Subgroups — C₂×D₄²×C₉ #11768183

Order 1152.

3804 conjugacy classes of subgroups.

The subgroup lattice has 264 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	4	yes	maximal, complemented
2	D₄²×C₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×#120	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×D₄×C₉	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×D₄²	C₃	1	1	yes	maximal, characteristic
8	C₂⁵×C₉	C₂²	1	4	yes	complemented
9	C₂²×D₄×C₉	—	2	32	no	—
10	C₂²×D₄×C₉	C₂²	1	2	yes	—
11	C₂²×D₄×C₉	C₂²	1	8	yes	complemented
12	C₂²×D₄×C₉	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₉	C₂²	1	2	yes	complemented
14	C₂²×D₄×C₉	C₂²	1	32	yes	complemented
15	C₂³×C₄×C₉	C₂²	1	4	yes	complemented
16	C₉×#127	C₂²	1	8	yes	complemented
17	C₉×#121	C₂²	1	32	yes	complemented
18	C₉×#120	C₂²	1	32	yes	complemented
19	C₄×D₄×C₉	C₂²	1	16	yes	complemented
20	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	complemented
21	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	complemented
22	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	complemented
23	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic, complemented
24	C₂³×C₃×D₄	C₂×C₃	1	4	yes	—
25	C₃×D₄²	C₂×C₃	1	16	yes	—
26	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#121	C₂×C₃	1	4	yes	—
28	C₂×C₃×#120	C₂×C₃	1	4	yes	—
29	C₂×C₃×C₄×D₄	C₂×C₃	1	2	yes	—
30	C₂⁴×C₉	—	2	32	no	—
31	C₂⁴×C₉	—	4	8	no	—
32	C₂⁴×C₉	C₂³	1	4	yes	—
33	C₂⁴×C₉	C₂³	1	4	yes	—
34	C₂⁴×C₉	C₂³	1	16	yes	complemented
35	C₂×D₄×C₉	—	2	32	no	—
36	C₂×D₄×C₉	—	2	128	no	—
37	C₂×D₄×C₉	—	2	16	no	—
38	C₂×D₄×C₉	—	4	4	no	—
39	C₂×D₄×C₉	D₄	1	16	yes	direct product factor
40	C₂×D₄×C₉	C₂³	1	8	yes	—
41	C₂×D₄×C₉	C₂³	1	32	yes	complemented
42	C₂×D₄×C₉	C₂³	1	16	yes	complemented
43	C₂×D₄×C₉	C₂³	1	8	yes	complemented
44	C₂²×C₄×C₉	—	2	16	no	—
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₉	C₂³	1	4	yes	—
47	C₂²×C₄×C₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₉	C₂³	1	16	yes	complemented
49	C₉×C₄⋊C₄	C₂³	1	8	yes	complemented
50	C₉×C₂²⋊C₄	C₂³	1	16	yes	complemented
51	C₉×C₂²⋊C₄	C₂³	1	16	yes	complemented
52	C₄²×C₉	C₂³	1	4	yes	complemented
53	C₂×D₄²	C₉	1	1	yes	characteristic, direct product factor, Sylow
54	C₂⁵×C₃	C₂²×C₃	1	4	yes	—
55	C₂²×C₃×D₄	—	2	32	no	—
56	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
57	C₂²×C₃×D₄	C₂²×C₃	1	8	yes	—
58	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
59	C₂²×C₃×D₄	C₂²×C₃	1	32	yes	—
60	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
61	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
62	C₃×#127	C₂²×C₃	1	8	yes	—
63	C₃×#121	C₂²×C₃	1	32	yes	—
64	C₃×#120	C₂²×C₃	1	32	yes	—
65	C₃×C₄×D₄	C₂²×C₃	1	16	yes	—
66	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
68	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
69	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₂³×C₉	—	2	8	no	—
71	C₂³×C₉	—	2	8	no	—
72	C₂³×C₉	—	2	16	no	—
73	C₂³×C₉	—	2	64	no	—
74	C₂³×C₉	—	2	8	no	—
75	C₂³×C₉	—	4	16	no	—
76	C₂³×C₉	—	4	64	no	—
77	C₂³×C₉	C₂×D₄	1	16	yes	complemented
78	C₂³×C₉	C₂⁴	1	8	yes	—
79	C₂³×C₉	C₂⁴	1	8	yes	—
80	C₂³×C₉	C₂⁴	1	1	yes	characteristic
81	D₄×C₉	—	2	64	no	—
82	D₄×C₉	—	2	32	no	—
83	D₄×C₉	—	4	8	no	—
84	D₄×C₉	C₂×D₄	1	32	yes	direct product factor
85	C₂×C₄×C₉	—	2	8	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	32	no	—
88	C₂×C₄×C₉	C₂×D₄	1	8	yes	complemented
89	C₂×C₄×C₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₉	C₂⁴	1	8	yes	—
91	C₂×C₄×C₉	C₂⁴	1	4	yes	—
92	C₂³×D₄	C₂×C₉	1	4	yes	complemented
93	D₄²	C₂×C₉	1	16	yes	direct product factor
94	C₂×#127	C₂×C₉	1	1	yes	characteristic, complemented
95	C₂×#121	C₂×C₉	1	4	yes	complemented
96	C₂×#120	C₂×C₉	1	4	yes	complemented
97	C₂×C₄×D₄	C₂×C₉	1	2	yes	complemented
98	C₂⁴×C₃	—	2	32	no	—
99	C₂⁴×C₃	—	4	8	no	—
100	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
101	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
102	C₂⁴×C₃	C₂³×C₃	1	16	yes	—
103	C₂×C₃×D₄	—	2	32	no	—
104	C₂×C₃×D₄	—	2	128	no	—
105	C₂×C₃×D₄	—	2	16	no	—
106	C₂×C₃×D₄	—	4	4	no	—
107	C₂×C₃×D₄	C₃×D₄	1	16	yes	—
108	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
109	C₂×C₃×D₄	C₂³×C₃	1	32	yes	—
110	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
111	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
112	C₂²×C₃×C₄	—	2	16	no	—
113	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
115	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
116	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
117	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
118	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
119	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
120	C₃×C₄²	C₂³×C₃	1	4	yes	—
121	C₂²×C₉	—	2	32	no	—
122	C₂²×C₉	—	2	8	no	—
123	C₂²×C₉	—	2	8	no	—
124	C₂²×C₉	—	2	16	no	—
125	C₂²×C₉	—	2	8	no	—
126	C₂²×C₉	—	4	64	no	—
127	C₂²×C₉	—	4	16	no	—
128	C₂²×C₉	C₂²×D₄	1	4	yes	—
129	C₂²×C₉	C₂²×D₄	1	16	yes	complemented
130	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
131	C₂²×C₉	C₂⁵	1	1	yes	characteristic
132	C₄×C₉	—	2	8	no	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	C₂²×D₄	1	8	yes	complemented
135	C₂⁵	C₂²×C₉	1	4	yes	complemented
136	C₂²×D₄	—	2	32	no	—
137	C₂²×D₄	C₂²×C₉	1	2	yes	—
138	C₂²×D₄	C₂²×C₉	1	8	yes	complemented
139	C₂²×D₄	C₂²×C₉	1	4	yes	complemented
140	C₂²×D₄	C₂²×C₉	1	2	yes	complemented
141	C₂²×D₄	C₂²×C₉	1	32	yes	complemented
142	C₂³×C₄	C₂²×C₉	1	4	yes	complemented
143	#127	C₂²×C₉	1	8	yes	complemented
144	#121	C₂²×C₉	1	32	yes	complemented
145	#120	C₂²×C₉	1	32	yes	complemented
146	C₄×D₄	C₂²×C₉	1	16	yes	complemented
147	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	complemented
148	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	complemented
149	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	complemented
150	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic, complemented
151	C₂³×C₃	—	2	8	no	—
152	C₂³×C₃	—	2	8	no	—
153	C₂³×C₃	—	2	16	no	—
154	C₂³×C₃	—	2	64	no	—
155	C₂³×C₃	—	2	8	no	—
156	C₂³×C₃	—	4	16	no	—
157	C₂³×C₃	—	4	64	no	—
158	C₂³×C₃	C₂×C₃×D₄	1	16	yes	—
159	C₂³×C₃	C₂⁴×C₃	1	8	yes	—
160	C₂³×C₃	C₂⁴×C₃	1	8	yes	—
161	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
162	C₃×D₄	—	2	64	no	—
163	C₃×D₄	—	2	32	no	—
164	C₃×D₄	—	4	8	no	—
165	C₃×D₄	C₂×C₃×D₄	1	32	yes	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	32	no	—
169	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
170	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
171	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
172	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
173	C₂×C₉	—	2	16	no	—
174	C₂×C₉	—	4	8	no	—
175	C₂×C₉	D₄²	1	4	yes	direct product factor
176	C₂×C₉	C₂³×D₄	1	2	yes	—
177	C₂×C₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
178	C₂⁴	—	2	32	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	C₂³×C₉	1	4	yes	—
181	C₂⁴	C₂³×C₉	1	4	yes	—
182	C₂⁴	C₂³×C₉	1	16	yes	complemented
183	C₂×D₄	—	2	32	no	—
184	C₂×D₄	—	2	128	no	—
185	C₂×D₄	—	2	16	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	D₄×C₉	1	16	yes	direct product factor
188	C₂×D₄	C₂³×C₉	1	8	yes	—
189	C₂×D₄	C₂³×C₉	1	32	yes	complemented
190	C₂×D₄	C₂³×C₉	1	16	yes	complemented
191	C₂×D₄	C₂³×C₉	1	8	yes	complemented
192	C₂²×C₄	—	2	16	no	—
193	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₉	1	4	yes	—
195	C₂²×C₄	C₂³×C₉	1	2	yes	—
196	C₂²×C₄	C₂³×C₉	1	16	yes	complemented
197	C₄⋊C₄	C₂³×C₉	1	8	yes	complemented
198	C₂²⋊C₄	C₂³×C₉	1	16	yes	complemented
199	C₂²⋊C₄	C₂³×C₉	1	16	yes	complemented
200	C₄²	C₂³×C₉	1	4	yes	complemented
201	C₂²×C₃	—	2	8	no	—
202	C₂²×C₃	—	2	8	no	—
203	C₂²×C₃	—	2	32	no	—
204	C₂²×C₃	—	2	16	no	—
205	C₂²×C₃	—	2	8	no	—
206	C₂²×C₃	—	4	64	no	—
207	C₂²×C₃	—	4	16	no	—
208	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
209	C₂²×C₃	C₂²×C₃×D₄	1	16	yes	—
210	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
211	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
212	C₃×C₄	—	2	8	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	C₂²×C₃×D₄	1	8	yes	—
215	C₉	C₂×D₄²	1	1	yes	characteristic, direct product factor, Sylow
216	C₂³	—	2	8	no	—
217	C₂³	—	2	8	no	—
218	C₂³	—	2	16	no	—
219	C₂³	—	2	64	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	4	16	no	—
222	C₂³	—	4	64	no	—
223	C₂³	C₂×D₄×C₉	1	16	yes	complemented
224	C₂³	C₂⁴×C₉	1	8	yes	—
225	C₂³	C₂⁴×C₉	1	8	yes	—
226	C₂³	C₂⁴×C₉	1	1	yes	characteristic
227	D₄	—	2	64	no	—
228	D₄	—	2	32	no	—
229	D₄	—	4	8	no	—
230	D₄	C₂×D₄×C₉	1	32	yes	direct product factor
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	32	no	—
234	C₂×C₄	C₂×D₄×C₉	1	8	yes	complemented
235	C₂×C₄	C₂⁴×C₉	1	2	yes	—
236	C₂×C₄	C₂⁴×C₉	1	8	yes	—
237	C₂×C₄	C₂⁴×C₉	1	4	yes	—
238	C₂×C₃	—	2	16	no	—
239	C₂×C₃	—	4	8	no	—
240	C₂×C₃	C₃×D₄²	1	4	yes	—
241	C₂×C₃	C₂³×C₃×D₄	1	2	yes	—
242	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	—	2	32	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	16	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	4	64	no	—
249	C₂²	—	4	16	no	—
250	C₂²	C₂²×D₄×C₉	1	4	yes	—
251	C₂²	C₂²×D₄×C₉	1	16	yes	complemented
252	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
253	C₂²	C₂⁵×C₉	1	1	yes	characteristic
254	C₄	—	2	8	no	—
255	C₄	—	2	2	no	—
256	C₄	C₂²×D₄×C₉	1	8	yes	complemented
257	C₃	C₂×C₃×D₄²	1	1	yes	characteristic
258	C₂	—	2	16	no	—
259	C₂	—	4	8	no	—
260	C₂	D₄²×C₉	1	4	yes	direct product factor
261	C₂	C₂³×D₄×C₉	1	2	yes	—
262	C₂	C₂×C₉×H(5,𝔽₂)	1	1	yes	characteristic

	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×D₄²	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. D₄²×C₉	C₂	direct (N × H)
3. C₂×C₉×#127	C₂	semidirect
4. C₂×C₉×#121	C₂	semidirect
5. C₂×C₉×#120	C₂	semidirect
6. C₂×C₄×D₄×C₉	C₂	semidirect
8. C₂⁵×C₉	C₂²	semidirect
11. C₂²×D₄×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. C₉×#127	C₂²	semidirect
17. C₉×#121	C₂²	semidirect
18. C₉×#120	C₂²	semidirect
19. C₄×D₄×C₉	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×C₄²×C₉	C₂²	semidirect
34. C₂⁴×C₉	C₂³	semidirect
39. C₂×D₄×C₉	D₄	direct (N × H)
41. C₂×D₄×C₉	C₂³	semidirect
42. C₂×D₄×C₉	C₂³	semidirect
43. C₂×D₄×C₉	C₂³	semidirect
48. C₂²×C₄×C₉	C₂³	semidirect
49. C₉×C₄⋊C₄	C₂³	semidirect
50. C₉×C₂²⋊C₄	C₂³	semidirect
51. C₉×C₂²⋊C₄	C₂³	semidirect
52. C₄²×C₉	C₂³	semidirect
53. C₂×D₄²	C₉	direct (N × H)
77. C₂³×C₉	C₂×D₄	semidirect
84. D₄×C₉	C₂×D₄	direct (N × H)
88. C₂×C₄×C₉	C₂×D₄	semidirect
92. C₂³×D₄	C₂×C₉	semidirect
93. D₄²	C₂×C₉	direct (N × H)
94. C₂×#127	C₂×C₉	semidirect
95. C₂×#121	C₂×C₉	semidirect
96. C₂×#120	C₂×C₉	semidirect
97. C₂×C₄×D₄	C₂×C₉	semidirect
129. C₂²×C₉	C₂²×D₄	semidirect
134. C₄×C₉	C₂²×D₄	semidirect
135. C₂⁵	C₂²×C₉	semidirect
138. C₂²×D₄	C₂²×C₉	semidirect
139. C₂²×D₄	C₂²×C₉	semidirect
140. C₂²×D₄	C₂²×C₉	semidirect
141. C₂²×D₄	C₂²×C₉	semidirect
142. C₂³×C₄	C₂²×C₉	semidirect
143. #127	C₂²×C₉	semidirect
144. #121	C₂²×C₉	semidirect
145. #120	C₂²×C₉	semidirect
146. C₄×D₄	C₂²×C₉	semidirect
147. C₂×C₄⋊C₄	C₂²×C₉	semidirect
148. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
149. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
150. C₂×C₄²	C₂²×C₉	semidirect
175. C₂×C₉	D₄²	direct (N × H)
182. C₂⁴	C₂³×C₉	semidirect
187. C₂×D₄	D₄×C₉	direct (N × H)
189. C₂×D₄	C₂³×C₉	semidirect
190. C₂×D₄	C₂³×C₉	semidirect
191. C₂×D₄	C₂³×C₉	semidirect
196. C₂²×C₄	C₂³×C₉	semidirect
197. C₄⋊C₄	C₂³×C₉	semidirect
198. C₂²⋊C₄	C₂³×C₉	semidirect
199. C₂²⋊C₄	C₂³×C₉	semidirect
200. C₄²	C₂³×C₉	semidirect
215. C₉	C₂×D₄²	direct (N × H)
223. C₂³	C₂×D₄×C₉	semidirect
230. D₄	C₂×D₄×C₉	direct (N × H)
234. C₂×C₄	C₂×D₄×C₉	semidirect
251. C₂²	C₂²×D₄×C₉	semidirect
256. C₄	C₂²×D₄×C₉	semidirect
260. C₂	D₄²×C₉	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
D₄²×C₉	2
C₂×C₉×#127	2
C₂×C₉×#121	2
C₂×C₉×#120	2
C₂×C₄×D₄×C₉	2
C₂×C₃×D₄²	3

Subgroups — C2×D42×C9 #11768183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄²×C₉ #11768183