Subgroups — #33982

Order 256.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3415	C₂	1	2	yes	maximal, complemented
2	#2263	C₂	1	6	yes	maximal, complemented
3	#2262	C₂	1	3	yes	maximal, complemented
4	C₂×D₃₂	C₂	1	3	yes	maximal, complemented
5	#2258	C₂	1	1	yes	maximal, characteristic, complemented
6	#576	—	2	2	no	—
7	#509	—	2	6	no	—
8	#508	C₂²	1	6	yes	complemented
9	C₂×D₁₆	C₂²	1	6	yes	complemented
10	#504	C₂²	1	1	yes	characteristic
11	Q₆₄	C₂²	1	1	yes	characteristic, complemented
12	SD₆₄	C₂²	1	6	yes	complemented
13	D₃₂	C₂²	1	3	yes	complemented
14	D₃₂	C₂²	1	6	yes	complemented
15	M₆₄	C₂²	1	3	yes	complemented
16	C₂×C₃₂	C₂²	1	3	yes	complemented
17	H(5,𝔽₂)	—	4	2	no	—
18	AGL(1,ℤ/8)	—	4	6	no	—
19	#135	—	2	6	no	—
20	C₂×D₈	—	2	6	no	—
21	#131	D₄	1	1	yes	characteristic
22	Q₃₂	C₂³	1	2	yes	—
23	SD₃₂	—	2	6	no	—
24	D₁₆	—	2	6	no	—
25	D₁₆	C₂³	1	6	yes	—
26	M₃₂	D₄	1	3	yes	—
27	C₂×C₁₆	C₂³	1	3	yes	—
28	C₃₂	C₂³	1	1	yes	characteristic
29	C₃₂	C₂³	1	3	yes	—
30	#41	—	4	6	no	—
31	#41	—	8	2	no	—
32	#41	D₈	1	1	yes	characteristic
33	C₂×D₄	—	4	6	no	—
34	C₂×D₄	—	8	6	no	—
35	Q₁₆	—	2	2	no	—
36	SD₁₆	—	4	6	no	—
37	D₈	—	2	6	no	—
38	D₈	—	4	6	no	—
39	M₁₆	D₈	1	3	yes	—
40	C₂×C₈	C₂×D₄	1	3	yes	—
41	C₁₆	C₂×D₄	1	3	yes	—
42	C₁₆	C₂⁴	1	1	yes	characteristic
43	C₂³	—	8	6	no	—
44	Q₈	—	4	2	no	—
45	Q₈	D₁₆	1	1	yes	characteristic
46	D₄	—	4	6	no	—
47	D₄	—	8	6	no	—
48	D₄	—	8	6	no	—
49	D₄	D₁₆	1	3	yes	—
50	C₂×C₄	—	8	6	no	—
51	C₂×C₄	C₂×D₈	1	3	yes	—
52	C₈	C₂×D₈	1	3	yes	—
53	C₈	C₂²×D₄	1	1	yes	characteristic
54	C₂²	—	8	6	no	—
55	C₂²	—	32	6	no	—
56	C₂²	C₂×D₁₆	1	3	yes	—
57	C₄	—	8	2	no	—
58	C₄	C₂×D₁₆	1	3	yes	—
59	C₄	C₂²×D₈	1	1	yes	characteristic
60	C₂	—	2	3	no	—
61	C₂	—	16	6	no	—
62	C₂	C₂²×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂²×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504 ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3415	C₂	semidirect
2. #2263	C₂	semidirect
3. #2262	C₂	semidirect
4. C₂×D₃₂	C₂	semidirect
5. #2258	C₂	semidirect
8. #508	C₂²	semidirect
9. C₂×D₁₆	C₂²	semidirect
11. Q₆₄	C₂²	semidirect
12. SD₆₄	C₂²	semidirect
13. D₃₂	C₂²	semidirect
14. D₃₂	C₂²	semidirect
15. M₆₄	C₂²	semidirect
16. C₂×C₃₂	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3415	2
#2263	2
#2262	2
C₂×D₃₂	2
#2258	2