Subgroups — #63448

Order 288.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#611	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₉)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄⋊Dic₉	C₂	1	2	yes	maximal, complemented
6	#886	—	3	1	no	maximal
7	C₂³×C₉	C₂²	1	1	yes	characteristic
8	C₂²×D₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₉	C₂²	1	2	yes	—
10	#611	—	2	2	no	—
11	C₂×Dic₉	—	2	2	no	—
12	C₂×Dic₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₉	C₂²	1	2	yes	—
14	C₂×#67	—	3	1	no	—
15	C₂²×Dic₃	—	3	1	no	—
16	C₃×C₂²⋊C₄	D₃	1	1	yes	characteristic
17	(C₂×D₃)⋊C₄	—	3	2	no	—
18	C₄⋊Dic₃	—	3	2	no	—
19	C₂²×C₉	—	2	2	no	—
20	C₂²×C₉	D₄	1	2	yes	—
21	C₂²×C₉	C₂³	1	1	yes	characteristic
22	C₂×D₉	—	2	1	no	—
23	C₂×D₉	—	2	2	no	—
24	C₄×C₉	—	2	2	no	—
25	Dic₉	—	2	2	no	—
26	Dic₉	—	2	1	no	—
27	#123	—	9	1	no	Sylow
28	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
29	C₂²×D₃	—	3	1	no	—
30	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
31	#67	—	6	2	no	—
32	C₂×Dic₃	—	3	1	no	—
33	C₂×Dic₃	—	3	2	no	—
34	C₂×Dic₃	—	6	2	no	—
35	C₂×C₉	—	2	2	no	—
36	C₂×C₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₉	#41	1	2	yes	—
38	D₉	—	4	1	no	—
39	C₂×D₄	—	9	1	no	—
40	C₂²×C₄	—	9	1	no	—
41	C₄⋊C₄	—	9	2	no	—
42	C₂²⋊C₄	—	9	2	no	—
43	C₂²⋊C₄	D₉	1	1	yes	characteristic, complemented
44	C₂²×C₃	—	2	2	no	—
45	C₂²×C₃	D₁₂	1	2	yes	—
46	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
47	C₂×D₃	—	6	1	no	—
48	C₂×D₃	—	6	2	no	—
49	C₃×C₄	—	2	2	no	—
50	Dic₃	—	6	2	no	—
51	Dic₃	—	6	1	no	—
52	C₉	#123	1	1	yes	characteristic, complemented, Sylow
53	C₂³	—	9	1	no	—
54	C₂³	C₂×D₉	1	1	yes	characteristic
55	D₄	—	18	2	no	—
56	C₂×C₄	—	9	2	no	—
57	C₂×C₄	—	9	1	no	—
58	C₂×C₄	—	18	2	no	—
59	C₂×C₄	C₂×D₉	1	2	yes	—
60	C₂×C₃	—	2	2	no	—
61	C₂×C₃	C₂×D₁₂	1	1	yes	characteristic
62	C₂×C₃	#237	1	2	yes	—
63	D₃	—	12	1	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	18	1	no	—
66	C₂²	—	18	2	no	—
67	C₂²	D₃₆	1	2	yes	—
68	C₂²	C₂²×D₉	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	18	2	no	—
71	C₄	—	18	1	no	—
72	C₃	#886	1	1	yes	characteristic
73	C₂	—	2	2	no	—
74	C₂	—	36	1	no	—
75	C₂	C₂×D₃₆	1	1	yes	characteristic
76	C₂	#3699	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂³
Frattini, socle	C₂²×C₃	C₂²×D₃
Fitting	C₉×C₂²⋊C₄	C₂
prime radical	C₂×#611	C₂
hypercenter	C₂²⋊C₄	D₉
hypocenter	C₉	#123
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₉	C₂	semidirect
3. C₉×C₂²⋊C₄	C₂	semidirect
4. (C₂×D₉)⋊C₄	C₂	semidirect
5. C₄⋊Dic₉	C₂	semidirect
43. C₂²⋊C₄	D₉	semidirect
52. C₉	#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×#611	2
C₂²×Dic₉	2
C₉×C₂²⋊C₄	2
(C₂×D₉)⋊C₄	2
C₄⋊Dic₉	2
#886	3