Subgroups — C₂×C₉×#534 #11768166

Order 1152.

2976 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	C₂³×D₄×C₉	C₂	1	2	yes	maximal, complemented
2	C₉×#534	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₉×#123	C₂	1	4	yes	maximal, complemented
4	C₂×C₉×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×#120	C₂	1	4	yes	maximal, complemented
6	C₂²×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#534	C₃	1	1	yes	maximal, characteristic
8	C₂⁵×C₉	C₂²	1	1	yes	characteristic
9	C₂⁵×C₉	C₂²	1	2	yes	complemented
10	C₂²×D₄×C₉	—	2	16	no	—
11	C₂²×D₄×C₉	C₂²	1	4	yes	—
12	C₂²×D₄×C₉	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₉	C₂²	1	4	yes	complemented
14	C₂²×D₄×C₉	C₂²	1	16	yes	complemented
15	C₂³×C₄×C₉	C₂²	1	2	yes	complemented
16	C₂³×C₄×C₉	C₂²	1	2	yes	complemented
17	C₉×#123	C₂²	1	32	yes	complemented
18	C₉×#121	C₂²	1	32	yes	complemented
19	C₉×#120	C₂²	1	32	yes	complemented
20	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	complemented
21	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	complemented
22	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	complemented
23	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	complemented
24	C₂³×C₃×D₄	C₂×C₃	1	2	yes	—
25	C₃×#534	C₂×C₃	1	16	yes	—
26	C₂×C₃×#123	C₂×C₃	1	4	yes	—
27	C₂×C₃×#121	C₂×C₃	1	4	yes	—
28	C₂×C₃×#120	C₂×C₃	1	4	yes	—
29	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
30	C₂⁴×C₉	—	2	8	no	—
31	C₂⁴×C₉	—	4	4	no	—
32	C₂⁴×C₉	—	4	8	no	—
33	C₂⁴×C₉	D₄	1	8	yes	complemented
34	C₂⁴×C₉	C₂³	1	4	yes	—
35	C₂⁴×C₉	C₂³	1	2	yes	—
36	C₂⁴×C₉	C₂³	1	1	yes	characteristic
37	C₂⁴×C₉	C₂³	1	4	yes	—
38	C₂⁴×C₉	C₂³	1	8	yes	complemented
39	C₂×D₄×C₉	—	2	16	no	—
40	C₂×D₄×C₉	—	2	16	no	—
41	C₂×D₄×C₉	—	2	64	no	—
42	C₂×D₄×C₉	—	4	8	no	—
43	C₂×D₄×C₉	C₂³	1	16	yes	—
44	C₂×D₄×C₉	C₂³	1	16	yes	complemented
45	C₂×D₄×C₉	C₂³	1	16	yes	complemented
46	C₂²×C₄×C₉	—	2	8	no	—
47	C₂²×C₄×C₉	—	4	4	no	—
48	C₂²×C₄×C₉	C₂³	1	4	yes	—
49	C₂²×C₄×C₉	C₂³	1	4	yes	—
50	C₂²×C₄×C₉	C₂³	1	8	yes	complemented
51	C₂²×C₄×C₉	C₂³	1	8	yes	complemented
52	C₉×C₄⋊C₄	C₂³	1	16	yes	complemented
53	C₉×C₂²⋊C₄	C₂³	1	16	yes	complemented
54	C₉×C₂²⋊C₄	C₂³	1	32	yes	complemented
55	C₂×#534	C₉	1	1	yes	characteristic, direct product factor, Sylow
56	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
57	C₂⁵×C₃	C₂²×C₃	1	2	yes	—
58	C₂²×C₃×D₄	—	2	16	no	—
59	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
60	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
61	C₂²×C₃×D₄	C₂²×C₃	1	16	yes	—
62	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
63	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
64	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
65	C₃×#123	C₂²×C₃	1	32	yes	—
66	C₃×#121	C₂²×C₃	1	32	yes	—
67	C₃×#120	C₂²×C₃	1	32	yes	—
68	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
69	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
70	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
71	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
72	C₂³×C₉	—	2	4	no	—
73	C₂³×C₉	—	2	8	no	—
74	C₂³×C₉	—	2	8	no	—
75	C₂³×C₉	—	2	16	no	—
76	C₂³×C₉	—	2	16	no	—
77	C₂³×C₉	—	4	16	no	—
78	C₂³×C₉	—	4	8	no	—
79	C₂³×C₉	—	4	16	no	—
80	C₂³×C₉	—	4	32	no	—
81	C₂³×C₉	C₂×D₄	1	8	yes	—
82	C₂³×C₉	C₂×D₄	1	4	yes	—
83	C₂³×C₉	C₂×D₄	1	16	yes	complemented
84	C₂³×C₉	C₂⁴	1	8	yes	—
85	C₂³×C₉	C₂⁴	1	4	yes	—
86	C₂³×C₉	C₂⁴	1	2	yes	—
87	C₂³×C₉	C₂⁴	1	1	yes	characteristic
88	D₄×C₉	—	2	32	no	—
89	D₄×C₉	—	2	32	no	—
90	D₄×C₉	—	4	16	no	—
91	C₂×C₄×C₉	—	2	8	no	—
92	C₂×C₄×C₉	—	2	16	no	—
93	C₂×C₄×C₉	—	2	8	no	—
94	C₂×C₄×C₉	—	4	8	no	—
95	C₂×C₄×C₉	C₂⁴	1	8	yes	—
96	C₂×C₄×C₉	C₂⁴	1	8	yes	—
97	C₂³×D₄	C₂×C₉	1	2	yes	complemented
98	#534	C₂×C₉	1	16	yes	direct product factor
99	C₂×#123	C₂×C₉	1	4	yes	complemented
100	C₂×#121	C₂×C₉	1	4	yes	complemented
101	C₂×#120	C₂×C₉	1	4	yes	complemented
102	C₂²×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂⁴×C₃	—	2	8	no	—
104	C₂⁴×C₃	—	4	4	no	—
105	C₂⁴×C₃	—	4	8	no	—
106	C₂⁴×C₃	C₃×D₄	1	8	yes	—
107	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
108	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
109	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
110	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
111	C₂⁴×C₃	C₂³×C₃	1	8	yes	—
112	C₂×C₃×D₄	—	2	16	no	—
113	C₂×C₃×D₄	—	2	16	no	—
114	C₂×C₃×D₄	—	2	64	no	—
115	C₂×C₃×D₄	—	4	8	no	—
116	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
117	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
118	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
119	C₂²×C₃×C₄	—	2	8	no	—
120	C₂²×C₃×C₄	—	4	4	no	—
121	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
122	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
123	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
124	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
125	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
126	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
127	C₃×C₂²⋊C₄	C₂³×C₃	1	32	yes	—
128	C₂²×C₉	—	2	16	no	—
129	C₂²×C₉	—	2	8	no	—
130	C₂²×C₉	—	2	4	no	—
131	C₂²×C₉	—	2	8	no	—
132	C₂²×C₉	—	2	8	no	—
133	C₂²×C₉	—	2	16	no	—
134	C₂²×C₉	—	4	16	no	—
135	C₂²×C₉	—	4	32	no	—
136	C₂²×C₉	—	4	16	no	—
137	C₂²×C₉	C₂²×D₄	1	8	yes	—
138	C₂²×C₉	C₂²×D₄	1	4	yes	—
139	C₂²×C₉	C₂²×D₄	1	2	yes	—
140	C₂²×C₉	H(5,𝔽₂)	1	4	yes	—
141	C₂²×C₉	C₂⁵	1	1	yes	characteristic
142	C₄×C₉	—	2	8	no	—
143	C₄×C₉	—	2	8	no	—
144	C₂⁵	C₂²×C₉	1	1	yes	characteristic
145	C₂⁵	C₂²×C₉	1	2	yes	complemented
146	C₂²×D₄	—	2	16	no	—
147	C₂²×D₄	C₂²×C₉	1	4	yes	—
148	C₂²×D₄	C₂²×C₉	1	4	yes	complemented
149	C₂²×D₄	C₂²×C₉	1	4	yes	complemented
150	C₂²×D₄	C₂²×C₉	1	16	yes	complemented
151	C₂³×C₄	C₂²×C₉	1	2	yes	complemented
152	C₂³×C₄	C₂²×C₉	1	2	yes	complemented
153	#123	C₂²×C₉	1	32	yes	complemented
154	#121	C₂²×C₉	1	32	yes	complemented
155	#120	C₂²×C₉	1	32	yes	complemented
156	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	complemented
157	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	complemented
158	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	complemented
159	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	complemented
160	C₂³×C₃	—	2	4	no	—
161	C₂³×C₃	—	2	8	no	—
162	C₂³×C₃	—	2	8	no	—
163	C₂³×C₃	—	2	16	no	—
164	C₂³×C₃	—	2	16	no	—
165	C₂³×C₃	—	4	16	no	—
166	C₂³×C₃	—	4	8	no	—
167	C₂³×C₃	—	4	16	no	—
168	C₂³×C₃	—	4	32	no	—
169	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
170	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
171	C₂³×C₃	C₂×C₃×D₄	1	16	yes	—
172	C₂³×C₃	C₂⁴×C₃	1	8	yes	—
173	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
174	C₂³×C₃	C₂⁴×C₃	1	2	yes	—
175	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
176	C₃×D₄	—	2	32	no	—
177	C₃×D₄	—	2	32	no	—
178	C₃×D₄	—	4	16	no	—
179	C₂×C₃×C₄	—	2	8	no	—
180	C₂×C₃×C₄	—	2	16	no	—
181	C₂×C₃×C₄	—	2	8	no	—
182	C₂×C₃×C₄	—	4	8	no	—
183	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
184	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
185	C₂×C₉	—	2	8	no	—
186	C₂×C₉	—	2	4	no	—
187	C₂×C₉	—	4	8	no	—
188	C₂×C₉	#534	1	4	yes	direct product factor
189	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
190	C₂×C₉	C₂×H(5,𝔽₂)	1	2	yes	—
191	C₂⁴	—	2	8	no	—
192	C₂⁴	—	4	4	no	—
193	C₂⁴	—	4	8	no	—
194	C₂⁴	D₄×C₉	1	8	yes	complemented
195	C₂⁴	C₂³×C₉	1	4	yes	—
196	C₂⁴	C₂³×C₉	1	2	yes	—
197	C₂⁴	C₂³×C₉	1	1	yes	characteristic
198	C₂⁴	C₂³×C₉	1	4	yes	—
199	C₂⁴	C₂³×C₉	1	8	yes	complemented
200	C₂×D₄	—	2	16	no	—
201	C₂×D₄	—	2	16	no	—
202	C₂×D₄	—	2	64	no	—
203	C₂×D₄	—	4	8	no	—
204	C₂×D₄	C₂³×C₉	1	16	yes	—
205	C₂×D₄	C₂³×C₉	1	16	yes	complemented
206	C₂×D₄	C₂³×C₉	1	16	yes	complemented
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	C₂³×C₉	1	4	yes	—
210	C₂²×C₄	C₂³×C₉	1	4	yes	—
211	C₂²×C₄	C₂³×C₉	1	8	yes	complemented
212	C₂²×C₄	C₂³×C₉	1	8	yes	complemented
213	C₄⋊C₄	C₂³×C₉	1	16	yes	complemented
214	C₂²⋊C₄	C₂³×C₉	1	16	yes	complemented
215	C₂²⋊C₄	C₂³×C₉	1	32	yes	complemented
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	2	8	no	—
218	C₂²×C₃	—	2	16	no	—
219	C₂²×C₃	—	2	8	no	—
220	C₂²×C₃	—	2	8	no	—
221	C₂²×C₃	—	2	16	no	—
222	C₂²×C₃	—	4	16	no	—
223	C₂²×C₃	—	4	32	no	—
224	C₂²×C₃	—	4	16	no	—
225	C₂²×C₃	C₂²×C₃×D₄	1	8	yes	—
226	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
227	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
228	C₂²×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
229	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
230	C₃×C₄	—	2	8	no	—
231	C₃×C₄	—	2	8	no	—
232	C₉	C₂×#534	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	2	4	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	16	no	—
237	C₂³	—	2	16	no	—
238	C₂³	—	4	16	no	—
239	C₂³	—	4	8	no	—
240	C₂³	—	4	16	no	—
241	C₂³	—	4	32	no	—
242	C₂³	C₂×D₄×C₉	1	8	yes	—
243	C₂³	C₂×D₄×C₉	1	4	yes	—
244	C₂³	C₂×D₄×C₉	1	16	yes	complemented
245	C₂³	C₂⁴×C₉	1	8	yes	—
246	C₂³	C₂⁴×C₉	1	4	yes	—
247	C₂³	C₂⁴×C₉	1	2	yes	—
248	C₂³	C₂⁴×C₉	1	1	yes	characteristic
249	D₄	—	2	32	no	—
250	D₄	—	2	32	no	—
251	D₄	—	4	16	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	16	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	C₂⁴×C₉	1	8	yes	—
257	C₂×C₄	C₂⁴×C₉	1	8	yes	—
258	C₂×C₃	—	2	8	no	—
259	C₂×C₃	—	2	4	no	—
260	C₂×C₃	—	4	8	no	—
261	C₂×C₃	C₃×#534	1	4	yes	—
262	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
263	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
264	C₂²	—	2	16	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	2	16	no	—
270	C₂²	—	4	16	no	—
271	C₂²	—	4	32	no	—
272	C₂²	—	4	16	no	—
273	C₂²	C₂²×D₄×C₉	1	8	yes	—
274	C₂²	C₂²×D₄×C₉	1	4	yes	—
275	C₂²	C₂²×D₄×C₉	1	2	yes	—
276	C₂²	C₉×H(5,𝔽₂)	1	4	yes	—
277	C₂²	C₂⁵×C₉	1	1	yes	characteristic
278	C₄	—	2	8	no	—
279	C₄	—	2	8	no	—
280	C₃	C₂×C₃×#534	1	1	yes	characteristic
281	C₂	—	2	8	no	—
282	C₂	—	2	4	no	—
283	C₂	—	4	8	no	—
284	C₂	C₉×#534	1	4	yes	direct product factor
285	C₂	C₂³×D₄×C₉	1	1	yes	characteristic
286	C₂	C₂×C₉×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#534	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₉	C₂	semidirect
2. C₉×#534	C₂	direct (N × H)
3. C₂×C₉×#123	C₂	semidirect
4. C₂×C₉×#121	C₂	semidirect
5. C₂×C₉×#120	C₂	semidirect
6. C₂²×C₉×C₂²⋊C₄	C₂	semidirect
9. C₂⁵×C₉	C₂²	semidirect
12. C₂²×D₄×C₉	C₂²	semidirect
13. C₂²×D₄×C₉	C₂²	semidirect
14. C₂²×D₄×C₉	C₂²	semidirect
15. C₂³×C₄×C₉	C₂²	semidirect
16. C₂³×C₄×C₉	C₂²	semidirect
17. C₉×#123	C₂²	semidirect
18. C₉×#121	C₂²	semidirect
19. C₉×#120	C₂²	semidirect
20. C₂×C₉×C₄⋊C₄	C₂²	semidirect
21. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
22. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
23. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
33. C₂⁴×C₉	D₄	semidirect
38. C₂⁴×C₉	C₂³	semidirect
44. C₂×D₄×C₉	C₂³	semidirect
45. C₂×D₄×C₉	C₂³	semidirect
50. C₂²×C₄×C₉	C₂³	semidirect
51. C₂²×C₄×C₉	C₂³	semidirect
52. C₉×C₄⋊C₄	C₂³	semidirect
53. C₉×C₂²⋊C₄	C₂³	semidirect
54. C₉×C₂²⋊C₄	C₂³	semidirect
55. C₂×#534	C₉	direct (N × H)
83. C₂³×C₉	C₂×D₄	semidirect
97. C₂³×D₄	C₂×C₉	semidirect
98. #534	C₂×C₉	direct (N × H)
99. C₂×#123	C₂×C₉	semidirect
100. C₂×#121	C₂×C₉	semidirect
101. C₂×#120	C₂×C₉	semidirect
102. C₂²×C₂²⋊C₄	C₂×C₉	semidirect
145. C₂⁵	C₂²×C₉	semidirect
148. C₂²×D₄	C₂²×C₉	semidirect
149. C₂²×D₄	C₂²×C₉	semidirect
150. C₂²×D₄	C₂²×C₉	semidirect
151. C₂³×C₄	C₂²×C₉	semidirect
152. C₂³×C₄	C₂²×C₉	semidirect
153. #123	C₂²×C₉	semidirect
154. #121	C₂²×C₉	semidirect
155. #120	C₂²×C₉	semidirect
156. C₂×C₄⋊C₄	C₂²×C₉	semidirect
157. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
158. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
159. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
188. C₂×C₉	#534	direct (N × H)
194. C₂⁴	D₄×C₉	semidirect
199. C₂⁴	C₂³×C₉	semidirect
205. C₂×D₄	C₂³×C₉	semidirect
206. C₂×D₄	C₂³×C₉	semidirect
211. C₂²×C₄	C₂³×C₉	semidirect
212. C₂²×C₄	C₂³×C₉	semidirect
213. C₄⋊C₄	C₂³×C₉	semidirect
214. C₂²⋊C₄	C₂³×C₉	semidirect
215. C₂²⋊C₄	C₂³×C₉	semidirect
232. C₉	C₂×#534	direct (N × H)
244. C₂³	C₂×D₄×C₉	semidirect
284. C₂	C₉×#534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₉	2
C₉×#534	2
C₂×C₉×#123	2
C₂×C₉×#121	2
C₂×C₉×#120	2
C₂²×C₉×C₂²⋊C₄	2
C₂×C₃×#534	3

Subgroups — C2×C9×#534 #11768166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#534 #11768166