Subgroups — #94008

Order 512.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13642	C₂	1	2	yes	maximal
2	#11402	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂	C₄	1	1	yes	characteristic
4	C₄×M₃₂	C₂²	1	1	yes	characteristic
5	C₂×C₈²	C₄	1	1	yes	characteristic
6	#1422	—	2	2	no	—
7	#1400	—	2	2	no	—
8	#1398	C₄	1	4	yes	—
9	#1312	—	2	2	no	—
10	C₂×M₃₂	C₈	1	2	yes	—
11	C₂×M₃₂	D₄	1	1	yes	characteristic
12	C₂×M₃₂	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₈	—	2	1	no	—
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	M₆₄	—	2	4	no	—
16	#346	—	2	1	no	—
17	#346	C₂×C₄	1	2	yes	—
18	C₄×C₁₆	—	2	1	no	—
19	C₄×C₁₆	D₄	1	2	yes	—
20	C₈²	—	2	1	no	—
21	C₈²	C₈	1	1	yes	characteristic
22	C₈²	C₈	1	1	yes	characteristic
23	C₂²×C₈	—	2	2	no	—
24	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄²	M₁₆	1	1	yes	characteristic
27	M₃₂	—	4	1	no	—
28	M₃₂	—	4	1	no	—
29	M₃₂	—	4	2	no	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
32	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	C₄²	1	1	yes	characteristic
37	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₈	C₂×C₈	1	1	yes	characteristic
39	C₄×C₈	M₁₆	1	1	yes	characteristic
40	C₃₂	—	4	4	no	—
41	C₂²×C₄	—	2	1	no	—
42	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	4	4	no	—
49	C₂×C₈	#95	1	1	yes	characteristic
50	C₂×C₈	C₄×C₈	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	#97	1	1	yes	characteristic
55	C₄²	C₂²⋊C₈	1	1	yes	characteristic
56	C₁₆	—	2	2	no	—
57	C₁₆	—	2	2	no	—
58	C₁₆	—	4	2	no	—
59	C₂³	#364	1	1	yes	characteristic
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	#336	1	1	yes	characteristic
66	C₂×C₄	#364	1	1	yes	characteristic
67	C₈	—	2	4	no	—
68	C₈	—	2	1	no	—
69	C₈	—	2	1	no	—
70	C₈	—	4	2	no	—
71	C₈	H(3,ℤ/4)	1	1	yes	characteristic
72	C₈	#338	1	1	yes	characteristic
73	C₈	#349	1	2	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	4	1	no	—
76	C₂²	#1376	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#1275	1	1	yes	characteristic
81	C₄	#1377	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	4	1	no	—
84	C₂	#7146	1	1	yes	characteristic

	subgroup	G/H
center	C₈	H(3,ℤ/4)
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄×M₃₂	C₂²
socle	C₂	#7146
prime radical	C₂³	#364
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×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
#13642	2
#11402	2