Subgroups — C₉×#3122 #11767843

Order 1152.

627 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#482	C₂	1	2	yes	maximal, complemented
4	C₉×#478	C₂	1	2	yes	maximal, complemented
5	C₉×#459	C₂	1	2	yes	maximal, complemented
6	C₉×#447	C₂	1	2	yes	maximal, complemented
7	C₉×#442	C₂	1	2	yes	maximal, complemented
8	C₄×C₉×D₈	C₂	1	2	yes	maximal, complemented
9	C₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3122	C₃	1	1	yes	maximal, characteristic
11	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×D₈	C₂²	1	2	yes	—
13	C₂×C₉×D₈	C₂²	1	2	yes	—
14	C₉×#127	C₂²	1	1	yes	characteristic, complemented
15	C₉×#125	C₂²	1	1	yes	characteristic, complemented
16	C₉×#123	—	2	2	no	—
17	C₉×#121	—	2	2	no	—
18	C₉×#121	C₂²	1	2	yes	—
19	C₄×D₄×C₉	—	2	2	no	—
20	C₄×D₄×C₉	C₂²	1	2	yes	—
21	C₄×D₄×C₉	C₂²	1	2	yes	—
22	C₄×D₄×C₉	C₂²	1	2	yes	complemented
23	C₉×#117	C₂²	1	1	yes	characteristic
24	C₉×#117	C₂²	1	1	yes	characteristic
25	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
26	C₉×#107	C₂²	1	2	yes	complemented
27	C₉×#106	C₂²	1	2	yes	complemented
28	C₉×C₄⋊C₈	C₂²	1	2	yes	complemented
29	C₉×#102	C₂²	1	2	yes	—
30	C₉×#102	C₂²	1	2	yes	—
31	C₉×#102	C₂²	1	2	yes	complemented
32	C₉×#102	C₂²	1	2	yes	complemented
33	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₉×#97	C₂²	1	1	yes	characteristic, complemented
35	C₄×C₈×C₉	C₂²	1	1	yes	characteristic, complemented
36	C₃×#540	C₂×C₃	1	1	yes	characteristic
37	C₃×#518	C₂×C₃	1	1	yes	characteristic
38	C₃×#482	C₂×C₃	1	2	yes	—
39	C₃×#478	C₂×C₃	1	2	yes	—
40	C₃×#459	C₂×C₃	1	2	yes	—
41	C₃×#447	C₂×C₃	1	2	yes	—
42	C₃×#442	C₂×C₃	1	2	yes	—
43	C₃×C₄×D₈	C₂×C₃	1	2	yes	—
44	C₃×#433	C₂×C₃	1	1	yes	characteristic
45	C₂⁴×C₉	—	2	1	no	—
46	C₂×D₄×C₉	—	2	4	no	—
47	C₂×D₄×C₉	—	2	2	no	—
48	C₂×D₄×C₉	—	2	1	no	—
49	C₂×D₄×C₉	—	2	2	no	—
50	C₂×D₄×C₉	C₂³	1	2	yes	—
51	C₂×D₄×C₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	—	2	2	no	—
54	C₂²×C₄×C₉	—	2	2	no	—
55	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
56	C₉×D₈	—	2	4	no	—
57	C₉×D₈	—	2	4	no	—
58	C₂×C₈×C₉	C₂³	1	2	yes	—
59	C₂×C₈×C₉	C₂³	1	2	yes	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	D₄	1	2	yes	—
62	C₉×C₄⋊C₄	C₂³	1	2	yes	—
63	C₉×C₄⋊C₄	C₂³	1	2	yes	—
64	C₉×C₂²⋊C₄	—	2	1	no	—
65	C₉×C₂²⋊C₄	—	2	2	no	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₉×C₂²⋊C₄	—	2	2	no	—
69	C₉×C₂²⋊C₄	D₄	1	2	yes	—
70	C₄²×C₉	—	2	1	no	—
71	C₄²×C₉	C₂³	1	1	yes	characteristic
72	C₄²×C₉	C₂³	1	1	yes	characteristic
73	#3122	C₉	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
76	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
77	C₃×#127	C₂²×C₃	1	1	yes	characteristic
78	C₃×#125	C₂²×C₃	1	1	yes	characteristic
79	C₃×#123	—	2	2	no	—
80	C₃×#121	—	2	2	no	—
81	C₃×#121	C₂²×C₃	1	2	yes	—
82	C₃×C₄×D₄	—	2	2	no	—
83	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
84	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
85	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
86	C₃×#117	C₂²×C₃	1	1	yes	characteristic
87	C₃×#117	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
89	C₃×#107	C₂²×C₃	1	2	yes	—
90	C₃×#106	C₂²×C₃	1	2	yes	—
91	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
92	C₃×#102	C₂²×C₃	1	2	yes	—
93	C₃×#102	C₂²×C₃	1	2	yes	—
94	C₃×#102	C₂²×C₃	1	2	yes	—
95	C₃×#102	C₂²×C₃	1	2	yes	—
96	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	C₃×#97	C₂²×C₃	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₂³×C₉	—	2	4	no	—
100	C₂³×C₉	—	2	2	no	—
101	C₂³×C₉	—	2	2	no	—
102	C₂³×C₉	—	8	2	no	—
103	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
104	D₄×C₉	—	2	4	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	2	2	no	—
107	D₄×C₉	—	2	2	no	—
108	D₄×C₉	—	8	2	no	—
109	D₄×C₉	—	8	2	no	—
110	D₄×C₉	#41	1	4	yes	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	2	2	no	—
114	C₂×C₄×C₉	—	8	2	no	—
115	C₂×C₄×C₉	—	8	2	no	—
116	C₂×C₄×C₉	—	8	2	no	—
117	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
121	C₈×C₉	—	2	2	no	—
122	C₈×C₉	—	2	2	no	—
123	#540	C₂×C₉	1	1	yes	characteristic, complemented
124	#518	C₂×C₉	1	1	yes	characteristic, complemented
125	#482	C₂×C₉	1	2	yes	complemented
126	#478	C₂×C₉	1	2	yes	complemented
127	#459	C₂×C₉	1	2	yes	complemented
128	#447	C₂×C₉	1	2	yes	complemented
129	#442	C₂×C₉	1	2	yes	complemented
130	C₄×D₈	C₂×C₉	1	2	yes	complemented
131	#433	C₂×C₉	1	1	yes	characteristic, complemented
132	C₂⁴×C₃	—	2	1	no	—
133	C₂×C₃×D₄	—	2	4	no	—
134	C₂×C₃×D₄	—	2	2	no	—
135	C₂×C₃×D₄	—	2	1	no	—
136	C₂×C₃×D₄	—	2	2	no	—
137	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
138	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₃×C₄	—	2	2	no	—
140	C₂²×C₃×C₄	—	2	2	no	—
141	C₂²×C₃×C₄	—	2	2	no	—
142	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₃×D₈	—	2	4	no	—
144	C₃×D₈	—	2	4	no	—
145	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
146	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
149	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
150	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₃×C₂²⋊C₄	—	2	1	no	—
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₂²⋊C₄	—	2	2	no	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₂²⋊C₄	—	2	2	no	—
156	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
157	C₃×C₄²	—	2	1	no	—
158	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
159	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	2	4	no	—
162	C₂²×C₉	—	2	2	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	—	4	2	no	—
166	C₂²×C₉	—	4	2	no	—
167	C₂²×C₉	—	4	2	no	—
168	C₂²×C₉	—	8	4	no	—
169	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	2	2	no	—
173	C₄×C₉	—	4	2	no	—
174	C₄×C₉	—	4	2	no	—
175	C₄×C₉	C₂×#41	1	2	yes	—
176	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
177	C₂×D₈	C₂²×C₉	1	2	yes	—
178	C₂×D₈	C₂²×C₉	1	2	yes	—
179	#127	C₂²×C₉	1	1	yes	characteristic, complemented
180	#125	C₂²×C₉	1	1	yes	characteristic, complemented
181	#123	—	2	2	no	—
182	#121	—	2	2	no	—
183	#121	C₂²×C₉	1	2	yes	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	C₂²×C₉	1	2	yes	—
186	C₄×D₄	C₂²×C₉	1	2	yes	—
187	C₄×D₄	C₂²×C₉	1	2	yes	complemented
188	#117	C₂²×C₉	1	1	yes	characteristic
189	#117	C₂²×C₉	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	#107	C₂²×C₉	1	2	yes	complemented
192	#106	C₂²×C₉	1	2	yes	complemented
193	C₄⋊C₈	C₂²×C₉	1	2	yes	complemented
194	#102	C₂²×C₉	1	2	yes	—
195	#102	C₂²×C₉	1	2	yes	—
196	#102	C₂²×C₉	1	2	yes	complemented
197	#102	C₂²×C₉	1	2	yes	complemented
198	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
199	#97	C₂²×C₉	1	1	yes	characteristic, complemented
200	C₄×C₈	C₂²×C₉	1	1	yes	characteristic, complemented
201	C₂³×C₃	—	2	4	no	—
202	C₂³×C₃	—	2	2	no	—
203	C₂³×C₃	—	2	2	no	—
204	C₂³×C₃	—	8	2	no	—
205	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
206	C₃×D₄	—	2	4	no	—
207	C₃×D₄	—	2	2	no	—
208	C₃×D₄	—	2	2	no	—
209	C₃×D₄	—	2	2	no	—
210	C₃×D₄	—	8	2	no	—
211	C₃×D₄	—	8	2	no	—
212	C₃×D₄	C₃×#41	1	4	yes	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	8	2	no	—
217	C₂×C₃×C₄	—	8	2	no	—
218	C₂×C₃×C₄	—	8	2	no	—
219	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
221	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
222	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
223	C₃×C₈	—	2	2	no	—
224	C₃×C₈	—	2	2	no	—
225	C₂×C₉	—	4	1	no	—
226	C₂×C₉	—	4	4	no	—
227	C₂×C₉	—	8	2	no	—
228	C₂×C₉	#525	1	1	yes	characteristic
229	C₂×C₉	#576	1	2	yes	—
230	C₂⁴	—	2	1	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	C₂³×C₉	1	2	yes	—
236	C₂×D₄	C₂³×C₉	1	2	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
241	D₈	—	2	4	no	—
242	D₈	—	2	4	no	—
243	C₂×C₈	C₂³×C₉	1	2	yes	—
244	C₂×C₈	C₂³×C₉	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	D₄×C₉	1	2	yes	—
247	C₄⋊C₄	C₂³×C₉	1	2	yes	—
248	C₄⋊C₄	C₂³×C₉	1	2	yes	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	D₄×C₉	1	2	yes	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂³×C₉	1	1	yes	characteristic
257	C₄²	C₂³×C₉	1	1	yes	characteristic
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	4	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	8	4	no	—
267	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	C₂×C₃×#41	1	2	yes	—
274	C₉	#3122	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	—	2	4	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	8	2	no	—
279	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	8	2	no	—
285	D₄	—	8	2	no	—
286	D₄	C₉×#41	1	4	yes	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
295	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
296	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₂×C₃	—	4	1	no	—
300	C₂×C₃	—	4	4	no	—
301	C₂×C₃	—	8	2	no	—
302	C₂×C₃	C₃×#525	1	1	yes	characteristic
303	C₂×C₃	C₃×#576	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	C₂×C₉×#41	1	2	yes	—
320	C₃	C₃×#3122	1	1	yes	characteristic
321	C₂	—	4	1	no	—
322	C₂	—	4	4	no	—
323	C₂	—	8	2	no	—
324	C₂	C₉×#525	1	1	yes	characteristic
325	C₂	C₉×#576	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#3122	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#540	C₂	semidirect
2. C₉×#518	C₂	semidirect
3. C₉×#482	C₂	semidirect
4. C₉×#478	C₂	semidirect
5. C₉×#459	C₂	semidirect
6. C₉×#447	C₂	semidirect
7. C₉×#442	C₂	semidirect
8. C₄×C₉×D₈	C₂	semidirect
9. C₉×#433	C₂	semidirect
14. C₉×#127	C₂²	semidirect
15. C₉×#125	C₂²	semidirect
22. C₄×D₄×C₉	C₂²	semidirect
26. C₉×#107	C₂²	semidirect
27. C₉×#106	C₂²	semidirect
28. C₉×C₄⋊C₈	C₂²	semidirect
31. C₉×#102	C₂²	semidirect
32. C₉×#102	C₂²	semidirect
34. C₉×#97	C₂²	semidirect
35. C₄×C₈×C₉	C₂²	semidirect
73. #3122	C₉	direct (N × H)
123. #540	C₂×C₉	semidirect
124. #518	C₂×C₉	semidirect
125. #482	C₂×C₉	semidirect
126. #478	C₂×C₉	semidirect
127. #459	C₂×C₉	semidirect
128. #447	C₂×C₉	semidirect
129. #442	C₂×C₉	semidirect
130. C₄×D₈	C₂×C₉	semidirect
131. #433	C₂×C₉	semidirect
179. #127	C₂²×C₉	semidirect
180. #125	C₂²×C₉	semidirect
187. C₄×D₄	C₂²×C₉	semidirect
191. #107	C₂²×C₉	semidirect
192. #106	C₂²×C₉	semidirect
193. C₄⋊C₈	C₂²×C₉	semidirect
196. #102	C₂²×C₉	semidirect
197. #102	C₂²×C₉	semidirect
199. #97	C₂²×C₉	semidirect
200. C₄×C₈	C₂²×C₉	semidirect
274. C₉	#3122	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#540	2
C₉×#518	2
C₉×#482	2
C₉×#478	2
C₉×#459	2
C₉×#447	2
C₉×#442	2
C₄×C₉×D₈	2
C₉×#433	2
C₃×#3122	3

Subgroups — C9×#3122 #11767843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3122 #11767843