Subgroups — C₉×#5116 #13143109

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#924	C₂	1	2	yes	maximal
3	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#818	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#816	C₂	1	2	yes	maximal, complemented
6	C₃×#5116	C₃	1	1	yes	maximal, characteristic
7	C₉×#467	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	—	2	2	no	—
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₉×#217	—	2	2	no	—
14	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₉×Dic₃⋊C₄	—	2	2	no	—
16	C₉×Dic₁₂	—	2	2	no	—
17	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×#924	C₂×C₃	1	2	yes	—
19	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
20	C₃×#818	C₂×C₃	1	1	yes	characteristic
21	C₃×#816	C₂×C₃	1	2	yes	—
22	C₂×C₉×Q₁₆	—	3	1	no	—
23	C₂²×C₈×C₉	—	2	1	no	—
24	C₂²×C₈×C₉	D₃	1	1	yes	characteristic
25	C₉×C₂²⋊Q₈	—	3	2	no	—
26	C₉×#107	—	3	1	no	—
27	C₉×#103	—	3	2	no	—
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₉×Dic₆	—	2	2	no	—
36	C₉×Dic₆	—	2	2	no	—
37	C₃×C₈×C₉	—	2	1	no	—
38	C₃×C₈×C₉	D₄	1	2	yes	—
39	C₃×#467	—	3	1	no	—
40	#5116	C₉	1	1	yes	characteristic, direct product factor
41	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃×#217	—	2	2	no	—
47	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
48	C₃×Dic₃⋊C₄	—	2	2	no	—
49	C₃×Dic₁₂	—	2	2	no	—
50	C₂×Q₈×C₉	—	3	2	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
53	C₉×Q₁₆	—	6	2	no	—
54	C₂×C₈×C₉	—	2	4	no	—
55	C₂×C₈×C₉	—	2	1	no	—
56	C₂×C₈×C₉	—	2	1	no	—
57	C₂×C₈×C₉	—	2	2	no	—
58	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
59	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
60	C₉×C₄⋊C₄	—	3	2	no	—
61	C₉×C₄⋊C₄	—	6	2	no	—
62	C₉×C₂²⋊C₄	—	6	2	no	—
63	C₂²×C₃×C₉	—	2	1	no	—
64	C₂²×C₃×C₉	—	2	1	no	—
65	C₂²×C₃×C₉	Q₁₆	1	2	yes	complemented
66	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
67	C₃×C₄×C₉	—	2	1	no	—
68	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₃×C₄×C₉	#41	1	1	yes	characteristic
70	C₉×Dic₃	—	4	2	no	—
71	C₉×Dic₃	—	4	2	no	—
72	C₂×C₃×Q₁₆	—	3	1	no	—
73	C₂²×C₃×C₈	—	2	1	no	—
74	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
75	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
76	C₃×C₂²⋊Q₈	—	3	2	no	—
77	#924	C₂×C₉	1	2	yes	—
78	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
79	C₃×#107	—	3	1	no	—
80	C₃×#103	—	3	2	no	—
81	#818	C₂×C₉	1	1	yes	characteristic, complemented
82	#816	C₂×C₉	1	2	yes	complemented
83	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	—	2	1	no	—
85	C₂×C₃²×C₄	—	2	1	no	—
86	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₃×Dic₆	—	2	2	no	—
91	C₃×Dic₆	—	2	2	no	—
92	C₂³×C₉	—	2	1	no	—
93	C₂³×C₉	D₁₂	1	1	yes	characteristic
94	C₃²×C₈	—	2	1	no	—
95	C₃²×C₈	C₃×D₄	1	2	yes	—
96	Q₈×C₉	—	6	2	no	—
97	Q₈×C₉	—	6	2	no	—
98	C₂×C₄×C₉	—	2	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	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	6	2	no	—
105	C₂×C₄×C₉	—	6	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	—	2	1	no	—
110	C₈×C₉	—	2	2	no	—
111	C₈×C₉	#67	1	2	yes	—
112	#467	—	3	1	no	Sylow
113	C₂×C₃×C₉	—	2	2	no	—
114	C₂×C₃×C₉	#121	1	1	yes	characteristic
115	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₃×C₉	#135	1	1	yes	characteristic
117	C₂×C₃×Q₈	—	3	2	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
121	C₂×Dic₆	C₂²×C₉	1	2	yes	—
122	C₃×Q₁₆	—	6	2	no	—
123	C₂×C₃×C₈	—	2	4	no	—
124	C₂×C₃×C₈	—	2	2	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
129	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
132	C₃×C₄⋊C₄	—	3	2	no	—
133	C₃×C₄⋊C₄	—	6	2	no	—
134	C₃×C₂²⋊C₄	—	6	2	no	—
135	#217	—	2	2	no	—
136	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
137	Dic₃⋊C₄	—	2	2	no	—
138	Dic₁₂	—	2	2	no	—
139	C₂²×C₃²	—	2	1	no	—
140	C₂²×C₃²	—	2	1	no	—
141	C₂²×C₃²	C₃×Q₁₆	1	2	yes	—
142	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
143	C₃²×C₄	—	2	1	no	—
144	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
145	C₃²×C₄	C₃×#41	1	1	yes	characteristic
146	C₃×Dic₃	—	4	2	no	—
147	C₃×Dic₃	—	4	2	no	—
148	C₂²×C₉	—	2	2	no	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	2	1	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	1	no	—
154	C₂²×C₉	Dic₁₂	1	2	yes	complemented
155	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
156	C₄×C₉	—	2	2	no	—
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	12	2	no	—
161	C₄×C₉	—	12	2	no	—
162	C₄×C₉	#235	1	1	yes	characteristic
163	C₄×C₉	C₂×#67	1	1	yes	characteristic
164	C₂×Q₁₆	—	3	1	no	—
165	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
166	C₂²⋊Q₈	—	3	2	no	—
167	#107	—	3	1	no	—
168	#103	—	3	2	no	—
169	C₃×C₉	#467	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
172	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
173	C₃×Q₈	—	6	2	no	—
174	C₃×Q₈	—	6	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	6	2	no	—
184	C₂×C₃×C₄	—	6	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	C₂×Dic₃	—	2	2	no	—
190	C₂×Dic₃	—	2	2	no	—
191	Dic₆	—	2	2	no	—
192	Dic₆	—	2	2	no	—
193	C₃×C₈	—	2	2	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	D₄×C₉	1	2	yes	—
198	C₃×C₈	C₃×#67	1	2	yes	—
199	C₂×C₃²	—	2	2	no	—
200	C₂×C₃²	C₃×#121	1	1	yes	characteristic
201	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
202	C₂×C₃²	C₃×#135	1	1	yes	characteristic
203	C₂×C₉	—	2	4	no	—
204	C₂×C₉	—	2	2	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	#904	1	1	yes	characteristic
209	C₂×C₉	C₂×Dic₁₂	1	1	yes	characteristic
210	C₂×C₉	#930	1	1	yes	characteristic
211	C₂×Q₈	—	3	2	no	—
212	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
213	Q₁₆	—	6	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
216	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	—	6	2	no	—
219	C₂²⋊C₄	—	6	2	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	C₉×Q₁₆	1	2	yes	complemented
229	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
230	C₂²×C₃	C₃×Dic₁₂	1	2	yes	—
231	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	12	2	no	—
238	C₃×C₄	—	12	2	no	—
239	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
240	C₃×C₄	C₉×#41	1	1	yes	characteristic
241	C₃×C₄	C₃×#235	1	1	yes	characteristic
242	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
243	Dic₃	—	4	2	no	—
244	Dic₃	—	4	2	no	—
245	C₃²	C₃×#467	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	#5116	1	1	yes	characteristic, direct product factor
248	C₂³	C₉×D₁₂	1	1	yes	characteristic
249	Q₈	—	6	2	no	—
250	Q₈	—	6	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	C₉×#67	1	2	yes	—
259	C₂×C₃	—	2	4	no	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	C₉×#121	1	1	yes	characteristic
266	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
267	C₂×C₃	C₉×#135	1	1	yes	characteristic
268	C₂×C₃	C₃×#904	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×Dic₁₂	1	1	yes	characteristic
270	C₂×C₃	C₃×#930	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₉×Dic₁₂	1	2	yes	complemented
274	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	12	2	no	—
277	C₄	—	12	2	no	—
278	C₄	C₉×#235	1	1	yes	characteristic
279	C₄	C₂×C₉×#67	1	1	yes	characteristic
280	C₃	—	2	1	no	—
281	C₃	C₉×#467	1	1	yes	characteristic, complemented
282	C₃	C₃×#5116	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	C₉×#904	1	1	yes	characteristic
285	C₂	C₂×C₉×Dic₁₂	1	1	yes	characteristic
286	C₂	C₉×#930	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	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂³×C₃²	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×C₈×C₉	D₃
hypocenter	C₃	C₉×#467
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
3. C₂×C₉×Dic₁₂	C₂	semidirect
4. C₉×#818	C₂	semidirect
5. C₉×#816	C₂	semidirect
40. #5116	C₉	direct (N × H)
65. C₂²×C₃×C₉	Q₁₆	semidirect
78. C₂×Dic₁₂	C₂×C₉	semidirect
81. #818	C₂×C₉	semidirect
82. #816	C₂×C₉	semidirect
154. C₂²×C₉	Dic₁₂	semidirect
169. C₃×C₉	#467	semidirect
228. C₂²×C₃	C₉×Q₁₆	semidirect
247. C₉	#5116	direct (N × H)
273. C₂²	C₉×Dic₁₂	semidirect
281. C₃	C₉×#467	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₉×#924	2
C₂×C₉×Dic₁₂	2
C₉×#818	2
C₉×#816	2
C₃×#5116	3
C₉×#467	3

Subgroups — C9×#5116 #13143109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5116 #13143109