Subgroups — #94692

Order 512.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13694	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#7472	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₂	C₄	1	1	yes	characteristic
5	C₂²×C₃₂	C₂²	1	1	yes	characteristic
6	#2215	—	2	1	no	—
7	C₂×#368	—	2	1	no	—
8	C₂×C₆₄	—	2	1	no	—
9	#1423	C₄	1	1	yes	characteristic
10	#1415	—	2	1	no	—
11	C₂×D₁₆	—	2	1	no	—
12	C₂²×C₁₆	D₄	1	1	yes	characteristic
13	#466	—	4	1	no	—
14	C₂×#108	—	4	1	no	—
15	D₃₂	—	4	1	no	—
16	C₂×C₃₂	—	4	1	no	—
17	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃₂	D₄	1	1	yes	characteristic
19	#368	—	2	2	no	—
20	#368	—	4	1	no	—
21	#366	—	2	1	no	—
22	#357	—	4	1	no	—
23	C₆₄	—	4	1	no	—
24	C₂×D₈	—	4	1	no	—
25	C₂×M₁₆	—	8	1	no	—
26	C₂²×C₈	D₈	1	1	yes	characteristic
27	#121	—	8	1	no	—
28	D₁₆	—	4	1	no	—
29	D₁₆	—	4	1	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
34	#108	—	4	2	no	—
35	#108	—	8	1	no	—
36	#107	—	4	1	no	—
37	#102	—	8	1	no	—
38	C₃₂	—	2	1	no	—
39	C₃₂	—	2	1	no	—
40	C₂×D₄	—	8	1	no	—
41	C₂×D₄	—	16	1	no	—
42	C₂²×C₄	D₁₆	1	1	yes	characteristic
43	D₈	—	8	1	no	—
44	D₈	—	8	1	no	—
45	M₁₆	—	8	2	no	—
46	M₁₆	—	16	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	16	1	no	—
50	C₂×C₈	#102	1	1	yes	characteristic
51	C₂×C₈	SD₃₂	1	1	yes	characteristic
52	C₄⋊C₄	—	8	1	no	—
53	C₂²⋊C₄	—	16	1	no	—
54	C₁₆	—	2	1	no	—
55	C₁₆	#99	1	1	yes	characteristic
56	C₁₆	#104	1	1	yes	characteristic
57	C₂³	—	16	1	no	—
58	C₂³	SD₆₄	1	1	yes	characteristic
59	D₄	—	16	1	no	—
60	D₄	—	16	1	no	—
61	D₄	—	32	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	16	1	no	—
65	C₂×C₄	#357	1	1	yes	characteristic
66	C₂×C₄	D₃₂	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	16	2	no	—
69	C₈	#327	1	1	yes	characteristic
70	C₈	#360	1	1	yes	characteristic
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	32	1	no	—
75	C₂²	—	32	1	no	—
76	C₂²	—	32	1	no	—
77	C₂²	#1415	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	32	1	no	—
80	C₄	#1347	1	1	yes	characteristic
81	C₄	#1418	1	1	yes	characteristic
82	C₂	—	4	1	no	—
83	C₂	—	64	1	no	—
84	C₂	#7418	1	1	yes	characteristic
85	C₂	#7539	1	1	yes	characteristic
86	C₂	#7541	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₂²×C₃₂	C₂²
prime radical	#13694	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₆₄	C₂	semidirect
3. #7472	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13694	2
C₂²⋊C₆₄	2
#7472	2