Subgroups — C₉×#2080 #11766801

Order 1152.

285 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#421	C₂	1	1	yes	maximal, characteristic
3	C₉×#416	C₂	1	1	yes	maximal, characteristic
4	C₉×#413	C₂	1	1	yes	maximal, characteristic
5	C₉×#344	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2080	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₉×#117	C₂²	1	1	yes	characteristic
15	C₉×#117	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊C₈	—	2	2	no	—
17	C₉×#104	—	2	2	no	—
18	C₉×#103	—	2	1	no	—
19	C₉×#102	—	2	1	no	—
20	C₉×#101	—	2	1	no	—
21	C₉×#100	—	2	1	no	—
22	C₃×#424	C₂×C₃	1	1	yes	characteristic
23	C₃×#421	C₂×C₃	1	1	yes	characteristic
24	C₃×#416	C₂×C₃	1	1	yes	characteristic
25	C₃×#413	C₂×C₃	1	1	yes	characteristic
26	C₃×#344	C₂×C₃	1	1	yes	characteristic
27	C₃×#341	C₂×C₃	1	1	yes	characteristic
28	C₃×H(3,ℤ/4)	C₂×C₃	1	1	yes	characteristic
29	C₉×#41	—	2	1	no	—
30	C₉×#41	—	2	1	no	—
31	C₉×#41	—	2	1	no	—
32	C₉×#41	—	2	1	no	—
33	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
34	C₂×D₄×C₉	—	2	1	no	—
35	C₂×D₄×C₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	—	2	1	no	—
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×M₁₆	—	2	1	no	—
39	C₉×M₁₆	—	2	1	no	—
40	C₉×M₁₆	—	2	1	no	—
41	C₉×M₁₆	—	2	1	no	—
42	C₉×M₁₆	—	2	1	no	—
43	C₉×M₁₆	—	2	1	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	—	2	1	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	—	2	1	no	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₄⋊C₄	—	2	1	no	—
52	C₉×C₂²⋊C₄	—	2	1	no	—
53	C₉×C₂²⋊C₄	—	2	1	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	D₄	1	2	yes	—
56	#2080	C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×#117	C₂²×C₃	1	1	yes	characteristic
63	C₃×#117	C₂²×C₃	1	1	yes	characteristic
64	C₃×C₄⋊C₈	—	2	2	no	—
65	C₃×#104	—	2	2	no	—
66	C₃×#103	—	2	1	no	—
67	C₃×#102	—	2	1	no	—
68	C₃×#101	—	2	1	no	—
69	C₃×#100	—	2	1	no	—
70	C₂³×C₉	—	2	1	no	—
71	C₂³×C₉	#41	1	1	yes	characteristic
72	Q₈×C₉	—	2	1	no	—
73	Q₈×C₉	—	2	1	no	—
74	D₄×C₉	—	2	1	no	—
75	D₄×C₉	—	2	1	no	—
76	D₄×C₉	—	4	1	no	—
77	D₄×C₉	—	4	1	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	4	1	no	—
81	C₂×C₄×C₉	—	4	1	no	—
82	C₂×C₄×C₉	—	4	1	no	—
83	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₉	#41	1	1	yes	characteristic
86	C₂×C₄×C₉	#41	1	1	yes	characteristic
87	C₂×C₄×C₉	#41	1	1	yes	characteristic
88	C₂×C₄×C₉	#41	1	1	yes	characteristic
89	C₈×C₉	—	4	1	no	—
90	C₈×C₉	—	4	1	no	—
91	C₈×C₉	—	4	1	no	—
92	C₈×C₉	—	4	1	no	—
93	#424	C₂×C₉	1	1	yes	characteristic, complemented
94	#421	C₂×C₉	1	1	yes	characteristic
95	#416	C₂×C₉	1	1	yes	characteristic
96	#413	C₂×C₉	1	1	yes	characteristic
97	#344	C₂×C₉	1	1	yes	characteristic, complemented
98	#341	C₂×C₉	1	1	yes	characteristic, complemented
99	H(3,ℤ/4)	C₂×C₉	1	1	yes	characteristic, complemented
100	C₃×#41	—	2	1	no	—
101	C₃×#41	—	2	1	no	—
102	C₃×#41	—	2	1	no	—
103	C₃×#41	—	2	1	no	—
104	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
105	C₂×C₃×D₄	—	2	1	no	—
106	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₃×M₁₆	—	2	1	no	—
110	C₃×M₁₆	—	2	1	no	—
111	C₃×M₁₆	—	2	1	no	—
112	C₃×M₁₆	—	2	1	no	—
113	C₃×M₁₆	—	2	1	no	—
114	C₃×M₁₆	—	2	1	no	—
115	C₂×C₃×C₈	—	2	1	no	—
116	C₂×C₃×C₈	—	2	1	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₃×C₄⋊C₄	—	2	1	no	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₂²⋊C₄	—	2	1	no	—
124	C₃×C₂²⋊C₄	—	2	1	no	—
125	C₃×C₄²	—	2	1	no	—
126	C₃×C₄²	C₃×D₄	1	2	yes	—
127	C₂²×C₉	—	4	1	no	—
128	C₂²×C₉	—	4	1	no	—
129	C₂²×C₉	—	8	1	no	—
130	C₂²×C₉	#123	1	1	yes	characteristic
131	C₂²×C₉	#123	1	1	yes	characteristic
132	C₂²×C₉	#126	1	1	yes	characteristic
133	C₄×C₉	—	4	2	no	—
134	C₄×C₉	—	4	1	no	—
135	C₄×C₉	—	8	1	no	—
136	C₄×C₉	#121	1	1	yes	characteristic
137	C₄×C₉	#123	1	1	yes	characteristic
138	C₄×C₉	#124	1	1	yes	characteristic
139	C₄×C₉	#126	1	1	yes	characteristic
140	C₂×#41	C₂²×C₉	1	1	yes	characteristic
141	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
142	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
143	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
144	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
145	#117	C₂²×C₉	1	1	yes	characteristic
146	#117	C₂²×C₉	1	1	yes	characteristic
147	C₄⋊C₈	—	2	2	no	—
148	#104	—	2	2	no	—
149	#103	—	2	1	no	—
150	#102	—	2	1	no	—
151	#101	—	2	1	no	—
152	#100	—	2	1	no	—
153	C₂³×C₃	—	2	1	no	—
154	C₂³×C₃	C₃×#41	1	1	yes	characteristic
155	C₃×Q₈	—	2	1	no	—
156	C₃×Q₈	—	2	1	no	—
157	C₃×D₄	—	2	1	no	—
158	C₃×D₄	—	2	1	no	—
159	C₃×D₄	—	4	1	no	—
160	C₃×D₄	—	4	1	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
170	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
171	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
172	C₃×C₈	—	4	1	no	—
173	C₃×C₈	—	4	1	no	—
174	C₃×C₈	—	4	1	no	—
175	C₃×C₈	—	4	1	no	—
176	C₂×C₉	—	2	1	no	—
177	C₂×C₉	—	2	1	no	—
178	C₂×C₉	—	2	1	no	—
179	C₂×C₉	—	8	1	no	—
180	C₂×C₉	#397	1	1	yes	characteristic
181	#41	—	2	1	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	2	1	no	—
185	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	D₄×C₉	1	1	yes	characteristic
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	1	no	—
192	M₁₆	—	2	1	no	—
193	M₁₆	—	2	1	no	—
194	M₁₆	—	2	1	no	—
195	M₁₆	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	D₄×C₉	1	2	yes	—
208	C₂²×C₃	—	4	1	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	—	8	1	no	—
211	C₂²×C₃	C₃×#123	1	1	yes	characteristic
212	C₂²×C₃	C₃×#123	1	1	yes	characteristic
213	C₂²×C₃	C₃×#126	1	1	yes	characteristic
214	C₃×C₄	—	4	2	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	—	8	1	no	—
217	C₃×C₄	C₃×#121	1	1	yes	characteristic
218	C₃×C₄	C₃×#123	1	1	yes	characteristic
219	C₃×C₄	C₃×#124	1	1	yes	characteristic
220	C₃×C₄	C₃×#126	1	1	yes	characteristic
221	C₉	#2080	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	—	2	1	no	—
223	C₂³	C₉×#41	1	1	yes	characteristic
224	Q₈	—	2	1	no	—
225	Q₈	—	2	1	no	—
226	D₄	—	2	1	no	—
227	D₄	—	2	1	no	—
228	D₄	—	4	1	no	—
229	D₄	—	4	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
237	C₂×C₄	C₉×#41	1	1	yes	characteristic
238	C₂×C₄	C₉×#41	1	1	yes	characteristic
239	C₂×C₄	C₉×#41	1	1	yes	characteristic
240	C₂×C₄	C₉×#41	1	1	yes	characteristic
241	C₈	—	4	1	no	—
242	C₈	—	4	1	no	—
243	C₈	—	4	1	no	—
244	C₈	—	4	1	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	8	1	no	—
249	C₂×C₃	C₃×#397	1	1	yes	characteristic
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	8	1	no	—
253	C₂²	C₉×#123	1	1	yes	characteristic
254	C₂²	C₉×#123	1	1	yes	characteristic
255	C₂²	C₉×#126	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	8	1	no	—
259	C₄	C₉×#121	1	1	yes	characteristic
260	C₄	C₉×#123	1	1	yes	characteristic
261	C₄	C₉×#124	1	1	yes	characteristic
262	C₄	C₉×#126	1	1	yes	characteristic
263	C₃	C₃×#2080	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	8	1	no	—
268	C₂	C₉×#397	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	#124
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#397
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#424	C₂	semidirect
5. C₉×#344	C₂	semidirect
6. C₉×#341	C₂	semidirect
7. C₉×H(3,ℤ/4)	C₂	semidirect
56. #2080	C₉	direct (N × H)
93. #424	C₂×C₉	semidirect
97. #344	C₂×C₉	semidirect
98. #341	C₂×C₉	semidirect
99. H(3,ℤ/4)	C₂×C₉	semidirect
221. C₉	#2080	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×#424	2
C₉×#421	2
C₉×#416	2
C₉×#413	2
C₉×#344	2
C₉×#341	2
C₉×H(3,ℤ/4)	2
C₃×#2080	3

Subgroups — C9×#2080 #11766801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2080 #11766801