Subgroups — #94608

Order 512.

77 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×C₆₄	C₂	1	1	yes	maximal, characteristic
3	#7460	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₆₄	C₄	1	1	yes	characteristic
5	C₄×Q₃₂	—	2	1	no	—
6	C₂×C₆₄	—	2	1	no	—
7	#1423	C₄	1	1	yes	characteristic
8	#1416	—	2	1	no	—
9	C₄×C₃₂	C₂²	1	1	yes	characteristic
10	#1371	—	2	1	no	—
11	C₂×Q₃₂	—	2	1	no	—
12	C₄×Q₁₆	—	4	1	no	—
13	Q₆₄	—	2	1	no	—
14	Q₆₄	C₈	1	2	yes	complemented
15	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃₂	D₄	1	1	yes	characteristic
17	#366	—	2	1	no	—
18	#358	—	4	1	no	—
19	C₄×C₁₆	D₄	1	1	yes	characteristic
20	#335	—	4	1	no	—
21	C₆₄	—	2	2	no	—
22	C₂×Q₁₆	—	4	1	no	—
23	C₄×Q₈	—	8	1	no	—
24	Q₃₂	—	2	2	no	—
25	Q₃₂	—	4	1	no	—
26	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
28	#107	—	4	1	no	—
29	C₄⋊C₈	—	8	1	no	—
30	#103	—	8	1	no	—
31	C₄×C₈	D₈	1	1	yes	characteristic
32	C₃₂	—	2	1	no	—
33	C₃₂	C₂×C₈	1	1	yes	characteristic
34	C₃₂	M₁₆	1	1	yes	characteristic
35	C₂×Q₈	—	8	1	no	—
36	Q₁₆	—	4	2	no	—
37	Q₁₆	—	8	1	no	—
38	C₂×C₈	—	16	1	no	—
39	C₂×C₈	#102	1	1	yes	characteristic
40	C₂×C₈	SD₃₂	1	1	yes	characteristic
41	C₄⋊C₄	—	8	1	no	—
42	C₄⋊C₄	—	16	1	no	—
43	C₄²	—	16	1	no	—
44	C₄²	D₁₆	1	1	yes	characteristic
45	C₁₆	—	2	1	no	—
46	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
47	C₁₆	#104	1	1	yes	characteristic
48	Q₈	—	8	2	no	—
49	Q₈	—	16	1	no	—
50	C₂×C₄	—	16	1	no	—
51	C₂×C₄	—	16	1	no	—
52	C₂×C₄	#357	1	1	yes	characteristic
53	C₂×C₄	D₃₂	1	1	yes	characteristic
54	C₂×C₄	SD₆₄	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	—	32	1	no	—
57	C₈	D₄⋊C₈	1	1	yes	characteristic
58	C₈	#359	1	1	yes	characteristic
59	C₂²	#1415	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	16	2	no	—
62	C₄	—	32	1	no	—
63	C₄	#1331	1	1	yes	characteristic
64	C₄	#1417	1	1	yes	characteristic
65	C₄	SD₁₂₈	1	1	yes	characteristic
66	C₄	Q₁₂₈	1	1	yes	characteristic
67	C₂	#7383	1	1	yes	characteristic
68	C₂	#7538	1	1	yes	characteristic
69	C₂	#7539	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃₂
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
socle, prime radical	C₂²	#1415
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. Q₆₄	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₆₄	2
C₄×C₆₄	2
#7460	2