Subgroups — #33981

Order 256.

233 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₄	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴
socle	C₂	C₂²×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ 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. C₂×#508	C₂	semidirect
2. #2264	C₂	semidirect
3. #2263	C₂	semidirect
4. #2262	C₂	semidirect
5. C₂×M₆₄	C₂	semidirect
10. #508	C₂²	semidirect
11. C₂×Q₃₂	C₂²	semidirect
13. C₂×D₁₆	C₂²	semidirect
15. Q₆₄	C₂²	semidirect
16. SD₆₄	C₂²	semidirect
17. D₃₂	C₂²	semidirect
18. M₆₄	C₂²	semidirect
19. 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
C₂×#508	2
#2264	2
#2263	2
#2262	2
C₂×M₆₄	2