Subgroups — #149495

Order 512.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₆₄	C₂	1	2	yes	maximal
2	C₂×#1311	C₂	1	1	yes	maximal, characteristic
3	#7330	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄	C₄	1	4	yes	—
5	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₈²	C₄	1	1	yes	characteristic
8	#1397	C₂²	1	4	yes	—
9	C₄×C₃₂	C₄	1	4	yes	—
10	#1311	C₄	1	2	yes	—
11	#1311	C₂²	1	2	yes	—
12	C₂²×C₁₆	C₈	1	2	yes	—
13	C₂²×C₁₆	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	C₈	1	2	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	M₆₄	C₈	1	16	yes	complemented
17	C₂×C₃₂	C₂×C₄	1	8	yes	—
18	C₄×C₁₆	C₈	1	2	yes	—
19	C₄×C₁₆	C₂×C₄	1	2	yes	—
20	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₄×C₁₆	C₂×C₄	1	2	yes	—
22	C₄×C₁₆	C₂³	1	1	yes	characteristic
23	C₈²	C₈	1	2	yes	—
24	C₈²	C₂×C₄	1	2	yes	—
25	C₂²×C₈	—	2	2	no	—
26	C₂²×C₈	C₄²	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
29	C₂×C₁₆	—	2	4	no	—
30	C₂×C₁₆	C₄²	1	2	yes	—
31	C₂×C₁₆	C₂×C₈	1	4	yes	—
32	C₂×C₁₆	C₂×C₈	1	8	yes	—
33	C₂×C₁₆	C₂²×C₄	1	2	yes	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	C₄²	1	1	yes	characteristic
36	C₄×C₈	C₂×C₈	1	1	yes	characteristic
37	C₄×C₈	C₂×C₈	1	4	yes	—
38	C₄×C₈	C₂×C₈	1	1	yes	characteristic
39	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₃₂	C₂×C₈	1	16	yes	complemented
41	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
42	C₂²×C₄	M₃₂	1	2	yes	—
43	C₂×C₈	—	2	4	no	—
44	C₂×C₈	—	4	4	no	—
45	C₂×C₈	C₄×C₈	1	1	yes	characteristic
46	C₂×C₈	C₄×C₈	1	4	yes	—
47	C₂×C₈	C₄×C₈	1	1	yes	characteristic
48	C₂×C₈	M₃₂	1	4	yes	—
49	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
51	C₄²	C₄×C₈	1	1	yes	characteristic
52	C₄²	M₃₂	1	2	yes	—
53	C₄²	C₂²×C₈	1	1	yes	characteristic
54	C₁₆	—	2	4	no	—
55	C₁₆	C₄×C₈	1	4	yes	—
56	C₁₆	C₂²×C₈	1	4	yes	—
57	C₂³	#346	1	1	yes	characteristic
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	C₈²	1	2	yes	—
60	C₂×C₄	#346	1	1	yes	characteristic
61	C₂×C₄	#346	1	2	yes	—
62	C₂×C₄	#346	1	2	yes	—
63	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂×C₄	C₂×M₃₂	1	2	yes	—
65	C₈	—	4	4	no	—
66	C₈	C₈²	1	2	yes	—
67	C₈	#346	1	2	yes	—
68	C₈	C₂×C₄×C₈	1	2	yes	—
69	C₈	C₂×M₃₂	1	2	yes	—
70	C₂²	—	2	2	no	—
71	C₂²	#1311	1	2	yes	—
72	C₂²	C₂×#346	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	#1311	1	2	yes	—
75	C₄	#1398	1	4	yes	—
76	C₄	C₂×C₈²	1	1	yes	characteristic
77	C₄	C₂×#346	1	1	yes	characteristic
78	C₂	—	2	2	no	—
79	C₂	C₂×#1311	1	1	yes	characteristic
80	C₂	C₂×#1398	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	#346
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #7330	C₂	semidirect
16. M₆₄	C₈	semidirect
40. C₃₂	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₆₄	2
C₂×#1311	2
#7330	2