Subgroups — #94697

Order 512.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13698	C₂	1	1	yes	maximal, characteristic
2	#7513	C₂	1	1	yes	maximal, characteristic
3	#7476	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₆₄	C₄	1	1	yes	characteristic
5	C₂×M₆₄	C₂²	1	1	yes	characteristic
6	#2216	—	2	1	no	—
7	C₂×#368	—	2	1	no	—
8	M₁₂₈	—	2	1	no	—
9	#1424	C₄	1	1	yes	characteristic
10	#1416	—	2	1	no	—
11	C₂×Q₃₂	—	2	1	no	—
12	C₂²×C₁₆	D₄	1	1	yes	characteristic
13	#467	—	4	1	no	—
14	C₂×#108	—	4	1	no	—
15	Q₆₄	—	4	1	no	—
16	M₆₄	—	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	#358	—	4	1	no	—
23	C₆₄	—	4	1	no	—
24	C₂×Q₁₆	—	4	1	no	—
25	C₂×M₁₆	—	8	1	no	—
26	C₂²×C₈	D₈	1	1	yes	characteristic
27	C₂²⋊Q₈	—	8	1	no	—
28	Q₃₂	—	4	1	no	—
29	Q₃₂	—	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	#103	—	8	1	no	—
38	C₃₂	—	2	1	no	—
39	C₃₂	—	2	1	no	—
40	C₂×Q₈	—	8	1	no	—
41	C₂²×C₄	D₁₆	1	1	yes	characteristic
42	Q₁₆	—	8	1	no	—
43	Q₁₆	—	8	1	no	—
44	M₁₆	—	8	2	no	—
45	M₁₆	—	16	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	16	1	no	—
49	C₂×C₈	#102	1	1	yes	characteristic
50	C₂×C₈	SD₃₂	1	1	yes	characteristic
51	C₄⋊C₄	—	8	1	no	—
52	C₄⋊C₄	—	16	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	SD₆₄	1	1	yes	characteristic
58	Q₈	—	16	1	no	—
59	Q₈	—	16	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	16	1	no	—
63	C₂×C₄	—	16	1	no	—
64	C₂×C₄	#357	1	1	yes	characteristic
65	C₂×C₄	D₃₂	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	16	2	no	—
68	C₈	#327	1	1	yes	characteristic
69	C₈	#360	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	#1415	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	32	1	no	—
75	C₄	—	32	1	no	—
76	C₄	#1347	1	1	yes	characteristic
77	C₄	#1418	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	#7418	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#7418
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₂×M₆₄	C₂²
prime radical	C₂³	SD₆₄
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#13698	2
#7513	2
#7476	2