Subgroups — #123405

Order 512.

264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12521	C₂	1	1	yes	maximal, characteristic, complemented
2	#12517	C₂	1	2	yes	maximal, complemented
3	#12515	C₂	1	1	yes	maximal, characteristic
4	#12509	C₂	1	2	yes	maximal, complemented
5	C₁₆²	C₂	1	1	yes	maximal, characteristic, complemented
6	#2252	—	2	2	no	—
7	#2248	—	2	4	no	—
8	#2246	—	2	2	no	—
9	#2240	—	2	4	no	—
10	#1713	C₂²	1	2	yes	—
11	#1712	C₂²	1	2	yes	—
12	C₈×C₁₆	C₂²	1	1	yes	characteristic
13	C₈×C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×SD₃₂	—	4	8	no	—
16	C₂×D₁₆	—	4	4	no	—
17	#500	—	2	2	no	—
18	#500	—	2	2	no	—
19	#494	—	2	2	no	—
20	#493	—	2	2	no	—
21	#493	—	2	2	no	—
22	#493	—	2	2	no	—
23	#366	—	4	4	no	—
24	#357	—	4	8	no	—
25	C₄×C₁₆	D₄	1	2	yes	—
26	C₄×C₁₆	D₄	1	2	yes	—
27	C₄×C₁₆	D₄	1	2	yes	—
28	C₈²	C₂³	1	1	yes	characteristic
29	C₂×Q₁₆	—	4	4	no	—
30	C₂×D₈	—	4	4	no	—
31	C₂×D₈	—	4	4	no	—
32	C₂×D₈	—	4	4	no	—
33	C₄⋊Q₈	—	4	2	no	—
34	#127	—	4	2	no	—
35	SD₃₂	—	8	16	no	—
36	D₁₆	—	8	8	no	—
37	C₂×C₁₆	D₈	1	4	yes	—
38	C₂×C₁₆	D₈	1	4	yes	—
39	C₂×C₁₆	D₈	1	4	yes	—
40	#107	—	4	4	no	—
41	#107	—	4	4	no	—
42	C₄×C₈	C₂×D₄	1	1	yes	characteristic
43	C₄×C₈	C₂×D₄	1	1	yes	characteristic
44	C₄×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×Q₈	—	8	2	no	—
46	C₂×D₄	—	8	2	no	—
47	C₂×D₄	—	8	2	no	—
48	C₂×D₄	—	8	2	no	—
49	Q₁₆	—	8	8	no	—
50	D₈	—	8	8	no	—
51	D₈	—	8	8	no	—
52	D₈	—	8	8	no	—
53	C₂×C₈	C₂×D₈	1	2	yes	—
54	C₂×C₈	C₂×D₈	1	2	yes	—
55	C₂×C₈	C₂×D₈	1	2	yes	—
56	C₄⋊C₄	—	8	2	no	—
57	C₄⋊C₄	—	8	2	no	—
58	C₄²	#127	1	1	yes	characteristic
59	C₁₆	—	2	4	no	—
60	C₁₆	D₁₆	1	8	yes	complemented
61	C₁₆	SD₃₂	1	8	yes	complemented
62	C₂³	—	16	2	no	—
63	Q₈	—	16	4	no	—
64	D₄	—	16	4	no	—
65	D₄	—	16	4	no	—
66	D₄	—	16	4	no	—
67	C₂×C₄	—	16	2	no	—
68	C₂×C₄	#493	1	1	yes	characteristic
69	C₂×C₄	#493	1	1	yes	characteristic
70	C₂×C₄	#493	1	1	yes	characteristic
71	C₈	C₂×D₁₆	1	4	yes	—
72	C₈	C₂×SD₃₂	1	4	yes	—
73	C₈	#508	1	4	yes	—
74	C₂²	—	32	2	no	—
75	C₂²	—	32	2	no	—
76	C₂²	—	32	2	no	—
77	C₂²	#1712	1	1	yes	characteristic
78	C₄	—	32	2	no	—
79	C₄	#2246	1	2	yes	—
80	C₄	#2248	1	2	yes	—
81	C₄	#2249	1	2	yes	—
82	C₂	—	64	2	no	—
83	C₂	#12515	1	1	yes	characteristic
84	C₂	#12517	1	1	yes	characteristic
85	C₂	#12518	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#12515	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #12521	C₂	semidirect
2. #12517	C₂	semidirect
4. #12509	C₂	semidirect
5. C₁₆²	C₂	semidirect
60. C₁₆	D₁₆	semidirect
61. C₁₆	SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
#12521	2
#12517	2
#12515	2
#12509	2
C₁₆²	2