Subgroups — C₉×#4878 #13142185

Order 1728.

294 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#890	C₂	1	2	yes	maximal
2	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic
3	C₉×#818	C₂	1	1	yes	maximal, characteristic
4	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₉×#808	C₂	1	2	yes	maximal
6	C₃×#4878	C₃	1	1	yes	maximal, characteristic
7	C₉×#499	—	3	1	no	maximal
8	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₉×C₄⋊Dic₃	—	2	2	no	—
11	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₉×Dic₃⋊C₄	—	2	2	no	—
13	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×#890	C₂×C₃	1	2	yes	—
16	C₃²×#107	C₂×C₃	1	1	yes	characteristic
17	C₃×#818	C₂×C₃	1	1	yes	characteristic
18	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₃×#808	C₂×C₃	1	2	yes	—
20	C₉×#125	—	3	2	no	—
21	C₉×#107	—	2	1	no	—
22	C₉×#107	—	3	1	no	—
23	C₉×#107	D₃	1	1	yes	characteristic
24	C₉×#106	—	3	2	no	—
25	C₄×C₈×C₉	—	3	1	no	—
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₉×Dic₃	—	2	2	no	—
29	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
30	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
31	C₃×C₈×C₉	Q₈	1	2	yes	—
32	C₉×C₃⋊C₈	Q₈	1	2	yes	—
33	C₃×#499	—	3	1	no	—
34	#4878	C₉	1	1	yes	characteristic, direct product factor
35	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
36	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
37	C₃×C₄⋊Dic₃	—	2	2	no	—
38	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
39	C₃×Dic₃⋊C₄	—	2	2	no	—
40	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₈×C₉	—	2	1	no	—
43	C₂×C₈×C₉	—	3	1	no	—
44	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
45	C₉×C₄⋊C₄	—	2	2	no	—
46	C₉×C₄⋊C₄	—	3	2	no	—
47	C₉×C₄⋊C₄	—	6	2	no	—
48	C₉×C₄⋊C₄	—	6	2	no	—
49	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₄²×C₉	—	3	1	no	—
51	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
52	C₃×C₄×C₉	—	4	2	no	—
53	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
54	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
55	C₉×Dic₃	—	2	1	no	—
56	C₉×Dic₃	—	2	1	no	—
57	C₉×Dic₃	—	4	2	no	—
58	C₃×#125	—	3	2	no	—
59	#890	C₂×C₉	1	2	yes	—
60	C₃×#107	—	2	1	no	—
61	C₃×#107	—	3	1	no	—
62	C₃×#107	C₂×C₉	1	1	yes	characteristic
63	C₃×#107	C₃×D₃	1	1	yes	characteristic
64	C₃×#106	—	3	2	no	—
65	C₃×C₄×C₈	—	3	1	no	—
66	#818	C₂×C₉	1	1	yes	characteristic
67	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic
68	#808	C₂×C₉	1	2	yes	—
69	C₂×C₃²×C₄	—	2	2	no	—
70	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
71	C₂×C₃×Dic₃	—	2	2	no	—
72	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
73	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
74	C₃²×C₈	C₃×Q₈	1	2	yes	—
75	C₃×C₃⋊C₈	C₃×Q₈	1	2	yes	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	1	no	—
78	C₂×C₄×C₉	—	3	1	no	—
79	C₂×C₄×C₉	—	3	1	no	—
80	C₂×C₄×C₉	—	4	2	no	—
81	C₂×C₄×C₉	—	6	2	no	—
82	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
83	C₈×C₉	—	2	2	no	—
84	C₈×C₉	—	3	2	no	—
85	C₈×C₉	Dic₆	1	2	yes	—
86	#499	—	3	1	no	Sylow
87	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₃×C₉	#135	1	1	yes	characteristic
89	C₂×C₃×C₉	#135	1	1	yes	characteristic
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	3	1	no	—
92	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
94	C₃×C₄⋊C₄	—	2	2	no	—
95	C₃×C₄⋊C₄	—	3	2	no	—
96	C₃×C₄⋊C₄	—	6	2	no	—
97	C₃×C₄⋊C₄	—	6	2	no	—
98	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
99	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
100	C₃×C₄²	—	3	1	no	—
101	C₄⋊Dic₃	—	2	2	no	—
102	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
103	Dic₃⋊C₄	—	2	2	no	—
104	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
105	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
106	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
107	C₃²×C₄	—	4	2	no	—
108	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
109	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
110	C₃×Dic₃	—	2	1	no	—
111	C₃×Dic₃	—	2	1	no	—
112	C₃×Dic₃	—	4	2	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
115	C₄×C₉	—	2	1	no	—
116	C₄×C₉	—	2	1	no	—
117	C₄×C₉	—	4	2	no	—
118	C₄×C₉	—	6	1	no	—
119	C₄×C₉	—	6	1	no	—
120	C₄×C₉	—	8	2	no	—
121	C₄×C₉	—	12	2	no	—
122	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
123	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
124	#125	—	3	2	no	—
125	#107	—	3	1	no	—
126	#107	D₃×C₉	1	1	yes	characteristic
127	#106	—	3	2	no	—
128	C₄×C₈	—	3	1	no	—
129	C₃×C₉	#499	1	1	yes	characteristic, complemented, Sylow
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	3	1	no	—
134	C₂×C₃×C₄	—	3	1	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	6	2	no	—
137	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
139	C₂×Dic₃	—	2	2	no	—
140	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
141	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	3	2	no	—
144	C₃×C₈	Q₈×C₉	1	2	yes	—
145	C₃×C₈	C₃×Dic₆	1	2	yes	—
146	C₃⋊C₈	Q₈×C₉	1	2	yes	—
147	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₃²	C₃×#135	1	1	yes	characteristic
149	C₂×C₃²	C₃×#135	1	1	yes	characteristic
150	C₂×C₉	—	2	1	no	—
151	C₂×C₉	—	2	1	no	—
152	C₂×C₉	—	2	1	no	—
153	C₂×C₉	#888	1	1	yes	characteristic
154	C₂×C₉	#912	1	1	yes	characteristic
155	C₂×C₉	#919	1	1	yes	characteristic
156	C₂×C₈	—	3	1	no	—
157	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
158	C₄⋊C₄	—	3	2	no	—
159	C₄⋊C₄	—	6	2	no	—
160	C₄⋊C₄	—	6	2	no	—
161	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
162	C₄²	—	3	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
165	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	4	2	no	—
169	C₃×C₄	—	4	2	no	—
170	C₃×C₄	—	6	1	no	—
171	C₃×C₄	—	6	1	no	—
172	C₃×C₄	—	8	2	no	—
173	C₃×C₄	—	12	2	no	—
174	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
175	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
176	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
177	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
178	Dic₃	—	2	1	no	—
179	Dic₃	—	2	1	no	—
180	Dic₃	—	4	2	no	—
181	C₃²	C₃×#499	1	1	yes	characteristic
182	C₉	—	2	1	no	—
183	C₉	#4878	1	1	yes	characteristic, direct product factor
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	3	1	no	—
186	C₂×C₄	—	3	1	no	—
187	C₂×C₄	—	6	2	no	—
188	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
189	C₈	—	3	2	no	—
190	C₈	C₉×Dic₆	1	2	yes	—
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×C₃	C₉×#135	1	1	yes	characteristic
196	C₂×C₃	C₉×#135	1	1	yes	characteristic
197	C₂×C₃	C₃×#888	1	1	yes	characteristic
198	C₂×C₃	C₃×#912	1	1	yes	characteristic
199	C₂×C₃	C₃×#919	1	1	yes	characteristic
200	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
201	C₄	—	4	2	no	—
202	C₄	—	6	1	no	—
203	C₄	—	6	1	no	—
204	C₄	—	12	2	no	—
205	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
206	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
207	C₃	—	2	1	no	—
208	C₃	C₉×#499	1	1	yes	characteristic, complemented
209	C₃	C₃×#4878	1	1	yes	characteristic
210	C₂	C₉×#888	1	1	yes	characteristic
211	C₂	C₉×#912	1	1	yes	characteristic
212	C₂	C₉×#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#107	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₉×#499
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#107

Complements

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

N	H ≅ G/N	type
34. #4878	C₉	direct (N × H)
129. C₃×C₉	#499	semidirect
183. C₉	#4878	direct (N × H)
208. C₃	C₉×#499	semidirect

Maximal subgroups

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

subgroup	index
C₉×#890	2
C₃×C₉×#107	2
C₉×#818	2
C₈×C₉×Dic₃	2
C₉×#808	2
C₃×#4878	3
C₉×#499	3

Subgroups — C9×#4878 #13142185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4878 #13142185