Subgroups — C₉×#2962 #11767683

Order 1152.

534 conjugacy classes of subgroups.

The subgroup lattice has 273 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
2	C₉×#446	C₂	1	2	yes	maximal, complemented
3	C₉×#446	C₂	1	4	yes	maximal, complemented
4	C₈×Q₈×C₉	C₂	1	2	yes	maximal, complemented
5	C₉×#424	C₂	1	2	yes	maximal
6	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₉×#405	C₂	1	2	yes	maximal
8	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₃×#2962	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
14	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
15	C₉×#125	C₄	1	2	yes	—
16	C₉×C₂²⋊Q₈	C₄	1	2	yes	—
17	C₉×C₂²⋊Q₈	C₄	1	2	yes	—
18	C₄×Q₈×C₉	C₂²	1	2	yes	—
19	C₄×Q₈×C₉	C₂²	1	2	yes	—
20	C₉×#117	C₂²	1	2	yes	—
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₉×C₄⋊C₈	C₂²	1	2	yes	—
25	C₉×C₄⋊C₈	C₂²	1	4	yes	—
26	C₉×C₄⋊C₈	C₂²	1	4	yes	—
27	C₉×#97	C₂²	1	2	yes	—
28	C₉×#97	C₂²	1	4	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	—
30	C₄×C₈×C₉	C₂²	1	4	yes	—
31	C₃×#533	C₂×C₃	1	1	yes	characteristic
32	C₃×#446	C₂×C₃	1	2	yes	—
33	C₃×#446	C₂×C₃	1	4	yes	—
34	C₃×C₈×Q₈	C₂×C₃	1	2	yes	—
35	C₃×#424	C₂×C₃	1	2	yes	—
36	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
37	C₃×#405	C₂×C₃	1	2	yes	—
38	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
39	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
40	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×M₁₆	—	2	2	no	—
45	C₉×M₁₆	Q₈	1	4	yes	complemented
46	C₂×C₈×C₉	—	2	4	no	—
47	C₂×C₈×C₉	C₂³	1	2	yes	—
48	C₂×C₈×C₉	C₂³	1	2	yes	—
49	C₂×C₈×C₉	C₂³	1	2	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×C₄⋊C₄	C₂×C₄	1	4	yes	—
54	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
55	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
58	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₉	C₂³	1	1	yes	characteristic
61	C₄²×C₉	C₂³	1	1	yes	characteristic
62	C₄²×C₉	C₂³	1	4	yes	—
63	#2962	C₉	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
67	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
68	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
69	C₃×#125	C₃×C₄	1	2	yes	—
70	C₃×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
71	C₃×C₂²⋊Q₈	C₃×C₄	1	2	yes	—
72	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
73	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
74	C₃×#117	C₂²×C₃	1	2	yes	—
75	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
76	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
77	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
78	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
79	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
80	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
81	C₃×#97	C₂²×C₃	1	2	yes	—
82	C₃×#97	C₂²×C₃	1	4	yes	—
83	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
85	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
86	Q₈×C₉	—	2	4	no	—
87	Q₈×C₉	—	4	2	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	4	1	no	—
90	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
95	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₉	#41	1	2	yes	—
99	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
100	C₈×C₉	—	2	2	no	—
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	—	2	2	no	—
103	C₈×C₉	C₂×Q₈	1	4	yes	—
104	#533	C₂×C₉	1	1	yes	characteristic
105	#446	C₂×C₉	1	2	yes	complemented
106	#446	C₂×C₉	1	4	yes	complemented
107	C₈×Q₈	C₂×C₉	1	2	yes	complemented
108	#424	C₂×C₉	1	2	yes	—
109	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
110	#405	C₂×C₉	1	2	yes	—
111	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic
112	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
114	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₃×M₁₆	—	2	2	no	—
118	C₃×M₁₆	C₃×Q₈	1	4	yes	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
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	2	yes	—
124	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
125	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
126	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
127	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
128	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
129	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
130	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
131	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
132	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
133	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
134	C₃×C₄²	C₂³×C₃	1	4	yes	—
135	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	—	2	1	no	—
138	C₂²×C₉	C₄×Q₈	1	2	yes	—
139	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	2	4	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	2	1	no	—
144	C₄×C₉	—	2	4	no	—
145	C₄×C₉	C₄×Q₈	1	2	yes	—
146	C₄×C₉	#131	1	2	yes	—
147	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
148	C₄×C₉	C₂×#41	1	1	yes	characteristic
149	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
150	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
151	C₂²×C₈	C₂²×C₉	1	2	yes	—
152	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
153	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
154	#125	C₄×C₉	1	2	yes	—
155	C₂²⋊Q₈	C₄×C₉	1	2	yes	—
156	C₂²⋊Q₈	C₄×C₉	1	2	yes	—
157	C₄×Q₈	C₂²×C₉	1	2	yes	—
158	C₄×Q₈	C₂²×C₉	1	2	yes	—
159	#117	C₂²×C₉	1	2	yes	—
160	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
161	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
162	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
163	C₄⋊C₈	C₂²×C₉	1	2	yes	—
164	C₄⋊C₈	C₂²×C₉	1	4	yes	—
165	C₄⋊C₈	C₂²×C₉	1	4	yes	—
166	#97	C₂²×C₉	1	2	yes	—
167	#97	C₂²×C₉	1	4	yes	—
168	C₄×C₈	C₂²×C₉	1	2	yes	—
169	C₄×C₈	C₂²×C₉	1	4	yes	—
170	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
171	C₃×Q₈	—	2	4	no	—
172	C₃×Q₈	—	4	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	4	1	no	—
175	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
179	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
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	C₂×C₃×Q₈	1	2	yes	—
183	C₂×C₃×C₄	C₃×#41	1	2	yes	—
184	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
189	C₂×C₉	—	2	2	no	—
190	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₂×C₉	C₂×#131	1	1	yes	characteristic
192	C₂×C₉	#568	1	1	yes	characteristic
193	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
194	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
195	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
198	M₁₆	—	2	2	no	—
199	M₁₆	Q₈×C₉	1	4	yes	complemented
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	C₂³×C₉	1	2	yes	—
202	C₂×C₈	C₂³×C₉	1	2	yes	—
203	C₂×C₈	C₂³×C₉	1	2	yes	—
204	C₂×C₈	C₂³×C₉	1	2	yes	—
205	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
206	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
207	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
208	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
209	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
210	C₄⋊C₄	C₂×C₄×C₉	1	2	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	4	yes	—
216	C₄²	C₂³×C₉	1	1	yes	characteristic
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
220	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	4	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
227	C₃×C₄	C₃×#131	1	2	yes	—
228	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
229	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
230	C₉	#2962	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
240	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
241	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
243	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
244	C₂×C₄	C₉×#41	1	2	yes	—
245	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	C₂×Q₈×C₉	1	4	yes	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
252	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
253	C₂×C₃	C₃×#568	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	C₄×Q₈×C₉	1	2	yes	—
257	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	4	no	—
263	C₄	C₄×Q₈×C₉	1	2	yes	—
264	C₄	C₉×#131	1	2	yes	—
265	C₄	C₂²×Q₈×C₉	1	1	yes	characteristic
266	C₄	C₂×C₉×#41	1	1	yes	characteristic
267	C₃	C₃×#2962	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	C₂×C₄×Q₈×C₉	1	1	yes	characteristic
270	C₂	C₂×C₉×#131	1	1	yes	characteristic
271	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	C₂³×C₃	C₂²×C₃×C₄
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×#446	C₂	semidirect
3. C₉×#446	C₂	semidirect
4. C₈×Q₈×C₉	C₂	semidirect
45. C₉×M₁₆	Q₈	semidirect
63. #2962	C₉	direct (N × H)
105. #446	C₂×C₉	semidirect
106. #446	C₂×C₉	semidirect
107. C₈×Q₈	C₂×C₉	semidirect
199. M₁₆	Q₈×C₉	semidirect
230. C₉	#2962	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
C₉×#446	2
C₈×Q₈×C₉	2
C₉×#424	2
C₂×C₉×C₄⋊C₈	2
C₉×#405	2
C₄×C₉×M₁₆	2
C₃×#2962	3

Subgroups — C9×#2962 #11767683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2962 #11767683