Subgroups — #94613

Order 512.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13656	C₂	1	1	yes	maximal, characteristic
2	#7511	C₂	1	1	yes	maximal, characteristic, complemented
3	#7463	C₂	1	1	yes	maximal, characteristic, complemented
4	#2262	C₄	1	1	yes	characteristic
5	#2178	—	2	1	no	—
6	M₁₂₈	—	2	1	no	—
7	#1425	C₄	1	1	yes	characteristic
8	#1417	—	2	1	no	—
9	#1397	C₂²	1	1	yes	characteristic
10	#1372	—	2	1	no	—
11	#508	—	2	1	no	—
12	#443	—	4	1	no	—
13	Q₆₄	C₈	1	1	yes	characteristic, complemented
14	SD₆₄	—	2	1	no	—
15	D₃₂	C₈	1	1	yes	characteristic, complemented
16	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃₂	D₄	1	1	yes	characteristic
18	#368	—	2	1	no	—
19	#359	—	4	1	no	—
20	C₄×C₁₆	D₄	1	1	yes	characteristic
21	#335	—	4	1	no	—
22	C₆₄	—	2	2	no	—
23	#135	—	4	1	no	—
24	#131	—	8	1	no	—
25	Q₃₂	—	2	1	no	—
26	SD₃₂	—	4	1	no	—
27	D₁₆	—	2	1	no	—
28	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
30	#108	—	4	1	no	—
31	C₄⋊C₈	—	8	1	no	—
32	#104	—	8	1	no	—
33	C₄×C₈	D₈	1	1	yes	characteristic
34	C₃₂	—	2	1	no	—
35	C₃₂	C₂×C₈	1	1	yes	characteristic
36	C₃₂	M₁₆	1	1	yes	characteristic
37	#41	—	8	1	no	—
38	Q₁₆	—	4	1	no	—
39	SD₁₆	—	8	1	no	—
40	D₈	—	4	1	no	—
41	M₁₆	—	8	1	no	—
42	M₁₆	—	16	1	no	—
43	C₂×C₈	—	16	1	no	—
44	C₂×C₈	—	16	1	no	—
45	C₂×C₈	#102	1	1	yes	characteristic
46	C₂×C₈	SD₃₂	1	1	yes	characteristic
47	C₄²	D₁₆	1	1	yes	characteristic
48	C₁₆	—	2	1	no	—
49	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
50	C₁₆	#104	1	1	yes	characteristic
51	Q₈	—	8	1	no	—
52	D₄	—	8	1	no	—
53	D₄	—	16	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	16	1	no	—
56	C₂×C₄	#357	1	1	yes	characteristic
57	C₈	—	2	1	no	—
58	C₈	—	16	1	no	—
59	C₈	—	32	1	no	—
60	C₈	D₄⋊C₈	1	1	yes	characteristic
61	C₈	#359	1	1	yes	characteristic
62	C₂²	—	16	1	no	—
63	C₂²	#1418	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	16	1	no	—
66	C₄	#1331	1	1	yes	characteristic
67	C₄	#1419	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	32	1	no	—
70	C₂	#7385	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#7385
commutator	C₃₂	C₂×C₈
Frattini, cosocle	#1397	C₂²
prime radical	#2262	C₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #7511	C₂	semidirect
3. #7463	C₂	semidirect
13. Q₆₄	C₈	semidirect
15. D₃₂	C₈	semidirect

Maximal subgroups

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

subgroup	index
#13656	2
#7511	2
#7463	2