Subgroups — #135671

Order 512.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12591	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	2	yes	maximal
3	#7156	C₂	1	4	yes	maximal
4	C₂×#346	C₂²	1	2	yes	—
5	#1831	—	2	2	no	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₂×#335	—	2	2	no	—
8	C₂×#334	C₂²	1	4	yes	—
9	#1374	C₄	1	4	yes	—
10	C₂²×C₁₆	C₂×C₄	1	4	yes	—
11	C₂×C₄⋊C₈	—	2	4	no	—
12	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₈	D₄	1	1	yes	characteristic
14	C₂×C₄×C₈	Q₈	1	2	yes	—
15	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
16	C₄³	D₄	1	1	yes	characteristic
17	#346	—	2	4	no	—
18	#336	—	4	4	no	—
19	#335	—	4	4	no	—
20	#334	—	2	8	no	—
21	C₂²×C₈	—	4	4	no	—
22	C₂²×C₈	C₄⋊C₄	1	2	yes	—
23	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄²	C₄⋊C₄	1	2	yes	—
26	C₂×C₄²	SD₁₆	1	4	yes	—
27	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
28	C₂×C₁₆	—	2	4	no	—
29	C₂×C₁₆	—	2	8	no	—
30	C₄⋊C₈	—	4	4	no	—
31	C₄⋊C₈	—	4	4	no	—
32	C₄×C₈	—	2	4	no	—
33	C₄×C₈	—	4	2	no	—
34	C₄×C₈	D₈	1	2	yes	—
35	C₄×C₈	Q₁₆	1	2	yes	—
36	C₄×C₈	#41	1	4	yes	—
37	C₂²×C₄	#106	1	4	yes	—
38	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	8	8	no	—
44	C₂×C₈	—	8	4	no	—
45	C₂×C₈	#107	1	4	yes	—
46	C₂×C₈	#117	1	2	yes	—
47	C₂×C₈	#125	1	2	yes	—
48	C₂×C₈	#131	1	4	yes	—
49	C₄²	—	2	2	no	—
50	C₄²	—	2	2	no	—
51	C₄²	—	4	4	no	—
52	C₄²	#124	1	2	yes	—
53	C₄²	C₂×D₈	1	1	yes	characteristic
54	C₄²	C₂×Q₁₆	1	1	yes	characteristic
55	C₁₆	—	4	4	no	—
56	C₂³	C₂×#106	1	1	yes	characteristic
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	—	2	8	no	—
60	C₂×C₄	#382	1	1	yes	characteristic
61	C₂×C₄	#424	1	2	yes	—
62	C₂×C₄	C₂×#107	1	1	yes	characteristic
63	C₂×C₄	#430	1	2	yes	—
64	C₂×C₄	#486	1	2	yes	—
65	C₂×C₄	#487	1	2	yes	—
66	C₈	—	2	4	no	—
67	C₈	—	2	2	no	—
68	C₈	—	16	4	no	—
69	C₈	#433	1	4	yes	—
70	C₂²	#1570	1	2	yes	—
71	C₂²	#1834	1	1	yes	characteristic
72	C₂²	#2155	1	4	yes	—
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	4	no	—
76	C₄	#1571	1	2	yes	—
77	C₄	#1832	1	1	yes	characteristic
78	C₄	#1838	1	1	yes	characteristic
79	C₂	#8787	1	1	yes	characteristic
80	C₂	#12031	1	4	yes	—
81	C₂	#12887	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12591	2
#8807	2
#7156	2