Subgroups — C₉×#4847 #13142039

Order 1728.

344 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	1	yes	maximal, characteristic
5	C₉×#821	C₂	1	1	yes	maximal, characteristic
6	C₉×#818	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#817	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4847	C₃	1	1	yes	maximal, characteristic
9	C₉×#478	—	3	1	no	maximal
10	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃×C₉	—	2	1	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₉×Dic₃⋊C₄	—	2	1	no	—
20	C₉×Dic₃⋊C₄	—	2	1	no	—
21	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
22	C₃×#870	C₂×C₃	1	1	yes	characteristic
23	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#821	C₂×C₃	1	1	yes	characteristic
25	C₃×#821	C₂×C₃	1	1	yes	characteristic
26	C₃×#818	C₂×C₃	1	1	yes	characteristic
27	C₃×#817	C₂×C₃	1	1	yes	characteristic
28	C₉×#125	—	3	1	no	—
29	C₄×D₄×C₉	—	3	1	no	—
30	C₉×#107	—	3	1	no	—
31	C₉×#106	—	3	1	no	—
32	C₉×C₄⋊C₈	—	2	1	no	—
33	C₉×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
34	C₉×#102	—	3	1	no	—
35	C₉×#102	—	3	1	no	—
36	C₂²×D₃×C₉	—	2	1	no	—
37	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₉×Dic₃	—	2	1	no	—
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₂×C₉×Dic₃	—	2	1	no	—
43	C₉×D₁₂	—	2	1	no	—
44	C₉×D₁₂	Q₈	1	2	yes	—
45	C₄×D₃×C₉	—	4	2	no	—
46	C₃×C₈×C₉	—	2	1	no	—
47	C₃×C₈×C₉	—	2	1	no	—
48	C₃×#478	—	3	1	no	—
49	#4847	C₉	1	1	yes	characteristic, direct product factor
50	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄×D₃	—	2	1	no	—
52	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
56	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
58	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
59	C₃×Dic₃⋊C₄	—	2	1	no	—
60	C₃×Dic₃⋊C₄	—	2	1	no	—
61	C₂×D₄×C₉	—	3	1	no	—
62	C₂²×C₄×C₉	—	6	1	no	—
63	C₂×C₈×C₉	—	2	1	no	—
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
66	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
67	C₉×C₄⋊C₄	—	3	1	no	—
68	C₉×C₄⋊C₄	—	3	1	no	—
69	C₉×C₄⋊C₄	—	3	1	no	—
70	C₉×C₄⋊C₄	—	6	1	no	—
71	C₉×C₄⋊C₄	—	6	1	no	—
72	C₉×C₂²⋊C₄	—	6	1	no	—
73	C₄²×C₉	—	2	1	no	—
74	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
75	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
76	C₂×D₃×C₉	—	2	2	no	—
77	C₂×D₃×C₉	—	4	1	no	—
78	C₂×D₃×C₉	—	4	1	no	—
79	C₃×C₄×C₉	—	2	1	no	—
80	C₃×C₄×C₉	—	2	1	no	—
81	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
82	C₃×C₄×C₉	#41	1	1	yes	characteristic
83	C₉×Dic₃	—	4	1	no	—
84	C₉×Dic₃	—	4	1	no	—
85	C₉×Dic₃	—	4	1	no	—
86	C₃×#125	—	3	1	no	—
87	C₃×C₄×D₄	—	3	1	no	—
88	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic
89	#870	C₂×C₉	1	1	yes	characteristic, complemented
90	C₃×#107	—	3	1	no	—
91	C₃×#106	—	3	1	no	—
92	C₃×C₄⋊C₈	—	2	1	no	—
93	C₃×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
94	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
95	C₃×#102	—	3	1	no	—
96	C₃×#102	—	3	1	no	—
97	#821	C₂×C₉	1	1	yes	characteristic
98	#821	C₂×C₉	1	1	yes	characteristic
99	#818	C₂×C₉	1	1	yes	characteristic, complemented
100	#817	C₂×C₉	1	1	yes	characteristic, complemented
101	C₂²×C₃×D₃	—	2	1	no	—
102	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₃×Dic₃	—	2	1	no	—
106	C₂×C₃×Dic₃	—	2	1	no	—
107	C₂×C₃×Dic₃	—	2	1	no	—
108	C₃×D₁₂	—	2	1	no	—
109	C₃×D₁₂	C₃×Q₈	1	2	yes	—
110	C₃×C₄×D₃	—	4	2	no	—
111	C₂³×C₉	—	6	1	no	—
112	C₃²×C₈	—	2	1	no	—
113	C₃²×C₈	—	2	1	no	—
114	D₄×C₉	—	3	2	no	—
115	D₄×C₉	—	6	1	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	6	1	no	—
120	C₂×C₄×C₉	—	6	1	no	—
121	C₂×C₄×C₉	—	6	1	no	—
122	C₂×C₄×C₉	—	12	2	no	—
123	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
124	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
125	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
126	C₈×C₉	—	2	1	no	—
127	C₈×C₉	—	2	1	no	—
128	C₈×C₉	—	4	1	no	—
129	C₈×C₉	—	4	1	no	—
130	#478	—	3	1	no	Sylow
131	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₃×C₉	#135	1	1	yes	characteristic
133	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	D₃×C₉	—	4	2	no	—
135	C₂×C₃×D₄	—	3	1	no	—
136	C₂²×C₃×C₄	—	6	1	no	—
137	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
138	C₂×C₄×D₃	—	2	1	no	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	2	1	no	—
141	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
145	C₃×C₄⋊C₄	—	3	1	no	—
146	C₃×C₄⋊C₄	—	3	1	no	—
147	C₃×C₄⋊C₄	—	3	1	no	—
148	C₃×C₄⋊C₄	—	6	1	no	—
149	C₃×C₄⋊C₄	—	6	1	no	—
150	C₃×C₂²⋊C₄	—	6	1	no	—
151	C₃×C₄²	—	2	1	no	—
152	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
153	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
154	(C₂×D₃)⋊C₄	—	2	1	no	—
155	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
156	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
157	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
158	Dic₃⋊C₄	—	2	1	no	—
159	Dic₃⋊C₄	—	2	1	no	—
160	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂×C₃×D₃	—	2	2	no	—
162	C₂×C₃×D₃	—	4	1	no	—
163	C₂×C₃×D₃	—	4	1	no	—
164	C₃²×C₄	—	2	1	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
167	C₃²×C₄	C₃×#41	1	1	yes	characteristic
168	C₃×Dic₃	—	4	1	no	—
169	C₃×Dic₃	—	4	1	no	—
170	C₃×Dic₃	—	4	1	no	—
171	C₂²×C₉	—	2	1	no	—
172	C₂²×C₉	—	6	2	no	—
173	C₂²×C₉	—	12	1	no	—
174	C₂²×C₉	—	12	1	no	—
175	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
176	C₄×C₉	—	2	2	no	—
177	C₄×C₉	—	2	1	no	—
178	C₄×C₉	—	2	1	no	—
179	C₄×C₉	—	2	1	no	—
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	4	1	no	—
182	C₄×C₉	—	12	1	no	—
183	C₄×C₉	—	12	1	no	—
184	C₄×C₉	—	12	1	no	—
185	C₄×C₉	#237	1	1	yes	characteristic
186	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
187	#125	—	3	1	no	—
188	C₄×D₄	—	3	1	no	—
189	#107	—	3	1	no	—
190	#106	—	3	1	no	—
191	C₄⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
192	#102	—	3	1	no	—
193	#102	—	3	1	no	—
194	C₃×C₉	#478	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₃	—	6	1	no	—
196	C₂²×D₃	—	2	1	no	—
197	C₃×D₄	—	3	2	no	—
198	C₃×D₄	—	6	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	6	1	no	—
203	C₂×C₃×C₄	—	6	1	no	—
204	C₂×C₃×C₄	—	6	1	no	—
205	C₂×C₃×C₄	—	12	2	no	—
206	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
207	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
209	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
210	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	C₂×Dic₃	—	2	1	no	—
213	C₂×Dic₃	—	2	1	no	—
214	C₂×Dic₃	—	2	1	no	—
215	D₁₂	—	2	1	no	—
216	D₁₂	Q₈×C₉	1	2	yes	—
217	C₄×D₃	—	4	2	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	2	1	no	—
220	C₃×C₈	—	2	1	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	4	1	no	—
223	C₃×C₈	—	4	1	no	—
224	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
225	C₂×C₃²	C₃×#135	1	1	yes	characteristic
226	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₃×D₃	—	4	2	no	—
228	C₂×C₉	—	2	1	no	—
229	C₂×C₉	—	2	1	no	—
230	C₂×C₉	—	2	1	no	—
231	C₂×C₉	—	12	2	no	—
232	C₂×C₉	#897	1	1	yes	characteristic
233	C₂×C₉	#904	1	1	yes	characteristic
234	C₂×C₉	#908	1	1	yes	characteristic
235	C₂×D₄	—	3	1	no	—
236	C₂²×C₄	—	6	1	no	—
237	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
239	C₄⋊C₄	—	3	1	no	—
240	C₄⋊C₄	—	3	1	no	—
241	C₄⋊C₄	—	3	1	no	—
242	C₄⋊C₄	—	6	1	no	—
243	C₄⋊C₄	—	6	1	no	—
244	C₂²⋊C₄	—	6	1	no	—
245	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	6	2	no	—
248	C₂²×C₃	—	12	1	no	—
249	C₂²×C₃	—	12	1	no	—
250	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
252	C₂×D₃	—	2	2	no	—
253	C₂×D₃	—	4	1	no	—
254	C₂×D₃	—	4	1	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	12	1	no	—
264	C₃×C₄	—	12	1	no	—
265	C₃×C₄	—	12	1	no	—
266	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
267	C₃×C₄	C₉×#41	1	1	yes	characteristic
268	C₃×C₄	C₃×#237	1	1	yes	characteristic
269	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
270	Dic₃	—	4	1	no	—
271	Dic₃	—	4	1	no	—
272	Dic₃	—	4	1	no	—
273	C₃²	C₃×#478	1	1	yes	characteristic
274	C₉	—	2	1	no	—
275	C₉	#4847	1	1	yes	characteristic, direct product factor
276	C₂³	—	6	1	no	—
277	D₄	—	3	2	no	—
278	D₄	—	6	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	12	2	no	—
283	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
284	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	12	2	no	—
292	C₂×C₃	C₉×C₂²⋊Q₈	1	1	yes	characteristic
293	C₂×C₃	C₉×#135	1	1	yes	characteristic
294	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₂×C₃	C₃×#897	1	1	yes	characteristic
296	C₂×C₃	C₃×#904	1	1	yes	characteristic
297	C₂×C₃	C₃×#908	1	1	yes	characteristic
298	D₃	—	4	2	no	—
299	C₂²	—	6	2	no	—
300	C₂²	—	12	1	no	—
301	C₂²	—	12	1	no	—
302	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	12	1	no	—
306	C₄	—	12	1	no	—
307	C₄	—	12	1	no	—
308	C₄	C₉×#237	1	1	yes	characteristic
309	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
310	C₃	—	2	1	no	—
311	C₃	C₉×#478	1	1	yes	characteristic, complemented
312	C₃	C₃×#4847	1	1	yes	characteristic
313	C₂	—	12	2	no	—
314	C₂	C₉×#897	1	1	yes	characteristic
315	C₂	C₉×#904	1	1	yes	characteristic
316	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×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₁₂	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₄⋊C₈	D₃
hypocenter	C₃	C₉×#478
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₉×#870	C₂	semidirect
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
6. C₉×#818	C₂	semidirect
7. C₉×#817	C₂	semidirect
33. C₉×C₄⋊C₈	D₃	semidirect
49. #4847	C₉	direct (N × H)
89. #870	C₂×C₉	semidirect
93. C₃×C₄⋊C₈	C₂×C₉	semidirect
99. #818	C₂×C₉	semidirect
100. #817	C₂×C₉	semidirect
191. C₄⋊C₈	D₃×C₉	semidirect
194. C₃×C₉	#478	semidirect
275. C₉	#4847	direct (N × H)
311. C₃	C₉×#478	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×D₁₂	2
C₉×#870	2
C₃×C₉×C₄⋊C₈	2
C₉×#821	2
C₉×#818	2
C₉×#817	2
C₃×#4847	3
C₉×#478	3

Subgroups — C9×#4847 #13142039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4847 #13142039