Subgroups — C₉×#3058 #11767779

Order 1152.

786 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#535	C₂	1	2	yes	maximal, complemented
4	C₉×#469	C₂	1	4	yes	maximal, complemented
5	C₉×#466	C₂	1	4	yes	maximal, complemented
6	C₉×#427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#418	C₂	1	2	yes	maximal, complemented
8	C₃×#3058	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	C₂²	1	2	yes	—
11	C₂×C₉×D₈	—	2	4	no	—
12	C₂×C₉×D₈	C₂²	1	2	yes	—
13	C₂×C₉×D₈	C₂²	1	2	yes	—
14	C₉×#131	—	2	4	no	—
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	2	yes	complemented
17	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₉	C₂²	1	2	yes	complemented
19	C₉×#127	—	2	2	no	—
20	C₉×#124	—	2	2	no	—
21	C₉×#121	C₂²	1	8	yes	complemented
22	C₉×#120	—	2	4	no	—
23	C₉×#117	C₂²	1	2	yes	—
24	C₉×#107	C₂²	1	2	yes	complemented
25	C₉×#106	C₂²	1	2	yes	complemented
26	C₉×#102	C₂²	1	8	yes	complemented
27	C₂²×C₃×D₈	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
29	C₃×#535	C₂×C₃	1	2	yes	—
30	C₃×#469	C₂×C₃	1	4	yes	—
31	C₃×#466	C₂×C₃	1	4	yes	—
32	C₃×#427	C₂×C₃	1	1	yes	characteristic
33	C₃×#418	C₂×C₃	1	2	yes	—
34	C₂⁴×C₉	—	2	2	no	—
35	C₉×#41	—	2	2	no	—
36	C₉×#41	—	2	2	no	—
37	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
38	C₂×D₄×C₉	—	2	8	no	—
39	C₂×D₄×C₉	—	2	4	no	—
40	C₂×D₄×C₉	—	2	2	no	—
41	C₂×D₄×C₉	—	2	4	no	—
42	C₂×D₄×C₉	D₄	1	2	yes	—
43	C₂×D₄×C₉	D₄	1	1	yes	characteristic
44	C₂×D₄×C₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×D₈	—	2	4	no	—
48	C₉×D₈	—	2	4	no	—
49	C₉×M₁₆	—	2	2	no	—
50	C₉×M₁₆	—	2	4	no	—
51	C₂×C₈×C₉	—	2	4	no	—
52	C₂×C₈×C₉	D₄	1	4	yes	complemented
53	C₂×C₈×C₉	C₂³	1	2	yes	—
54	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
56	C₉×C₄⋊C₄	C₂³	1	4	yes	—
57	C₉×C₂²⋊C₄	—	2	2	no	—
58	C₉×C₂²⋊C₄	—	2	8	no	—
59	C₄²×C₉	—	2	2	no	—
60	#3058	C₉	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
63	C₂×C₃×D₈	—	2	4	no	—
64	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
65	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
66	C₃×#131	—	2	4	no	—
67	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
69	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
71	C₃×#127	—	2	2	no	—
72	C₃×#124	—	2	2	no	—
73	C₃×#121	C₂²×C₃	1	8	yes	—
74	C₃×#120	—	2	4	no	—
75	C₃×#117	C₂²×C₃	1	2	yes	—
76	C₃×#107	C₂²×C₃	1	2	yes	—
77	C₃×#106	C₂²×C₃	1	2	yes	—
78	C₃×#102	C₂²×C₃	1	8	yes	—
79	C₂³×C₉	—	2	4	no	—
80	C₂³×C₉	—	4	4	no	—
81	C₂³×C₉	—	8	4	no	—
82	C₂³×C₉	C₂×D₄	1	2	yes	—
83	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₉	—	2	2	no	—
85	D₄×C₉	—	2	4	no	—
86	D₄×C₉	—	2	2	no	—
87	D₄×C₉	—	2	4	no	—
88	D₄×C₉	—	2	4	no	—
89	D₄×C₉	—	4	4	no	—
90	D₄×C₉	—	8	8	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	4	no	—
94	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₉	#41	1	2	yes	—
98	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	C₂×D₄	1	2	yes	—
101	C₈×C₉	C₂×D₄	1	2	yes	—
102	C₂²×D₈	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂×#131	C₂×C₉	1	1	yes	characteristic, complemented
104	#535	C₂×C₉	1	2	yes	complemented
105	#469	C₂×C₉	1	4	yes	complemented
106	#466	C₂×C₉	1	4	yes	complemented
107	#427	C₂×C₉	1	1	yes	characteristic, complemented
108	#418	C₂×C₉	1	2	yes	complemented
109	C₂⁴×C₃	—	2	2	no	—
110	C₃×#41	—	2	2	no	—
111	C₃×#41	—	2	2	no	—
112	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
113	C₂×C₃×D₄	—	2	8	no	—
114	C₂×C₃×D₄	—	2	4	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	—	2	4	no	—
117	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
118	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
119	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₃×D₈	—	2	4	no	—
123	C₃×D₈	—	2	4	no	—
124	C₃×M₁₆	—	2	2	no	—
125	C₃×M₁₆	—	2	4	no	—
126	C₂×C₃×C₈	—	2	4	no	—
127	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
128	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
129	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
131	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
132	C₃×C₂²⋊C₄	—	2	2	no	—
133	C₃×C₂²⋊C₄	—	2	8	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₂²×C₉	—	2	2	no	—
136	C₂²×C₉	—	2	2	no	—
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	—	4	4	no	—
139	C₂²×C₉	—	4	8	no	—
140	C₂²×C₉	—	8	8	no	—
141	C₂²×C₉	#121	1	2	yes	—
142	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	4	4	no	—
145	C₄×C₉	#121	1	2	yes	—
146	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₉	C₂×#41	1	1	yes	characteristic
148	C₂×#41	C₂²×C₉	1	1	yes	characteristic
149	C₂²×D₄	C₂²×C₉	1	2	yes	—
150	C₂×D₈	—	2	4	no	—
151	C₂×D₈	C₂²×C₉	1	2	yes	—
152	C₂×D₈	C₂²×C₉	1	2	yes	—
153	#131	—	2	4	no	—
154	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
155	C₂×M₁₆	C₂²×C₉	1	2	yes	complemented
156	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
157	C₂²×C₈	C₂²×C₉	1	2	yes	complemented
158	#127	—	2	2	no	—
159	#124	—	2	2	no	—
160	#121	C₂²×C₉	1	8	yes	complemented
161	#120	—	2	4	no	—
162	#117	C₂²×C₉	1	2	yes	—
163	#107	C₂²×C₉	1	2	yes	complemented
164	#106	C₂²×C₉	1	2	yes	complemented
165	#102	C₂²×C₉	1	8	yes	complemented
166	C₂³×C₃	—	2	4	no	—
167	C₂³×C₃	—	4	4	no	—
168	C₂³×C₃	—	8	4	no	—
169	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
170	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃×Q₈	—	2	2	no	—
172	C₃×D₄	—	2	4	no	—
173	C₃×D₄	—	2	2	no	—
174	C₃×D₄	—	2	4	no	—
175	C₃×D₄	—	2	4	no	—
176	C₃×D₄	—	4	4	no	—
177	C₃×D₄	—	8	8	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₃×C₄	C₃×#41	1	2	yes	—
185	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
188	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₉	—	2	2	no	—
190	C₂×C₉	—	4	2	no	—
191	C₂×C₉	—	8	4	no	—
192	C₂×C₉	C₂×#121	1	1	yes	characteristic
193	C₂×C₉	#576	1	2	yes	—
194	C₂⁴	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	D₄×C₉	1	2	yes	—
203	C₂×D₄	D₄×C₉	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₉	1	4	yes	—
205	C₂²×C₄	C₂³×C₉	1	2	yes	—
206	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
207	D₈	—	2	4	no	—
208	D₈	—	2	4	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	D₄×C₉	1	4	yes	complemented
213	C₂×C₈	C₂³×C₉	1	2	yes	—
214	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
215	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
216	C₄⋊C₄	C₂³×C₉	1	4	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	8	no	—
219	C₄²	—	2	2	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	4	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	4	8	no	—
224	C₂²×C₃	—	4	4	no	—
225	C₂²×C₃	—	8	8	no	—
226	C₂²×C₃	C₃×#121	1	2	yes	—
227	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	4	4	no	—
230	C₃×C₄	C₃×#121	1	2	yes	—
231	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
232	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
233	C₉	#3058	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	—	2	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	8	4	no	—
237	C₂³	C₂×D₄×C₉	1	2	yes	—
238	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	4	no	—
245	D₄	—	8	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
250	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
252	C₂×C₄	C₉×#41	1	2	yes	—
253	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	C₂×D₄×C₉	1	2	yes	—
256	C₈	C₂×D₄×C₉	1	2	yes	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	8	4	no	—
260	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
261	C₂×C₃	C₃×#576	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	4	no	—
266	C₂²	—	4	8	no	—
267	C₂²	—	8	8	no	—
268	C₂²	C₉×#121	1	2	yes	—
269	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	4	4	no	—
272	C₄	C₉×#121	1	2	yes	—
273	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
274	C₄	C₂×C₉×#41	1	1	yes	characteristic
275	C₃	C₃×#3058	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	4	2	no	—
278	C₂	—	8	4	no	—
279	C₂	C₂×C₉×#121	1	1	yes	characteristic
280	C₂	C₉×#576	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	C₃×#3058	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
1. C₂²×C₉×D₈	C₂	semidirect
2. C₂×C₉×#131	C₂	semidirect
3. C₉×#535	C₂	semidirect
4. C₉×#469	C₂	semidirect
5. C₉×#466	C₂	semidirect
6. C₉×#427	C₂	semidirect
7. C₉×#418	C₂	semidirect
16. C₂×C₉×M₁₆	C₂²	semidirect
18. C₂²×C₈×C₉	C₂²	semidirect
21. C₉×#121	C₂²	semidirect
24. C₉×#107	C₂²	semidirect
25. C₉×#106	C₂²	semidirect
26. C₉×#102	C₂²	semidirect
52. C₂×C₈×C₉	D₄	semidirect
60. #3058	C₉	direct (N × H)
102. C₂²×D₈	C₂×C₉	semidirect
103. C₂×#131	C₂×C₉	semidirect
104. #535	C₂×C₉	semidirect
105. #469	C₂×C₉	semidirect
106. #466	C₂×C₉	semidirect
107. #427	C₂×C₉	semidirect
108. #418	C₂×C₉	semidirect
155. C₂×M₁₆	C₂²×C₉	semidirect
157. C₂²×C₈	C₂²×C₉	semidirect
160. #121	C₂²×C₉	semidirect
163. #107	C₂²×C₉	semidirect
164. #106	C₂²×C₉	semidirect
165. #102	C₂²×C₉	semidirect
212. C₂×C₈	D₄×C₉	semidirect
233. C₉	#3058	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₉×D₈	2
C₂×C₉×#131	2
C₉×#535	2
C₉×#469	2
C₉×#466	2
C₉×#427	2
C₉×#418	2
C₃×#3058	3

Subgroups — C9×#3058 #11767779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3058 #11767779