Subgroups — C₉×#3140 #11767861

Order 1152.

600 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic
3	C₉×#489	C₂	1	4	yes	maximal, complemented
4	C₉×#488	C₂	1	4	yes	maximal, complemented
5	C₉×#419	C₂	1	2	yes	maximal, complemented
6	C₉×#418	C₂	1	2	yes	maximal
7	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3140	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
13	C₉×#125	C₂²	1	2	yes	complemented
14	C₉×#124	—	2	2	no	—
15	C₉×#124	C₂²	1	2	yes	—
16	C₉×#124	C₂²	1	2	yes	—
17	C₉×C₂²⋊Q₈	—	2	2	no	—
18	C₄×Q₈×C₉	—	2	2	no	—
19	C₉×#117	C₂²	1	2	yes	—
20	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₉×#103	C₂²	1	8	yes	complemented
23	C₉×#102	C₂²	1	8	yes	—
24	C₉×#97	C₂²	1	2	yes	complemented
25	C₄×C₈×C₉	C₂²	1	2	yes	complemented
26	C₃×#533	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
28	C₃×#489	C₂×C₃	1	4	yes	—
29	C₃×#488	C₂×C₃	1	4	yes	—
30	C₃×#419	C₂×C₃	1	2	yes	—
31	C₃×#418	C₂×C₃	1	2	yes	—
32	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
33	C₂⁴×C₉	—	2	1	no	—
34	C₂×Q₈×C₉	—	2	2	no	—
35	C₂×Q₈×C₉	—	2	1	no	—
36	C₂×Q₈×C₉	—	2	4	no	—
37	C₂×Q₈×C₉	C₂³	1	2	yes	—
38	C₂×D₄×C₉	—	2	1	no	—
39	C₂×D₄×C₉	—	2	4	no	—
40	C₂×D₄×C₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	D₄	1	2	yes	—
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×M₁₆	—	2	4	no	—
45	C₂×C₈×C₉	C₂³	1	4	yes	—
46	C₉×C₄⋊C₄	—	2	4	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	C₂³	1	4	yes	—
49	C₉×C₂²⋊C₄	—	2	2	no	—
50	C₉×C₂²⋊C₄	—	2	4	no	—
51	C₉×C₂²⋊C₄	—	2	4	no	—
52	C₄²×C₉	—	2	2	no	—
53	C₄²×C₉	D₄	1	2	yes	—
54	C₄²×C₉	C₂³	1	1	yes	characteristic
55	C₄²×C₉	C₂³	1	1	yes	characteristic
56	#3140	C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
60	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
61	C₃×#125	C₂²×C₃	1	2	yes	—
62	C₃×#124	—	2	2	no	—
63	C₃×#124	C₂²×C₃	1	2	yes	—
64	C₃×#124	C₂²×C₃	1	2	yes	—
65	C₃×C₂²⋊Q₈	—	2	2	no	—
66	C₃×C₄×Q₈	—	2	2	no	—
67	C₃×#117	C₂²×C₃	1	2	yes	—
68	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
69	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₃×#103	C₂²×C₃	1	8	yes	—
71	C₃×#102	C₂²×C₃	1	8	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	—	4	4	no	—
76	C₂³×C₉	—	4	2	no	—
77	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₉	—	2	4	no	—
79	Q₈×C₉	—	4	4	no	—
80	Q₈×C₉	—	4	2	no	—
81	D₄×C₉	—	2	4	no	—
82	D₄×C₉	—	4	2	no	—
83	C₂×C₄×C₉	—	2	4	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	4	2	no	—
86	C₂×C₄×C₉	—	4	2	no	—
87	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
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	4	yes	—
91	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
92	C₈×C₉	—	2	4	no	—
93	#533	C₂×C₉	1	1	yes	characteristic, complemented
94	C₂×#124	C₂×C₉	1	1	yes	characteristic
95	#489	C₂×C₉	1	4	yes	complemented
96	#488	C₂×C₉	1	4	yes	complemented
97	#419	C₂×C₉	1	2	yes	complemented
98	#418	C₂×C₉	1	2	yes	—
99	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
100	C₂⁴×C₃	—	2	1	no	—
101	C₂×C₃×Q₈	—	2	2	no	—
102	C₂×C₃×Q₈	—	2	1	no	—
103	C₂×C₃×Q₈	—	2	4	no	—
104	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
105	C₂×C₃×D₄	—	2	1	no	—
106	C₂×C₃×D₄	—	2	4	no	—
107	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
110	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₃×M₁₆	—	2	4	no	—
112	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
116	C₃×C₂²⋊C₄	—	2	2	no	—
117	C₃×C₂²⋊C₄	—	2	4	no	—
118	C₃×C₂²⋊C₄	—	2	4	no	—
119	C₃×C₄²	—	2	2	no	—
120	C₃×C₄²	C₃×D₄	1	2	yes	—
121	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
122	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	4	2	no	—
125	C₂²×C₉	—	4	4	no	—
126	C₂²×C₉	—	8	4	no	—
127	C₂²×C₉	#124	1	2	yes	—
128	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₉	—	2	2	no	—
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	—	4	2	no	—
132	C₄×C₉	—	4	4	no	—
133	C₄×C₉	#124	1	2	yes	—
134	C₄×C₉	C₂×#41	1	2	yes	—
135	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
136	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₉	1	2	yes	—
138	C₄⋊Q₈	C₂²×C₉	1	2	yes	complemented
139	#125	C₂²×C₉	1	2	yes	complemented
140	#124	—	2	2	no	—
141	#124	C₂²×C₉	1	2	yes	—
142	#124	C₂²×C₉	1	2	yes	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	#117	C₂²×C₉	1	2	yes	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
148	#103	C₂²×C₉	1	8	yes	complemented
149	#102	C₂²×C₉	1	8	yes	—
150	#97	C₂²×C₉	1	2	yes	complemented
151	C₄×C₈	C₂²×C₉	1	2	yes	complemented
152	C₂³×C₃	—	2	2	no	—
153	C₂³×C₃	—	4	4	no	—
154	C₂³×C₃	—	4	2	no	—
155	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
156	C₃×Q₈	—	2	4	no	—
157	C₃×Q₈	—	4	4	no	—
158	C₃×Q₈	—	4	2	no	—
159	C₃×D₄	—	2	4	no	—
160	C₃×D₄	—	4	2	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×#41	1	4	yes	—
169	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
170	C₃×C₈	—	2	4	no	—
171	C₂×C₉	—	2	2	no	—
172	C₂×C₉	—	8	2	no	—
173	C₂×C₉	C₂×#124	1	1	yes	characteristic
174	C₂×C₉	#575	1	2	yes	—
175	C₂⁴	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	C₂³×C₉	1	2	yes	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	C₂³×C₉	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	D₄×C₉	1	2	yes	—
185	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
186	M₁₆	—	2	4	no	—
187	C₂×C₈	C₂³×C₉	1	4	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	C₂³×C₉	1	4	yes	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	D₄×C₉	1	2	yes	—
196	C₄²	C₂³×C₉	1	1	yes	characteristic
197	C₄²	C₂³×C₉	1	1	yes	characteristic
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	4	4	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	—	8	4	no	—
202	C₂²×C₃	C₃×#124	1	2	yes	—
203	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	4	4	no	—
207	C₃×C₄	—	4	2	no	—
208	C₃×C₄	C₃×#124	1	2	yes	—
209	C₃×C₄	C₂×C₃×#41	1	2	yes	—
210	C₉	#3140	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³	—	2	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	4	2	no	—
218	D₄	—	2	4	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
225	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
227	C₂×C₄	C₉×#41	1	4	yes	—
228	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₂×C₃	—	2	2	no	—
231	C₂×C₃	—	8	2	no	—
232	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
233	C₂×C₃	C₃×#575	1	2	yes	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	8	4	no	—
238	C₂²	C₉×#124	1	2	yes	—
239	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	4	no	—
244	C₄	C₉×#124	1	2	yes	—
245	C₄	C₂×C₉×#41	1	2	yes	—
246	C₃	C₃×#3140	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	8	2	no	—
249	C₂	C₂×C₉×#124	1	1	yes	characteristic
250	C₂	C₉×#575	1	2	yes	—

	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	C₂²×C₃×D₄	C₂²×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
1. C₉×#533	C₂	semidirect
3. C₉×#489	C₂	semidirect
4. C₉×#488	C₂	semidirect
5. C₉×#419	C₂	semidirect
7. C₄×C₉×M₁₆	C₂	semidirect
12. C₉×C₄⋊Q₈	C₂²	semidirect
13. C₉×#125	C₂²	semidirect
22. C₉×#103	C₂²	semidirect
24. C₉×#97	C₂²	semidirect
25. C₄×C₈×C₉	C₂²	semidirect
56. #3140	C₉	direct (N × H)
93. #533	C₂×C₉	semidirect
95. #489	C₂×C₉	semidirect
96. #488	C₂×C₉	semidirect
97. #419	C₂×C₉	semidirect
99. C₄×M₁₆	C₂×C₉	semidirect
138. C₄⋊Q₈	C₂²×C₉	semidirect
139. #125	C₂²×C₉	semidirect
148. #103	C₂²×C₉	semidirect
150. #97	C₂²×C₉	semidirect
151. C₄×C₈	C₂²×C₉	semidirect
210. C₉	#3140	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×#533	2
C₂×C₉×#124	2
C₉×#489	2
C₉×#488	2
C₉×#419	2
C₉×#418	2
C₄×C₉×M₁₆	2
C₃×#3140	3

Subgroups — C9×#3140 #11767861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3140 #11767861