Subgroups — #63365

Order 288.

78 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	C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#3676	C₂	1	1	yes	maximal, characteristic, complemented
4	#806	—	3	1	no	maximal
5	C₂³×C₉	C₂²	1	1	yes	characteristic
6	C₂²×D₉	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×C₉	—	2	1	no	—
8	#611	—	2	1	no	—
9	#611	—	2	1	no	—
10	C₂×Dic₉	—	2	1	no	—
11	C₂×Dic₉	C₄	1	1	yes	characteristic, complemented
12	C₂×#67	—	3	1	no	—
13	C₃×C₂²⋊C₄	D₃	1	1	yes	characteristic
14	#217	—	3	1	no	—
15	C₂²×C₉	—	4	1	no	—
16	C₂²×C₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₉	D₄	1	1	yes	characteristic
18	C₂²×C₉	D₄	1	1	yes	characteristic
19	C₂×D₉	—	2	1	no	—
20	C₂×D₉	—	4	1	no	—
21	C₄×C₉	—	4	1	no	—
22	Dic₉	—	2	1	no	—
23	Dic₉	—	4	1	no	—
24	#99	—	9	1	no	Sylow
25	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
26	C₂²×D₃	—	3	1	no	—
27	C₂×C₃×C₄	—	2	1	no	—
28	#67	—	6	1	no	—
29	#67	—	6	1	no	—
30	C₂×Dic₃	—	3	1	no	—
31	C₂×Dic₃	—	6	1	no	—
32	C₂×C₉	—	2	1	no	—
33	C₂×C₉	—	2	1	no	—
34	C₂×C₉	—	2	1	no	—
35	C₂×C₉	C₂²⋊C₄	1	1	yes	characteristic
36	D₉	—	4	1	no	—
37	C₂×D₄	—	9	1	no	—
38	C₂²⋊C₄	—	9	1	no	—
39	C₂²⋊C₄	D₉	1	1	yes	characteristic, complemented
40	C₂²×C₃	—	4	1	no	—
41	C₂²×C₃	C₄×D₃	1	1	yes	characteristic
42	C₂²×C₃	D₁₂	1	1	yes	characteristic
43	C₂²×C₃	#67	1	1	yes	characteristic
44	C₂×D₃	—	6	1	no	—
45	C₂×D₃	—	12	1	no	—
46	C₃×C₄	—	4	1	no	—
47	Dic₃	—	6	1	no	—
48	Dic₃	—	12	1	no	—
49	C₉	#99	1	1	yes	characteristic, complemented, Sylow
50	C₂³	—	9	1	no	—
51	C₂³	C₂×D₉	1	1	yes	characteristic
52	D₄	—	18	1	no	—
53	D₄	—	18	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	9	1	no	—
56	C₂×C₄	—	18	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
61	D₃	—	12	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	18	1	no	—
64	C₂²	—	36	1	no	—
65	C₂²	C₄×D₉	1	1	yes	characteristic
66	C₂²	D₃₆	1	1	yes	characteristic
67	C₂²	#611	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	—	18	1	no	—
70	C₄	—	36	1	no	—
71	C₃	#806	1	1	yes	characteristic
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	—	2	1	no	—
75	C₂	—	36	1	no	—
76	C₂	(C₂×D₉)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×D₉)⋊C₄
commutator	C₂²×C₉	C₂×C₄
Frattini	C₂³×C₃	C₂×D₃
Fitting	C₉×C₂²⋊C₄	C₂
socle	C₂×C₃	(C₂×D₃)⋊C₄
prime radical	C₂×#611	C₂
cosocle	C₂³×C₉	C₂²
hypercenter	C₂²⋊C₄	D₉
hypocenter	C₉	#99
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₉×C₂²⋊C₄	C₂	semidirect
3. #3676	C₂	semidirect
6. C₂²×D₉	C₄	semidirect
11. C₂×Dic₉	C₄	semidirect
39. C₂²⋊C₄	D₉	semidirect
49. C₉	#99	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
C₉×C₂²⋊C₄	2
#3676	2
#806	3