Subgroups — #94512

Order 512.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13553	C₂	1	1	yes	maximal, characteristic
2	#7513	C₂	1	2	yes	maximal, complemented
3	C₂³×C₁₆	C₄	1	1	yes	characteristic
4	C₂×M₆₄	C₄	1	1	yes	characteristic
5	C₂×M₆₄	C₂²	1	1	yes	characteristic
6	M₁₂₈	—	2	2	no	—
7	C₂²⋊C₃₂	—	2	2	no	—
8	C₂³×C₈	C₈	1	1	yes	characteristic
9	C₂²×C₁₆	—	2	1	no	—
10	C₂²×C₁₆	—	2	1	no	—
11	C₂²×C₁₆	—	4	1	no	—
12	C₂²×C₁₆	—	4	1	no	—
13	C₂²×C₁₆	C₈	1	1	yes	characteristic
14	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
15	M₆₄	—	4	1	no	—
16	M₆₄	—	4	1	no	—
17	C₂×C₃₂	—	2	1	no	—
18	C₂×C₃₂	D₄	1	2	yes	—
19	C₆₄	—	4	2	no	—
20	C₂³×C₄	C₁₆	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	—	4	1	no	—
24	C₂²×C₈	—	4	1	no	—
25	C₂²×C₈	C₁₆	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	—	2	1	no	—
29	C₂×C₁₆	—	2	1	no	—
30	C₂×C₁₆	—	4	2	no	—
31	C₂×C₁₆	—	4	1	no	—
32	C₂×C₁₆	—	4	1	no	—
33	C₂×C₁₆	—	4	1	no	—
34	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₁₆	M₁₆	1	1	yes	characteristic
36	C₃₂	—	2	2	no	—
37	C₃₂	—	4	1	no	—
38	C₂⁴	C₃₂	1	1	yes	characteristic
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	—	2	1	no	—
41	C₂²×C₄	—	4	1	no	—
42	C₂²×C₄	—	4	1	no	—
43	C₂²×C₄	C₃₂	1	1	yes	characteristic
44	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₈	M₃₂	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	4	1	no	—
56	C₁₆	#99	1	1	yes	characteristic
57	C₁₆	#100	1	1	yes	characteristic
58	C₂³	—	2	1	no	—
59	C₂³	—	4	1	no	—
60	C₂³	—	4	2	no	—
61	C₂³	C₂×C₃₂	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₂×C₄	M₆₄	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	4	1	no	—
73	C₈	#323	1	1	yes	characteristic
74	C₈	#349	1	1	yes	characteristic
75	C₂²	—	2	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	2	no	—
80	C₂²	—	4	2	no	—
81	C₂²	—	4	1	no	—
82	C₂²	C₂²⋊C₃₂	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	#1314	1	1	yes	characteristic
86	C₄	#1400	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	4	2	no	—
89	C₂	—	4	1	no	—
90	C₂	#7334	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#99
commutator	C₂³	C₂×C₃₂
Frattini, cosocle	C₂×M₆₄	C₂²
socle	C₂	#7334
prime radical	C₂⁴	C₃₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #7513	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13553	2
#7513	2