Subgroups — C₉×#2952 #11767673

Order 1152.

726 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₉×#520	C₂	1	2	yes	maximal, complemented
3	C₉×#442	C₂	1	2	yes	maximal, complemented
4	C₉×#441	C₂	1	2	yes	maximal, complemented
5	C₄×C₉×SD₁₆	C₂	1	4	yes	maximal, complemented
6	C₂×C₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#417	C₂	1	2	yes	maximal, complemented
8	C₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2952	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₂²	1	2	yes	—
11	C₉×#135	C₄	1	8	yes	complemented
12	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₉	C₂²	1	4	yes	—
18	C₄×D₄×C₉	—	2	4	no	—
19	C₄×D₄×C₉	C₂²	1	4	yes	—
20	C₉×#117	—	2	2	no	—
21	C₉×#117	C₂²	1	1	yes	characteristic
22	C₂×C₉×C₄⋊C₄	—	2	2	no	—
23	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
24	C₉×#106	C₂²	1	2	yes	—
25	C₉×#106	C₂²	1	2	yes	—
26	C₉×#103	C₂²	1	4	yes	—
27	C₉×#102	C₂²	1	4	yes	—
28	C₉×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	—
30	C₂×C₃×#135	C₂×C₃	1	1	yes	characteristic
31	C₃×#520	C₂×C₃	1	2	yes	—
32	C₃×#442	C₂×C₃	1	2	yes	—
33	C₃×#441	C₂×C₃	1	2	yes	—
34	C₃×C₄×SD₁₆	C₂×C₃	1	4	yes	—
35	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
36	C₃×#417	C₂×C₃	1	2	yes	—
37	C₃×#405	C₂×C₃	1	1	yes	characteristic
38	C₉×#41	—	2	4	no	—
39	C₉×#41	C₂×C₄	1	8	yes	complemented
40	C₂×Q₈×C₉	C₂³	1	2	yes	—
41	C₂×D₄×C₉	—	2	2	no	—
42	C₂×D₄×C₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₉	—	2	2	no	—
44	C₂²×C₄×C₉	—	2	2	no	—
45	C₂²×C₄×C₉	—	2	4	no	—
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×Q₁₆	C₂×C₄	1	4	yes	complemented
48	C₉×SD₁₆	C₂×C₄	1	8	yes	complemented
49	C₉×D₈	C₂×C₄	1	4	yes	complemented
50	C₉×M₁₆	—	2	2	no	—
51	C₂×C₈×C₉	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₉	C₂³	1	2	yes	—
53	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	4	no	—
57	C₉×C₄⋊C₄	D₄	1	2	yes	—
58	C₉×C₄⋊C₄	C₂³	1	4	yes	—
59	C₉×C₂²⋊C₄	—	2	4	no	—
60	C₉×C₂²⋊C₄	D₄	1	2	yes	—
61	C₄²×C₉	—	2	4	no	—
62	C₄²×C₉	C₂³	1	2	yes	—
63	#2952	C₉	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
65	C₃×#135	C₃×C₄	1	8	yes	—
66	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
68	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
72	C₃×C₄×D₄	—	2	4	no	—
73	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
74	C₃×#117	—	2	2	no	—
75	C₃×#117	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	—	2	2	no	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₃×#106	C₂²×C₃	1	2	yes	—
79	C₃×#106	C₂²×C₃	1	2	yes	—
80	C₃×#103	C₂²×C₃	1	4	yes	—
81	C₃×#102	C₂²×C₃	1	4	yes	—
82	C₃×#97	C₂²×C₃	1	2	yes	—
83	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
84	C₂³×C₉	—	2	2	no	—
85	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₉	—	2	2	no	—
87	Q₈×C₉	C₂²×C₄	1	4	yes	—
88	D₄×C₉	—	2	8	no	—
89	D₄×C₉	—	2	2	no	—
90	D₄×C₉	C₂²×C₄	1	4	yes	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	8	no	—
93	C₂×C₄×C₉	—	2	4	no	—
94	C₂×C₄×C₉	—	8	4	no	—
95	C₂×C₄×C₉	—	8	2	no	—
96	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	#41	1	2	yes	—
100	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	C₂²×C₄	1	2	yes	—
103	C₈×C₉	C₂²×C₄	1	2	yes	—
104	C₂×#135	C₂×C₉	1	1	yes	characteristic
105	#520	C₂×C₉	1	2	yes	complemented
106	#442	C₂×C₉	1	2	yes	complemented
107	#441	C₂×C₉	1	2	yes	complemented
108	C₄×SD₁₆	C₂×C₉	1	4	yes	complemented
109	C₂×#106	C₂×C₉	1	1	yes	characteristic, complemented
110	#417	C₂×C₉	1	2	yes	complemented
111	#405	C₂×C₉	1	1	yes	characteristic, complemented
112	C₃×#41	—	2	4	no	—
113	C₃×#41	C₂×C₃×C₄	1	8	yes	—
114	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
117	C₂²×C₃×C₄	—	2	2	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	—	2	4	no	—
120	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
121	C₃×Q₁₆	C₂×C₃×C₄	1	4	yes	—
122	C₃×SD₁₆	C₂×C₃×C₄	1	8	yes	—
123	C₃×D₈	C₂×C₃×C₄	1	4	yes	—
124	C₃×M₁₆	—	2	2	no	—
125	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
126	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
127	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄⋊C₄	—	2	4	no	—
131	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
132	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
133	C₃×C₂²⋊C₄	—	2	4	no	—
134	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
135	C₃×C₄²	—	2	4	no	—
136	C₃×C₄²	C₂³×C₃	1	2	yes	—
137	C₂²×C₉	—	2	2	no	—
138	C₂²×C₉	—	2	4	no	—
139	C₂²×C₉	—	4	2	no	—
140	C₂²×C₉	—	4	2	no	—
141	C₂²×C₉	C₄×D₄	1	2	yes	—
142	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₉	—	2	4	no	—
144	C₄×C₉	—	2	4	no	—
145	C₄×C₉	—	4	4	no	—
146	C₄×C₉	C₄×D₄	1	2	yes	—
147	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
148	C₄×C₉	C₂×#41	1	1	yes	characteristic
149	C₂×#41	C₂²×C₉	1	2	yes	—
150	#135	C₄×C₉	1	8	yes	complemented
151	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
152	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
153	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
154	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
155	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
156	C₄×Q₈	C₂²×C₉	1	4	yes	—
157	C₄×D₄	—	2	4	no	—
158	C₄×D₄	C₂²×C₉	1	4	yes	—
159	#117	—	2	2	no	—
160	#117	C₂²×C₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
163	#106	C₂²×C₉	1	2	yes	—
164	#106	C₂²×C₉	1	2	yes	—
165	#103	C₂²×C₉	1	4	yes	—
166	#102	C₂²×C₉	1	4	yes	—
167	#97	C₂²×C₉	1	2	yes	—
168	C₄×C₈	C₂²×C₉	1	2	yes	—
169	C₂³×C₃	—	2	2	no	—
170	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃×Q₈	—	2	2	no	—
172	C₃×Q₈	C₂²×C₃×C₄	1	4	yes	—
173	C₃×D₄	—	2	8	no	—
174	C₃×D₄	—	2	2	no	—
175	C₃×D₄	C₂²×C₃×C₄	1	4	yes	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	8	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	8	4	no	—
180	C₂×C₃×C₄	—	8	2	no	—
181	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₃×#41	1	2	yes	—
185	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
188	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
189	C₂×C₉	—	2	2	no	—
190	C₂×C₉	—	4	4	no	—
191	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂×C₉	#577	1	2	yes	—
193	#41	—	2	4	no	—
194	#41	C₂×C₄×C₉	1	8	yes	complemented
195	C₂×Q₈	C₂³×C₉	1	2	yes	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	C₂³×C₉	1	2	yes	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
202	Q₁₆	C₂×C₄×C₉	1	4	yes	complemented
203	SD₁₆	C₂×C₄×C₉	1	8	yes	complemented
204	D₈	C₂×C₄×C₉	1	4	yes	complemented
205	M₁₆	—	2	2	no	—
206	C₂×C₈	C₂×C₄×C₉	1	4	yes	complemented
207	C₂×C₈	C₂³×C₉	1	2	yes	—
208	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
209	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	D₄×C₉	1	2	yes	—
213	C₄⋊C₄	C₂³×C₉	1	4	yes	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	D₄×C₉	1	2	yes	—
216	C₄²	—	2	4	no	—
217	C₄²	C₂³×C₉	1	2	yes	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	4	2	no	—
222	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
223	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	4	4	no	—
227	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
228	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
229	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
230	C₉	#2952	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	2	2	no	—
232	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	Q₈	C₂²×C₄×C₉	1	4	yes	—
235	D₄	—	2	8	no	—
236	D₄	—	2	2	no	—
237	D₄	C₂²×C₄×C₉	1	4	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
245	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
246	C₂×C₄	C₉×#41	1	2	yes	—
247	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂²×C₄×C₉	1	2	yes	—
250	C₈	C₂²×C₄×C₉	1	2	yes	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
254	C₂×C₃	C₃×#577	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	C₄×D₄×C₉	1	2	yes	—
260	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	4	4	no	—
264	C₄	C₄×D₄×C₉	1	2	yes	—
265	C₄	C₂³×C₄×C₉	1	1	yes	characteristic
266	C₄	C₂×C₉×#41	1	1	yes	characteristic
267	C₃	C₃×#2952	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic
271	C₂	C₉×#577	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂×C₃×#135	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#520	C₂	semidirect
3. C₉×#442	C₂	semidirect
4. C₉×#441	C₂	semidirect
5. C₄×C₉×SD₁₆	C₂	semidirect
6. C₂×C₉×#106	C₂	semidirect
7. C₉×#417	C₂	semidirect
8. C₉×#405	C₂	semidirect
11. C₉×#135	C₄	semidirect
39. C₉×#41	C₂×C₄	semidirect
47. C₉×Q₁₆	C₂×C₄	semidirect
48. C₉×SD₁₆	C₂×C₄	semidirect
49. C₉×D₈	C₂×C₄	semidirect
51. C₂×C₈×C₉	C₂×C₄	semidirect
63. #2952	C₉	direct (N × H)
105. #520	C₂×C₉	semidirect
106. #442	C₂×C₉	semidirect
107. #441	C₂×C₉	semidirect
108. C₄×SD₁₆	C₂×C₉	semidirect
109. C₂×#106	C₂×C₉	semidirect
110. #417	C₂×C₉	semidirect
111. #405	C₂×C₉	semidirect
150. #135	C₄×C₉	semidirect
194. #41	C₂×C₄×C₉	semidirect
202. Q₁₆	C₂×C₄×C₉	semidirect
203. SD₁₆	C₂×C₄×C₉	semidirect
204. D₈	C₂×C₄×C₉	semidirect
206. C₂×C₈	C₂×C₄×C₉	semidirect
230. C₉	#2952	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#135	2
C₉×#520	2
C₉×#442	2
C₉×#441	2
C₄×C₉×SD₁₆	2
C₂×C₉×#106	2
C₉×#417	2
C₉×#405	2
C₃×#2952	3

Subgroups — C9×#2952 #11767673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2952 #11767673