Subgroups — C₉×#1980 #11766701

Order 1152.

504 conjugacy classes of subgroups.

The subgroup lattice has 279 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×#104	C₂	1	4	yes	maximal, complemented
3	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1980	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×#41	C₂²	1	2	yes	—
7	C₂×C₉×M₁₆	C₂²	1	2	yes	—
8	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	C₉×#127	C₄	1	1	yes	characteristic, complemented
10	C₉×#124	C₄	1	2	yes	complemented
11	C₉×#121	—	2	1	no	—
12	C₉×#121	—	2	1	no	—
13	C₄×D₄×C₉	—	2	1	no	—
14	C₄×D₄×C₉	—	2	1	no	—
15	C₂×C₉×C₄⋊C₄	—	2	1	no	—
16	C₂×C₄²×C₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₉×#104	—	2	8	no	—
19	C₉×#97	—	2	1	no	—
20	C₄×C₈×C₉	—	2	1	no	—
21	C₉×#95	—	2	1	no	—
22	C₃×#532	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#104	C₂×C₃	1	4	yes	—
24	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
25	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₉×#41	—	2	4	no	—
27	C₉×#41	—	2	4	no	—
28	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₉	—	2	2	no	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×M₁₆	—	2	2	no	—
39	C₉×M₁₆	D₄	1	4	yes	complemented
40	C₂×C₈×C₉	—	2	2	no	—
41	C₉×C₄⋊C₄	—	2	4	no	—
42	C₉×C₄⋊C₄	—	2	1	no	—
43	C₉×C₄⋊C₄	—	2	1	no	—
44	C₉×C₂²⋊C₄	—	2	2	no	—
45	C₉×C₂²⋊C₄	—	2	2	no	—
46	C₄²×C₉	—	2	4	no	—
47	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₉	D₄	1	1	yes	characteristic
50	C₄²×C₉	D₄	1	1	yes	characteristic
51	#1980	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
53	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
54	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
55	C₃×#127	C₃×C₄	1	1	yes	characteristic
56	C₃×#124	C₃×C₄	1	2	yes	—
57	C₃×#121	—	2	1	no	—
58	C₃×#121	—	2	1	no	—
59	C₃×C₄×D₄	—	2	1	no	—
60	C₃×C₄×D₄	—	2	1	no	—
61	C₂×C₃×C₄⋊C₄	—	2	1	no	—
62	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
63	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
64	C₃×#104	—	2	8	no	—
65	C₃×#97	—	2	1	no	—
66	C₃×C₄×C₈	—	2	1	no	—
67	C₃×#95	—	2	1	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₉	—	2	4	no	—
71	D₄×C₉	—	2	4	no	—
72	D₄×C₉	—	4	2	no	—
73	D₄×C₉	—	4	2	no	—
74	D₄×C₉	—	4	2	no	—
75	D₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	4	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	4	2	no	—
81	C₂×C₄×C₉	—	4	2	no	—
82	C₂×C₄×C₉	—	4	4	no	—
83	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₉	#41	1	2	yes	—
91	C₈×C₉	—	2	4	no	—
92	#532	C₂×C₉	1	1	yes	characteristic
93	C₂×#104	C₂×C₉	1	4	yes	complemented
94	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
95	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
96	C₃×#41	—	2	4	no	—
97	C₃×#41	—	2	4	no	—
98	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₃×D₄	—	2	2	no	—
100	C₂×C₃×D₄	—	2	1	no	—
101	C₂×C₃×D₄	—	2	1	no	—
102	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	—	2	2	no	—
105	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₃×M₁₆	—	2	2	no	—
109	C₃×M₁₆	C₃×D₄	1	4	yes	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	4	no	—
112	C₃×C₄⋊C₄	—	2	1	no	—
113	C₃×C₄⋊C₄	—	2	1	no	—
114	C₃×C₂²⋊C₄	—	2	2	no	—
115	C₃×C₂²⋊C₄	—	2	2	no	—
116	C₃×C₄²	—	2	4	no	—
117	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
118	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
119	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
120	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
121	C₂²×C₉	—	2	1	no	—
122	C₂²×C₉	—	2	1	no	—
123	C₂²×C₉	—	4	2	no	—
124	C₂²×C₉	—	4	2	no	—
125	C₂²×C₉	—	4	2	no	—
126	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₉	#124	1	1	yes	characteristic
128	C₂²×C₉	#127	1	1	yes	characteristic
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	—	2	1	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	—	4	4	no	—
133	C₄×C₉	—	4	2	no	—
134	C₄×C₉	#104	1	2	yes	complemented
135	C₄×C₉	C₄×D₄	1	2	yes	—
136	C₄×C₉	#121	1	2	yes	—
137	C₂×#41	C₂²×C₉	1	2	yes	—
138	C₂×M₁₆	C₂²×C₉	1	2	yes	—
139	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic, complemented
140	#127	C₄×C₉	1	1	yes	characteristic, complemented
141	#124	C₄×C₉	1	2	yes	complemented
142	#121	—	2	1	no	—
143	#121	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄²	C₂²×C₉	1	2	yes	—
148	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
149	#104	—	2	8	no	—
150	#97	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	#95	—	2	1	no	—
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
155	C₃×Q₈	—	2	4	no	—
156	C₃×D₄	—	2	4	no	—
157	C₃×D₄	—	4	2	no	—
158	C₃×D₄	—	4	2	no	—
159	C₃×D₄	—	4	2	no	—
160	C₃×D₄	—	4	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	—	4	2	no	—
167	C₂×C₃×C₄	—	4	4	no	—
168	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₃×#41	1	2	yes	—
176	C₃×C₈	—	2	4	no	—
177	C₂×C₉	—	2	1	no	—
178	C₂×C₉	—	2	1	no	—
179	C₂×C₉	—	8	2	no	—
180	C₂×C₉	#390	1	1	yes	characteristic
181	C₂×C₉	C₂×#104	1	1	yes	characteristic
182	C₂×C₉	#421	1	1	yes	characteristic
183	#41	—	2	4	no	—
184	#41	—	2	4	no	—
185	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
193	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	D₄×C₉	1	4	yes	complemented
197	C₂×C₈	—	2	2	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
205	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
206	C₄²	D₄×C₉	1	1	yes	characteristic
207	C₄²	D₄×C₉	1	1	yes	characteristic
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	4	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₃	C₃×#124	1	1	yes	characteristic
215	C₂²×C₃	C₃×#127	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	4	4	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	C₃×#104	1	2	yes	—
222	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
223	C₃×C₄	C₃×#121	1	2	yes	—
224	C₉	#1980	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	2	2	no	—
226	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
246	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
247	C₂×C₄	C₉×#41	1	2	yes	—
248	C₈	—	2	4	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	8	2	no	—
252	C₂×C₃	C₃×#390	1	1	yes	characteristic
253	C₂×C₃	C₂×C₃×#104	1	1	yes	characteristic
254	C₂×C₃	C₃×#421	1	1	yes	characteristic
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
261	C₂²	C₉×#124	1	1	yes	characteristic
262	C₂²	C₉×#127	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	2	no	—
268	C₄	C₉×#104	1	2	yes	complemented
269	C₄	C₄×D₄×C₉	1	2	yes	—
270	C₄	C₉×#121	1	2	yes	—
271	C₃	C₃×#1980	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	C₉×#390	1	1	yes	characteristic
276	C₂	C₂×C₉×#104	1	1	yes	characteristic
277	C₂	C₉×#421	1	1	yes	characteristic

	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×#532	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#104	C₂	semidirect
3. C₄×C₉×M₁₆	C₂	semidirect
4. C₄×C₉×C₄⋊C₄	C₂	semidirect
8. C₉×C₄⋊Q₈	C₄	semidirect
9. C₉×#127	C₄	semidirect
10. C₉×#124	C₄	semidirect
39. C₉×M₁₆	D₄	semidirect
51. #1980	C₉	direct (N × H)
93. C₂×#104	C₂×C₉	semidirect
94. C₄×M₁₆	C₂×C₉	semidirect
95. C₄×C₄⋊C₄	C₂×C₉	semidirect
134. C₄×C₉	#104	semidirect
139. C₄⋊Q₈	C₄×C₉	semidirect
140. #127	C₄×C₉	semidirect
141. #124	C₄×C₉	semidirect
196. M₁₆	D₄×C₉	semidirect
224. C₉	#1980	direct (N × H)
268. C₄	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×#532	2
C₂×C₉×#104	2
C₄×C₉×M₁₆	2
C₄×C₉×C₄⋊C₄	2
C₃×#1980	3

Subgroups — C9×#1980 #11766701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1980 #11766701