Subgroups — C₉×#3262 #11767983

Order 1152.

519 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	C₉×#542	C₂	1	2	yes	maximal, complemented
2	C₉×#477	C₂	1	2	yes	maximal, complemented
3	C₉×#474	C₂	1	2	yes	maximal, complemented
4	C₉×#468	C₂	1	2	yes	maximal, complemented
5	C₉×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#466	C₂	1	1	yes	maximal, characteristic
7	C₉×#464	C₂	1	2	yes	maximal, complemented
8	C₉×#463	C₂	1	2	yes	maximal, complemented
9	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3262	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₉×#126	—	2	2	no	—
18	C₉×#124	—	2	2	no	—
19	C₉×#124	C₂²	1	2	yes	—
20	C₉×#123	—	2	2	no	—
21	C₉×C₂²⋊Q₈	—	2	2	no	—
22	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₉×#121	C₂²	1	2	yes	—
24	C₄×Q₈×C₉	C₂²	1	2	yes	—
25	C₄×D₄×C₉	C₂²	1	2	yes	—
26	C₉×#117	C₂²	1	1	yes	characteristic
27	C₉×#107	C₂²	1	1	yes	characteristic
28	C₉×#107	C₂²	1	1	yes	characteristic
29	C₉×#107	C₂²	1	2	yes	—
30	C₉×C₄⋊C₈	C₂²	1	2	yes	—
31	C₉×C₄⋊C₈	C₂²	1	2	yes	—
32	C₉×#103	C₂²	1	2	yes	—
33	C₉×#103	C₂²	1	2	yes	—
34	C₉×#102	C₂²	1	2	yes	—
35	C₉×#102	C₂²	1	2	yes	—
36	C₃×#542	C₂×C₃	1	2	yes	—
37	C₃×#477	C₂×C₃	1	2	yes	—
38	C₃×#474	C₂×C₃	1	2	yes	—
39	C₃×#468	C₂×C₃	1	2	yes	—
40	C₃×#467	C₂×C₃	1	1	yes	characteristic
41	C₃×#466	C₂×C₃	1	1	yes	characteristic
42	C₃×#464	C₂×C₃	1	2	yes	—
43	C₃×#463	C₂×C₃	1	2	yes	—
44	C₃×#424	C₂×C₃	1	1	yes	characteristic
45	C₂×Q₈×C₉	—	2	2	no	—
46	C₂×Q₈×C₉	C₂³	1	2	yes	—
47	C₂×D₄×C₉	—	2	2	no	—
48	C₂×D₄×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×Q₁₆	—	4	1	no	—
52	C₉×SD₁₆	—	4	2	no	—
53	C₉×D₈	—	4	1	no	—
54	C₉×M₁₆	—	4	1	no	—
55	C₂×C₈×C₉	—	4	1	no	—
56	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₉	C₂³	1	2	yes	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₉×C₄⋊C₄	C₂³	1	2	yes	—
65	C₉×C₄⋊C₄	C₂³	1	2	yes	—
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	—	2	2	no	—
70	C₉×C₂²⋊C₄	—	2	2	no	—
71	C₉×C₂²⋊C₄	D₄	1	2	yes	—
72	C₄²×C₉	—	2	2	no	—
73	C₄²×C₉	C₂³	1	2	yes	—
74	#3262	C₉	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
77	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
80	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
81	C₃×#126	—	2	2	no	—
82	C₃×#124	—	2	2	no	—
83	C₃×#124	C₂²×C₃	1	2	yes	—
84	C₃×#123	—	2	2	no	—
85	C₃×C₂²⋊Q₈	—	2	2	no	—
86	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
87	C₃×#121	C₂²×C₃	1	2	yes	—
88	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
89	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
90	C₃×#117	C₂²×C₃	1	1	yes	characteristic
91	C₃×#107	C₂²×C₃	1	1	yes	characteristic
92	C₃×#107	C₂²×C₃	1	1	yes	characteristic
93	C₃×#107	C₂²×C₃	1	2	yes	—
94	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
95	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
96	C₃×#103	C₂²×C₃	1	2	yes	—
97	C₃×#103	C₂²×C₃	1	2	yes	—
98	C₃×#102	C₂²×C₃	1	2	yes	—
99	C₃×#102	C₂²×C₃	1	2	yes	—
100	C₂³×C₉	—	2	2	no	—
101	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₉	—	2	2	no	—
103	Q₈×C₉	—	2	2	no	—
104	Q₈×C₉	—	8	2	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	2	2	no	—
107	D₄×C₉	—	8	2	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	8	2	no	—
114	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
118	C₈×C₉	—	2	1	no	—
119	C₈×C₉	—	2	1	no	—
120	C₈×C₉	—	2	2	no	—
121	#542	C₂×C₉	1	2	yes	complemented
122	#477	C₂×C₉	1	2	yes	complemented
123	#474	C₂×C₉	1	2	yes	complemented
124	#468	C₂×C₉	1	2	yes	complemented
125	#467	C₂×C₉	1	1	yes	characteristic, complemented
126	#466	C₂×C₉	1	1	yes	characteristic
127	#464	C₂×C₉	1	2	yes	complemented
128	#463	C₂×C₉	1	2	yes	complemented
129	#424	C₂×C₉	1	1	yes	characteristic, complemented
130	C₂×C₃×Q₈	—	2	2	no	—
131	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₃×Q₁₆	—	4	1	no	—
137	C₃×SD₁₆	—	4	2	no	—
138	C₃×D₈	—	4	1	no	—
139	C₃×M₁₆	—	4	1	no	—
140	C₂×C₃×C₈	—	4	1	no	—
141	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
143	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
148	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
149	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
150	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₃×C₂²⋊C₄	—	2	2	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	2	no	—
158	C₃×C₄²	C₂³×C₃	1	2	yes	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	4	2	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	—	4	2	no	—
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	—	4	2	no	—
172	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
173	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
174	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
175	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
176	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
177	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
178	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
179	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
180	#126	—	2	2	no	—
181	#124	—	2	2	no	—
182	#124	C₂²×C₉	1	2	yes	—
183	#123	—	2	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
186	#121	C₂²×C₉	1	2	yes	—
187	C₄×Q₈	C₂²×C₉	1	2	yes	—
188	C₄×D₄	C₂²×C₉	1	2	yes	—
189	#117	C₂²×C₉	1	1	yes	characteristic
190	#107	C₂²×C₉	1	1	yes	characteristic
191	#107	C₂²×C₉	1	1	yes	characteristic
192	#107	C₂²×C₉	1	2	yes	—
193	C₄⋊C₈	C₂²×C₉	1	2	yes	—
194	C₄⋊C₈	C₂²×C₉	1	2	yes	—
195	#103	C₂²×C₉	1	2	yes	—
196	#103	C₂²×C₉	1	2	yes	—
197	#102	C₂²×C₉	1	2	yes	—
198	#102	C₂²×C₉	1	2	yes	—
199	C₂³×C₃	—	2	2	no	—
200	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
201	C₃×Q₈	—	2	2	no	—
202	C₃×Q₈	—	2	2	no	—
203	C₃×Q₈	—	8	2	no	—
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	2	2	no	—
206	C₃×D₄	—	8	2	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	8	2	no	—
213	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
215	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	2	2	no	—
220	C₂×C₉	—	4	1	no	—
221	C₂×C₉	—	8	2	no	—
222	C₂×C₉	#537	1	1	yes	characteristic
223	C₂×C₉	#576	1	1	yes	characteristic
224	C₂×C₉	#578	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	C₂³×C₉	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂³×C₉	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
231	Q₁₆	—	4	1	no	—
232	SD₁₆	—	4	2	no	—
233	D₈	—	4	1	no	—
234	M₁₆	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
237	C₂×C₈	C₂³×C₉	1	2	yes	—
238	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
243	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂³×C₉	1	2	yes	—
245	C₄⋊C₄	C₂³×C₉	1	2	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	D₄×C₉	1	2	yes	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂³×C₉	1	2	yes	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	4	2	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	4	2	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	4	2	no	—
267	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
269	C₉	#3262	1	1	yes	characteristic, direct product factor, Sylow
270	C₂³	—	2	2	no	—
271	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	8	2	no	—
275	D₄	—	2	2	no	—
276	D₄	—	2	2	no	—
277	D₄	—	8	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
285	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
286	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
287	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	2	no	—
291	C₂×C₃	—	4	1	no	—
292	C₂×C₃	—	8	2	no	—
293	C₂×C₃	C₃×#537	1	1	yes	characteristic
294	C₂×C₃	C₃×#576	1	1	yes	characteristic
295	C₂×C₃	C₃×#578	1	1	yes	characteristic
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
310	C₄	C₉×2₋¹⁺⁴	1	1	yes	characteristic
311	C₃	C₃×#3262	1	1	yes	characteristic
312	C₂	—	4	1	no	—
313	C₂	—	8	2	no	—
314	C₂	C₉×#537	1	1	yes	characteristic
315	C₂	C₉×#576	1	1	yes	characteristic
316	C₂	C₉×#578	1	1	yes	characteristic

	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×#466	C₂×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×#542	C₂	semidirect
2. C₉×#477	C₂	semidirect
3. C₉×#474	C₂	semidirect
4. C₉×#468	C₂	semidirect
5. C₉×#467	C₂	semidirect
7. C₉×#464	C₂	semidirect
8. C₉×#463	C₂	semidirect
9. C₉×#424	C₂	semidirect
74. #3262	C₉	direct (N × H)
121. #542	C₂×C₉	semidirect
122. #477	C₂×C₉	semidirect
123. #474	C₂×C₉	semidirect
124. #468	C₂×C₉	semidirect
125. #467	C₂×C₉	semidirect
127. #464	C₂×C₉	semidirect
128. #463	C₂×C₉	semidirect
129. #424	C₂×C₉	semidirect
269. C₉	#3262	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×#542	2
C₉×#477	2
C₉×#474	2
C₉×#468	2
C₉×#467	2
C₉×#466	2
C₉×#464	2
C₉×#463	2
C₉×#424	2
C₃×#3262	3

Subgroups — C9×#3262 #11767983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3262 #11767983