Subgroups — C₉×#4735 #13141595

Order 1728.

392 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#930	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	2	yes	maximal, complemented
5	C₉×#817	C₂	1	2	yes	maximal, complemented
6	C₃×#4735	C₃	1	1	yes	maximal, characteristic
7	C₉×#481	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×#67	—	2	1	no	—
10	C₂²×C₉×Dic₃	—	2	1	no	—
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
13	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
14	C₉×C₄⋊Dic₃	—	2	1	no	—
15	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₃×#930	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₃×#821	C₂×C₃	1	2	yes	—
21	C₃×#817	C₂×C₃	1	2	yes	—
22	C₉×#121	—	3	1	no	—
23	C₂×C₉×C₄⋊C₄	—	3	1	no	—
24	C₉×#106	—	3	2	no	—
25	C₉×#102	—	3	2	no	—
26	C₉×C₂²⋊C₈	—	2	1	no	—
27	C₉×C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂²×D₃×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×#67	—	4	2	no	—
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₂×C₉×Dic₃	—	2	1	no	—
36	C₂×C₉×Dic₃	—	4	2	no	—
37	C₉×D₁₂	—	2	2	no	—
38	C₃×C₈×C₉	—	2	2	no	—
39	C₃×#481	—	3	1	no	—
40	#4735	C₉	1	1	yes	characteristic, direct product factor
41	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×#67	—	2	1	no	—
43	C₂²×C₃×Dic₃	—	2	1	no	—
44	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
46	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
47	C₃×C₄⋊Dic₃	—	2	1	no	—
48	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
49	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
50	C₂×D₄×C₉	—	3	1	no	—
51	C₂×D₄×C₉	—	6	1	no	—
52	C₂²×C₄×C₉	—	2	1	no	—
53	C₂²×C₄×C₉	—	6	1	no	—
54	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
55	C₂×C₈×C₉	—	2	2	no	—
56	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
57	C₉×C₄⋊C₄	—	3	2	no	—
58	C₉×C₄⋊C₄	—	3	1	no	—
59	C₉×C₄⋊C₄	—	6	1	no	—
60	C₉×C₂²⋊C₄	—	6	1	no	—
61	C₂²×C₃×C₉	—	2	1	no	—
62	C₂²×C₃×C₉	—	2	1	no	—
63	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×D₃×C₉	—	4	1	no	—
66	C₂×D₃×C₉	—	4	1	no	—
67	C₂×D₃×C₉	—	4	1	no	—
68	C₃×C₄×C₉	—	2	1	no	—
69	C₃×C₄×C₉	#41	1	2	yes	—
70	C₉×Dic₃	—	4	1	no	—
71	C₉×Dic₃	—	4	2	no	—
72	C₃×#121	—	3	1	no	—
73	C₂×C₃×C₄⋊C₄	—	3	1	no	—
74	#930	C₂×C₉	1	1	yes	characteristic
75	C₂×C₄⋊Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
76	C₃×#106	—	3	2	no	—
77	C₃×#102	—	3	2	no	—
78	C₃×C₂²⋊C₈	—	2	1	no	—
79	C₃×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
80	C₃×C₂²⋊C₈	C₃×D₃	1	1	yes	characteristic
81	#821	C₂×C₉	1	2	yes	complemented
82	#817	C₂×C₉	1	2	yes	complemented
83	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
84	C₂²×C₃×D₃	—	2	1	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₃×#67	—	4	2	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	—	2	1	no	—
91	C₂×C₃×Dic₃	—	4	2	no	—
92	C₃×D₁₂	—	2	2	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	—	6	1	no	—
95	C₂³×C₉	D₁₂	1	1	yes	characteristic
96	C₃²×C₈	—	2	2	no	—
97	D₄×C₉	—	6	2	no	—
98	D₄×C₉	—	12	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	4	2	no	—
103	C₂×C₄×C₉	—	6	1	no	—
104	C₂×C₄×C₉	—	6	2	no	—
105	C₂×C₄×C₉	—	12	2	no	—
106	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
108	C₈×C₉	—	2	2	no	—
109	C₈×C₉	—	4	2	no	—
110	#481	—	3	1	no	Sylow
111	C₂×C₃×C₉	—	2	2	no	—
112	C₂×C₃×C₉	#123	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	D₃×C₉	—	8	1	no	—
116	C₂×C₃×D₄	—	3	1	no	—
117	C₂×C₃×D₄	—	6	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	—	6	1	no	—
120	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
122	C₂×#67	—	2	1	no	—
123	C₂²×Dic₃	—	2	1	no	—
124	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
127	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
128	C₃×C₄⋊C₄	—	3	2	no	—
129	C₃×C₄⋊C₄	—	3	1	no	—
130	C₃×C₄⋊C₄	—	6	1	no	—
131	C₃×C₂²⋊C₄	—	6	1	no	—
132	(C₂×D₃)⋊C₄	—	2	1	no	—
133	C₄⋊Dic₃	—	2	1	no	—
134	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
135	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
136	C₂²×C₃²	—	2	1	no	—
137	C₂²×C₃²	—	2	1	no	—
138	C₂²×C₃²	C₃×SD₁₆	1	2	yes	—
139	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×C₃×D₃	—	4	1	no	—
141	C₂×C₃×D₃	—	4	1	no	—
142	C₂×C₃×D₃	—	4	1	no	—
143	C₃²×C₄	—	2	1	no	—
144	C₃²×C₄	C₃×#41	1	2	yes	—
145	C₃×Dic₃	—	4	1	no	—
146	C₃×Dic₃	—	4	2	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	4	1	no	—
152	C₂²×C₉	—	4	1	no	—
153	C₂²×C₉	—	12	1	no	—
154	C₂²×C₉	—	12	1	no	—
155	C₂²×C₉	—	12	1	no	—
156	C₂²×C₉	#204	1	2	yes	—
157	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	4	1	no	—
161	C₄×C₉	—	12	1	no	—
162	C₄×C₉	—	12	2	no	—
163	C₄×C₉	#237	1	2	yes	—
164	#121	—	3	1	no	—
165	C₂×C₄⋊C₄	—	3	1	no	—
166	#106	—	3	2	no	—
167	#102	—	3	2	no	—
168	C₂²⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
169	C₃×C₉	#481	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	—	6	1	no	—
172	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
173	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
174	C₂²×D₃	—	2	1	no	—
175	C₃×D₄	—	6	2	no	—
176	C₃×D₄	—	12	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	6	1	no	—
183	C₂×C₃×C₄	—	6	2	no	—
184	C₂×C₃×C₄	—	12	2	no	—
185	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
187	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
189	#67	—	4	2	no	—
190	C₂×Dic₃	—	2	2	no	—
191	C₂×Dic₃	—	2	1	no	—
192	C₂×Dic₃	—	4	2	no	—
193	D₁₂	—	2	2	no	—
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	4	2	no	—
197	C₂×C₃²	—	2	2	no	—
198	C₂×C₃²	C₃×#123	1	1	yes	characteristic
199	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
200	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₃×D₃	—	8	1	no	—
202	C₂×C₉	—	2	2	no	—
203	C₂×C₉	—	2	1	no	—
204	C₂×C₉	—	2	1	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	4	2	no	—
207	C₂×C₉	—	24	1	no	—
208	C₂×C₉	#886	1	1	yes	characteristic
209	C₂×C₉	C₂×#204	1	1	yes	characteristic
210	C₂×C₉	#908	1	1	yes	characteristic
211	C₂×D₄	—	3	1	no	—
212	C₂×D₄	—	6	1	no	—
213	C₂²×C₄	—	6	1	no	—
214	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
215	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
216	C₄⋊C₄	—	3	2	no	—
217	C₄⋊C₄	—	3	1	no	—
218	C₄⋊C₄	—	6	1	no	—
219	C₂²⋊C₄	—	6	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	4	1	no	—
227	C₂²×C₃	—	4	1	no	—
228	C₂²×C₃	—	12	1	no	—
229	C₂²×C₃	—	12	1	no	—
230	C₂²×C₃	—	12	1	no	—
231	C₂²×C₃	C₉×SD₁₆	1	2	yes	—
232	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
233	C₂²×C₃	C₃×#204	1	2	yes	—
234	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
235	C₂×D₃	—	4	1	no	—
236	C₂×D₃	—	4	1	no	—
237	C₂×D₃	—	4	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	4	1	no	—
242	C₃×C₄	—	12	1	no	—
243	C₃×C₄	—	12	2	no	—
244	C₃×C₄	C₉×#41	1	2	yes	—
245	C₃×C₄	C₃×#237	1	2	yes	—
246	Dic₃	—	4	1	no	—
247	Dic₃	—	4	2	no	—
248	C₃²	C₃×#481	1	1	yes	characteristic
249	C₉	—	2	1	no	—
250	C₉	#4735	1	1	yes	characteristic, direct product factor
251	C₂³	—	6	1	no	—
252	C₂³	C₉×D₁₂	1	1	yes	characteristic
253	D₄	—	6	2	no	—
254	D₄	—	12	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	6	1	no	—
257	C₂×C₄	—	6	2	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	4	2	no	—
268	C₂×C₃	—	24	1	no	—
269	C₂×C₃	C₉×#123	1	1	yes	characteristic
270	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
271	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂×C₃	C₃×#886	1	1	yes	characteristic
273	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
274	C₂×C₃	C₃×#908	1	1	yes	characteristic
275	D₃	—	8	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	12	1	no	—
279	C₂²	—	12	1	no	—
280	C₂²	—	12	1	no	—
281	C₂²	C₉×#204	1	2	yes	—
282	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	12	1	no	—
285	C₄	—	12	2	no	—
286	C₄	C₉×#237	1	2	yes	—
287	C₃	—	2	1	no	—
288	C₃	C₉×#481	1	1	yes	characteristic, complemented
289	C₃	C₃×#4735	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	24	1	no	—
292	C₂	C₉×#886	1	1	yes	characteristic
293	C₂	C₂×C₉×#204	1	1	yes	characteristic
294	C₂	C₉×#908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₂²⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#930	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₂²⋊C₈	D₃
hypocenter	C₃	C₉×#481
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₉×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊Dic₃	C₂	semidirect
3. C₃×C₉×C₂²⋊C₈	C₂	semidirect
4. C₉×#821	C₂	semidirect
5. C₉×#817	C₂	semidirect
27. C₉×C₂²⋊C₈	D₃	semidirect
40. #4735	C₉	direct (N × H)
75. C₂×C₄⋊Dic₃	C₂×C₉	semidirect
79. C₃×C₂²⋊C₈	C₂×C₉	semidirect
81. #821	C₂×C₉	semidirect
82. #817	C₂×C₉	semidirect
168. C₂²⋊C₈	D₃×C₉	semidirect
169. C₃×C₉	#481	semidirect
250. C₉	#4735	direct (N × H)
288. C₃	C₉×#481	semidirect

Maximal subgroups

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

subgroup	index
C₉×#930	2
C₂×C₉×C₄⋊Dic₃	2
C₃×C₉×C₂²⋊C₈	2
C₉×#821	2
C₉×#817	2
C₃×#4735	3
C₉×#481	3

Subgroups — C9×#4735 #13141595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4735 #13141595