Subgroups — C₂×C₉×#460 #11767753

Order 1152.

876 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#460	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#460	C₃	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₉	—	2	1	no	—
10	C₂²×D₄×C₉	—	2	1	no	—
11	C₂²×D₄×C₉	C₂²	1	2	yes	—
12	C₂×C₉×SD₁₆	—	2	8	no	—
13	C₂×C₉×SD₁₆	C₂²	1	8	yes	complemented
14	C₂²×C₈×C₉	C₂²	1	2	yes	—
15	C₉×#127	—	2	2	no	—
16	C₉×#127	C₂²	1	4	yes	—
17	C₄×Q₈×C₉	—	2	2	no	—
18	C₄×Q₈×C₉	C₂²	1	4	yes	complemented
19	C₂×C₉×C₄⋊C₄	—	2	1	no	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₉	—	2	1	no	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
24	C₉×#102	C₂²	1	8	yes	complemented
25	C₂²×C₃×SD₁₆	C₂×C₃	1	2	yes	—
26	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
28	C₃×#460	C₂×C₃	1	8	yes	—
29	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#102	C₂×C₃	1	2	yes	—
31	C₂⁴×C₉	—	2	2	no	—
32	C₂×Q₈×C₉	—	2	1	no	—
33	C₂×Q₈×C₉	—	2	2	no	—
34	C₂×Q₈×C₉	D₄	1	4	yes	complemented
35	C₂×Q₈×C₉	C₂³	1	2	yes	—
36	C₂×D₄×C₉	—	2	4	no	—
37	C₂×D₄×C₉	—	2	2	no	—
38	C₂×D₄×C₉	—	2	4	no	—
39	C₂×D₄×C₉	—	2	4	no	—
40	C₂×D₄×C₉	—	2	4	no	—
41	C₂×D₄×C₉	—	4	4	no	—
42	C₂×D₄×C₉	—	4	4	no	—
43	C₂×D₄×C₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₉×SD₁₆	—	2	16	no	—
50	C₂×C₈×C₉	—	2	4	no	—
51	C₂×C₈×C₉	C₂³	1	4	yes	—
52	C₉×C₄⋊C₄	—	2	4	no	—
53	C₉×C₄⋊C₄	—	2	1	no	—
54	C₉×C₄⋊C₄	C₂³	1	2	yes	—
55	C₄²×C₉	—	2	4	no	—
56	C₄²×C₉	D₄	1	2	yes	—
57	C₄²×C₉	C₂³	1	2	yes	—
58	C₂×#460	C₉	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃×D₄	—	2	1	no	—
61	C₂²×C₃×D₄	—	2	1	no	—
62	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
63	C₂×C₃×SD₁₆	—	2	8	no	—
64	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	—
65	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
66	C₃×#127	—	2	2	no	—
67	C₃×#127	C₂²×C₃	1	4	yes	—
68	C₃×C₄×Q₈	—	2	2	no	—
69	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
70	C₂×C₃×C₄⋊C₄	—	2	1	no	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₃×C₄²	—	2	1	no	—
73	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
74	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
75	C₃×#102	C₂²×C₃	1	8	yes	—
76	C₂³×C₉	—	2	4	no	—
77	C₂³×C₉	—	4	4	no	—
78	C₂³×C₉	—	4	4	no	—
79	C₂³×C₉	—	4	4	no	—
80	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
81	Q₈×C₉	—	2	4	no	—
82	Q₈×C₉	—	2	2	no	—
83	Q₈×C₉	C₂×D₄	1	4	yes	complemented
84	D₄×C₉	—	2	4	no	—
85	D₄×C₉	—	2	4	no	—
86	D₄×C₉	—	4	8	no	—
87	D₄×C₉	—	4	4	no	—
88	D₄×C₉	—	4	8	no	—
89	C₂×C₄×C₉	—	2	4	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	4	2	no	—
94	C₂×C₄×C₉	SD₁₆	1	4	yes	complemented
95	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₉	#41	1	2	yes	—
100	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
101	C₈×C₉	—	2	4	no	—
102	C₂²×SD₁₆	C₂×C₉	1	2	yes	complemented
103	C₂×#127	C₂×C₉	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
105	#460	C₂×C₉	1	8	yes	direct product factor
106	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
107	C₂×#102	C₂×C₉	1	2	yes	complemented
108	C₂⁴×C₃	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	1	no	—
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
112	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
113	C₂×C₃×D₄	—	2	4	no	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	—	2	4	no	—
116	C₂×C₃×D₄	—	2	4	no	—
117	C₂×C₃×D₄	—	2	4	no	—
118	C₂×C₃×D₄	—	4	4	no	—
119	C₂×C₃×D₄	—	4	4	no	—
120	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₃×SD₁₆	—	2	16	no	—
127	C₂×C₃×C₈	—	2	4	no	—
128	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
129	C₃×C₄⋊C₄	—	2	4	no	—
130	C₃×C₄⋊C₄	—	2	1	no	—
131	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
132	C₃×C₄²	—	2	4	no	—
133	C₃×C₄²	C₃×D₄	1	2	yes	—
134	C₃×C₄²	C₂³×C₃	1	2	yes	—
135	C₂²×C₉	—	4	4	no	—
136	C₂²×C₉	—	4	4	no	—
137	C₂²×C₉	—	4	4	no	—
138	C₂²×C₉	—	8	8	no	—
139	C₂²×C₉	#121	1	2	yes	—
140	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
141	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
142	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₉	—	2	4	no	—
144	C₄×C₉	—	2	2	no	—
145	C₄×C₉	—	4	2	no	—
146	C₄×C₉	#121	1	2	yes	—
147	C₄×C₉	C₂×SD₁₆	1	4	yes	complemented
148	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₉	C₂×#41	1	1	yes	characteristic
150	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
151	C₂²×D₄	—	2	1	no	—
152	C₂²×D₄	—	2	1	no	—
153	C₂²×D₄	C₂²×C₉	1	2	yes	—
154	C₂×SD₁₆	—	2	8	no	—
155	C₂×SD₁₆	C₂²×C₉	1	8	yes	complemented
156	C₂²×C₈	C₂²×C₉	1	2	yes	—
157	#127	—	2	2	no	—
158	#127	C₂²×C₉	1	4	yes	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	C₂²×C₉	1	4	yes	complemented
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
165	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
166	#102	C₂²×C₉	1	8	yes	complemented
167	C₂³×C₃	—	2	4	no	—
168	C₂³×C₃	—	4	4	no	—
169	C₂³×C₃	—	4	4	no	—
170	C₂³×C₃	—	4	4	no	—
171	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
172	C₃×Q₈	—	2	4	no	—
173	C₃×Q₈	—	2	2	no	—
174	C₃×Q₈	C₂×C₃×D₄	1	4	yes	—
175	C₃×D₄	—	2	4	no	—
176	C₃×D₄	—	2	4	no	—
177	C₃×D₄	—	4	8	no	—
178	C₃×D₄	—	4	4	no	—
179	C₃×D₄	—	4	8	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
186	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
187	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
190	C₂×C₃×C₄	C₃×#41	1	2	yes	—
191	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
192	C₃×C₈	—	2	4	no	—
193	C₂×C₉	—	8	4	no	—
194	C₂×C₉	#460	1	4	yes	direct product factor
195	C₂×C₉	C₂×#121	1	1	yes	characteristic
196	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
197	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₂⁴	—	2	2	no	—
199	C₂×Q₈	—	2	1	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	D₄×C₉	1	4	yes	complemented
202	C₂×Q₈	C₂³×C₉	1	2	yes	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	C₂³×C₉	1	4	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
214	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
216	SD₁₆	—	2	16	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	C₂³×C₉	1	4	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	C₂³×C₉	1	2	yes	—
222	C₄²	—	2	4	no	—
223	C₄²	D₄×C₉	1	2	yes	—
224	C₄²	C₂³×C₉	1	2	yes	—
225	C₂²×C₃	—	4	4	no	—
226	C₂²×C₃	—	4	4	no	—
227	C₂²×C₃	—	4	4	no	—
228	C₂²×C₃	—	8	8	no	—
229	C₂²×C₃	C₃×#121	1	2	yes	—
230	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
231	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	4	2	no	—
236	C₃×C₄	C₃×#121	1	2	yes	—
237	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
238	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
240	C₉	C₂×#460	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	4	no	—
242	C₂³	—	4	4	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	4	no	—
245	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	2	2	no	—
248	Q₈	C₂×D₄×C₉	1	4	yes	complemented
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	8	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₉×SD₁₆	1	4	yes	complemented
260	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
263	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
264	C₂×C₄	C₉×#41	1	2	yes	—
265	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₂×C₃	—	8	4	no	—
268	C₂×C₃	C₃×#460	1	4	yes	—
269	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
270	C₂×C₃	C₂²×C₃×SD₁₆	1	1	yes	characteristic
271	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂²	—	4	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	8	no	—
276	C₂²	C₉×#121	1	2	yes	—
277	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
278	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
279	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	C₉×#121	1	2	yes	—
284	C₄	C₂×C₉×SD₁₆	1	4	yes	complemented
285	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
286	C₄	C₂×C₉×#41	1	1	yes	characteristic
287	C₃	C₂×C₃×#460	1	1	yes	characteristic
288	C₂	—	8	4	no	—
289	C₂	C₉×#460	1	4	yes	direct product factor
290	C₂	C₂×C₉×#121	1	1	yes	characteristic
291	C₂	C₂²×C₉×SD₁₆	1	1	yes	characteristic
292	C₂	C₂×C₉×AGL(1,ℤ/8)	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×#127	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×SD₁₆	C₂	semidirect
3. C₂×C₄×Q₈×C₉	C₂	semidirect
4. C₉×#460	C₂	direct (N × H)
5. C₂×C₉×C₄⋊C₈	C₂	semidirect
6. C₂×C₉×#102	C₂	semidirect
13. C₂×C₉×SD₁₆	C₂²	semidirect
18. C₄×Q₈×C₉	C₂²	semidirect
23. C₉×C₄⋊C₈	C₂²	semidirect
24. C₉×#102	C₂²	semidirect
34. C₂×Q₈×C₉	D₄	semidirect
58. C₂×#460	C₉	direct (N × H)
83. Q₈×C₉	C₂×D₄	semidirect
94. C₂×C₄×C₉	SD₁₆	semidirect
102. C₂²×SD₁₆	C₂×C₉	semidirect
104. C₂×C₄×Q₈	C₂×C₉	semidirect
105. #460	C₂×C₉	direct (N × H)
106. C₂×C₄⋊C₈	C₂×C₉	semidirect
107. C₂×#102	C₂×C₉	semidirect
147. C₄×C₉	C₂×SD₁₆	semidirect
155. C₂×SD₁₆	C₂²×C₉	semidirect
160. C₄×Q₈	C₂²×C₉	semidirect
165. C₄⋊C₈	C₂²×C₉	semidirect
166. #102	C₂²×C₉	semidirect
194. C₂×C₉	#460	direct (N × H)
201. C₂×Q₈	D₄×C₉	semidirect
240. C₉	C₂×#460	direct (N × H)
248. Q₈	C₂×D₄×C₉	semidirect
259. C₂×C₄	C₉×SD₁₆	semidirect
284. C₄	C₂×C₉×SD₁₆	semidirect
289. C₂	C₉×#460	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×SD₁₆	2
C₂×C₉×#127	2
C₂×C₄×Q₈×C₉	2
C₉×#460	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#102	2
C₂×C₃×#460	3

Subgroups — C2×C9×#460 #11767753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#460 #11767753