₂²×C₄	C₂⁴	1	96	yes	—
48	C₂²×C₄	C₂⁴	1	6	yes	—
49	C₄⋊C₄	C₂⁴	1	48	yes	—
50	C₄⋊C₄	C₂⁴	1	192	yes	—
51	C₂²⋊C₄	C₂⁴	1	48	yes	—
52	C₂²⋊C₄	C₂⁴	1	96	yes	—
53	C₄²	C₂⁴	1	48	yes	—
54	C₂³	—	4	24	no	—
55	C₂³	—	4	2	no	—
56	C₂³	—	8	16	no	—
57	C₂³	C₂²×Q₈	1	4	yes	—
58	C₂³	C₂²×Q₈	1	48	yes	—
59	C₂³	C₂⁵	1	3	yes	—
60	C₂³	C₂⁵	1	8	yes	—
61	Q₈	—	4	64	no	—
62	D₄	C₂²×Q₈	1	64	yes	—
63	C₂×C₄	—	4	24	no	—
64	C₂×C₄	—	4	96	no	—
65	C₂×C₄	C₂²×Q₈	1	24	yes	—
66	C₂×C₄	C₂⁵	1	48	yes	—
67	C₂×C₄	C₂⁵	1	4	yes	—
68	C₂²	—	2	16	no	—
69	C₂²	—	2	6	no	—
70	C₂²	—	4	4	no	—
71	C₂²	—	4	48	no	—
72	C₂²	C₂³×Q₈	1	6	yes	—
73	C₂²	C₂³×Q₈	1	16	yes	—
74	C₂²	C₂⁶	1	1	yes	characteristic
75	C₄	—	2	48	no	—
76	C₄	C₂³×Q₈	1	8	yes	—
77	C₂	—	2	6	no	—
78	C₂	—	2	16	no	—
79	C₂	C₂⁴×Q₈	1	1	yes	characteristic
80	C₂	H(7,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂² ▷ 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
1. #3501	C₂	semidirect
2. #3458	C₂	semidirect
3. C₂×#543	C₂	semidirect
6. #554	C₂²	semidirect
7. D₄×Q₈	C₂²	semidirect
8. #544	C₂²	semidirect
10. #543	C₂²	semidirect
11. #533	C₂²	semidirect
12. C₂×C₂²⋊Q₈	C₂²	semidirect
18. C₂²×Q₈	C₂³	semidirect
19. C₂²×D₄	Q₈	semidirect
23. C₄⋊Q₈	C₂³	semidirect
24. #125	C₂³	semidirect
26. C₂²⋊Q₈	C₂³	semidirect
27. C₄×Q₈	C₂³	semidirect
31. C₂×C₄⋊C₄	C₂³	semidirect
37. C₂⁴	C₂×Q₈	semidirect
42. C₂×D₄	C₂×Q₈	semidirect
45. C₂²×C₄	C₂×Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3501	2
#3458	2
C₂×#543	2
C₂×#518	2