Subgroups — C₉×#3206 #11767927

Order 1152.

642 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#492	C₂	1	2	yes	maximal, complemented
4	C₉×#487	C₂	1	2	yes	maximal, complemented
5	C₉×#461	C₂	1	4	yes	maximal, complemented
6	C₉×#448	C₂	1	4	yes	maximal, complemented
7	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3206	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	—	2	1	no	—
10	C₂²×Q₈×C₉	C₂²	1	2	yes	—
11	C₂×C₉×SD₁₆	C₂²	1	8	yes	—
12	C₉×C₄⋊Q₈	—	2	1	no	—
13	C₉×C₄⋊Q₈	—	2	1	no	—
14	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₉×#127	C₂²	1	1	yes	characteristic
18	C₉×#124	—	2	1	no	—
19	C₉×#121	—	2	1	no	—
20	C₉×#121	C₂²	1	2	yes	—
21	C₄×D₄×C₉	—	2	1	no	—
22	C₄×D₄×C₉	C₂²	1	2	yes	—
23	C₂×C₉×C₄⋊C₄	—	2	1	no	—
24	C₂×C₉×C₄⋊C₄	—	2	1	no	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₉×#103	C₂²	1	8	yes	—
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	—
30	C₃×#532	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	C₃×#492	C₂×C₃	1	2	yes	—
33	C₃×#487	C₂×C₃	1	2	yes	—
34	C₃×#461	C₂×C₃	1	4	yes	—
35	C₃×#448	C₂×C₃	1	4	yes	—
36	C₃×#431	C₂×C₃	1	1	yes	characteristic
37	C₉×#41	—	4	4	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	—	2	2	no	—
40	C₂×Q₈×C₉	—	4	4	no	—
41	C₂×Q₈×C₉	C₂³	1	4	yes	—
42	C₂×D₄×C₉	—	2	2	no	—
43	C₂×D₄×C₉	—	2	1	no	—
44	C₂×D₄×C₉	—	2	1	no	—
45	C₂×D₄×C₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₉	—	2	2	no	—
47	C₂²×C₄×C₉	—	2	2	no	—
48	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×SD₁₆	—	4	8	no	—
52	C₂×C₈×C₉	C₂³	1	4	yes	—
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	—	2	2	no	—
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	1	no	—
58	C₉×C₄⋊C₄	C₂³	1	2	yes	—
59	C₉×C₂²⋊C₄	—	2	2	no	—
60	C₉×C₂²⋊C₄	—	2	2	no	—
61	C₄²×C₉	D₄	1	1	yes	characteristic
62	C₄²×C₉	D₄	1	1	yes	characteristic
63	C₄²×C₉	C₂³	1	1	yes	characteristic
64	C₄²×C₉	C₂³	1	1	yes	characteristic
65	#3206	C₉	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃×#41	—	2	1	no	—
67	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
68	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	—
69	C₃×C₄⋊Q₈	—	2	1	no	—
70	C₃×C₄⋊Q₈	—	2	1	no	—
71	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
72	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
73	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×#127	C₂²×C₃	1	1	yes	characteristic
75	C₃×#124	—	2	1	no	—
76	C₃×#121	—	2	1	no	—
77	C₃×#121	C₂²×C₃	1	2	yes	—
78	C₃×C₄×D₄	—	2	1	no	—
79	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	—	2	1	no	—
81	C₂×C₃×C₄⋊C₄	—	2	1	no	—
82	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
84	C₃×#103	C₂²×C₃	1	8	yes	—
85	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
86	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
87	C₂³×C₉	—	2	2	no	—
88	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₉	—	2	8	no	—
90	Q₈×C₉	—	4	4	no	—
91	Q₈×C₉	—	4	2	no	—
92	D₄×C₉	—	2	4	no	—
93	D₄×C₉	—	4	4	no	—
94	D₄×C₉	—	4	2	no	—
95	D₄×C₉	—	4	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	4	4	no	—
101	C₂×C₄×C₉	—	4	4	no	—
102	C₂×C₄×C₉	SD₁₆	1	4	yes	complemented
103	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
109	C₈×C₉	—	2	4	no	—
110	#532	C₂×C₉	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
112	#492	C₂×C₉	1	2	yes	complemented
113	#487	C₂×C₉	1	2	yes	complemented
114	#461	C₂×C₉	1	4	yes	complemented
115	#448	C₂×C₉	1	4	yes	complemented
116	#431	C₂×C₉	1	1	yes	characteristic, complemented
117	C₃×#41	—	4	4	no	—
118	C₂×C₃×Q₈	—	2	1	no	—
119	C₂×C₃×Q₈	—	2	2	no	—
120	C₂×C₃×Q₈	—	4	4	no	—
121	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
122	C₂×C₃×D₄	—	2	2	no	—
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	—	2	1	no	—
125	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
131	C₃×SD₁₆	—	4	8	no	—
132	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
139	C₃×C₂²⋊C₄	—	2	2	no	—
140	C₃×C₂²⋊C₄	—	2	2	no	—
141	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
142	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
143	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
144	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
145	C₂²×C₉	—	2	1	no	—
146	C₂²×C₉	—	2	1	no	—
147	C₂²×C₉	—	4	2	no	—
148	C₂²×C₉	—	4	2	no	—
149	C₂²×C₉	—	4	2	no	—
150	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
151	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	2	1	no	—
154	C₄×C₉	—	4	4	no	—
155	C₄×C₉	—	4	2	no	—
156	C₄×C₉	C₂×SD₁₆	1	2	yes	—
157	C₄×C₉	C₂×SD₁₆	1	2	yes	—
158	C₄×C₉	#137	1	2	yes	—
159	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
160	C₂×#41	—	2	1	no	—
161	C₂²×Q₈	C₂²×C₉	1	2	yes	—
162	C₂×SD₁₆	C₂²×C₉	1	8	yes	—
163	C₄⋊Q₈	—	2	1	no	—
164	C₄⋊Q₈	—	2	1	no	—
165	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
166	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
167	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
168	#127	C₂²×C₉	1	1	yes	characteristic
169	#124	—	2	1	no	—
170	#121	—	2	1	no	—
171	#121	C₂²×C₉	1	2	yes	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	C₂²×C₉	1	2	yes	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
177	C₄⋊C₈	C₂²×C₉	1	2	yes	—
178	#103	C₂²×C₉	1	8	yes	—
179	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
180	C₄×C₈	C₂²×C₉	1	2	yes	—
181	C₂³×C₃	—	2	2	no	—
182	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
183	C₃×Q₈	—	2	8	no	—
184	C₃×Q₈	—	4	4	no	—
185	C₃×Q₈	—	4	2	no	—
186	C₃×D₄	—	2	4	no	—
187	C₃×D₄	—	4	4	no	—
188	C₃×D₄	—	4	2	no	—
189	C₃×D₄	—	4	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	4	4	no	—
195	C₂×C₃×C₄	—	4	4	no	—
196	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
197	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
203	C₃×C₈	—	2	4	no	—
204	C₂×C₉	—	2	2	no	—
205	C₂×C₉	—	8	2	no	—
206	C₂×C₉	#535	1	1	yes	characteristic
207	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
208	C₂×C₉	C₂×#137	1	1	yes	characteristic
209	#41	—	4	4	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	C₂³×C₉	1	4	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₂³×C₉	1	2	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
221	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
223	SD₁₆	—	4	8	no	—
224	C₂×C₈	C₂³×C₉	1	4	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	C₂³×C₉	1	2	yes	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₄²	D₄×C₉	1	1	yes	characteristic
234	C₄²	D₄×C₉	1	1	yes	characteristic
235	C₄²	C₂³×C₉	1	1	yes	characteristic
236	C₄²	C₂³×C₉	1	1	yes	characteristic
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	—	4	2	no	—
241	C₂²×C₃	—	4	2	no	—
242	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
243	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	4	2	no	—
247	C₃×C₄	—	4	4	no	—
248	C₃×C₄	C₂×C₃×SD₁₆	1	2	yes	—
249	C₃×C₄	C₂×C₃×SD₁₆	1	2	yes	—
250	C₃×C₄	C₃×#137	1	2	yes	—
251	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
252	C₉	#3206	1	1	yes	characteristic, direct product factor, Sylow
253	C₂³	—	2	2	no	—
254	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
255	Q₈	—	2	8	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	4	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₉×SD₁₆	1	4	yes	complemented
269	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
274	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₂×C₃	—	2	2	no	—
277	C₂×C₃	—	8	2	no	—
278	C₂×C₃	C₃×#535	1	1	yes	characteristic
279	C₂×C₃	C₂²×C₃×SD₁₆	1	1	yes	characteristic
280	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
287	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	2	no	—
292	C₄	C₂×C₉×SD₁₆	1	2	yes	—
293	C₄	C₂×C₉×SD₁₆	1	2	yes	—
294	C₄	C₉×#137	1	2	yes	—
295	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
296	C₃	C₃×#3206	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	2	no	—
299	C₂	C₉×#535	1	1	yes	characteristic
300	C₂	C₂²×C₉×SD₁₆	1	1	yes	characteristic
301	C₂	C₂×C₉×#137	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×#532	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊Q₈	C₂	semidirect
3. C₉×#492	C₂	semidirect
4. C₉×#487	C₂	semidirect
5. C₉×#461	C₂	semidirect
6. C₉×#448	C₂	semidirect
7. C₉×#431	C₂	semidirect
65. #3206	C₉	direct (N × H)
102. C₂×C₄×C₉	SD₁₆	semidirect
111. C₂×C₄⋊Q₈	C₂×C₉	semidirect
112. #492	C₂×C₉	semidirect
113. #487	C₂×C₉	semidirect
114. #461	C₂×C₉	semidirect
115. #448	C₂×C₉	semidirect
116. #431	C₂×C₉	semidirect
252. C₉	#3206	direct (N × H)
268. C₂×C₄	C₉×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₉×#532	2
C₂×C₉×C₄⋊Q₈	2
C₉×#492	2
C₉×#487	2
C₉×#461	2
C₉×#448	2
C₉×#431	2
C₃×#3206	3

Subgroups — C9×#3206 #11767927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3206 #11767927