Subgroups — #63444

Order 288.

195 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#611	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃₆	C₂	1	2	yes	maximal, complemented
4	C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₉)⋊C₄	C₂	1	2	yes	maximal, complemented
6	#882	—	3	1	no	maximal
7	C₂³×C₉	C₂²	1	1	yes	characteristic
8	C₂²×D₉	—	2	2	no	—
9	C₂²×D₉	—	2	4	no	—
10	C₂²×D₉	C₂²	1	2	yes	—
11	C₂²×D₉	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₄×C₉	C₂²	1	2	yes	complemented
13	#611	—	2	2	no	—
14	C₂×Dic₉	C₂²	1	1	yes	characteristic, complemented
15	D₃₆	—	2	4	no	—
16	C₂³×D₃	—	3	1	no	—
17	C₂×#67	—	3	1	no	—
18	C₂×D₁₂	—	3	2	no	—
19	C₃×C₂²⋊C₄	D₃	1	1	yes	characteristic
20	(C₂×D₃)⋊C₄	—	3	2	no	—
21	C₂²×C₉	—	2	2	no	—
22	C₂²×C₉	D₄	1	2	yes	complemented
23	C₂²×C₉	C₂³	1	1	yes	characteristic
24	C₂×D₉	—	2	8	no	—
25	C₂×D₉	—	2	2	no	—
26	C₂×D₉	—	2	2	no	—
27	C₂×D₉	—	2	2	no	—
28	C₂×D₉	—	2	1	no	—
29	C₂×D₉	D₄	1	4	yes	complemented
30	C₄×C₉	—	2	2	no	—
31	Dic₉	—	2	1	no	—
32	#120	—	9	1	no	Sylow
33	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
34	C₂²×D₃	—	3	2	no	—
35	C₂²×D₃	—	3	1	no	—
36	C₂²×D₃	—	6	4	no	—
37	C₂²×D₃	—	6	2	no	—
38	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
39	#67	—	6	2	no	—
40	C₂×Dic₃	—	3	1	no	—
41	D₁₂	—	6	4	no	—
42	C₂×C₉	—	2	2	no	—
43	C₂×C₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₉	C₂×D₄	1	2	yes	—
45	D₉	—	2	4	no	—
46	D₉	—	4	1	no	—
47	C₂⁴	—	9	1	no	—
48	C₂×D₄	—	9	1	no	—
49	C₂×D₄	—	9	2	no	—
50	C₂²⋊C₄	—	9	2	no	—
51	C₂²⋊C₄	D₉	1	1	yes	characteristic, complemented
52	C₂²×C₃	—	2	2	no	—
53	C₂²×C₃	D₁₂	1	2	yes	—
54	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
55	C₂×D₃	—	3	4	no	—
56	C₂×D₃	—	6	8	no	—
57	C₂×D₃	—	6	2	no	—
58	C₂×D₃	—	6	2	no	—
59	C₂×D₃	—	6	1	no	—
60	C₂×D₃	—	6	2	no	—
61	C₃×C₄	—	2	2	no	—
62	Dic₃	—	6	1	no	—
63	C₉	#120	1	1	yes	characteristic, complemented, Sylow
64	C₂³	—	9	2	no	—
65	C₂³	—	9	1	no	—
66	C₂³	—	18	4	no	—
67	C₂³	—	18	2	no	—
68	C₂³	C₂×D₉	1	1	yes	characteristic
69	D₄	—	18	4	no	—
70	D₄	—	18	2	no	—
71	C₂×C₄	—	9	1	no	—
72	C₂×C₄	C₂×D₉	1	2	yes	complemented
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	C₂×D₁₂	1	1	yes	characteristic
75	C₂×C₃	D₃×D₄	1	2	yes	—
76	D₃	—	6	4	no	—
77	D₃	—	12	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	9	4	no	—
80	C₂²	—	18	8	no	—
81	C₂²	—	18	2	no	—
82	C₂²	—	18	2	no	—
83	C₂²	—	18	1	no	—
84	C₂²	—	18	2	no	—
85	C₂²	D₃₆	1	2	yes	complemented
86	C₂²	C₂²×D₉	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	18	1	no	—
89	C₃	#882	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	—	18	4	no	—
92	C₂	—	36	1	no	—
93	C₂	C₂×D₃₆	1	1	yes	characteristic
94	C₂	D₄×D₉	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
hypercenter	C₂²⋊C₄	D₉
hypocenter	C₉	#120
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
1. C₂³×D₉	C₂	semidirect
2. C₂×#611	C₂	semidirect
3. C₂×D₃₆	C₂	semidirect
4. C₉×C₂²⋊C₄	C₂	semidirect
5. (C₂×D₉)⋊C₄	C₂	semidirect
11. C₂²×D₉	C₂²	semidirect
12. C₂×C₄×C₉	C₂²	semidirect
14. C₂×Dic₉	C₂²	semidirect
22. C₂²×C₉	D₄	semidirect
29. C₂×D₉	D₄	semidirect
51. C₂²⋊C₄	D₉	semidirect
63. C₉	#120	semidirect
72. C₂×C₄	C₂×D₉	semidirect
85. C₂²	D₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₉	2
C₂×#611	2
C₂×D₃₆	2
C₉×C₂²⋊C₄	2
(C₂×D₉)⋊C₄	2
#882	3