Subgroups — C₉×#2955 #11767676

Order 1152.

684 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#575	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#131	C₂	1	2	yes	maximal, complemented
3	C₉×#444	C₂	1	4	yes	maximal, complemented
4	C₉×#443	C₂	1	4	yes	maximal, complemented
5	C₉×#430	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#421	C₂	1	2	yes	maximal, complemented
7	C₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2955	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	2	yes	—
10	C₉×#137	C₄	1	4	yes	complemented
11	C₉×AGL(1,ℤ/8)	C₄	1	4	yes	complemented
12	C₉×#135	C₂²	1	4	yes	—
13	C₉×#131	—	2	2	no	—
14	C₉×#131	—	2	2	no	—
15	C₉×#131	C₂²	1	4	yes	complemented
16	C₉×#131	C₂²	1	4	yes	complemented
17	C₂×C₉×M₁₆	—	2	2	no	—
18	C₂×C₉×M₁₆	C₂²	1	2	yes	—
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₉	—	2	2	no	—
21	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
22	C₉×#117	C₂²	1	1	yes	characteristic
23	C₉×#108	C₂²	1	4	yes	complemented
24	C₉×#104	C₂²	1	8	yes	complemented
25	C₉×#97	C₂²	1	2	yes	complemented
26	C₄×C₈×C₉	C₂²	1	2	yes	complemented
27	C₃×#575	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×#131	C₂×C₃	1	2	yes	—
29	C₃×#444	C₂×C₃	1	4	yes	—
30	C₃×#443	C₂×C₃	1	4	yes	—
31	C₃×#430	C₂×C₃	1	1	yes	characteristic
32	C₃×#421	C₂×C₃	1	2	yes	—
33	C₃×#405	C₂×C₃	1	1	yes	characteristic
34	C₉×#41	—	2	2	no	—
35	C₉×#41	—	2	2	no	—
36	C₉×#41	C₂×C₄	1	4	yes	—
37	C₉×#41	C₂³	1	4	yes	—
38	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₉	—	2	2	no	—
40	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₉	—	2	2	no	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₉×Q₁₆	C₂×C₄	1	4	yes	—
44	C₉×SD₁₆	C₂×C₄	1	8	yes	—
45	C₉×D₈	C₂×C₄	1	4	yes	—
46	C₉×M₁₆	—	2	2	no	—
47	C₉×M₁₆	—	2	4	no	—
48	C₉×M₁₆	—	2	4	no	—
49	C₉×M₁₆	C₂×C₄	1	4	yes	—
50	C₉×M₁₆	C₂³	1	4	yes	—
51	C₂×C₈×C₉	—	2	2	no	—
52	C₂×C₈×C₉	—	2	4	no	—
53	C₂×C₈×C₉	—	2	4	no	—
54	C₂×C₈×C₉	D₄	1	2	yes	—
55	C₂×C₈×C₉	D₄	1	2	yes	—
56	C₂×C₈×C₉	C₂³	1	2	yes	—
57	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₂²⋊C₄	—	2	1	no	—
61	C₄²×C₉	C₂³	1	2	yes	—
62	#2955	C₉	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
64	C₃×#137	C₃×C₄	1	4	yes	—
65	C₃×AGL(1,ℤ/8)	C₃×C₄	1	4	yes	—
66	C₃×#135	C₂²×C₃	1	4	yes	—
67	C₃×#131	—	2	2	no	—
68	C₃×#131	—	2	2	no	—
69	C₃×#131	C₂²×C₃	1	4	yes	—
70	C₃×#131	C₂²×C₃	1	4	yes	—
71	C₂×C₃×M₁₆	—	2	2	no	—
72	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
73	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₃×C₈	—	2	2	no	—
75	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×#117	C₂²×C₃	1	1	yes	characteristic
77	C₃×#108	C₂²×C₃	1	4	yes	—
78	C₃×#104	C₂²×C₃	1	8	yes	—
79	C₃×#97	C₂²×C₃	1	2	yes	—
80	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
81	C₂³×C₉	—	2	2	no	—
82	C₂³×C₉	#41	1	1	yes	characteristic
83	Q₈×C₉	—	2	2	no	—
84	Q₈×C₉	C₂²×C₄	1	4	yes	—
85	D₄×C₉	—	2	4	no	—
86	D₄×C₉	—	2	4	no	—
87	D₄×C₉	—	2	2	no	—
88	D₄×C₉	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	4	no	—
91	C₂×C₄×C₉	—	2	4	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉	#41	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
97	C₈×C₉	—	2	4	no	—
98	C₈×C₉	C₂²×C₄	1	4	yes	—
99	C₈×C₉	C₂×D₄	1	2	yes	—
100	C₈×C₉	C₂×D₄	1	1	yes	characteristic
101	C₈×C₉	C₂×D₄	1	1	yes	characteristic
102	#575	C₂×C₉	1	1	yes	characteristic
103	C₂×#131	C₂×C₉	1	2	yes	complemented
104	#444	C₂×C₉	1	4	yes	complemented
105	#443	C₂×C₉	1	4	yes	complemented
106	#430	C₂×C₉	1	1	yes	characteristic, complemented
107	#421	C₂×C₉	1	2	yes	complemented
108	#405	C₂×C₉	1	1	yes	characteristic, complemented
109	C₃×#41	—	2	2	no	—
110	C₃×#41	—	2	2	no	—
111	C₃×#41	C₂×C₃×C₄	1	4	yes	—
112	C₃×#41	C₂³×C₃	1	4	yes	—
113	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₃×Q₁₆	C₂×C₃×C₄	1	4	yes	—
119	C₃×SD₁₆	C₂×C₃×C₄	1	8	yes	—
120	C₃×D₈	C₂×C₃×C₄	1	4	yes	—
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	2	4	no	—
123	C₃×M₁₆	—	2	4	no	—
124	C₃×M₁₆	C₂×C₃×C₄	1	4	yes	—
125	C₃×M₁₆	C₂³×C₃	1	4	yes	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	4	no	—
128	C₂×C₃×C₈	—	2	4	no	—
129	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
130	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
131	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
132	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₂²⋊C₄	—	2	1	no	—
136	C₃×C₄²	C₂³×C₃	1	2	yes	—
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	—	4	1	no	—
139	C₂²×C₉	—	4	4	no	—
140	C₂²×C₉	C₄×D₄	1	2	yes	—
141	C₂²×C₉	C₂×#41	1	1	yes	characteristic
142	C₄×C₉	—	2	4	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	C₄×D₄	1	2	yes	—
145	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
146	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×#41	C₂²×C₉	1	2	yes	—
148	#137	C₄×C₉	1	4	yes	complemented
149	AGL(1,ℤ/8)	C₄×C₉	1	4	yes	complemented
150	#135	C₂²×C₉	1	4	yes	—
151	#131	—	2	2	no	—
152	#131	—	2	2	no	—
153	#131	C₂²×C₉	1	4	yes	complemented
154	#131	C₂²×C₉	1	4	yes	complemented
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₂²×C₉	1	2	yes	—
157	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
160	#117	C₂²×C₉	1	1	yes	characteristic
161	#108	C₂²×C₉	1	4	yes	complemented
162	#104	C₂²×C₉	1	8	yes	complemented
163	#97	C₂²×C₉	1	2	yes	complemented
164	C₄×C₈	C₂²×C₉	1	2	yes	complemented
165	C₂³×C₃	—	2	2	no	—
166	C₂³×C₃	C₃×#41	1	1	yes	characteristic
167	C₃×Q₈	—	2	2	no	—
168	C₃×Q₈	C₂²×C₃×C₄	1	4	yes	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	2	4	no	—
171	C₃×D₄	—	2	2	no	—
172	C₃×D₄	C₂²×C₃×C₄	1	4	yes	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
181	C₃×C₈	—	2	4	no	—
182	C₃×C₈	C₂²×C₃×C₄	1	4	yes	—
183	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
184	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
185	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	—	2	2	no	—
188	C₂×C₉	—	4	4	no	—
189	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	C₂×C₄×C₉	1	4	yes	—
193	#41	C₂³×C₉	1	4	yes	—
194	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
199	Q₁₆	C₂×C₄×C₉	1	4	yes	—
200	SD₁₆	C₂×C₄×C₉	1	8	yes	—
201	D₈	C₂×C₄×C₉	1	4	yes	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	C₂×C₄×C₉	1	4	yes	—
206	M₁₆	C₂³×C₉	1	4	yes	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	D₄×C₉	1	2	yes	—
211	C₂×C₈	D₄×C₉	1	2	yes	—
212	C₂×C₈	C₂³×C₉	1	2	yes	—
213	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
214	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₄²	C₂³×C₉	1	2	yes	—
218	C₂²×C₃	—	2	4	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	4	no	—
221	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
222	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	4	2	no	—
225	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
226	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
227	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₉	#2955	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	2	2	no	—
230	C₂³	C₉×#41	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	Q₈	C₂²×C₄×C₉	1	4	yes	—
233	D₄	—	2	4	no	—
234	D₄	—	2	4	no	—
235	D₄	—	2	2	no	—
236	D₄	C₂²×C₄×C₉	1	4	yes	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
243	C₂×C₄	C₉×#41	1	1	yes	characteristic
244	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
245	C₈	—	2	4	no	—
246	C₈	C₂²×C₄×C₉	1	4	yes	—
247	C₈	C₂×D₄×C₉	1	1	yes	characteristic
248	C₈	C₂×D₄×C₉	1	2	yes	—
249	C₈	C₂×D₄×C₉	1	1	yes	characteristic
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₄×D₄×C₉	1	2	yes	—
258	C₂²	C₂×C₉×#41	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	C₄×D₄×C₉	1	2	yes	—
262	C₄	C₂³×C₄×C₉	1	1	yes	characteristic
263	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
264	C₃	C₃×#2955	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	2	no	—
267	C₂	—	4	4	no	—
268	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₉	C₂×D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂×C₃×C₄×D₄
prime radical	C₃×#575	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₉ ◁ C₂²×C₈×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#131	C₂	semidirect
3. C₉×#444	C₂	semidirect
4. C₉×#443	C₂	semidirect
5. C₉×#430	C₂	semidirect
6. C₉×#421	C₂	semidirect
7. C₉×#405	C₂	semidirect
10. C₉×#137	C₄	semidirect
11. C₉×AGL(1,ℤ/8)	C₄	semidirect
15. C₉×#131	C₂²	semidirect
16. C₉×#131	C₂²	semidirect
23. C₉×#108	C₂²	semidirect
24. C₉×#104	C₂²	semidirect
25. C₉×#97	C₂²	semidirect
26. C₄×C₈×C₉	C₂²	semidirect
62. #2955	C₉	direct (N × H)
103. C₂×#131	C₂×C₉	semidirect
104. #444	C₂×C₉	semidirect
105. #443	C₂×C₉	semidirect
106. #430	C₂×C₉	semidirect
107. #421	C₂×C₉	semidirect
108. #405	C₂×C₉	semidirect
148. #137	C₄×C₉	semidirect
149. AGL(1,ℤ/8)	C₄×C₉	semidirect
153. #131	C₂²×C₉	semidirect
154. #131	C₂²×C₉	semidirect
161. #108	C₂²×C₉	semidirect
162. #104	C₂²×C₉	semidirect
163. #97	C₂²×C₉	semidirect
164. C₄×C₈	C₂²×C₉	semidirect
228. C₉	#2955	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#575	2
C₂×C₉×#131	2
C₉×#444	2
C₉×#443	2
C₉×#430	2
C₉×#421	2
C₉×#405	2
C₃×#2955	3

Subgroups — C9×#2955 #11767676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2955 #11767676