Subgroups — C₉×#4932 #13142392

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉×Dic₃	C₂	1	2	yes	maximal
3	C₉×C₃⋊#95	C₂	1	4	yes	maximal
4	C₃×#4932	C₃	1	1	yes	maximal, characteristic
5	C₉×#376	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₉×Dic₃	C₂²	1	4	yes	—
9	C₃×C₄²×C₉	C₄	1	4	yes	complemented
10	C₄×C₉×Dic₃	C₄	1	8	yes	complemented
11	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₃	C₂×C₃	1	2	yes	—
13	C₃×C₃⋊#95	C₂×C₃	1	4	yes	—
14	C₂×C₄²×C₉	—	2	1	no	—
15	C₂×C₄²×C₉	—	3	2	no	—
16	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
17	C₉×#95	—	3	4	no	—
18	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
19	C₂×C₃×C₄×C₉	—	2	4	no	—
20	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
23	C₂×C₉×Dic₃	—	2	8	no	—
24	C₂×C₉×Dic₃	C₂×C₄	1	8	yes	—
25	C₃×#376	—	3	1	no	—
26	#4932	C₉	1	1	yes	characteristic, direct product factor
27	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
28	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	—
30	C₃²×C₄²	C₃×C₄	1	4	yes	—
31	C₃×C₄×Dic₃	C₃×C₄	1	8	yes	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	—	3	4	no	—
35	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
36	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
37	C₄²×C₉	—	2	4	no	—
38	C₄²×C₉	—	3	8	no	—
39	C₄²×C₉	Dic₃	1	4	yes	complemented
40	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
41	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₉	#41	1	4	yes	—
43	C₃×C₄×C₉	—	2	4	no	—
44	C₃×C₄×C₉	C₄²	1	4	yes	—
45	C₉×Dic₃	—	2	8	no	—
46	C₂×C₃×C₄²	—	2	1	no	—
47	C₂×C₃×C₄²	—	3	2	no	—
48	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
50	C₂×C₄×Dic₃	C₂×C₉	1	2	yes	—
51	C₃×#95	—	3	4	no	—
52	C₃⋊#95	C₂×C₉	1	4	yes	—
53	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
54	C₂×C₃²×C₄	—	2	4	no	—
55	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
56	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
57	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
58	C₂×C₃×Dic₃	—	2	8	no	—
59	C₂×C₃×Dic₃	C₂×C₃×C₄	1	8	yes	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
62	C₂×C₄×C₉	—	2	8	no	—
63	C₂×C₄×C₉	—	2	4	no	—
64	C₂×C₄×C₉	—	2	4	no	—
65	C₂×C₄×C₉	—	2	4	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	3	8	no	—
68	C₂×C₄×C₉	—	6	8	no	—
69	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
70	C₂×C₄×C₉	C₂×Dic₃	1	4	yes	—
71	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
72	#376	—	3	1	no	Sylow
73	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₃×C₉	#117	1	4	yes	—
75	C₂×C₃×C₉	#117	1	2	yes	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	3	4	no	—
79	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
80	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
82	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
83	C₂²×Dic₃	C₂²×C₉	1	4	yes	—
84	C₃×C₄²	—	2	4	no	—
85	C₃×C₄²	—	3	8	no	—
86	C₃×C₄²	C₄×C₉	1	4	yes	complemented
87	C₃×C₄²	C₃×Dic₃	1	4	yes	—
88	C₄×Dic₃	C₄×C₉	1	8	yes	complemented
89	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
90	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
91	C₂²×C₃²	C₃×#41	1	4	yes	—
92	C₃²×C₄	—	2	4	no	—
93	C₃²×C₄	C₃×C₄²	1	4	yes	—
94	C₃×Dic₃	—	2	8	no	—
95	C₂²×C₉	—	2	2	no	—
96	C₂²×C₉	—	2	4	no	—
97	C₂²×C₉	—	2	1	no	—
98	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
99	C₂²×C₉	#235	1	4	yes	—
100	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
101	C₄×C₉	—	2	8	no	—
102	C₄×C₉	—	2	4	no	—
103	C₄×C₉	—	2	4	no	—
104	C₄×C₉	—	6	8	no	—
105	C₄×C₉	C₄×Dic₃	1	4	yes	—
106	C₂×C₄²	—	3	2	no	—
107	C₂×C₄²	D₃×C₉	1	1	yes	characteristic
108	#95	—	3	4	no	—
109	C₃×C₉	#376	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₃	—	2	1	no	—
111	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
112	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	8	no	—
114	C₂×C₃×C₄	—	2	4	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	3	8	no	—
120	C₂×C₃×C₄	—	6	8	no	—
121	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
122	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
123	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
124	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
125	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
126	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
127	C₂×Dic₃	—	2	8	no	—
128	C₂×Dic₃	C₂×C₄×C₉	1	8	yes	—
129	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
130	C₂×C₃²	C₃×#117	1	2	yes	—
131	C₂×C₃²	C₃×#117	1	4	yes	—
132	C₂×C₉	—	2	1	no	—
133	C₂×C₉	—	2	2	no	—
134	C₂×C₉	—	2	4	no	—
135	C₂×C₉	#872	1	4	yes	—
136	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
137	C₂×C₉	#926	1	2	yes	—
138	C₂²×C₄	—	3	4	no	—
139	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
140	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
141	C₄²	—	3	8	no	—
142	C₄²	C₉×Dic₃	1	4	yes	complemented
143	C₂²×C₃	—	2	2	no	—
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	C₂²×C₄×C₉	1	2	yes	—
147	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
148	C₂²×C₃	C₉×#41	1	4	yes	—
149	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
150	C₂²×C₃	C₃×#235	1	4	yes	—
151	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
152	C₃×C₄	—	2	8	no	—
153	C₃×C₄	—	2	4	no	—
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	6	8	no	—
157	C₃×C₄	C₄²×C₉	1	4	yes	—
158	C₃×C₄	C₃×C₄×Dic₃	1	4	yes	—
159	Dic₃	—	2	8	no	—
160	C₃²	C₃×#376	1	1	yes	characteristic
161	C₉	—	2	1	no	—
162	C₉	#4932	1	1	yes	characteristic, direct product factor
163	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	3	8	no	—
166	C₂×C₄	—	6	8	no	—
167	C₂×C₄	C₄×D₃×C₉	1	4	yes	—
168	C₂×C₄	C₂×C₉×Dic₃	1	4	yes	—
169	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	—	2	2	no	—
172	C₂×C₃	—	2	4	no	—
173	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
174	C₂×C₃	C₉×#117	1	2	yes	—
175	C₂×C₃	C₉×#117	1	4	yes	—
176	C₂×C₃	C₃×#872	1	4	yes	—
177	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
178	C₂×C₃	C₃×#926	1	2	yes	—
179	C₂²	C₂×C₄×D₃×C₉	1	2	yes	—
180	C₂²	C₉×#235	1	4	yes	—
181	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	6	8	no	—
184	C₄	C₄×C₉×Dic₃	1	4	yes	—
185	C₃	—	2	1	no	—
186	C₃	C₉×#376	1	1	yes	characteristic, complemented
187	C₃	C₃×#4932	1	1	yes	characteristic
188	C₂	C₉×#872	1	4	yes	—
189	C₂	C₂×C₄×C₉×Dic₃	1	1	yes	characteristic
190	C₂	C₉×#926	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂×C₄²×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₄²×C₉	D₃
hypocenter	C₃	C₉×#376
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
9. C₃×C₄²×C₉	C₄	semidirect
10. C₄×C₉×Dic₃	C₄	semidirect
26. #4932	C₉	direct (N × H)
39. C₄²×C₉	Dic₃	semidirect
86. C₃×C₄²	C₄×C₉	semidirect
88. C₄×Dic₃	C₄×C₉	semidirect
109. C₃×C₉	#376	semidirect
142. C₄²	C₉×Dic₃	semidirect
162. C₉	#4932	direct (N × H)
186. C₃	C₉×#376	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₄×C₉×Dic₃	2
C₉×C₃⋊#95	2
C₃×#4932	3
C₉×#376	3

Subgroups — C9×#4932 #13142392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4932 #13142392