Subgroups — C₉×#2956 #11767677

Order 1152.

687 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#575	C₂	1	1	yes	maximal, characteristic
2	C₉×#568	C₂	1	2	yes	maximal, complemented
3	C₉×#444	C₂	1	4	yes	maximal, complemented
4	C₉×#443	C₂	1	4	yes	maximal, complemented
5	C₉×#430	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×#104	C₂	1	2	yes	maximal, complemented
7	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2956	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	2	yes	—
10	C₉×#137	C₄	1	4	yes	complemented
11	C₉×AGL(1,ℤ/8)	C₄	1	4	yes	complemented
12	C₉×#135	C₂²	1	4	yes	—
13	C₉×#131	—	2	4	no	—
14	C₉×#131	C₂²	1	8	yes	complemented
15	C₂×C₉×M₁₆	—	2	4	no	—
16	C₂×C₉×M₁₆	C₂²	1	2	yes	—
17	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₉×#108	C₂²	1	4	yes	complemented
21	C₉×#104	C₂²	1	4	yes	complemented
22	C₉×#104	C₂²	1	4	yes	complemented
23	C₉×#97	C₂²	1	2	yes	complemented
24	C₄×C₈×C₉	C₂²	1	2	yes	complemented
25	C₃×#575	C₂×C₃	1	1	yes	characteristic
26	C₃×#568	C₂×C₃	1	2	yes	—
27	C₃×#444	C₂×C₃	1	4	yes	—
28	C₃×#443	C₂×C₃	1	4	yes	—
29	C₃×#430	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#104	C₂×C₃	1	2	yes	—
31	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
32	C₉×#41	—	2	2	no	—
33	C₉×#41	—	2	2	no	—
34	C₉×#41	C₂×C₄	1	4	yes	—
35	C₉×#41	C₂³	1	4	yes	—
36	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₉	—	2	2	no	—
38	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₉	—	2	2	no	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₉×Q₁₆	C₂×C₄	1	4	yes	complemented
43	C₉×SD₁₆	C₂×C₄	1	8	yes	complemented
44	C₉×D₈	C₂×C₄	1	4	yes	complemented
45	C₉×M₁₆	—	2	2	no	—
46	C₉×M₁₆	—	2	8	no	—
47	C₉×M₁₆	C₂×C₄	1	4	yes	—
48	C₉×M₁₆	D₄	1	4	yes	—
49	C₉×M₁₆	C₂³	1	4	yes	—
50	C₂×C₈×C₉	—	2	2	no	—
51	C₂×C₈×C₉	—	2	8	no	—
52	C₂×C₈×C₉	C₂³	1	2	yes	—
53	C₂×C₈×C₉	C₂³	1	2	yes	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	C₂³	1	1	yes	characteristic
56	C₄²×C₉	C₂³	1	1	yes	characteristic
57	#2956	C₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
59	C₃×#137	C₃×C₄	1	4	yes	—
60	C₃×AGL(1,ℤ/8)	C₃×C₄	1	4	yes	—
61	C₃×#135	C₂²×C₃	1	4	yes	—
62	C₃×#131	—	2	4	no	—
63	C₃×#131	C₂²×C₃	1	8	yes	—
64	C₂×C₃×M₁₆	—	2	4	no	—
65	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
66	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
69	C₃×#108	C₂²×C₃	1	4	yes	—
70	C₃×#104	C₂²×C₃	1	4	yes	—
71	C₃×#104	C₂²×C₃	1	4	yes	—
72	C₃×#97	C₂²×C₃	1	2	yes	—
73	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
74	C₂³×C₉	—	2	2	no	—
75	C₂³×C₉	#41	1	1	yes	characteristic
76	Q₈×C₉	—	2	2	no	—
77	Q₈×C₉	C₂²×C₄	1	4	yes	—
78	D₄×C₉	—	2	4	no	—
79	D₄×C₉	—	2	4	no	—
80	D₄×C₉	—	2	2	no	—
81	D₄×C₉	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	4	2	no	—
88	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₉	#41	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
92	C₈×C₉	—	2	4	no	—
93	C₈×C₉	C₂²×C₄	1	4	yes	—
94	C₈×C₉	C₂×D₄	1	4	yes	—
95	#575	C₂×C₉	1	1	yes	characteristic
96	#568	C₂×C₉	1	2	yes	complemented
97	#444	C₂×C₉	1	4	yes	complemented
98	#443	C₂×C₉	1	4	yes	complemented
99	#430	C₂×C₉	1	1	yes	characteristic, complemented
100	C₂×#104	C₂×C₉	1	2	yes	complemented
101	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
102	C₃×#41	—	2	2	no	—
103	C₃×#41	—	2	2	no	—
104	C₃×#41	C₂×C₃×C₄	1	4	yes	—
105	C₃×#41	C₂³×C₃	1	4	yes	—
106	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₃×D₄	—	2	2	no	—
108	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₃×Q₁₆	C₂×C₃×C₄	1	4	yes	—
113	C₃×SD₁₆	C₂×C₃×C₄	1	8	yes	—
114	C₃×D₈	C₂×C₃×C₄	1	4	yes	—
115	C₃×M₁₆	—	2	2	no	—
116	C₃×M₁₆	—	2	8	no	—
117	C₃×M₁₆	C₂×C₃×C₄	1	4	yes	—
118	C₃×M₁₆	C₃×D₄	1	4	yes	—
119	C₃×M₁₆	C₂³×C₃	1	4	yes	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	2	8	no	—
122	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
123	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
126	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
127	C₂²×C₉	—	2	4	no	—
128	C₂²×C₉	—	4	1	no	—
129	C₂²×C₉	—	8	2	no	—
130	C₂²×C₉	C₄×D₄	1	2	yes	—
131	C₂²×C₉	C₂×#41	1	1	yes	characteristic
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	4	no	—
134	C₄×C₉	—	4	1	no	—
135	C₄×C₉	C₄×D₄	1	2	yes	—
136	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
137	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×#41	C₂²×C₉	1	2	yes	—
139	#137	C₄×C₉	1	4	yes	complemented
140	AGL(1,ℤ/8)	C₄×C₉	1	4	yes	complemented
141	#135	C₂²×C₉	1	4	yes	—
142	#131	—	2	4	no	—
143	#131	C₂²×C₉	1	8	yes	complemented
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	C₂²×C₉	1	2	yes	—
146	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
147	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
148	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
149	#108	C₂²×C₉	1	4	yes	complemented
150	#104	C₂²×C₉	1	4	yes	complemented
151	#104	C₂²×C₉	1	4	yes	complemented
152	#97	C₂²×C₉	1	2	yes	complemented
153	C₄×C₈	C₂²×C₉	1	2	yes	complemented
154	C₂³×C₃	—	2	2	no	—
155	C₂³×C₃	C₃×#41	1	1	yes	characteristic
156	C₃×Q₈	—	2	2	no	—
157	C₃×Q₈	C₂²×C₃×C₄	1	4	yes	—
158	C₃×D₄	—	2	4	no	—
159	C₃×D₄	—	2	4	no	—
160	C₃×D₄	—	2	2	no	—
161	C₃×D₄	C₂²×C₃×C₄	1	4	yes	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	4	2	no	—
168	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
170	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
172	C₃×C₈	—	2	4	no	—
173	C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
174	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
175	C₂×C₉	—	2	1	no	—
176	C₂×C₉	—	2	2	no	—
177	C₂×C₉	—	4	4	no	—
178	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	#41	C₂×C₄×C₉	1	4	yes	—
182	#41	C₂³×C₉	1	4	yes	—
183	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
189	Q₁₆	C₂×C₄×C₉	1	4	yes	complemented
190	SD₁₆	C₂×C₄×C₉	1	8	yes	complemented
191	D₈	C₂×C₄×C₉	1	4	yes	complemented
192	M₁₆	—	2	2	no	—
193	M₁₆	—	2	8	no	—
194	M₁₆	C₂×C₄×C₉	1	4	yes	—
195	M₁₆	D₄×C₉	1	4	yes	—
196	M₁₆	C₂³×C₉	1	4	yes	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	C₂³×C₉	1	2	yes	—
200	C₂×C₈	C₂³×C₉	1	2	yes	—
201	C₄²	—	2	1	no	—
202	C₄²	C₂³×C₉	1	1	yes	characteristic
203	C₄²	C₂³×C₉	1	1	yes	characteristic
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	—	8	2	no	—
207	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
208	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	4	no	—
211	C₃×C₄	—	4	1	no	—
212	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
213	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
214	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
215	C₉	#2956	1	1	yes	characteristic, direct product factor, Sylow
216	C₂³	—	2	2	no	—
217	C₂³	C₉×#41	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	C₂²×C₄×C₉	1	4	yes	—
220	D₄	—	2	4	no	—
221	D₄	—	2	4	no	—
222	D₄	—	2	2	no	—
223	D₄	C₂²×C₄×C₉	1	4	yes	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
231	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
232	C₂×C₄	C₉×#41	1	1	yes	characteristic
233	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₈	C₂²×C₄×C₉	1	4	yes	—
236	C₈	C₂×D₄×C₉	1	4	yes	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	—	4	4	no	—
240	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	8	2	no	—
244	C₂²	C₄×D₄×C₉	1	2	yes	—
245	C₂²	C₂×C₉×#41	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	1	no	—
249	C₄	C₄×D₄×C₉	1	2	yes	—
250	C₄	C₂³×C₄×C₉	1	1	yes	characteristic
251	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
252	C₃	C₃×#2956	1	1	yes	characteristic
253	C₂	—	2	1	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	4	no	—
256	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	C₂²×D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂×C₃×C₄×D₄
prime radical	C₃×#575	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₉×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#568	C₂	semidirect
3. C₉×#444	C₂	semidirect
4. C₉×#443	C₂	semidirect
5. C₉×#430	C₂	semidirect
6. C₂×C₉×#104	C₂	semidirect
7. C₄×C₉×M₁₆	C₂	semidirect
10. C₉×#137	C₄	semidirect
11. C₉×AGL(1,ℤ/8)	C₄	semidirect
14. C₉×#131	C₂²	semidirect
20. C₉×#108	C₂²	semidirect
21. C₉×#104	C₂²	semidirect
22. C₉×#104	C₂²	semidirect
23. C₉×#97	C₂²	semidirect
24. C₄×C₈×C₉	C₂²	semidirect
42. C₉×Q₁₆	C₂×C₄	semidirect
43. C₉×SD₁₆	C₂×C₄	semidirect
44. C₉×D₈	C₂×C₄	semidirect
57. #2956	C₉	direct (N × H)
96. #568	C₂×C₉	semidirect
97. #444	C₂×C₉	semidirect
98. #443	C₂×C₉	semidirect
99. #430	C₂×C₉	semidirect
100. C₂×#104	C₂×C₉	semidirect
101. C₄×M₁₆	C₂×C₉	semidirect
139. #137	C₄×C₉	semidirect
140. AGL(1,ℤ/8)	C₄×C₉	semidirect
143. #131	C₂²×C₉	semidirect
149. #108	C₂²×C₉	semidirect
150. #104	C₂²×C₉	semidirect
151. #104	C₂²×C₉	semidirect
152. #97	C₂²×C₉	semidirect
153. C₄×C₈	C₂²×C₉	semidirect
189. Q₁₆	C₂×C₄×C₉	semidirect
190. SD₁₆	C₂×C₄×C₉	semidirect
191. D₈	C₂×C₄×C₉	semidirect
215. C₉	#2956	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×#575	2
C₉×#568	2
C₉×#444	2
C₉×#443	2
C₉×#430	2
C₂×C₉×#104	2
C₄×C₉×M₁₆	2
C₃×#2956	3

Subgroups — C9×#2956 #11767677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2956 #11767677