Subgroups — C₉×#3202 #11767923

Order 1152.

567 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×#536	C₂	1	2	yes	maximal
2	C₉×#485	C₂	1	4	yes	maximal, complemented
3	C₉×#470	C₂	1	2	yes	maximal, complemented
4	C₉×#467	C₂	1	2	yes	maximal, complemented
5	C₉×#451	C₂	1	4	yes	maximal, complemented
6	C₉×#408	C₂	1	1	yes	maximal, characteristic
7	C₃×#3202	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊Q₈	—	2	2	no	—
15	C₉×#124	—	2	2	no	—
16	C₉×#123	—	2	2	no	—
17	C₉×#123	—	2	2	no	—
18	C₉×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₉×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₉×#117	C₂²	1	2	yes	—
21	C₉×#107	C₂²	1	2	yes	—
22	C₉×#106	C₂²	1	2	yes	—
23	C₉×#103	C₂²	1	4	yes	—
24	C₉×#103	C₂²	1	4	yes	—
25	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
26	C₃×#536	C₂×C₃	1	2	yes	—
27	C₃×#485	C₂×C₃	1	4	yes	—
28	C₃×#470	C₂×C₃	1	2	yes	—
29	C₃×#467	C₂×C₃	1	2	yes	—
30	C₃×#451	C₂×C₃	1	4	yes	—
31	C₃×#408	C₂×C₃	1	1	yes	characteristic
32	C₉×#41	—	4	1	no	—
33	C₉×#41	—	4	1	no	—
34	C₂×Q₈×C₉	—	2	2	no	—
35	C₂×Q₈×C₉	—	4	2	no	—
36	C₂×Q₈×C₉	—	4	2	no	—
37	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
38	C₂×Q₈×C₉	C₂³	1	4	yes	—
39	C₂×D₄×C₉	D₄	1	1	yes	characteristic
40	C₂×D₄×C₉	D₄	1	2	yes	—
41	C₂²×C₄×C₉	—	2	2	no	—
42	C₂²×C₄×C₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×Q₁₆	—	4	2	no	—
45	C₉×Q₁₆	—	4	2	no	—
46	C₉×M₁₆	—	4	1	no	—
47	C₂×C₈×C₉	—	4	1	no	—
48	C₂×C₈×C₉	C₂³	1	2	yes	—
49	C₂×C₈×C₉	C₂³	1	2	yes	—
50	C₉×C₄⋊C₄	—	2	4	no	—
51	C₉×C₄⋊C₄	—	2	4	no	—
52	C₉×C₄⋊C₄	C₂³	1	4	yes	—
53	C₉×C₂²⋊C₄	—	2	2	no	—
54	C₉×C₂²⋊C₄	—	2	4	no	—
55	C₉×C₂²⋊C₄	—	2	4	no	—
56	C₄²×C₉	—	2	2	no	—
57	#3202	C₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
60	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
61	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
62	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₃×C₄⋊Q₈	—	2	2	no	—
65	C₃×#124	—	2	2	no	—
66	C₃×#123	—	2	2	no	—
67	C₃×#123	—	2	2	no	—
68	C₃×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
69	C₃×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
70	C₃×#117	C₂²×C₃	1	2	yes	—
71	C₃×#107	C₂²×C₃	1	2	yes	—
72	C₃×#106	C₂²×C₃	1	2	yes	—
73	C₃×#103	C₂²×C₃	1	4	yes	—
74	C₃×#103	C₂²×C₃	1	4	yes	—
75	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
76	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₉	C₂×D₄	1	2	yes	—
78	Q₈×C₉	—	2	4	no	—
79	Q₈×C₉	—	2	4	no	—
80	Q₈×C₉	—	4	1	no	—
81	Q₈×C₉	—	4	4	no	—
82	D₄×C₉	—	4	1	no	—
83	D₄×C₉	—	4	2	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	2	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	—	4	4	no	—
91	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₉	—	2	2	no	—
95	C₈×C₉	—	2	2	no	—
96	#536	C₂×C₉	1	2	yes	—
97	#485	C₂×C₉	1	4	yes	complemented
98	#470	C₂×C₉	1	2	yes	complemented
99	#467	C₂×C₉	1	2	yes	complemented
100	#451	C₂×C₉	1	4	yes	complemented
101	#408	C₂×C₉	1	1	yes	characteristic
102	C₃×#41	—	4	1	no	—
103	C₃×#41	—	4	1	no	—
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₂×C₃×Q₈	—	4	2	no	—
106	C₂×C₃×Q₈	—	4	2	no	—
107	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
108	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
109	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
111	C₂²×C₃×C₄	—	2	2	no	—
112	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₃×Q₁₆	—	4	2	no	—
115	C₃×Q₁₆	—	4	2	no	—
116	C₃×M₁₆	—	4	1	no	—
117	C₂×C₃×C₈	—	4	1	no	—
118	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
119	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
120	C₃×C₄⋊C₄	—	2	4	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
123	C₃×C₂²⋊C₄	—	2	2	no	—
124	C₃×C₂²⋊C₄	—	2	4	no	—
125	C₃×C₂²⋊C₄	—	2	4	no	—
126	C₃×C₄²	—	2	2	no	—
127	C₂²×C₉	—	2	2	no	—
128	C₂²×C₉	—	2	1	no	—
129	C₂²×C₉	—	2	2	no	—
130	C₂²×C₉	—	2	4	no	—
131	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
132	C₄×C₉	—	2	1	no	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	4	4	no	—
135	C₄×C₉	—	4	4	no	—
136	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×#41	C₂²×C₉	1	1	yes	characteristic
139	C₂²×Q₈	C₂²×C₉	1	2	yes	—
140	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
141	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
142	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
144	C₄⋊Q₈	—	2	2	no	—
145	#124	—	2	2	no	—
146	#123	—	2	2	no	—
147	#123	—	2	2	no	—
148	C₂²⋊Q₈	C₂²×C₉	1	4	yes	—
149	C₂²⋊Q₈	C₂²×C₉	1	4	yes	—
150	#117	C₂²×C₉	1	2	yes	—
151	#107	C₂²×C₉	1	2	yes	—
152	#106	C₂²×C₉	1	2	yes	—
153	#103	C₂²×C₉	1	4	yes	—
154	#103	C₂²×C₉	1	4	yes	—
155	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
156	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
158	C₃×Q₈	—	2	4	no	—
159	C₃×Q₈	—	2	4	no	—
160	C₃×Q₈	—	4	1	no	—
161	C₃×Q₈	—	4	4	no	—
162	C₃×D₄	—	4	2	no	—
163	C₃×D₄	—	4	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	4	4	no	—
171	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
174	C₃×C₈	—	2	2	no	—
175	C₃×C₈	—	2	2	no	—
176	C₂×C₉	—	4	2	no	—
177	C₂×C₉	—	4	1	no	—
178	C₂×C₉	#534	1	1	yes	characteristic
179	C₂×C₉	#578	1	2	yes	—
180	#41	—	4	1	no	—
181	#41	—	4	1	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
186	C₂×Q₈	C₂³×C₉	1	4	yes	—
187	C₂×D₄	D₄×C₉	1	1	yes	characteristic
188	C₂×D₄	D₄×C₉	1	2	yes	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂³×C₉	1	2	yes	—
191	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
192	Q₁₆	—	4	2	no	—
193	Q₁₆	—	4	2	no	—
194	M₁₆	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	C₂³×C₉	1	2	yes	—
197	C₂×C₈	C₂³×C₉	1	2	yes	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	C₂³×C₉	1	4	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	4	4	no	—
213	C₃×C₄	—	4	4	no	—
214	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
215	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₉	#3202	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
218	C₂³	C₂×D₄×C₉	1	2	yes	—
219	Q₈	—	2	4	no	—
220	Q₈	—	2	4	no	—
221	Q₈	—	4	1	no	—
222	Q₈	—	4	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
233	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
234	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	—	2	2	no	—
237	C₂×C₃	—	4	2	no	—
238	C₂×C₃	—	4	1	no	—
239	C₂×C₃	C₃×#534	1	1	yes	characteristic
240	C₂×C₃	C₃×#578	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	4	no	—
249	C₄	—	4	4	no	—
250	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
251	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
252	C₃	C₃×#3202	1	1	yes	characteristic
253	C₂	—	4	2	no	—
254	C₂	—	4	1	no	—
255	C₂	C₉×#534	1	1	yes	characteristic
256	C₂	C₉×#578	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	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₉×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#485	C₂	semidirect
3. C₉×#470	C₂	semidirect
4. C₉×#467	C₂	semidirect
5. C₉×#451	C₂	semidirect
57. #3202	C₉	direct (N × H)
97. #485	C₂×C₉	semidirect
98. #470	C₂×C₉	semidirect
99. #467	C₂×C₉	semidirect
100. #451	C₂×C₉	semidirect
216. C₉	#3202	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×#536	2
C₉×#485	2
C₉×#470	2
C₉×#467	2
C₉×#451	2
C₉×#408	2
C₃×#3202	3

Subgroups — C9×#3202 #11767923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3202 #11767923