Subgroups — C₂×C₉×#495 #11767867

Order 1152.

888 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²×C₉×D₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₉×#124	C₂	1	2	yes	maximal, complemented
5	C₉×#495	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#495	C₃	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₉	C₂²	1	2	yes	—
9	C₂²×D₄×C₉	C₂²	1	2	yes	—
10	C₂×C₉×Q₁₆	—	2	4	no	—
11	C₂×C₉×Q₁₆	C₂²	1	4	yes	complemented
12	C₂×C₉×SD₁₆	—	2	8	no	—
13	C₂×C₉×SD₁₆	C₂²	1	8	yes	complemented
14	C₂×C₉×D₈	—	2	4	no	—
15	C₂×C₉×D₈	C₂²	1	4	yes	—
16	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₉×#124	—	2	4	no	—
19	C₉×#124	C₂²	1	8	yes	complemented
20	C₂×C₉×C₂²⋊C₄	—	2	4	no	—
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₉	C₂²	1	4	yes	complemented
23	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃×SD₁₆	C₂×C₃	1	2	yes	—
25	C₂²×C₃×D₈	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×#124	C₂×C₃	1	2	yes	—
27	C₃×#495	C₂×C₃	1	8	yes	—
28	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
29	C₂⁴×C₉	—	2	2	no	—
30	C₂×Q₈×C₉	—	2	2	no	—
31	C₂×Q₈×C₉	—	2	4	no	—
32	C₂×Q₈×C₉	—	2	4	no	—
33	C₂×Q₈×C₉	C₂³	1	4	yes	—
34	C₂×D₄×C₉	—	2	4	no	—
35	C₂×D₄×C₉	—	2	4	no	—
36	C₂×D₄×C₉	—	2	2	no	—
37	C₂×D₄×C₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	D₄	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₉×Q₁₆	—	2	8	no	—
42	C₉×SD₁₆	—	2	16	no	—
43	C₉×D₈	—	2	8	no	—
44	C₂×C₈×C₉	D₄	1	4	yes	—
45	C₂×C₈×C₉	D₄	1	4	yes	—
46	C₂×C₈×C₉	C₂³	1	2	yes	—
47	C₂×C₈×C₉	C₂³	1	2	yes	—
48	C₉×C₂²⋊C₄	—	2	16	no	—
49	C₄²×C₉	D₄	1	2	yes	—
50	C₄²×C₉	C₂³	1	2	yes	—
51	C₂×#495	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
53	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
54	C₂×C₃×Q₁₆	—	2	4	no	—
55	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
56	C₂×C₃×SD₁₆	—	2	8	no	—
57	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	—
58	C₂×C₃×D₈	—	2	4	no	—
59	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
60	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×#124	—	2	4	no	—
63	C₃×#124	C₂²×C₃	1	8	yes	—
64	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
65	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
67	C₂³×C₉	—	2	4	no	—
68	C₂³×C₉	—	4	4	no	—
69	C₂³×C₉	—	4	8	no	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₉	—	2	4	no	—
72	Q₈×C₉	—	2	4	no	—
73	Q₈×C₉	—	4	4	no	—
74	D₄×C₉	—	2	4	no	—
75	D₄×C₉	—	2	4	no	—
76	D₄×C₉	—	4	4	no	—
77	C₂×C₄×C₉	—	2	4	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	—	4	4	no	—
80	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
81	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
85	C₈×C₉	C₂×D₄	1	4	yes	—
86	C₈×C₉	C₂×D₄	1	4	yes	—
87	C₂²×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
88	C₂²×SD₁₆	C₂×C₉	1	2	yes	complemented
89	C₂²×D₈	C₂×C₉	1	1	yes	characteristic
90	C₂×#124	C₂×C₉	1	2	yes	complemented
91	#495	C₂×C₉	1	8	yes	direct product factor
92	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
93	C₂⁴×C₃	—	2	2	no	—
94	C₂×C₃×Q₈	—	2	2	no	—
95	C₂×C₃×Q₈	—	2	4	no	—
96	C₂×C₃×Q₈	—	2	4	no	—
97	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
98	C₂×C₃×D₄	—	2	4	no	—
99	C₂×C₃×D₄	—	2	4	no	—
100	C₂×C₃×D₄	—	2	2	no	—
101	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
104	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₃×Q₁₆	—	2	8	no	—
106	C₃×SD₁₆	—	2	16	no	—
107	C₃×D₈	—	2	8	no	—
108	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
109	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
110	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
111	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
112	C₃×C₂²⋊C₄	—	2	16	no	—
113	C₃×C₄²	C₃×D₄	1	2	yes	—
114	C₃×C₄²	C₂³×C₃	1	2	yes	—
115	C₂²×C₉	—	4	4	no	—
116	C₂²×C₉	—	4	8	no	—
117	C₂²×C₉	—	8	8	no	—
118	C₂²×C₉	#127	1	2	yes	—
119	C₂²×C₉	#135	1	4	yes	—
120	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₉	—	2	4	no	—
122	C₄×C₉	—	4	4	no	—
123	C₄×C₉	#127	1	2	yes	—
124	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²×Q₈	C₂²×C₉	1	2	yes	—
127	C₂²×D₄	C₂²×C₉	1	2	yes	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	C₂²×C₉	1	4	yes	complemented
130	C₂×SD₁₆	—	2	8	no	—
131	C₂×SD₁₆	C₂²×C₉	1	8	yes	complemented
132	C₂×D₈	—	2	4	no	—
133	C₂×D₈	C₂²×C₉	1	4	yes	—
134	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
136	#124	—	2	4	no	—
137	#124	C₂²×C₉	1	8	yes	complemented
138	C₂×C₂²⋊C₄	—	2	4	no	—
139	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₉	1	4	yes	complemented
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	—	4	4	no	—
143	C₂³×C₃	—	4	8	no	—
144	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
145	C₃×Q₈	—	2	4	no	—
146	C₃×Q₈	—	2	4	no	—
147	C₃×Q₈	—	4	4	no	—
148	C₃×D₄	—	2	4	no	—
149	C₃×D₄	—	2	4	no	—
150	C₃×D₄	—	4	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	4	4	no	—
154	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
155	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
158	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
159	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
160	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
161	C₂×C₉	—	8	4	no	—
162	C₂×C₉	#495	1	4	yes	direct product factor
163	C₂×C₉	C₂×#127	1	1	yes	characteristic
164	C₂×C₉	C₂×#135	1	2	yes	—
165	C₂⁴	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	C₂³×C₉	1	4	yes	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	C₂³×C₉	1	4	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	D₄×C₉	1	2	yes	—
176	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
177	Q₁₆	—	2	8	no	—
178	SD₁₆	—	2	16	no	—
179	D₈	—	2	8	no	—
180	C₂×C₈	D₄×C₉	1	4	yes	—
181	C₂×C₈	D₄×C₉	1	4	yes	—
182	C₂×C₈	C₂³×C₉	1	2	yes	—
183	C₂×C₈	C₂³×C₉	1	2	yes	—
184	C₂²⋊C₄	—	2	16	no	—
185	C₄²	D₄×C₉	1	2	yes	—
186	C₄²	C₂³×C₉	1	2	yes	—
187	C₂²×C₃	—	4	8	no	—
188	C₂²×C₃	—	4	4	no	—
189	C₂²×C₃	—	8	8	no	—
190	C₂²×C₃	C₃×#127	1	2	yes	—
191	C₂²×C₃	C₃×#135	1	4	yes	—
192	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	4	4	no	—
195	C₃×C₄	C₃×#127	1	2	yes	—
196	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
197	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₉	C₂×#495	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	2	4	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	8	no	—
202	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	2	4	no	—
205	Q₈	—	4	4	no	—
206	D₄	—	2	4	no	—
207	D₄	—	2	4	no	—
208	D₄	—	4	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
213	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
216	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
217	C₈	C₂×D₄×C₉	1	4	yes	—
218	C₈	C₂×D₄×C₉	1	4	yes	—
219	C₂×C₃	—	8	4	no	—
220	C₂×C₃	C₃×#495	1	4	yes	—
221	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
222	C₂×C₃	C₂×C₃×#135	1	2	yes	—
223	C₂²	—	4	4	no	—
224	C₂²	—	4	8	no	—
225	C₂²	—	8	8	no	—
226	C₂²	C₉×#127	1	2	yes	—
227	C₂²	C₉×#135	1	4	yes	—
228	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	4	4	no	—
231	C₄	C₉×#127	1	2	yes	—
232	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
233	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
234	C₃	C₂×C₃×#495	1	1	yes	characteristic
235	C₂	—	8	4	no	—
236	C₂	C₉×#495	1	4	yes	direct product factor
237	C₂	C₂×C₉×#127	1	1	yes	characteristic
238	C₂	C₂×C₉×#135	1	2	yes	—

	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	C₂²×C₃×D₈	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×Q₁₆	C₂	semidirect
2. C₂²×C₉×SD₁₆	C₂	semidirect
4. C₂×C₉×#124	C₂	semidirect
5. C₉×#495	C₂	direct (N × H)
6. C₂×C₄×C₈×C₉	C₂	semidirect
11. C₂×C₉×Q₁₆	C₂²	semidirect
13. C₂×C₉×SD₁₆	C₂²	semidirect
19. C₉×#124	C₂²	semidirect
22. C₄×C₈×C₉	C₂²	semidirect
51. C₂×#495	C₉	direct (N × H)
87. C₂²×Q₁₆	C₂×C₉	semidirect
88. C₂²×SD₁₆	C₂×C₉	semidirect
90. C₂×#124	C₂×C₉	semidirect
91. #495	C₂×C₉	direct (N × H)
92. C₂×C₄×C₈	C₂×C₉	semidirect
129. C₂×Q₁₆	C₂²×C₉	semidirect
131. C₂×SD₁₆	C₂²×C₉	semidirect
137. #124	C₂²×C₉	semidirect
140. C₄×C₈	C₂²×C₉	semidirect
162. C₂×C₉	#495	direct (N × H)
198. C₉	C₂×#495	direct (N × H)
236. C₂	C₉×#495	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×Q₁₆	2
C₂²×C₉×SD₁₆	2
C₂²×C₉×D₈	2
C₂×C₉×#124	2
C₉×#495	2
C₂×C₄×C₈×C₉	2
C₂×C₃×#495	3

Subgroups — C2×C9×#495 #11767867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#495 #11767867