Subgroups — #33983

Order 256.

231 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3417	C₂	1	1	yes	maximal, characteristic, complemented
2	#3415	C₂	1	1	yes	maximal, characteristic, complemented
3	#2264	C₂	1	3	yes	maximal, complemented
4	#2263	C₂	1	3	yes	maximal, complemented
5	#2262	C₂	1	3	yes	maximal, complemented
6	C₂×SD₆₄	C₂	1	3	yes	maximal, complemented
7	#2258	C₂	1	1	yes	maximal, characteristic, complemented
8	#578	—	2	1	no	—
9	#576	—	2	1	no	—
10	#510	—	2	3	no	—
11	#509	—	2	3	no	—
12	#508	C₂²	1	3	yes	—
13	#508	C₂²	1	3	yes	complemented
14	C₂×Q₃₂	C₂²	1	3	yes	complemented
15	C₂×D₁₆	C₂²	1	3	yes	—
16	#504	C₂²	1	1	yes	characteristic
17	Q₆₄	C₂²	1	3	yes	complemented
18	SD₆₄	C₂²	1	3	yes	complemented
19	SD₆₄	C₂²	1	6	yes	complemented
20	SD₆₄	C₂²	1	1	yes	characteristic, complemented
21	D₃₂	C₂²	1	3	yes	complemented
22	M₆₄	C₂²	1	3	yes	complemented
23	C₂×C₃₂	C₂²	1	3	yes	complemented
24	2₋¹⁺⁴	—	4	1	no	—
25	H(5,𝔽₂)	—	4	1	no	—
26	#137	—	4	3	no	—
27	AGL(1,ℤ/8)	—	4	3	no	—
28	#135	—	2	3	no	—
29	#135	—	2	3	no	—
30	C₂×Q₁₆	—	2	3	no	—
31	C₂×D₈	—	2	3	no	—
32	#131	D₄	1	1	yes	characteristic
33	Q₃₂	—	2	3	no	—
34	Q₃₂	C₂³	1	3	yes	—
35	Q₃₂	C₂³	1	1	yes	characteristic
36	SD₃₂	—	2	3	no	—
37	SD₃₂	—	2	3	no	—
38	D₁₆	—	2	3	no	—
39	D₁₆	C₂³	1	1	yes	characteristic
40	D₁₆	C₂³	1	3	yes	—
41	M₃₂	D₄	1	3	yes	—
42	C₂×C₁₆	C₂³	1	3	yes	—
43	C₃₂	C₂³	1	1	yes	characteristic
44	C₃₂	C₂³	1	3	yes	—
45	#41	—	4	3	no	—
46	#41	—	4	3	no	—
47	#41	—	8	1	no	—
48	#41	—	8	3	no	—
49	#41	D₈	1	1	yes	characteristic
50	C₂×Q₈	—	4	3	no	—
51	C₂×Q₈	—	8	1	no	—
52	C₂×D₄	—	4	3	no	—
53	C₂×D₄	—	8	3	no	—
54	Q₁₆	—	2	1	no	—
55	Q₁₆	—	2	3	no	—
56	Q₁₆	—	4	3	no	—
57	SD₁₆	—	4	3	no	—
58	SD₁₆	—	4	3	no	—
59	D₈	—	2	3	no	—
60	D₈	—	2	1	no	—
61	D₈	—	4	3	no	—
62	M₁₆	D₈	1	3	yes	—
63	C₂×C₈	C₂×D₄	1	3	yes	—
64	C₁₆	C₂×D₄	1	3	yes	—
65	C₁₆	C₂⁴	1	1	yes	characteristic
66	C₂³	—	8	3	no	—
67	Q₈	—	4	1	no	—
68	Q₈	—	4	3	no	—
69	Q₈	—	8	3	no	—
70	Q₈	D₁₆	1	1	yes	characteristic
71	D₄	—	4	3	no	—
72	D₄	—	4	1	no	—
73	D₄	—	8	3	no	—
74	D₄	—	8	3	no	—
75	D₄	—	8	3	no	—
76	D₄	D₁₆	1	3	yes	—
77	C₂×C₄	—	8	3	no	—
78	C₂×C₄	—	8	3	no	—
79	C₂×C₄	—	8	3	no	—
80	C₂×C₄	C₂×D₈	1	3	yes	—
81	C₈	C₂×D₈	1	3	yes	—
82	C₈	C₂²×D₄	1	1	yes	characteristic
83	C₂²	—	8	1	no	—
84	C₂²	—	8	3	no	—
85	C₂²	—	32	3	no	—
86	C₂²	C₂×D₁₆	1	3	yes	—
87	C₄	—	8	1	no	—
88	C₄	—	8	3	no	—
89	C₄	C₂×D₁₆	1	3	yes	—
90	C₄	C₂²×D₈	1	1	yes	characteristic
91	C₂	—	2	3	no	—
92	C₂	—	16	1	no	—
93	C₂	—	16	3	no	—
94	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. #3417	C₂	semidirect
2. #3415	C₂	semidirect
3. #2264	C₂	semidirect
4. #2263	C₂	semidirect
5. #2262	C₂	semidirect
6. C₂×SD₆₄	C₂	semidirect
7. #2258	C₂	semidirect
13. #508	C₂²	semidirect
14. C₂×Q₃₂	C₂²	semidirect
17. Q₆₄	C₂²	semidirect
18. SD₆₄	C₂²	semidirect
19. SD₆₄	C₂²	semidirect
20. SD₆₄	C₂²	semidirect
21. D₃₂	C₂²	semidirect
22. M₆₄	C₂²	semidirect
23. 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
#3417	2
#3415	2
#2264	2
#2263	2
#2262	2
C₂×SD₆₄	2
#2258	2