Subgroups — #94118

Order 512.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13640	C₂	1	2	yes	maximal
2	#11401	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂	C₄	1	1	yes	characteristic
4	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
5	#1450	C₄	1	1	yes	characteristic
6	C₄⋊C₃₂	—	2	2	no	—
7	C₂²⋊C₃₂	C₄	1	4	yes	—
8	#1397	—	2	2	no	—
9	#1312	—	2	2	no	—
10	C₂×M₃₂	C₈	1	2	yes	—
11	C₂²×C₁₆	C₂×C₄	1	2	yes	—
12	#431	C₈	1	2	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃₂	—	2	4	no	—
15	#346	—	2	1	no	—
16	C₄×C₁₆	—	2	1	no	—
17	C₄×C₁₆	D₄	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	C₈²	—	2	1	no	—
22	C₂²×C₈	C₄²	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
25	M₃₂	—	4	2	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	—	4	2	no	—
28	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
29	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
30	C₄⋊C₈	—	2	2	no	—
31	C₂²⋊C₈	C₁₆	1	4	yes	—
32	C₄×C₈	—	2	2	no	—
33	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₈	M₁₆	1	1	yes	characteristic
36	C₄×C₈	M₁₆	1	1	yes	characteristic
37	C₃₂	—	4	4	no	—
38	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
39	C₂²×C₄	C₂×C₁₆	1	2	yes	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	#95	1	1	yes	characteristic
44	C₂×C₈	#97	1	1	yes	characteristic
45	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
46	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
47	C₄²	C₂²⋊C₈	1	1	yes	characteristic
48	C₄²	C₄⋊C₈	1	1	yes	characteristic
49	C₄²	M₃₂	1	2	yes	—
50	C₁₆	—	2	2	no	—
51	C₁₆	—	2	2	no	—
52	C₁₆	—	4	2	no	—
53	C₂³	C₄×C₁₆	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	#336	1	1	yes	characteristic
58	C₂×C₄	#346	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
60	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
61	C₈	—	2	1	no	—
62	C₈	—	2	1	no	—
63	C₈	—	4	4	no	—
64	C₈	#344	1	2	yes	—
65	C₈	#349	1	2	yes	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	#1374	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	#1281	1	1	yes	characteristic
73	C₄	#1377	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	#7195	1	1	yes	characteristic
76	C₂	#7469	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#95
commutator, prime radical	C₂³	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ 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
#13640	2
#11401	2