Subgroups — C₄×C₉×#120 #11767020

Order 1152.

1836 conjugacy classes of subgroups.

The subgroup lattice has 288 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, complemented
2	C₂×C₉×#120	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₄×C₉	C₂	1	3	yes	maximal, complemented
4	C₉×#386	C₂	1	3	yes	maximal, complemented
5	C₉×#385	C₂	1	3	yes	maximal, complemented
6	C₉×#379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₉×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
8	C₃×C₄×#120	C₃	1	1	yes	maximal, characteristic
9	C₂⁵×C₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	C₂²	1	3	yes	—
11	C₂³×C₄×C₉	—	2	12	no	—
12	C₂³×C₄×C₉	C₂²	1	3	yes	—
13	C₂³×C₄×C₉	C₂²	1	3	yes	—
14	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic, complemented
15	C₉×#120	C₄	1	8	yes	direct product factor
16	C₄×D₄×C₉	—	2	12	no	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	3	yes	complemented
18	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
19	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
20	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	complemented
22	C₂×C₄²×C₉	C₂²	1	3	yes	complemented
23	C₉×#95	C₂²	1	6	yes	complemented
24	C₂⁴×C₃×C₄	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×#120	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×C₄×D₄	C₂×C₃	1	3	yes	—
27	C₃×#386	C₂×C₃	1	3	yes	—
28	C₃×#385	C₂×C₃	1	3	yes	—
29	C₃×#379	C₂×C₃	1	1	yes	characteristic
30	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	3	yes	—
31	C₂⁴×C₉	—	2	6	no	—
32	C₂⁴×C₉	—	2	6	no	—
33	C₂⁴×C₉	C₂×C₄	1	4	yes	complemented
34	C₂⁴×C₉	C₂³	1	1	yes	characteristic
35	C₂⁴×C₉	C₂³	1	3	yes	—
36	C₂×D₄×C₉	—	2	6	no	—
37	C₂×D₄×C₉	—	2	6	no	—
38	C₂×D₄×C₉	C₂×C₄	1	12	yes	complemented
39	C₂²×C₄×C₉	—	2	32	no	—
40	C₂²×C₄×C₉	—	2	6	no	—
41	C₂²×C₄×C₉	—	2	12	no	—
42	C₂²×C₄×C₉	—	2	12	no	—
43	C₂²×C₄×C₉	—	2	6	no	—
44	C₂²×C₄×C₉	D₄	1	12	yes	complemented
45	C₂²×C₄×C₉	C₂³	1	3	yes	—
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂³	1	3	yes	—
48	C₂²×C₄×C₉	C₂³	1	3	yes	—
49	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₉×C₄⋊C₄	—	2	6	no	—
51	C₉×C₂²⋊C₄	—	2	6	no	—
52	C₉×C₂²⋊C₄	—	2	6	no	—
53	C₉×C₂²⋊C₄	—	2	6	no	—
54	C₉×C₂²⋊C₄	C₂×C₄	1	12	yes	complemented
55	C₄²×C₉	—	2	6	no	—
56	C₄×#120	C₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₃×D₄	C₂²×C₃	1	3	yes	—
59	C₂³×C₃×C₄	—	2	12	no	—
60	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
61	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
62	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
63	C₃×#120	C₃×C₄	1	8	yes	—
64	C₃×C₄×D₄	—	2	12	no	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	3	yes	—
66	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
67	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
68	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
69	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
70	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
71	C₃×#95	C₂²×C₃	1	6	yes	—
72	C₂³×C₉	—	2	8	no	—
73	C₂³×C₉	—	2	24	no	—
74	C₂³×C₉	—	2	24	no	—
75	C₂³×C₉	—	2	6	no	—
76	C₂³×C₉	—	2	6	no	—
77	C₂³×C₉	—	2	3	no	—
78	C₂³×C₉	—	2	3	no	—
79	C₂³×C₉	C₂²×C₄	1	6	yes	—
80	C₂³×C₉	C₂²×C₄	1	2	yes	complemented
81	C₂³×C₉	C₂×D₄	1	6	yes	—
82	C₂³×C₉	#41	1	6	yes	—
83	C₂³×C₉	C₂⁴	1	1	yes	characteristic
84	D₄×C₉	—	2	24	no	—
85	C₂×C₄×C₉	—	2	24	no	—
86	C₂×C₄×C₉	—	2	3	no	—
87	C₂×C₄×C₉	—	2	6	no	—
88	C₂×C₄×C₉	—	2	3	no	—
89	C₂×C₄×C₉	—	2	24	no	—
90	C₂×C₄×C₉	—	2	3	no	—
91	C₂×C₄×C₉	—	2	3	no	—
92	C₂×C₄×C₉	—	2	6	no	—
93	C₂×C₄×C₉	—	4	4	no	—
94	C₂×C₄×C₉	C₂²×C₄	1	6	yes	complemented
95	C₂×C₄×C₉	C₂×D₄	1	6	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	6	yes	—
97	C₂⁴×C₄	C₂×C₉	1	1	yes	characteristic, complemented
98	C₂×#120	C₂×C₉	1	1	yes	characteristic
99	C₂×C₄×D₄	C₂×C₉	1	3	yes	complemented
100	#386	C₂×C₉	1	3	yes	complemented
101	#385	C₂×C₉	1	3	yes	complemented
102	#379	C₂×C₉	1	1	yes	characteristic, complemented
103	C₄×C₂²⋊C₄	C₂×C₉	1	3	yes	complemented
104	C₂⁴×C₃	—	2	6	no	—
105	C₂⁴×C₃	—	2	6	no	—
106	C₂⁴×C₃	C₂×C₃×C₄	1	4	yes	—
107	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
108	C₂⁴×C₃	C₂³×C₃	1	3	yes	—
109	C₂×C₃×D₄	—	2	6	no	—
110	C₂×C₃×D₄	—	2	6	no	—
111	C₂×C₃×D₄	C₂×C₃×C₄	1	12	yes	—
112	C₂²×C₃×C₄	—	2	32	no	—
113	C₂²×C₃×C₄	—	2	6	no	—
114	C₂²×C₃×C₄	—	2	12	no	—
115	C₂²×C₃×C₄	—	2	12	no	—
116	C₂²×C₃×C₄	—	2	6	no	—
117	C₂²×C₃×C₄	C₃×D₄	1	12	yes	—
118	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
119	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
122	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₃×C₄⋊C₄	—	2	6	no	—
124	C₃×C₂²⋊C₄	—	2	6	no	—
125	C₃×C₂²⋊C₄	—	2	6	no	—
126	C₃×C₂²⋊C₄	—	2	6	no	—
127	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	12	yes	—
128	C₃×C₄²	—	2	6	no	—
129	C₂²×C₉	—	2	32	no	—
130	C₂²×C₉	—	2	12	no	—
131	C₂²×C₉	—	2	6	no	—
132	C₂²×C₉	—	2	6	no	—
133	C₂²×C₉	—	2	12	no	—
134	C₂²×C₉	—	2	6	no	—
135	C₂²×C₉	—	4	1	no	—
136	C₂²×C₉	—	4	3	no	—
137	C₂²×C₉	C₄×D₄	1	12	yes	complemented
138	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
139	C₂²×C₉	C₂²×D₄	1	3	yes	—
140	C₂²×C₉	C₂×#41	1	3	yes	—
141	C₄×C₉	—	2	6	no	—
142	C₄×C₉	—	2	6	no	—
143	C₄×C₉	—	4	1	no	—
144	C₄×C₉	—	4	3	no	—
145	C₄×C₉	#120	1	4	yes	direct product factor
146	C₂⁵	C₂²×C₉	1	1	yes	characteristic
147	C₂²×D₄	C₂²×C₉	1	3	yes	—
148	C₂³×C₄	—	2	12	no	—
149	C₂³×C₄	C₂²×C₉	1	3	yes	—
150	C₂³×C₄	C₂²×C₉	1	3	yes	—
151	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic, complemented
152	#120	C₄×C₉	1	8	yes	direct product factor
153	C₄×D₄	—	2	12	no	—
154	C₂×C₄⋊C₄	C₂²×C₉	1	3	yes	complemented
155	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	complemented
159	C₂×C₄²	C₂²×C₉	1	3	yes	complemented
160	#95	C₂²×C₉	1	6	yes	complemented
161	C₂³×C₃	—	2	8	no	—
162	C₂³×C₃	—	2	24	no	—
163	C₂³×C₃	—	2	24	no	—
164	C₂³×C₃	—	2	6	no	—
165	C₂³×C₃	—	2	6	no	—
166	C₂³×C₃	—	2	3	no	—
167	C₂³×C₃	—	2	3	no	—
168	C₂³×C₃	C₂²×C₃×C₄	1	6	yes	—
169	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
170	C₂³×C₃	C₂×C₃×D₄	1	6	yes	—
171	C₂³×C₃	C₃×#41	1	6	yes	—
172	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
173	C₃×D₄	—	2	24	no	—
174	C₂×C₃×C₄	—	2	24	no	—
175	C₂×C₃×C₄	—	2	3	no	—
176	C₂×C₃×C₄	—	2	6	no	—
177	C₂×C₃×C₄	—	2	3	no	—
178	C₂×C₃×C₄	—	2	24	no	—
179	C₂×C₃×C₄	—	2	3	no	—
180	C₂×C₃×C₄	—	2	3	no	—
181	C₂×C₃×C₄	—	2	6	no	—
182	C₂×C₃×C₄	—	4	4	no	—
183	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
184	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
185	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
186	C₂×C₉	—	2	12	no	—
187	C₂×C₉	—	4	2	no	—
188	C₂×C₉	C₂×C₄×D₄	1	3	yes	—
189	C₂×C₉	C₂×#120	1	1	yes	characteristic
190	C₂×C₉	#525	1	3	yes	—
191	C₂⁴	—	2	6	no	—
192	C₂⁴	—	2	6	no	—
193	C₂⁴	C₂×C₄×C₉	1	4	yes	complemented
194	C₂⁴	C₂³×C₉	1	1	yes	characteristic
195	C₂⁴	C₂³×C₉	1	3	yes	—
196	C₂×D₄	—	2	6	no	—
197	C₂×D₄	—	2	6	no	—
198	C₂×D₄	C₂×C₄×C₉	1	12	yes	complemented
199	C₂²×C₄	—	2	32	no	—
200	C₂²×C₄	—	2	6	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	6	no	—
204	C₂²×C₄	D₄×C₉	1	12	yes	complemented
205	C₂²×C₄	C₂³×C₉	1	3	yes	—
206	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
207	C₂²×C₄	C₂³×C₉	1	3	yes	—
208	C₂²×C₄	C₂³×C₉	1	3	yes	—
209	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
210	C₄⋊C₄	—	2	6	no	—
211	C₂²⋊C₄	—	2	6	no	—
212	C₂²⋊C₄	—	2	6	no	—
213	C₂²⋊C₄	—	2	6	no	—
214	C₂²⋊C₄	C₂×C₄×C₉	1	12	yes	complemented
215	C₄²	—	2	6	no	—
216	C₂²×C₃	—	2	32	no	—
217	C₂²×C₃	—	2	12	no	—
218	C₂²×C₃	—	2	6	no	—
219	C₂²×C₃	—	2	6	no	—
220	C₂²×C₃	—	2	12	no	—
221	C₂²×C₃	—	2	6	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	—	4	3	no	—
224	C₂²×C₃	C₃×C₄×D₄	1	12	yes	—
225	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
226	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
227	C₂²×C₃	C₂×C₃×#41	1	3	yes	—
228	C₃×C₄	—	2	6	no	—
229	C₃×C₄	—	2	6	no	—
230	C₃×C₄	—	4	1	no	—
231	C₃×C₄	—	4	3	no	—
232	C₃×C₄	C₃×#120	1	4	yes	—
233	C₉	C₄×#120	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	—	2	8	no	—
235	C₂³	—	2	24	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	2	6	no	—
239	C₂³	—	2	3	no	—
240	C₂³	—	2	3	no	—
241	C₂³	C₂²×C₄×C₉	1	6	yes	—
242	C₂³	C₂²×C₄×C₉	1	2	yes	complemented
243	C₂³	C₂×D₄×C₉	1	6	yes	—
244	C₂³	C₉×#41	1	6	yes	—
245	C₂³	C₂⁴×C₉	1	1	yes	characteristic
246	D₄	—	2	24	no	—
247	C₂×C₄	—	2	24	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	6	no	—
250	C₂×C₄	—	2	3	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	3	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	2	6	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₂²×C₄×C₉	1	6	yes	complemented
257	C₂×C₄	C₂×D₄×C₉	1	6	yes	—
258	C₂×C₄	C₂×D₄×C₉	1	6	yes	—
259	C₂×C₃	—	2	12	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	C₂×C₃×C₄×D₄	1	3	yes	—
262	C₂×C₃	C₂×C₃×#120	1	1	yes	characteristic
263	C₂×C₃	C₃×#525	1	3	yes	—
264	C₂²	—	2	32	no	—
265	C₂²	—	2	6	no	—
266	C₂²	—	2	6	no	—
267	C₂²	—	2	12	no	—
268	C₂²	—	2	12	no	—
269	C₂²	—	2	6	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	3	no	—
272	C₂²	C₄×D₄×C₉	1	12	yes	complemented
273	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
274	C₂²	C₂²×D₄×C₉	1	3	yes	—
275	C₂²	C₂×C₉×#41	1	3	yes	—
276	C₄	—	2	6	no	—
277	C₄	—	2	6	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	3	no	—
280	C₄	C₉×#120	1	4	yes	direct product factor
281	C₃	C₃×C₄×#120	1	1	yes	characteristic
282	C₂	—	2	12	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×D₄×C₉	1	3	yes	—
285	C₂	C₂×C₉×#120	1	1	yes	characteristic
286	C₂	C₉×#525	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#120	C₂×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
1. C₂⁴×C₄×C₉	C₂	semidirect
3. C₂×C₄×D₄×C₉	C₂	semidirect
4. C₉×#386	C₂	semidirect
5. C₉×#385	C₂	semidirect
6. C₉×#379	C₂	semidirect
7. C₄×C₉×C₂²⋊C₄	C₂	semidirect
14. C₂³×C₄×C₉	C₂²	semidirect
15. C₉×#120	C₄	direct (N × H)
17. C₂×C₉×C₄⋊C₄	C₂²	semidirect
21. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
22. C₂×C₄²×C₉	C₂²	semidirect
23. C₉×#95	C₂²	semidirect
33. C₂⁴×C₉	C₂×C₄	semidirect
38. C₂×D₄×C₉	C₂×C₄	semidirect
44. C₂²×C₄×C₉	D₄	semidirect
54. C₉×C₂²⋊C₄	C₂×C₄	semidirect
56. C₄×#120	C₉	direct (N × H)
80. C₂³×C₉	C₂²×C₄	semidirect
94. C₂×C₄×C₉	C₂²×C₄	semidirect
97. C₂⁴×C₄	C₂×C₉	semidirect
99. C₂×C₄×D₄	C₂×C₉	semidirect
100. #386	C₂×C₉	semidirect
101. #385	C₂×C₉	semidirect
102. #379	C₂×C₉	semidirect
103. C₄×C₂²⋊C₄	C₂×C₉	semidirect
137. C₂²×C₉	C₄×D₄	semidirect
145. C₄×C₉	#120	direct (N × H)
151. C₂³×C₄	C₂²×C₉	semidirect
152. #120	C₄×C₉	direct (N × H)
154. C₂×C₄⋊C₄	C₂²×C₉	semidirect
158. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
159. C₂×C₄²	C₂²×C₉	semidirect
160. #95	C₂²×C₉	semidirect
193. C₂⁴	C₂×C₄×C₉	semidirect
198. C₂×D₄	C₂×C₄×C₉	semidirect
204. C₂²×C₄	D₄×C₉	semidirect
214. C₂²⋊C₄	C₂×C₄×C₉	semidirect
233. C₉	C₄×#120	direct (N × H)
242. C₂³	C₂²×C₄×C₉	semidirect
256. C₂×C₄	C₂²×C₄×C₉	semidirect
272. C₂²	C₄×D₄×C₉	semidirect
280. C₄	C₉×#120	direct (N × H)

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×#120	2
C₂×C₄×D₄×C₉	2
C₉×#386	2
C₉×#385	2
C₉×#379	2
C₄×C₉×C₂²⋊C₄	2
C₃×C₄×#120	3

Subgroups — C4×C9×#120 #11767020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₉×#120 #11767020