Subgroups — C₂×C₉×#480 #11767806

Order 1152.

768 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	C₂×C₉×#121	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#480	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₉×#107	C₂	1	2	yes	maximal, complemented
5	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#480	C₃	1	1	yes	maximal, characteristic
8	C₂²×D₄×C₉	—	2	1	no	—
9	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₉	—	2	1	no	—
11	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₉×#121	—	2	2	no	—
14	C₉×#121	C₂²	1	4	yes	—
15	C₂×C₉×C₄⋊C₄	—	2	2	no	—
16	C₂×C₉×C₄⋊C₄	—	2	1	no	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	complemented
20	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
21	C₉×#107	C₂²	1	8	yes	complemented
22	C₉×#102	C₂²	1	8	yes	complemented
23	C₉×C₂²⋊C₈	C₂²	1	4	yes	complemented
24	C₂×C₃×#121	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₃×#480	C₂×C₃	1	8	yes	—
27	C₂×C₃×#107	C₂×C₃	1	2	yes	—
28	C₂×C₃×#102	C₂×C₃	1	2	yes	—
29	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
30	C₂⁴×C₉	—	2	1	no	—
31	C₂⁴×C₉	D₄	1	1	yes	characteristic
32	C₂×D₄×C₉	—	2	4	no	—
33	C₂×D₄×C₉	—	2	1	no	—
34	C₂×D₄×C₉	—	2	4	no	—
35	C₂×D₄×C₉	—	4	4	no	—
36	C₂×D₄×C₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₉	—	2	4	no	—
38	C₂²×C₄×C₉	—	2	4	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	—	2	2	no	—
41	C₂²×C₄×C₉	—	4	4	no	—
42	C₂²×C₄×C₉	D₄	1	2	yes	—
43	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₉	—	2	4	no	—
47	C₂×C₈×C₉	C₂³	1	4	yes	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	4	no	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₄⋊C₄	C₂³	1	4	yes	—
52	C₉×C₄⋊C₄	C₂³	1	2	yes	—
53	C₉×C₂²⋊C₄	—	2	4	no	—
54	C₂×#480	C₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₃×D₄	—	2	1	no	—
56	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂³×C₃×C₄	—	2	1	no	—
58	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
60	C₃×#121	—	2	2	no	—
61	C₃×#121	C₂²×C₃	1	4	yes	—
62	C₂×C₃×C₄⋊C₄	—	2	1	no	—
63	C₂×C₃×C₄⋊C₄	—	2	2	no	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
67	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
68	C₃×#107	C₂²×C₃	1	8	yes	—
69	C₃×#102	C₂²×C₃	1	8	yes	—
70	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
71	C₂³×C₉	—	2	2	no	—
72	C₂³×C₉	—	2	2	no	—
73	C₂³×C₉	—	2	2	no	—
74	C₂³×C₉	—	4	2	no	—
75	C₂³×C₉	—	4	2	no	—
76	C₂³×C₉	—	4	2	no	—
77	C₂³×C₉	D₈	1	4	yes	—
78	C₂³×C₉	C₂×D₄	1	2	yes	—
79	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
80	D₄×C₉	—	2	4	no	—
81	D₄×C₉	—	4	8	no	—
82	D₄×C₉	—	4	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	4	8	no	—
89	C₂×C₄×C₉	—	4	4	no	—
90	C₂×C₄×C₉	—	4	2	no	—
91	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉	#41	1	4	yes	—
94	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
95	C₈×C₉	—	2	4	no	—
96	C₂×#121	C₂×C₉	1	1	yes	characteristic
97	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
98	#480	C₂×C₉	1	8	yes	direct product factor
99	C₂×#107	C₂×C₉	1	2	yes	complemented
100	C₂×#102	C₂×C₉	1	2	yes	complemented
101	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
102	C₂⁴×C₃	—	2	1	no	—
103	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×D₄	—	2	4	no	—
105	C₂×C₃×D₄	—	2	1	no	—
106	C₂×C₃×D₄	—	2	4	no	—
107	C₂×C₃×D₄	—	4	4	no	—
108	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
109	C₂²×C₃×C₄	—	2	4	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	2	2	no	—
112	C₂²×C₃×C₄	—	2	4	no	—
113	C₂²×C₃×C₄	—	4	4	no	—
114	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
116	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₃×C₈	—	2	4	no	—
119	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
124	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
125	C₃×C₂²⋊C₄	—	2	4	no	—
126	C₂²×C₉	—	2	2	no	—
127	C₂²×C₉	—	2	2	no	—
128	C₂²×C₉	—	2	8	no	—
129	C₂²×C₉	—	4	2	no	—
130	C₂²×C₉	—	4	2	no	—
131	C₂²×C₉	—	4	2	no	—
132	C₂²×C₉	—	8	4	no	—
133	C₂²×C₉	#123	1	2	yes	—
134	C₂²×C₉	C₂×D₈	1	4	yes	—
135	C₂²×C₉	C₂×D₈	1	2	yes	—
136	C₂²×C₉	#137	1	2	yes	—
137	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	4	4	no	—
140	C₄×C₉	—	4	2	no	—
141	C₄×C₉	#123	1	2	yes	—
142	C₄×C₉	C₂×#41	1	2	yes	—
143	C₂²×D₄	—	2	1	no	—
144	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
145	C₂³×C₄	—	2	1	no	—
146	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
147	C₂²×C₈	C₂²×C₉	1	2	yes	—
148	#121	—	2	2	no	—
149	#121	C₂²×C₉	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
153	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	complemented
155	C₂×C₂²⋊C₄	—	2	1	no	—
156	#107	C₂²×C₉	1	8	yes	complemented
157	#102	C₂²×C₉	1	8	yes	complemented
158	C₂²⋊C₈	C₂²×C₉	1	4	yes	complemented
159	C₂³×C₃	—	2	2	no	—
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	—	4	2	no	—
163	C₂³×C₃	—	4	2	no	—
164	C₂³×C₃	—	4	2	no	—
165	C₂³×C₃	C₃×D₈	1	4	yes	—
166	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
167	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
168	C₃×D₄	—	2	4	no	—
169	C₃×D₄	—	4	8	no	—
170	C₃×D₄	—	4	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	4	8	no	—
177	C₂×C₃×C₄	—	4	4	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₃×#41	1	4	yes	—
182	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
183	C₃×C₈	—	2	4	no	—
184	C₂×C₉	—	2	4	no	—
185	C₂×C₉	—	8	2	no	—
186	C₂×C₉	#480	1	4	yes	direct product factor
187	C₂×C₉	C₂×#123	1	1	yes	characteristic
188	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
189	C₂×C₉	C₂×#137	1	1	yes	characteristic
190	C₂⁴	—	2	1	no	—
191	C₂⁴	D₄×C₉	1	1	yes	characteristic
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	C₂³×C₉	1	2	yes	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
203	C₂²×C₄	D₄×C₉	1	2	yes	—
204	C₂²×C₄	C₂³×C₉	1	2	yes	—
205	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂³×C₉	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₂³×C₉	1	4	yes	—
212	C₄⋊C₄	C₂³×C₉	1	2	yes	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	8	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	—	4	2	no	—
220	C₂²×C₃	—	8	4	no	—
221	C₂²×C₃	C₃×#123	1	2	yes	—
222	C₂²×C₃	C₂×C₃×D₈	1	4	yes	—
223	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
224	C₂²×C₃	C₃×#137	1	2	yes	—
225	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	4	4	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	C₃×#123	1	2	yes	—
230	C₃×C₄	C₂×C₃×#41	1	2	yes	—
231	C₉	C₂×#480	1	1	yes	characteristic, direct product factor, Sylow
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₉×D₈	1	4	yes	—
239	C₂³	C₂×D₄×C₉	1	2	yes	—
240	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
241	D₄	—	2	4	no	—
242	D₄	—	4	8	no	—
243	D₄	—	4	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
253	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
254	C₂×C₄	C₉×#41	1	4	yes	—
255	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₂×C₃	—	2	4	no	—
258	C₂×C₃	—	8	2	no	—
259	C₂×C₃	C₃×#480	1	4	yes	—
260	C₂×C₃	C₂×C₃×#123	1	1	yes	characteristic
261	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
262	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	4	no	—
270	C₂²	C₉×#123	1	2	yes	—
271	C₂²	C₂×C₉×D₈	1	4	yes	—
272	C₂²	C₂×C₉×D₈	1	2	yes	—
273	C₂²	C₉×#137	1	2	yes	—
274	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	2	no	—
278	C₄	C₉×#123	1	2	yes	—
279	C₄	C₂×C₉×#41	1	2	yes	—
280	C₃	C₂×C₃×#480	1	1	yes	characteristic
281	C₂	—	2	4	no	—
282	C₂	—	8	2	no	—
283	C₂	C₉×#480	1	4	yes	direct product factor
284	C₂	C₂×C₉×#123	1	1	yes	characteristic
285	C₂	C₂²×C₉×D₈	1	1	yes	characteristic
286	C₂	C₂×C₉×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#121	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉×C₄⋊C₄	C₂	semidirect
3. C₉×#480	C₂	direct (N × H)
4. C₂×C₉×#107	C₂	semidirect
5. C₂×C₉×#102	C₂	semidirect
6. C₂×C₉×C₂²⋊C₈	C₂	semidirect
19. C₂×C₉×C₄⋊C₄	C₂²	semidirect
21. C₉×#107	C₂²	semidirect
22. C₉×#102	C₂²	semidirect
23. C₉×C₂²⋊C₈	C₂²	semidirect
54. C₂×#480	C₉	direct (N × H)
97. C₂²×C₄⋊C₄	C₂×C₉	semidirect
98. #480	C₂×C₉	direct (N × H)
99. C₂×#107	C₂×C₉	semidirect
100. C₂×#102	C₂×C₉	semidirect
101. C₂×C₂²⋊C₈	C₂×C₉	semidirect
154. C₂×C₄⋊C₄	C₂²×C₉	semidirect
156. #107	C₂²×C₉	semidirect
157. #102	C₂²×C₉	semidirect
158. C₂²⋊C₈	C₂²×C₉	semidirect
186. C₂×C₉	#480	direct (N × H)
231. C₉	C₂×#480	direct (N × H)
283. C₂	C₉×#480	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#121	2
C₂²×C₉×C₄⋊C₄	2
C₉×#480	2
C₂×C₉×#107	2
C₂×C₉×#102	2
C₂×C₉×C₂²⋊C₈	2
C₂×C₃×#480	3

Subgroups — C2×C9×#480 #11767806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#480 #11767806