Subgroups — #94121

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	#11404	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂	C₄	1	1	yes	characteristic
4	C₄×M₃₂	C₂²	1	1	yes	characteristic
5	C₂×#322	C₄	1	1	yes	characteristic
6	#1422	—	2	2	no	—
7	#1400	C₄	1	4	yes	—
8	#1398	—	2	2	no	—
9	#1313	—	2	2	no	—
10	C₂×M₃₂	—	2	1	no	—
11	C₂×M₃₂	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₈	—	2	1	no	—
13	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
14	M₆₄	—	2	4	no	—
15	#346	—	2	1	no	—
16	#346	D₄	1	2	yes	—
17	C₄×C₁₆	C₈	1	2	yes	—
18	C₄×C₁₆	D₄	1	1	yes	characteristic
19	C₄×C₁₆	Q₈	1	1	yes	characteristic
20	#322	—	2	1	no	—
21	#322	C₈	1	2	yes	—
22	C₂²×C₈	—	4	1	no	—
23	C₂²×C₈	C₄²	1	1	yes	characteristic
24	C₂²×C₈	M₁₆	1	1	yes	characteristic
25	C₂×C₄²	M₁₆	1	1	yes	characteristic
26	M₃₂	—	2	4	no	—
27	M₃₂	—	4	2	no	—
28	C₂×C₁₆	—	2	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	C₄×C₈	—	4	1	no	—
33	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₈	C₂×C₈	1	1	yes	characteristic
36	C₄×C₈	C₂×C₈	1	1	yes	characteristic
37	C₃₂	—	4	4	no	—
38	C₂²×C₄	—	2	1	no	—
39	C₂²×C₄	#97	1	1	yes	characteristic
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	—	8	2	no	—
44	C₂×C₈	#95	1	1	yes	characteristic
45	C₂×C₈	C₄×C₈	1	1	yes	characteristic
46	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
48	C₄²	—	2	1	no	—
49	C₄²	C₂²⋊C₈	1	1	yes	characteristic
50	C₄²	C₄⋊C₈	1	1	yes	characteristic
51	C₁₆	—	2	4	no	—
52	C₁₆	—	2	2	no	—
53	C₁₆	—	2	2	no	—
54	C₂³	#347	1	1	yes	characteristic
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	#336	1	1	yes	characteristic
61	C₂×C₄	#347	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	—	2	1	no	—
64	C₈	—	8	2	no	—
65	C₈	#344	1	2	yes	—
66	C₈	#364	1	2	yes	—
67	C₂²	—	2	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	#1375	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	#1281	1	1	yes	characteristic
74	C₄	#1376	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	4	1	no	—
77	C₂	#7196	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1281
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄×M₃₂	C₂²
socle	C₂	#7196
prime radical	C₂³	#347
LCS	G ▷ C₂×C₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ C₄×M₃₂ ◁ 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
#11404	2