Subgroups — #92909

Order 512.

199 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11402	C₂	1	1	yes	maximal, characteristic
2	C₈×M₃₂	C₂	1	2	yes	maximal
3	C₄×M₃₂	C₄	1	2	yes	—
4	C₄×M₃₂	C₄	1	4	yes	—
5	C₂×C₈²	C₂²	1	1	yes	characteristic
6	#1312	—	2	2	no	—
7	#1311	—	2	2	no	—
8	C₈×C₁₆	—	2	2	no	—
9	C₂×M₃₂	—	2	4	no	—
10	C₂×M₃₂	C₈	1	4	yes	—
11	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
13	#346	—	2	2	no	—
14	#346	C₈	1	8	yes	—
15	C₄×C₁₆	—	2	4	no	—
16	C₄×C₁₆	—	2	2	no	—
17	C₈²	D₄	1	1	yes	characteristic
18	C₈²	D₄	1	2	yes	—
19	C₈²	Q₈	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₈	1	2	yes	—
21	C₂²×C₈	M₁₆	1	4	yes	—
22	C₂×C₄²	C₄²	1	1	yes	characteristic
23	M₃₂	—	2	16	no	—
24	M₃₂	—	4	4	no	—
25	C₂×C₁₆	—	2	8	no	—
26	C₂×C₁₆	—	2	4	no	—
27	C₄×C₈	C₂²⋊C₄	1	2	yes	—
28	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₈	C₄⋊C₄	1	2	yes	—
31	C₄×C₈	C₂×C₈	1	4	yes	—
32	C₄×C₈	M₁₆	1	2	yes	—
33	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
34	C₂²×C₄	#97	1	2	yes	—
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	—	2	8	no	—
37	C₂×C₈	C₂²⋊C₈	1	4	yes	—
38	C₂×C₈	C₂²⋊C₈	1	2	yes	—
39	C₂×C₈	C₄⋊C₈	1	2	yes	—
40	C₂×C₈	C₄⋊C₈	1	4	yes	—
41	C₄²	#95	1	1	yes	characteristic
42	C₄²	C₄×C₈	1	2	yes	—
43	C₄²	#97	1	1	yes	characteristic
44	C₁₆	—	2	16	no	—
45	C₁₆	—	4	4	no	—
46	C₂³	#322	1	1	yes	characteristic
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	C₈²	1	1	yes	characteristic
50	C₂×C₄	#322	1	2	yes	—
51	C₂×C₄	#336	1	2	yes	—
52	C₂×C₄	#336	1	1	yes	characteristic
53	C₈	—	2	4	no	—
54	C₈	—	2	2	no	—
55	C₈	#349	1	4	yes	—
56	C₈	#364	1	8	yes	—
57	C₂²	—	2	2	no	—
58	C₂²	—	2	1	no	—
59	C₂²	#1273	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	2	1	no	—
62	C₄	#1376	1	4	yes	—
63	C₄	#1377	1	2	yes	—
64	C₂	—	4	1	no	—
65	C₂	#7138	1	2	yes	—
66	C₂	#7139	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
prime radical	C₂³	#322
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#11402	2
C₈×M₃₂	2