Subgroups — #10588180

Order 576.

87 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#63378	C₂	1	1	yes	maximal, characteristic
4	#4503	—	3	1	no	maximal
5	C₂×Dic₁₈	—	2	1	no	—
6	C₂×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₉	—	2	1	no	—
8	Dic₃₆	—	2	1	no	—
9	Dic₃₆	C₄	1	2	yes	complemented
10	C₉×C₁₆	—	2	1	no	—
11	C₂×Dic₁₂	—	3	1	no	—
12	C₂×C₃×C₁₆	D₃	1	1	yes	characteristic
13	#818	—	3	1	no	—
14	C₂×C₄×C₉	D₄	1	1	yes	characteristic
15	C₂×Dic₉	—	4	1	no	—
16	C₂×Dic₉	—	4	1	no	—
17	Dic₁₈	—	2	2	no	—
18	Dic₁₈	—	4	1	no	—
19	C₈×C₉	C₂×C₄	1	1	yes	characteristic
20	C₈×C₉	D₄	1	1	yes	characteristic
21	#358	—	9	1	no	Sylow
22	C₂×Dic₆	—	6	1	no	—
23	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
24	C₄⋊Dic₃	—	6	1	no	—
25	Dic₁₂	—	3	2	no	—
26	Dic₁₂	—	6	1	no	—
27	C₃×C₁₆	—	2	1	no	—
28	C₂²×C₉	D₈	1	1	yes	characteristic
29	C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₉	SD₁₆	1	1	yes	characteristic
31	Dic₉	—	4	2	no	—
32	Dic₉	—	8	1	no	—
33	C₂×Q₁₆	—	9	1	no	—
34	C₂×C₁₆	D₉	1	1	yes	characteristic
35	#107	—	9	1	no	—
36	C₂×C₃×C₄	D₁₂	1	1	yes	characteristic
37	C₂×Dic₃	—	12	1	no	—
38	C₂×Dic₃	—	12	1	no	—
39	Dic₆	—	6	2	no	—
40	Dic₆	—	12	1	no	—
41	C₃×C₈	C₄×D₃	1	1	yes	characteristic
42	C₃×C₈	#67	1	1	yes	characteristic
43	C₂×C₉	#102	1	1	yes	characteristic
44	C₂×C₉	SD₃₂	1	1	yes	characteristic
45	C₂×C₉	Q₃₂	1	1	yes	characteristic
46	C₂×Q₈	—	18	1	no	—
47	Q₁₆	—	9	2	no	—
48	Q₁₆	—	18	1	no	—
49	C₂×C₈	C₂×D₉	1	1	yes	characteristic
50	C₄⋊C₄	—	18	1	no	—
51	C₁₆	—	2	1	no	—
52	C₂²×C₃	D₂₄	1	1	yes	characteristic
53	C₃×C₄	#204	1	1	yes	characteristic
54	C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
55	Dic₃	—	12	2	no	—
56	Dic₃	—	24	1	no	—
57	C₉	#358	1	1	yes	characteristic, complemented, Sylow
58	Q₈	—	18	2	no	—
59	Q₈	—	36	1	no	—
60	C₂×C₄	—	36	1	no	—
61	C₂×C₄	—	36	1	no	—
62	C₂×C₄	D₃₆	1	1	yes	characteristic
63	C₈	C₄×D₉	1	1	yes	characteristic
64	C₈	#611	1	1	yes	characteristic
65	C₂×C₃	#800	1	1	yes	characteristic
66	C₂×C₃	Dic₂₄	1	1	yes	characteristic
67	C₂×C₃	#821	1	1	yes	characteristic
68	C₂²	D₇₂	1	1	yes	characteristic
69	C₄	—	36	2	no	—
70	C₄	—	72	1	no	—
71	C₄	#3664	1	1	yes	characteristic
72	C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
73	C₃	#4503	1	1	yes	characteristic
74	C₂	#63359	1	1	yes	characteristic
75	C₂	Dic₇₂	1	1	yes	characteristic
76	C₂	#63380	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₇₂
commutator	C₈×C₉	C₂×C₄
Frattini	C₂×C₃×C₈	C₂×D₃
Fitting	C₂×C₉×C₁₆	C₂
socle, prime radical	C₂²×C₃	D₂₄
cosocle	C₂×C₈×C₉	C₂²
hypercenter	C₂×C₁₆	D₉
hypocenter	C₉	#358
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
9. Dic₃₆	C₄	semidirect
57. C₉	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₃₆	2
C₂×C₉×C₁₆	2
#63378	2
#4503	3