Subgroups — C₉×#1896 #11766617

Order 1152.

663 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×#525	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#104	C₂	1	2	yes	maximal, complemented
4	C₉×#407	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#339	C₂	1	2	yes	maximal, complemented
6	C₃×#1896	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉	—	2	1	no	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	2	yes	—
11	C₉×#123	—	2	1	no	—
12	C₉×C₂²⋊Q₈	C₄	1	2	yes	complemented
13	C₉×#121	C₄	1	2	yes	complemented
14	C₉×#120	—	2	1	no	—
15	C₄×D₄×C₉	—	2	2	no	—
16	C₉×#117	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₉	—	2	1	no	—
18	C₂×C₄²×C₉	—	2	4	no	—
19	C₂×C₄²×C₉	C₂²	1	2	yes	—
20	C₉×#104	—	2	4	no	—
21	C₉×C₂²⋊C₈	—	2	2	no	—
22	C₃×#525	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#104	C₂×C₃	1	2	yes	—
25	C₃×#407	C₂×C₃	1	1	yes	characteristic
26	C₃×#339	C₂×C₃	1	2	yes	—
27	C₂⁴×C₉	D₄	1	1	yes	characteristic
28	C₉×#41	—	2	2	no	—
29	C₉×#41	—	2	2	no	—
30	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic, complemented
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic, complemented
34	C₂²×C₄×C₉	—	2	4	no	—
35	C₂²×C₄×C₉	—	2	8	no	—
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	—	2	2	no	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	D₄	1	2	yes	—
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×M₁₆	—	2	2	no	—
45	C₉×M₁₆	—	2	2	no	—
46	C₂×C₈×C₉	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
49	C₉×C₂²⋊C₄	—	2	2	no	—
50	C₉×C₂²⋊C₄	—	2	1	no	—
51	C₉×C₂²⋊C₄	—	2	2	no	—
52	C₄²×C₉	—	2	4	no	—
53	C₄²×C₉	—	2	2	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	D₄	1	4	yes	complemented
56	#1896	C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
58	C₂³×C₃×C₄	—	2	1	no	—
59	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
61	C₃×#123	—	2	1	no	—
62	C₃×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
63	C₃×#121	C₃×C₄	1	2	yes	—
64	C₃×#120	—	2	1	no	—
65	C₃×C₄×D₄	—	2	2	no	—
66	C₃×#117	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₄²	—	2	1	no	—
68	C₂×C₃×C₄²	—	2	4	no	—
69	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
70	C₃×#104	—	2	4	no	—
71	C₃×C₂²⋊C₈	—	2	2	no	—
72	C₂³×C₉	—	2	4	no	—
73	C₂³×C₉	—	2	1	no	—
74	C₂³×C₉	—	2	1	no	—
75	C₂³×C₉	—	2	1	no	—
76	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
77	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₉	—	2	2	no	—
79	D₄×C₉	—	2	2	no	—
80	D₄×C₉	—	4	4	no	—
81	D₄×C₉	—	4	1	no	—
82	D₄×C₉	—	4	1	no	—
83	C₂×C₄×C₉	—	2	16	no	—
84	C₂×C₄×C₉	—	2	8	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	2	4	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	4	2	no	—
93	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	#41	1	2	yes	—
98	C₈×C₉	—	4	2	no	—
99	#525	C₂×C₉	1	1	yes	characteristic
100	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic, complemented
101	C₂×#104	C₂×C₉	1	2	yes	complemented
102	#407	C₂×C₉	1	1	yes	characteristic, complemented
103	#339	C₂×C₉	1	2	yes	complemented
104	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
105	C₃×#41	—	2	2	no	—
106	C₃×#41	—	2	2	no	—
107	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₃×D₄	—	2	2	no	—
109	C₂×C₃×D₄	—	2	1	no	—
110	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	2	4	no	—
112	C₂²×C₃×C₄	—	2	8	no	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
118	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	2	2	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
126	C₃×C₂²⋊C₄	—	2	2	no	—
127	C₃×C₂²⋊C₄	—	2	1	no	—
128	C₃×C₂²⋊C₄	—	2	2	no	—
129	C₃×C₄²	—	2	4	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	C₃×D₄	1	4	yes	—
133	C₂²×C₉	—	2	4	no	—
134	C₂²×C₉	—	2	4	no	—
135	C₂²×C₉	—	2	2	no	—
136	C₂²×C₉	—	2	2	no	—
137	C₂²×C₉	—	2	2	no	—
138	C₂²×C₉	—	4	1	no	—
139	C₂²×C₉	—	4	2	no	—
140	C₂²×C₉	#104	1	4	yes	complemented
141	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₉	#120	1	1	yes	characteristic
143	C₂²×C₉	#123	1	1	yes	characteristic
144	C₄×C₉	—	2	8	no	—
145	C₄×C₉	—	2	2	no	—
146	C₄×C₉	—	4	1	no	—
147	C₄×C₉	—	4	2	no	—
148	C₄×C₉	C₄×D₄	1	2	yes	—
149	C₄×C₉	#121	1	2	yes	—
150	C₂×#41	C₂²×C₉	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
153	C₂×M₁₆	C₂²×C₉	1	2	yes	—
154	#123	—	2	1	no	—
155	C₂²⋊Q₈	C₄×C₉	1	2	yes	complemented
156	#121	C₄×C₉	1	2	yes	complemented
157	#120	—	2	1	no	—
158	C₄×D₄	—	2	2	no	—
159	#117	C₂²×C₉	1	1	yes	characteristic
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	C₂²×C₉	1	2	yes	—
163	#104	—	2	4	no	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₂³×C₃	—	2	4	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	—	2	1	no	—
168	C₂³×C₃	—	2	1	no	—
169	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
170	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃×Q₈	—	2	2	no	—
172	C₃×D₄	—	2	2	no	—
173	C₃×D₄	—	4	4	no	—
174	C₃×D₄	—	4	1	no	—
175	C₃×D₄	—	4	1	no	—
176	C₂×C₃×C₄	—	2	16	no	—
177	C₂×C₃×C₄	—	2	8	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
187	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₃×#41	1	2	yes	—
191	C₃×C₈	—	4	2	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	2	4	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	—	8	1	no	—
196	C₂×C₉	#386	1	1	yes	characteristic
197	C₂×C₉	C₂×#104	1	2	yes	—
198	C₂⁴	D₄×C₉	1	1	yes	characteristic
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic, complemented
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic, complemented
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
212	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
213	C₂²×C₄	D₄×C₉	1	2	yes	—
214	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	complemented
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₄²	—	2	4	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	1	no	—
226	C₄²	D₄×C₉	1	4	yes	complemented
227	C₂²×C₃	—	2	4	no	—
228	C₂²×C₃	—	2	4	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	4	1	no	—
233	C₂²×C₃	—	4	2	no	—
234	C₂²×C₃	C₃×#104	1	4	yes	—
235	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
236	C₂²×C₃	C₃×#120	1	1	yes	characteristic
237	C₂²×C₃	C₃×#123	1	1	yes	characteristic
238	C₃×C₄	—	2	8	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
243	C₃×C₄	C₃×#121	1	2	yes	—
244	C₉	#1896	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
250	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	D₄	—	2	2	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
267	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
269	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
270	C₂×C₄	C₉×#41	1	2	yes	—
271	C₈	—	4	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	4	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	8	1	no	—
276	C₂×C₃	C₃×#386	1	1	yes	characteristic
277	C₂×C₃	C₂×C₃×#104	1	2	yes	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₉×#104	1	4	yes	complemented
286	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
287	C₂²	C₉×#120	1	1	yes	characteristic
288	C₂²	C₉×#123	1	1	yes	characteristic
289	C₄	—	2	8	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	2	no	—
293	C₄	C₄×D₄×C₉	1	2	yes	—
294	C₄	C₉×#121	1	2	yes	—
295	C₃	C₃×#1896	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	4	no	—
299	C₂	—	8	1	no	—
300	C₂	C₉×#386	1	1	yes	characteristic
301	C₂	C₂×C₉×#104	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₃×#525	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×C₉	C₂	semidirect
3. C₂×C₉×#104	C₂	semidirect
4. C₉×#407	C₂	semidirect
5. C₉×#339	C₂	semidirect
12. C₉×C₂²⋊Q₈	C₄	semidirect
13. C₉×#121	C₄	semidirect
30. C₂×Q₈×C₉	C₂×C₄	semidirect
33. C₂×D₄×C₉	C₂×C₄	semidirect
48. C₉×C₄⋊C₄	C₂×C₄	semidirect
55. C₄²×C₉	D₄	semidirect
56. #1896	C₉	direct (N × H)
100. C₂²×C₄²	C₂×C₉	semidirect
101. C₂×#104	C₂×C₉	semidirect
102. #407	C₂×C₉	semidirect
103. #339	C₂×C₉	semidirect
140. C₂²×C₉	#104	semidirect
155. C₂²⋊Q₈	C₄×C₉	semidirect
156. #121	C₄×C₉	semidirect
201. C₂×Q₈	C₂×C₄×C₉	semidirect
204. C₂×D₄	C₂×C₄×C₉	semidirect
219. C₄⋊C₄	C₂×C₄×C₉	semidirect
226. C₄²	D₄×C₉	semidirect
244. C₉	#1896	direct (N × H)
285. C₂²	C₉×#104	semidirect

Maximal subgroups

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

subgroup	index
C₉×#525	2
C₂²×C₄²×C₉	2
C₂×C₉×#104	2
C₉×#407	2
C₉×#339	2
C₃×#1896	3

Subgroups — C9×#1896 #11766617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1896 #11766617