Subgroups — C₂×C₉×#461 #11767751

Order 1152.

816 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	C₂²×C₉×SD₁₆	C₂	1	2	yes	maximal
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
4	C₉×#461	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×#103	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#461	C₃	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₉	C₂²	1	2	yes	—
9	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₉	—	2	1	no	—
11	C₂×C₉×SD₁₆	—	2	8	no	—
12	C₂×C₉×SD₁₆	C₂²	1	8	yes	—
13	C₂²×C₈×C₉	C₂²	1	2	yes	—
14	C₉×C₄⋊Q₈	—	2	2	no	—
15	C₉×C₄⋊Q₈	C₂²	1	4	yes	complemented
16	C₄×D₄×C₉	—	2	2	no	—
17	C₄×D₄×C₉	C₂²	1	4	yes	—
18	C₂×C₉×C₄⋊C₄	—	2	1	no	—
19	C₂×C₉×C₄⋊C₄	—	2	1	no	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
24	C₉×#103	C₂²	1	8	yes	complemented
25	C₂²×C₃×SD₁₆	C₂×C₃	1	2	yes	—
26	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
28	C₃×#461	C₂×C₃	1	8	yes	—
29	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#103	C₂×C₃	1	2	yes	—
31	C₂⁴×C₉	—	2	1	no	—
32	C₂×Q₈×C₉	—	2	2	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	—	2	4	no	—
35	C₂×Q₈×C₉	C₂³	1	4	yes	—
36	C₂×D₄×C₉	—	2	1	no	—
37	C₂×D₄×C₉	—	2	2	no	—
38	C₂×D₄×C₉	D₄	1	4	yes	—
39	C₂×D₄×C₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	—	2	2	no	—
42	C₂²×C₄×C₉	—	2	4	no	—
43	C₂²×C₄×C₉	—	4	4	no	—
44	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×SD₁₆	—	2	16	no	—
48	C₂×C₈×C₉	—	2	4	no	—
49	C₂×C₈×C₉	C₂³	1	4	yes	—
50	C₉×C₄⋊C₄	—	2	4	no	—
51	C₉×C₄⋊C₄	—	2	4	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	C₂³	1	2	yes	—
54	C₉×C₂²⋊C₄	—	2	4	no	—
55	C₄²×C₉	D₄	1	2	yes	—
56	C₄²×C₉	C₂³	1	2	yes	—
57	C₂×#461	C₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
59	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
60	C₂³×C₃×C₄	—	2	1	no	—
61	C₂×C₃×SD₁₆	—	2	8	no	—
62	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	—
63	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
64	C₃×C₄⋊Q₈	—	2	2	no	—
65	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
66	C₃×C₄×D₄	—	2	2	no	—
67	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
68	C₂×C₃×C₄⋊C₄	—	2	1	no	—
69	C₂×C₃×C₄⋊C₄	—	2	1	no	—
70	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
72	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
73	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
74	C₃×#103	C₂²×C₃	1	8	yes	—
75	C₂³×C₉	—	2	4	no	—
76	C₂³×C₉	—	2	2	no	—
77	C₂³×C₉	—	4	2	no	—
78	C₂³×C₉	—	4	2	no	—
79	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₉	—	2	4	no	—
81	Q₈×C₉	—	2	4	no	—
82	Q₈×C₉	—	4	4	no	—
83	D₄×C₉	—	2	4	no	—
84	D₄×C₉	—	2	2	no	—
85	D₄×C₉	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	4	4	no	—
90	C₂×C₄×C₉	—	4	2	no	—
91	C₂×C₄×C₉	—	4	8	no	—
92	C₂×C₄×C₉	SD₁₆	1	4	yes	complemented
93	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₉	#41	1	2	yes	—
98	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
99	C₈×C₉	—	2	4	no	—
100	C₂²×SD₁₆	C₂×C₉	1	2	yes	—
101	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
102	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
103	#461	C₂×C₉	1	8	yes	direct product factor
104	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
105	C₂×#103	C₂×C₉	1	2	yes	complemented
106	C₂⁴×C₃	—	2	1	no	—
107	C₂×C₃×Q₈	—	2	2	no	—
108	C₂×C₃×Q₈	—	2	4	no	—
109	C₂×C₃×Q₈	—	2	4	no	—
110	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
111	C₂×C₃×D₄	—	2	1	no	—
112	C₂×C₃×D₄	—	2	2	no	—
113	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
114	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	—	2	4	no	—
118	C₂²×C₃×C₄	—	4	4	no	—
119	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₃×SD₁₆	—	2	16	no	—
123	C₂×C₃×C₈	—	2	4	no	—
124	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
125	C₃×C₄⋊C₄	—	2	4	no	—
126	C₃×C₄⋊C₄	—	2	4	no	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
129	C₃×C₂²⋊C₄	—	2	4	no	—
130	C₃×C₄²	C₃×D₄	1	2	yes	—
131	C₃×C₄²	C₂³×C₃	1	2	yes	—
132	C₂²×C₉	—	2	4	no	—
133	C₂²×C₉	—	4	2	no	—
134	C₂²×C₉	—	4	2	no	—
135	C₂²×C₉	—	4	8	no	—
136	C₂²×C₉	#121	1	2	yes	—
137	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
138	C₂²×C₉	#137	1	2	yes	—
139	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₉	—	2	2	no	—
141	C₄×C₉	—	4	4	no	—
142	C₄×C₉	—	4	2	no	—
143	C₄×C₉	#121	1	2	yes	—
144	C₄×C₉	C₂×SD₁₆	1	4	yes	complemented
145	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₉	C₂×#41	1	1	yes	characteristic
147	C₂²×Q₈	C₂²×C₉	1	2	yes	—
148	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
149	C₂³×C₄	—	2	1	no	—
150	C₂×SD₁₆	—	2	8	no	—
151	C₂×SD₁₆	C₂²×C₉	1	8	yes	—
152	C₂²×C₈	C₂²×C₉	1	2	yes	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	C₂²×C₉	1	4	yes	complemented
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	C₂²×C₉	1	4	yes	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
162	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
163	#103	C₂²×C₉	1	8	yes	complemented
164	C₂³×C₃	—	2	4	no	—
165	C₂³×C₃	—	2	2	no	—
166	C₂³×C₃	—	4	2	no	—
167	C₂³×C₃	—	4	2	no	—
168	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
169	C₃×Q₈	—	2	4	no	—
170	C₃×Q₈	—	2	4	no	—
171	C₃×Q₈	—	4	4	no	—
172	C₃×D₄	—	2	4	no	—
173	C₃×D₄	—	2	2	no	—
174	C₃×D₄	C₂×C₃×D₄	1	4	yes	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	4	4	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	—	4	8	no	—
181	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
185	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₃×C₄	C₃×#41	1	2	yes	—
187	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
188	C₃×C₈	—	2	4	no	—
189	C₂×C₉	—	4	4	no	—
190	C₂×C₉	#461	1	4	yes	direct product factor
191	C₂×C₉	C₂×#121	1	1	yes	characteristic
192	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
193	C₂×C₉	C₂×#137	1	1	yes	characteristic
194	C₂⁴	—	2	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₂³×C₉	1	4	yes	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	D₄×C₉	1	4	yes	—
202	C₂×D₄	C₂³×C₉	1	2	yes	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
208	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
209	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
210	SD₁₆	—	2	16	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂³×C₉	1	4	yes	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	C₂³×C₉	1	2	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	D₄×C₉	1	2	yes	—
219	C₄²	C₂³×C₉	1	2	yes	—
220	C₂²×C₃	—	2	4	no	—
221	C₂²×C₃	—	4	2	no	—
222	C₂²×C₃	—	4	2	no	—
223	C₂²×C₃	—	4	8	no	—
224	C₂²×C₃	C₃×#121	1	2	yes	—
225	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
226	C₂²×C₃	C₃×#137	1	2	yes	—
227	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	4	2	no	—
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	C₃×#121	1	2	yes	—
232	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
233	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
235	C₉	C₂×#461	1	1	yes	characteristic, direct product factor, Sylow
236	C₂³	—	2	4	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	4	no	—
244	D₄	—	2	4	no	—
245	D₄	—	2	2	no	—
246	D₄	C₂×D₄×C₉	1	4	yes	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	C₉×SD₁₆	1	4	yes	complemented
254	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
258	C₂×C₄	C₉×#41	1	2	yes	—
259	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₂×C₃	—	4	4	no	—
262	C₂×C₃	C₃×#461	1	4	yes	—
263	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
264	C₂×C₃	C₂²×C₃×SD₁₆	1	1	yes	characteristic
265	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	8	no	—
270	C₂²	C₉×#121	1	2	yes	—
271	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
272	C₂²	C₉×#137	1	2	yes	—
273	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	2	no	—
277	C₄	C₉×#121	1	2	yes	—
278	C₄	C₂×C₉×SD₁₆	1	4	yes	complemented
279	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
280	C₄	C₂×C₉×#41	1	1	yes	characteristic
281	C₃	C₂×C₃×#461	1	1	yes	characteristic
282	C₂	—	4	4	no	—
283	C₂	C₉×#461	1	4	yes	direct product factor
284	C₂	C₂×C₉×#121	1	1	yes	characteristic
285	C₂	C₂²×C₉×SD₁₆	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	C₂²×C₃×D₄	C₂²×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
2. C₂×C₉×C₄⋊Q₈	C₂	semidirect
4. C₉×#461	C₂	direct (N × H)
5. C₂×C₉×C₄⋊C₈	C₂	semidirect
6. C₂×C₉×#103	C₂	semidirect
15. C₉×C₄⋊Q₈	C₂²	semidirect
23. C₉×C₄⋊C₈	C₂²	semidirect
24. C₉×#103	C₂²	semidirect
57. C₂×#461	C₉	direct (N × H)
92. C₂×C₄×C₉	SD₁₆	semidirect
101. C₂×C₄⋊Q₈	C₂×C₉	semidirect
103. #461	C₂×C₉	direct (N × H)
104. C₂×C₄⋊C₈	C₂×C₉	semidirect
105. C₂×#103	C₂×C₉	semidirect
144. C₄×C₉	C₂×SD₁₆	semidirect
154. C₄⋊Q₈	C₂²×C₉	semidirect
162. C₄⋊C₈	C₂²×C₉	semidirect
163. #103	C₂²×C₉	semidirect
190. C₂×C₉	#461	direct (N × H)
235. C₉	C₂×#461	direct (N × H)
253. C₂×C₄	C₉×SD₁₆	semidirect
278. C₄	C₂×C₉×SD₁₆	semidirect
283. C₂	C₉×#461	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×C₄⋊Q₈	2
C₂×C₄×D₄×C₉	2
C₉×#461	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#103	2
C₂×C₃×#461	3

Subgroups — C2×C9×#461 #11767751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#461 #11767751