Subgroups — C₉×#2280 #11767001

Order 1152.

1632 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	C₂⁴×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#117	C₂	1	2	yes	maximal, complemented
3	C₉×#381	C₂	1	2	yes	maximal, complemented
4	C₉×#380	C₂	1	4	yes	maximal, complemented
5	C₉×#379	C₂	1	2	yes	maximal
6	C₄×C₉×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
7	C₃×#2280	C₃	1	1	yes	maximal, characteristic
8	C₂⁵×C₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₉	—	2	8	no	—
10	C₂³×C₄×C₉	C₄	1	8	yes	complemented
11	C₂³×C₄×C₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
15	C₉×#117	—	2	8	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	complemented
17	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄²×C₉	C₂²	1	4	yes	complemented
20	C₉×#95	C₂²	1	8	yes	complemented
21	C₂⁴×C₃×C₄	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×#117	C₂×C₃	1	2	yes	—
23	C₃×#381	C₂×C₃	1	2	yes	—
24	C₃×#380	C₂×C₃	1	4	yes	—
25	C₃×#379	C₂×C₃	1	2	yes	—
26	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	4	yes	—
27	C₂⁴×C₉	—	2	4	no	—
28	C₂⁴×C₉	—	2	8	no	—
29	C₂⁴×C₉	C₂×C₄	1	4	yes	—
30	C₂⁴×C₉	C₂³	1	1	yes	characteristic
31	C₂⁴×C₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₉	—	2	4	no	—
33	C₂²×C₄×C₉	—	2	32	no	—
34	C₂²×C₄×C₉	—	2	8	no	—
35	C₂²×C₄×C₉	—	2	8	no	—
36	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₉	C₂×C₄	1	8	yes	—
39	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₉	D₄	1	8	yes	—
41	C₂²×C₄×C₉	C₂³	1	8	yes	—
42	C₂²×C₄×C₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₉×C₄⋊C₄	—	2	8	no	—
46	C₉×C₂²⋊C₄	—	2	8	no	—
47	C₉×C₂²⋊C₄	—	2	16	no	—
48	C₄²×C₉	—	2	8	no	—
49	#2280	C₉	1	1	yes	characteristic, direct product factor, Sylow
50	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
51	C₂³×C₃×C₄	—	2	8	no	—
52	C₂³×C₃×C₄	C₃×C₄	1	8	yes	—
53	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
55	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃×#117	—	2	8	no	—
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
60	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
61	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
62	C₃×#95	C₂²×C₃	1	8	yes	—
63	C₂³×C₉	—	2	32	no	—
64	C₂³×C₉	—	2	8	no	—
65	C₂³×C₉	—	2	16	no	—
66	C₂³×C₉	—	2	4	no	—
67	C₂³×C₉	—	2	4	no	—
68	C₂³×C₉	—	2	4	no	—
69	C₂³×C₉	C₂²×C₄	1	2	yes	—
70	C₂³×C₉	C₂²×C₄	1	4	yes	—
71	C₂³×C₉	C₂×D₄	1	4	yes	—
72	C₂³×C₉	#41	1	8	yes	—
73	C₂³×C₉	C₂⁴	1	1	yes	characteristic
74	C₂×C₄×C₉	—	2	8	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	—	2	16	no	—
77	C₂×C₄×C₉	—	2	16	no	—
78	C₂×C₄×C₉	—	2	8	no	—
79	C₂×C₄×C₉	—	2	16	no	—
80	C₂×C₄×C₉	C₂²⋊C₄	1	16	yes	complemented
81	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
86	C₂⁴×C₄	C₂×C₉	1	1	yes	characteristic
87	C₂×#117	C₂×C₉	1	2	yes	complemented
88	#381	C₂×C₉	1	2	yes	complemented
89	#380	C₂×C₉	1	4	yes	complemented
90	#379	C₂×C₉	1	2	yes	—
91	C₄×C₂²⋊C₄	C₂×C₉	1	4	yes	complemented
92	C₂⁴×C₃	—	2	4	no	—
93	C₂⁴×C₃	—	2	8	no	—
94	C₂⁴×C₃	C₂×C₃×C₄	1	4	yes	—
95	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
96	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
97	C₂²×C₃×C₄	—	2	4	no	—
98	C₂²×C₃×C₄	—	2	32	no	—
99	C₂²×C₃×C₄	—	2	8	no	—
100	C₂²×C₃×C₄	—	2	8	no	—
101	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
102	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
103	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
104	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
105	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
106	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
108	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	2	8	no	—
111	C₃×C₂²⋊C₄	—	2	8	no	—
112	C₃×C₂²⋊C₄	—	2	16	no	—
113	C₃×C₄²	—	2	8	no	—
114	C₂²×C₉	—	2	32	no	—
115	C₂²×C₉	—	2	4	no	—
116	C₂²×C₉	—	2	4	no	—
117	C₂²×C₉	—	2	4	no	—
118	C₂²×C₉	—	2	8	no	—
119	C₂²×C₉	—	2	16	no	—
120	C₂²×C₉	C₂×C₂²⋊C₄	1	4	yes	—
121	C₂²×C₉	#117	1	8	yes	complemented
122	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
123	C₂²×C₉	C₂²×D₄	1	2	yes	—
124	C₂²×C₉	C₂×#41	1	4	yes	—
125	C₄×C₉	—	2	4	no	—
126	C₄×C₉	—	4	8	no	—
127	C₄×C₉	C₂×C₂²⋊C₄	1	4	yes	—
128	C₄×C₉	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂⁵	C₂²×C₉	1	1	yes	characteristic
130	C₂³×C₄	—	2	8	no	—
131	C₂³×C₄	C₄×C₉	1	8	yes	complemented
132	C₂³×C₄	C₂²×C₉	1	2	yes	—
133	C₂³×C₄	C₂²×C₉	1	2	yes	—
134	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
135	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
136	#117	—	2	8	no	—
137	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	complemented
138	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
140	C₂×C₄²	C₂²×C₉	1	4	yes	complemented
141	#95	C₂²×C₉	1	8	yes	complemented
142	C₂³×C₃	—	2	32	no	—
143	C₂³×C₃	—	2	8	no	—
144	C₂³×C₃	—	2	16	no	—
145	C₂³×C₃	—	2	4	no	—
146	C₂³×C₃	—	2	4	no	—
147	C₂³×C₃	—	2	4	no	—
148	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
149	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
150	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
151	C₂³×C₃	C₃×#41	1	8	yes	—
152	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
153	C₂×C₃×C₄	—	2	8	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	16	no	—
156	C₂×C₃×C₄	—	2	16	no	—
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	16	no	—
159	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	16	yes	—
160	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
161	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
162	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
164	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
165	C₂×C₉	—	2	4	no	—
166	C₂×C₉	—	2	8	no	—
167	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₉	C₂×#117	1	2	yes	—
169	C₂×C₉	#525	1	4	yes	—
170	C₂⁴	—	2	4	no	—
171	C₂⁴	—	2	8	no	—
172	C₂⁴	C₂×C₄×C₉	1	4	yes	—
173	C₂⁴	C₂³×C₉	1	1	yes	characteristic
174	C₂⁴	C₂³×C₉	1	2	yes	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	32	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
180	C₂²×C₄	C₂×C₄×C₉	1	8	yes	—
181	C₂²×C₄	C₂×C₄×C₉	1	8	yes	—
182	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
183	C₂²×C₄	D₄×C₉	1	8	yes	—
184	C₂²×C₄	C₂³×C₉	1	8	yes	—
185	C₂²×C₄	C₂³×C₉	1	2	yes	—
186	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
188	C₄⋊C₄	—	2	8	no	—
189	C₂²⋊C₄	—	2	8	no	—
190	C₂²⋊C₄	—	2	16	no	—
191	C₄²	—	2	8	no	—
192	C₂²×C₃	—	2	32	no	—
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	—	2	8	no	—
197	C₂²×C₃	—	2	16	no	—
198	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	4	yes	—
199	C₂²×C₃	C₃×#117	1	8	yes	—
200	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
201	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
202	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	4	8	no	—
205	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
206	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
207	C₉	#2280	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	—	2	32	no	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	16	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	4	no	—
214	C₂³	C₂²×C₄×C₉	1	2	yes	—
215	C₂³	C₂²×C₄×C₉	1	4	yes	—
216	C₂³	C₂×D₄×C₉	1	4	yes	—
217	C₂³	C₉×#41	1	8	yes	—
218	C₂³	C₂⁴×C₉	1	1	yes	characteristic
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	C₉×C₂²⋊C₄	1	16	yes	complemented
226	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
227	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
228	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
229	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
230	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
231	C₂×C₃	—	2	4	no	—
232	C₂×C₃	—	2	8	no	—
233	C₂×C₃	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₃	C₂×C₃×#117	1	2	yes	—
235	C₂×C₃	C₃×#525	1	4	yes	—
236	C₂²	—	2	32	no	—
237	C₂²	—	2	8	no	—
238	C₂²	—	2	16	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	C₂×C₉×C₂²⋊C₄	1	4	yes	—
243	C₂²	C₉×#117	1	8	yes	complemented
244	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
245	C₂²	C₂²×D₄×C₉	1	2	yes	—
246	C₂²	C₂×C₉×#41	1	4	yes	—
247	C₄	—	2	4	no	—
248	C₄	—	4	8	no	—
249	C₄	C₂×C₉×C₂²⋊C₄	1	4	yes	—
250	C₄	C₂×C₉×C₂²⋊C₄	1	4	yes	—
251	C₃	C₃×#2280	1	1	yes	characteristic
252	C₂	—	2	4	no	—
253	C₂	—	2	8	no	—
254	C₂	C₂²×C₉×C₂²⋊C₄	1	1	yes	characteristic
255	C₂	C₂×C₉×#117	1	2	yes	—
256	C₂	C₉×#525	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#117	C₂	semidirect
3. C₉×#381	C₂	semidirect
4. C₉×#380	C₂	semidirect
6. C₄×C₉×C₂²⋊C₄	C₂	semidirect
10. C₂³×C₄×C₉	C₄	semidirect
16. C₂×C₉×C₄⋊C₄	C₂²	semidirect
19. C₂×C₄²×C₉	C₂²	semidirect
20. C₉×#95	C₂²	semidirect
49. #2280	C₉	direct (N × H)
80. C₂×C₄×C₉	C₂²⋊C₄	semidirect
87. C₂×#117	C₂×C₉	semidirect
88. #381	C₂×C₉	semidirect
89. #380	C₂×C₉	semidirect
91. C₄×C₂²⋊C₄	C₂×C₉	semidirect
121. C₂²×C₉	#117	semidirect
131. C₂³×C₄	C₄×C₉	semidirect
137. C₂×C₄⋊C₄	C₂²×C₉	semidirect
140. C₂×C₄²	C₂²×C₉	semidirect
141. #95	C₂²×C₉	semidirect
207. C₉	#2280	direct (N × H)
225. C₂×C₄	C₉×C₂²⋊C₄	semidirect
243. C₂²	C₉×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₉	2
C₂×C₉×#117	2
C₉×#381	2
C₉×#380	2
C₉×#379	2
C₄×C₉×C₂²⋊C₄	2
C₃×#2280	3

Subgroups — C9×#2280 #11767001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2280 #11767001