Subgroups — #94009

Order 512.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13642	C₂	1	2	yes	maximal
2	#11401	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂	C₂²	1	1	yes	characteristic
4	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
5	#1450	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	D₄	1	1	yes	characteristic
11	C₂×M₃₂	Q₈	1	1	yes	characteristic
12	C₂²×C₁₆	C₈	1	2	yes	—
13	#431	—	2	1	no	—
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	M₆₄	—	2	4	no	—
16	#346	C₂×C₄	1	2	yes	—
17	C₄×C₁₆	—	2	1	no	—
18	C₄×C₁₆	—	2	1	no	—
19	C₄×C₁₆	D₄	1	2	yes	—
20	#322	C₈	1	2	yes	—
21	C₈²	—	2	1	no	—
22	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄²	M₁₆	1	1	yes	characteristic
25	M₃₂	—	4	1	no	—
26	M₃₂	—	4	1	no	—
27	C₂×C₁₆	—	2	2	no	—
28	C₂×C₁₆	—	4	2	no	—
29	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
30	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
31	C₄⋊C₈	—	2	2	no	—
32	C₂²⋊C₈	—	2	2	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	C₄²	1	1	yes	characteristic
35	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₈	C₂×C₈	1	1	yes	characteristic
37	C₄×C₈	M₁₆	1	1	yes	characteristic
38	C₃₂	—	4	4	no	—
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	#95	1	1	yes	characteristic
46	C₂×C₈	C₄×C₈	1	1	yes	characteristic
47	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
49	C₄²	—	2	1	no	—
50	C₄²	#97	1	1	yes	characteristic
51	C₄²	C₂²⋊C₈	1	1	yes	characteristic
52	C₁₆	—	2	2	no	—
53	C₁₆	—	2	2	no	—
54	C₁₆	—	4	2	no	—
55	C₂³	#364	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	#336	1	1	yes	characteristic
62	C₂×C₄	#364	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₈	—	4	4	no	—
66	C₈	H(3,ℤ/4)	1	1	yes	characteristic
67	C₈	#338	1	1	yes	characteristic
68	C₈	#349	1	2	yes	—
69	C₂²	—	2	1	no	—
70	C₂²	—	4	1	no	—
71	C₂²	#1376	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	2	1	no	—
74	C₄	—	4	1	no	—
75	C₄	#1275	1	1	yes	characteristic
76	C₄	#1377	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	4	1	no	—
79	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
#11401	2