Subgroups — C₉×#2981 #11767702

Order 1152.

603 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	C₉×#436	C₂	1	2	yes	maximal, complemented
3	C₉×#435	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₉	C₂	1	2	yes	maximal, complemented
5	C₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#408	C₂	1	4	yes	maximal, complemented
8	C₃×#2981	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	2	yes	—
10	C₂×C₉×M₁₆	C₂²	1	4	yes	—
11	C₂²×C₈×C₉	C₂²	1	4	yes	—
12	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
13	C₉×#127	C₄	1	1	yes	characteristic
14	C₉×#124	C₄	1	2	yes	—
15	C₉×#121	C₄	1	2	yes	—
16	C₉×#121	C₄	1	2	yes	—
17	C₄×D₄×C₉	C₂²	1	2	yes	—
18	C₄×D₄×C₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₉×C₄⋊C₈	C₂²	1	2	yes	—
22	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
25	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
26	C₉×#97	C₂²	1	2	yes	—
27	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
29	C₃×#532	C₂×C₃	1	1	yes	characteristic
30	C₃×#436	C₂×C₃	1	2	yes	—
31	C₃×#435	C₂×C₃	1	4	yes	—
32	C₃×C₈×D₄	C₂×C₃	1	2	yes	—
33	C₃×#432	C₂×C₃	1	1	yes	characteristic
34	C₃×#431	C₂×C₃	1	1	yes	characteristic
35	C₃×#408	C₂×C₃	1	4	yes	—
36	C₉×#41	—	4	4	no	—
37	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
40	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
41	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂³	1	4	yes	—
46	C₉×M₁₆	—	4	4	no	—
47	C₂×C₈×C₉	—	4	4	no	—
48	C₂×C₈×C₉	C₂³	1	2	yes	—
49	C₂×C₈×C₉	C₂³	1	4	yes	—
50	C₂×C₈×C₉	C₂³	1	2	yes	—
51	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
54	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
55	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₉	C₂³	1	1	yes	characteristic
58	C₄²×C₉	C₂³	1	1	yes	characteristic
59	#2981	C₉	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
61	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
62	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
63	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
64	C₃×#127	C₃×C₄	1	1	yes	characteristic
65	C₃×#124	C₃×C₄	1	2	yes	—
66	C₃×#121	C₃×C₄	1	2	yes	—
67	C₃×#121	C₃×C₄	1	2	yes	—
68	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
69	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
70	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
71	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
72	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
73	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
74	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
75	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
76	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
77	C₃×#97	C₂²×C₃	1	2	yes	—
78	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
79	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
80	C₂³×C₉	C₂²×C₄	1	4	yes	—
81	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
82	Q₈×C₉	—	4	2	no	—
83	D₄×C₉	—	2	4	no	—
84	D₄×C₉	—	4	2	no	—
85	D₄×C₉	—	4	2	no	—
86	D₄×C₉	—	4	4	no	—
87	C₂×C₄×C₉	—	2	2	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	—	4	1	no	—
92	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
97	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	2	4	no	—
101	C₈×C₉	—	2	2	no	—
102	#532	C₂×C₉	1	1	yes	characteristic
103	#436	C₂×C₉	1	2	yes	complemented
104	#435	C₂×C₉	1	4	yes	complemented
105	C₈×D₄	C₂×C₉	1	2	yes	complemented
106	#432	C₂×C₉	1	1	yes	characteristic, complemented
107	#431	C₂×C₉	1	1	yes	characteristic, complemented
108	#408	C₂×C₉	1	4	yes	complemented
109	C₃×#41	—	4	4	no	—
110	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
111	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
112	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
114	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
119	C₃×M₁₆	—	4	4	no	—
120	C₂×C₃×C₈	—	4	4	no	—
121	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
122	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
123	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
124	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
125	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
126	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
127	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
128	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
129	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
130	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
131	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₉	—	2	1	no	—
133	C₂²×C₉	—	2	4	no	—
134	C₂²×C₉	—	2	1	no	—
135	C₂²×C₉	—	2	4	no	—
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₉	—	2	1	no	—
140	C₄×C₉	—	2	4	no	—
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	2	1	no	—
144	C₄×C₉	#131	1	2	yes	—
145	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
146	C₂×#41	C₂²×C₉	1	2	yes	—
147	C₂×M₁₆	C₂²×C₉	1	4	yes	—
148	C₂²×C₈	C₂²×C₉	1	4	yes	—
149	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
150	#127	C₄×C₉	1	1	yes	characteristic
151	#124	C₄×C₉	1	2	yes	—
152	#121	C₄×C₉	1	2	yes	—
153	#121	C₄×C₉	1	2	yes	—
154	C₄×D₄	C₂²×C₉	1	2	yes	—
155	C₄×D₄	C₂²×C₉	1	2	yes	—
156	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
157	C₄⋊C₈	C₂²×C₉	1	2	yes	—
158	C₄⋊C₈	C₂²×C₉	1	2	yes	—
159	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
160	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
161	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
162	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
163	#97	C₂²×C₉	1	2	yes	—
164	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
165	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
166	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
167	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
168	C₃×Q₈	—	4	2	no	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	4	2	no	—
171	C₃×D₄	—	4	2	no	—
172	C₃×D₄	—	4	4	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
183	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	4	no	—
187	C₃×C₈	—	2	2	no	—
188	C₂×C₉	—	4	1	no	—
189	C₂×C₉	—	4	4	no	—
190	C₂×C₉	#518	1	1	yes	characteristic
191	C₂×C₉	C₂×#131	1	1	yes	characteristic
192	C₂×C₉	#568	1	1	yes	characteristic
193	#41	—	4	4	no	—
194	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
195	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
196	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
197	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
198	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
199	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₉	1	4	yes	—
203	M₁₆	—	4	4	no	—
204	C₂×C₈	—	4	4	no	—
205	C₂×C₈	C₂³×C₉	1	2	yes	—
206	C₂×C₈	C₂³×C₉	1	2	yes	—
207	C₂×C₈	C₂³×C₉	1	4	yes	—
208	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
209	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
210	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
211	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
212	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
213	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
214	C₄²	C₂³×C₉	1	1	yes	characteristic
215	C₄²	C₂³×C₉	1	1	yes	characteristic
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	4	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	4	no	—
222	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	C₃×#131	1	2	yes	—
229	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
230	C₉	#2981	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	C₂²×C₄×C₉	1	4	yes	—
232	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
233	Q₈	—	4	2	no	—
234	D₄	—	2	4	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
248	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
249	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	4	no	—
252	C₈	—	2	2	no	—
253	C₂×C₃	—	4	1	no	—
254	C₂×C₃	—	4	4	no	—
255	C₂×C₃	C₃×#518	1	1	yes	characteristic
256	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
257	C₂×C₃	C₃×#568	1	1	yes	characteristic
258	C₂²	—	2	1	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	4	no	—
264	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	4	no	—
270	C₄	C₉×#131	1	2	yes	—
271	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
272	C₃	C₃×#2981	1	1	yes	characteristic
273	C₂	—	4	1	no	—
274	C₂	—	4	4	no	—
275	C₂	C₉×#518	1	1	yes	characteristic
276	C₂	C₂×C₉×#131	1	1	yes	characteristic
277	C₂	C₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₃×#532	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#436	C₂	semidirect
3. C₉×#435	C₂	semidirect
4. C₈×D₄×C₉	C₂	semidirect
5. C₉×#432	C₂	semidirect
6. C₉×#431	C₂	semidirect
7. C₉×#408	C₂	semidirect
59. #2981	C₉	direct (N × H)
103. #436	C₂×C₉	semidirect
104. #435	C₂×C₉	semidirect
105. C₈×D₄	C₂×C₉	semidirect
106. #432	C₂×C₉	semidirect
107. #431	C₂×C₉	semidirect
108. #408	C₂×C₉	semidirect
230. C₉	#2981	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×#532	2
C₉×#436	2
C₉×#435	2
C₈×D₄×C₉	2
C₉×#432	2
C₉×#431	2
C₉×#408	2
C₃×#2981	3

Subgroups — C9×#2981 #11767702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2981 #11767702