Subgroups — C₉×#4883 #13142203

Order 1728.

438 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#895	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#809	C₂	1	2	yes	maximal, complemented
6	C₃×#4883	C₃	1	1	yes	maximal, characteristic
7	C₉×#466	—	3	1	no	maximal
8	C₂×C₉×D₁₂	—	2	2	no	—
9	C₂×C₉×D₁₂	C₂²	1	2	yes	—
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₉×(C₂×D₃)⋊C₄	—	2	2	no	—
14	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₉×D₂₄	—	2	2	no	—
16	D₃×C₈×C₉	—	2	2	no	—
17	C₂×C₃×D₂₄	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×D₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×#895	C₂×C₃	1	2	yes	—
20	C₃²×#107	C₂×C₃	1	1	yes	characteristic
21	C₃×#809	C₂×C₃	1	2	yes	—
22	C₂×C₉×D₈	—	3	1	no	—
23	C₂²×C₈×C₉	—	3	1	no	—
24	C₉×#121	—	3	2	no	—
25	C₉×#107	—	2	1	no	—
26	C₉×#107	D₃	1	1	yes	characteristic, complemented
27	C₉×#102	—	3	2	no	—
28	C₂²×D₃×C₉	—	2	2	no	—
29	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
33	C₉×D₁₂	—	2	2	no	—
34	C₉×D₁₂	—	2	2	no	—
35	C₉×D₁₂	—	4	4	no	—
36	C₄×D₃×C₉	—	2	1	no	—
37	C₄×D₃×C₉	—	2	1	no	—
38	C₃×C₈×C₉	D₄	1	2	yes	complemented
39	C₉×C₃⋊C₈	—	2	1	no	—
40	C₃×#466	—	3	1	no	—
41	#4883	C₉	1	1	yes	characteristic, direct product factor
42	C₂×C₃×D₁₂	—	2	2	no	—
43	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
44	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
49	C₃×D₂₄	—	2	2	no	—
50	C₃×D₃×C₈	—	2	2	no	—
51	C₂×D₄×C₉	—	3	2	no	—
52	C₂×D₄×C₉	—	6	2	no	—
53	C₂²×C₄×C₉	—	3	1	no	—
54	C₉×D₈	—	6	2	no	—
55	C₂×C₈×C₉	—	2	1	no	—
56	C₂×C₈×C₉	—	3	1	no	—
57	C₂×C₈×C₉	—	6	2	no	—
58	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
61	C₉×C₂²⋊C₄	—	6	2	no	—
62	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×D₃×C₉	—	2	1	no	—
64	C₂×D₃×C₉	—	2	1	no	—
65	C₂×D₃×C₉	—	4	2	no	—
66	C₂×D₃×C₉	—	4	2	no	—
67	C₂×D₃×C₉	—	4	2	no	—
68	C₂×D₃×C₉	D₈	1	2	yes	complemented
69	C₃×C₄×C₉	—	4	2	no	—
70	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₉	#41	1	1	yes	characteristic
72	C₉×Dic₃	—	2	1	no	—
73	C₂×C₃×D₈	—	3	1	no	—
74	C₂²×C₃×C₈	—	3	1	no	—
75	C₃×#121	—	3	2	no	—
76	C₂×D₂₄	C₂×C₉	1	1	yes	characteristic, complemented
77	C₂×D₃×C₈	C₂×C₉	1	1	yes	characteristic, complemented
78	#895	C₂×C₉	1	2	yes	complemented
79	C₃×#107	—	2	1	no	—
80	C₃×#107	C₂×C₉	1	1	yes	characteristic, complemented
81	C₃×#107	C₃×D₃	1	1	yes	characteristic
82	C₃×#102	—	3	2	no	—
83	#809	C₂×C₉	1	2	yes	complemented
84	C₂²×C₃×D₃	—	2	2	no	—
85	C₂²×C₃×D₃	C₃×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
89	C₃×D₁₂	—	2	2	no	—
90	C₃×D₁₂	—	2	2	no	—
91	C₃×D₁₂	—	4	4	no	—
92	C₃×C₄×D₃	—	2	1	no	—
93	C₃×C₄×D₃	—	2	1	no	—
94	C₂³×C₉	—	3	1	no	—
95	C₂³×C₉	—	6	2	no	—
96	C₃²×C₈	C₃×D₄	1	2	yes	—
97	C₃×C₃⋊C₈	—	2	1	no	—
98	D₄×C₉	—	6	2	no	—
99	D₄×C₉	—	6	2	no	—
100	D₄×C₉	—	12	4	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	3	1	no	—
104	C₂×C₄×C₉	—	4	2	no	—
105	C₂×C₄×C₉	—	6	1	no	—
106	C₂×C₄×C₉	—	6	1	no	—
107	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
108	C₈×C₉	—	2	2	no	—
109	C₈×C₉	—	6	1	no	—
110	C₈×C₉	D₁₂	1	2	yes	complemented
111	#466	—	3	1	no	Sylow
112	C₂×C₃×C₉	#121	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
114	C₂×C₃×C₉	#135	1	1	yes	characteristic
115	D₃×C₉	—	2	2	no	—
116	D₃×C₉	—	8	2	no	—
117	C₂×C₃×D₄	—	3	2	no	—
118	C₂×C₃×D₄	—	6	2	no	—
119	C₂²×C₃×C₄	—	3	1	no	—
120	C₂×D₁₂	—	2	2	no	—
121	C₂×D₁₂	C₂²×C₉	1	2	yes	—
122	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
123	C₃×D₈	—	6	2	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	3	1	no	—
126	C₂×C₃×C₈	—	6	2	no	—
127	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
131	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
132	C₃×C₂²⋊C₄	—	6	2	no	—
133	(C₂×D₃)⋊C₄	—	2	2	no	—
134	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
135	D₂₄	—	2	2	no	—
136	D₃×C₈	—	2	2	no	—
137	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₃×D₃	—	2	1	no	—
139	C₂×C₃×D₃	—	2	1	no	—
140	C₂×C₃×D₃	—	4	2	no	—
141	C₂×C₃×D₃	—	4	2	no	—
142	C₂×C₃×D₃	—	4	2	no	—
143	C₂×C₃×D₃	C₃×D₈	1	2	yes	—
144	C₃²×C₄	—	4	2	no	—
145	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
146	C₃²×C₄	C₃×#41	1	1	yes	characteristic
147	C₃×Dic₃	—	2	1	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	—	3	2	no	—
150	C₂²×C₉	—	6	1	no	—
151	C₂²×C₉	—	6	1	no	—
152	C₂²×C₉	—	12	2	no	—
153	C₂²×C₉	—	12	2	no	—
154	C₂²×C₉	—	12	2	no	—
155	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
156	C₄×C₉	—	2	1	no	—
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	—	4	2	no	—
159	C₄×C₉	—	6	1	no	—
160	C₄×C₉	—	8	2	no	—
161	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
162	C₄×C₉	#239	1	1	yes	characteristic
163	C₂×D₈	—	3	1	no	—
164	C₂²×C₈	—	3	1	no	—
165	#121	—	3	2	no	—
166	#107	D₃×C₉	1	1	yes	characteristic, complemented
167	#102	—	3	2	no	—
168	C₃×C₉	#466	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₃	—	3	1	no	—
170	C₂³×C₃	—	6	2	no	—
171	C₂²×D₃	—	2	2	no	—
172	C₂²×D₃	D₄×C₉	1	1	yes	characteristic
173	C₃×D₄	—	6	2	no	—
174	C₃×D₄	—	6	2	no	—
175	C₃×D₄	—	12	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	3	1	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	6	1	no	—
182	C₂×C₃×C₄	—	6	1	no	—
183	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
185	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
186	D₁₂	—	2	2	no	—
187	D₁₂	—	2	2	no	—
188	D₁₂	—	4	4	no	—
189	C₄×D₃	—	2	1	no	—
190	C₄×D₃	—	2	1	no	—
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	—	6	1	no	—
193	C₃×C₈	D₄×C₉	1	2	yes	complemented
194	C₃×C₈	C₃×D₁₂	1	2	yes	—
195	C₃⋊C₈	—	2	1	no	—
196	C₂×C₃²	C₃×#121	1	1	yes	characteristic
197	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
198	C₂×C₃²	C₃×#135	1	1	yes	characteristic
199	C₃×D₃	—	2	2	no	—
200	C₃×D₃	—	8	2	no	—
201	C₂×C₉	—	2	1	no	—
202	C₂×C₉	—	2	1	no	—
203	C₂×C₉	—	2	1	no	—
204	C₂×C₉	—	6	2	no	—
205	C₂×C₉	—	24	2	no	—
206	C₂×C₉	#895	1	1	yes	characteristic
207	C₂×C₉	D₃×D₈	1	1	yes	characteristic
208	C₂×C₉	#919	1	1	yes	characteristic
209	C₂×D₄	—	3	2	no	—
210	C₂×D₄	—	6	2	no	—
211	C₂²×C₄	—	3	1	no	—
212	D₈	—	6	2	no	—
213	C₂×C₈	—	3	1	no	—
214	C₂×C₈	—	6	2	no	—
215	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
216	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
217	C₂²⋊C₄	—	6	2	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	3	2	no	—
220	C₂²×C₃	—	6	1	no	—
221	C₂²×C₃	—	6	1	no	—
222	C₂²×C₃	—	12	2	no	—
223	C₂²×C₃	—	12	2	no	—
224	C₂²×C₃	—	12	2	no	—
225	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
226	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
227	C₂×D₃	—	2	1	no	—
228	C₂×D₃	—	2	1	no	—
229	C₂×D₃	—	4	2	no	—
230	C₂×D₃	—	4	2	no	—
231	C₂×D₃	—	4	2	no	—
232	C₂×D₃	C₉×D₈	1	2	yes	complemented
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	4	2	no	—
236	C₃×C₄	—	4	2	no	—
237	C₃×C₄	—	6	1	no	—
238	C₃×C₄	—	8	2	no	—
239	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
240	C₃×C₄	C₉×#41	1	1	yes	characteristic
241	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
242	C₃×C₄	C₃×#239	1	1	yes	characteristic
243	Dic₃	—	2	1	no	—
244	C₃²	C₃×#466	1	1	yes	characteristic
245	C₉	—	2	1	no	—
246	C₉	#4883	1	1	yes	characteristic, direct product factor
247	C₂³	—	3	1	no	—
248	C₂³	—	6	2	no	—
249	D₄	—	6	2	no	—
250	D₄	—	6	2	no	—
251	D₄	—	12	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	6	1	no	—
255	C₂×C₄	—	6	1	no	—
256	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
257	C₈	—	6	1	no	—
258	C₈	C₉×D₁₂	1	2	yes	complemented
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	6	2	no	—
263	C₂×C₃	—	24	2	no	—
264	C₂×C₃	C₉×#121	1	1	yes	characteristic
265	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
266	C₂×C₃	C₉×#135	1	1	yes	characteristic
267	C₂×C₃	C₃×#895	1	1	yes	characteristic
268	C₂×C₃	C₃×D₃×D₈	1	1	yes	characteristic
269	C₂×C₃	C₃×#919	1	1	yes	characteristic
270	D₃	—	2	2	no	—
271	D₃	—	8	2	no	—
272	C₂²	—	3	2	no	—
273	C₂²	—	6	1	no	—
274	C₂²	—	6	1	no	—
275	C₂²	—	12	2	no	—
276	C₂²	—	12	2	no	—
277	C₂²	—	12	2	no	—
278	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	6	1	no	—
281	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
282	C₄	C₉×#239	1	1	yes	characteristic
283	C₃	—	2	1	no	—
284	C₃	C₉×#466	1	1	yes	characteristic, complemented
285	C₃	C₃×#4883	1	1	yes	characteristic
286	C₂	—	6	2	no	—
287	C₂	—	24	2	no	—
288	C₂	C₉×#895	1	1	yes	characteristic
289	C₂	D₃×C₉×D₈	1	1	yes	characteristic
290	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	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#4883	C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#107	D₃
hypocenter	C₃	C₉×#466
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
1. C₂×C₉×D₂₄	C₂	semidirect
2. C₂×D₃×C₈×C₉	C₂	semidirect
3. C₉×#895	C₂	semidirect
4. C₃×C₉×#107	C₂	semidirect
5. C₉×#809	C₂	semidirect
26. C₉×#107	D₃	semidirect
38. C₃×C₈×C₉	D₄	semidirect
41. #4883	C₉	direct (N × H)
68. C₂×D₃×C₉	D₈	semidirect
76. C₂×D₂₄	C₂×C₉	semidirect
77. C₂×D₃×C₈	C₂×C₉	semidirect
78. #895	C₂×C₉	semidirect
80. C₃×#107	C₂×C₉	semidirect
83. #809	C₂×C₉	semidirect
110. C₈×C₉	D₁₂	semidirect
166. #107	D₃×C₉	semidirect
168. C₃×C₉	#466	semidirect
193. C₃×C₈	D₄×C₉	semidirect
232. C₂×D₃	C₉×D₈	semidirect
246. C₉	#4883	direct (N × H)
258. C₈	C₉×D₁₂	semidirect
284. C₃	C₉×#466	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×D₂₄	2
C₂×D₃×C₈×C₉	2
C₉×#895	2
C₃×C₉×#107	2
C₉×#809	2
C₃×#4883	3
C₉×#466	3

Subgroups — C9×#4883 #13142203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4883 #13142203