Subgroups — #128207

Order 512.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#346	C₂	1	2	yes	maximal
2	#7841	C₂	1	1	yes	maximal, characteristic
3	#7137	C₂	1	4	yes	maximal
4	C₂×#346	C₄	1	4	yes	—
5	C₂×#346	C₂²	1	4	yes	—
6	C₂×C₄×C₁₆	C₄	1	4	yes	—
7	#1744	C₄	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#322	C₂²	1	2	yes	—
10	C₂×C₈²	C₄	1	2	yes	—
11	C₂²×C₁₆	C₂×C₄	1	8	yes	—
12	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
17	C₄³	C₂×C₄	1	1	yes	characteristic
18	#346	—	2	8	no	—
19	#346	C₈	1	16	yes	complemented
20	C₄×C₁₆	—	2	8	no	—
21	#336	—	2	4	no	—
22	#322	—	2	4	no	—
23	C₈²	C₈	1	8	yes	—
24	C₂²×C₈	—	2	4	no	—
25	C₂²×C₈	C₄²	1	1	yes	characteristic
26	C₂²×C₈	C₄²	1	1	yes	characteristic
27	C₂²×C₈	C₂²×C₄	1	2	yes	—
28	C₂×C₄²	C₄²	1	2	yes	—
29	C₂×C₄²	M₁₆	1	4	yes	—
30	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₁₆	—	2	8	no	—
32	C₂×C₁₆	—	2	8	no	—
33	C₂×C₁₆	C₂×C₈	1	16	yes	—
34	C₄×C₈	—	2	4	no	—
35	C₄×C₈	C₂×C₈	1	8	yes	—
36	C₄×C₈	C₂×C₈	1	8	yes	—
37	C₄×C₈	C₂×C₈	1	4	yes	—
38	C₄×C₈	M₁₆	1	8	yes	—
39	C₄×C₈	#41	1	4	yes	—
40	C₂²×C₄	#97	1	4	yes	—
41	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
42	C₂²×C₄	C₂×M₁₆	1	2	yes	—
43	C₂×C₈	—	2	8	no	—
44	C₂×C₈	—	4	8	no	—
45	C₂×C₈	C₄×C₈	1	2	yes	—
46	C₂×C₈	C₄×C₈	1	4	yes	—
47	C₂×C₈	C₄×C₈	1	2	yes	—
48	C₂×C₈	#97	1	4	yes	—
49	C₂×C₈	#117	1	4	yes	—
50	C₂×C₈	C₂²×C₈	1	4	yes	—
51	C₂×C₈	C₂×M₁₆	1	4	yes	—
52	C₄²	—	4	4	no	—
53	C₄²	C₄×C₈	1	4	yes	—
54	C₄²	#97	1	4	yes	—
55	C₄²	#117	1	2	yes	—
56	C₄²	C₂²×C₈	1	2	yes	—
57	C₁₆	—	2	16	no	—
58	C₂³	C₂×#97	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	8	no	—
61	C₂×C₄	#322	1	8	yes	—
62	C₂×C₄	#376	1	1	yes	characteristic
63	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
65	C₂×C₄	C₂×#97	1	2	yes	—
66	C₂×C₄	#431	1	4	yes	—
67	C₈	—	4	8	no	—
68	C₈	C₈²	1	4	yes	—
69	C₈	#322	1	4	yes	—
70	C₈	#431	1	8	yes	—
71	C₂²	C₂×#322	1	2	yes	—
72	C₂²	#1744	1	1	yes	characteristic
73	C₂²	#2110	1	4	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	2	8	no	—
76	C₄	C₂×C₈²	1	1	yes	characteristic
77	C₄	C₂×#322	1	1	yes	characteristic
78	C₄	#1744	1	2	yes	—
79	C₂	#8131	1	1	yes	characteristic
80	C₂	#11406	1	4	yes	—
81	C₂	#12593	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₄²
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
19. #346	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#346	2
#7841	2
#7137	2