Subgroups — C₉×#3192 #11767913

Order 1152.

771 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#121	C₂	1	2	yes	maximal, complemented
2	C₉×#481	C₂	1	4	yes	maximal, complemented
3	C₉×#469	C₂	1	4	yes	maximal, complemented
4	C₉×#447	C₂	1	4	yes	maximal, complemented
5	C₉×#407	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#3192	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	—	2	2	no	—
8	C₂²×D₄×C₉	C₂²	1	2	yes	complemented
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×D₈	C₂²	1	4	yes	complemented
11	C₂×C₉×M₁₆	C₂²	1	2	yes	complemented
12	C₉×#121	—	2	2	no	—
13	C₉×#121	—	2	2	no	—
14	C₉×#121	C₂²	1	4	yes	—
15	C₉×#121	C₂²	1	4	yes	complemented
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	complemented
17	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
18	C₉×#106	C₂²	1	4	yes	complemented
19	C₉×#102	C₂²	1	8	yes	complemented
20	C₉×C₂²⋊C₈	C₂²	1	4	yes	complemented
21	C₂×C₃×#121	C₂×C₃	1	2	yes	—
22	C₃×#481	C₂×C₃	1	4	yes	—
23	C₃×#469	C₂×C₃	1	4	yes	—
24	C₃×#447	C₂×C₃	1	4	yes	—
25	C₃×#407	C₂×C₃	1	1	yes	characteristic
26	C₂⁴×C₉	—	2	2	no	—
27	C₂⁴×C₉	D₄	1	1	yes	characteristic
28	C₂×D₄×C₉	—	2	4	no	—
29	C₂×D₄×C₉	—	2	4	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	4	4	no	—
32	C₂×D₄×C₉	—	4	8	no	—
33	C₂×D₄×C₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	—	4	2	no	—
36	C₂²×C₄×C₉	D₄	1	2	yes	—
37	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₉×D₈	—	4	4	no	—
41	C₉×M₁₆	—	4	2	no	—
42	C₂×C₈×C₉	C₂³	1	4	yes	—
43	C₉×C₄⋊C₄	—	2	2	no	—
44	C₉×C₄⋊C₄	C₂³	1	4	yes	—
45	C₉×C₂²⋊C₄	—	2	4	no	—
46	C₉×C₂²⋊C₄	—	2	4	no	—
47	#3192	C₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₃×D₄	—	2	2	no	—
49	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
50	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
52	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
53	C₃×#121	—	2	2	no	—
54	C₃×#121	—	2	2	no	—
55	C₃×#121	C₂²×C₃	1	4	yes	—
56	C₃×#121	C₂²×C₃	1	4	yes	—
57	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
59	C₃×#106	C₂²×C₃	1	4	yes	—
60	C₃×#102	C₂²×C₃	1	8	yes	—
61	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
62	C₂³×C₉	—	2	4	no	—
63	C₂³×C₉	—	2	4	no	—
64	C₂³×C₉	—	4	1	no	—
65	C₂³×C₉	—	4	4	no	—
66	C₂³×C₉	—	4	4	no	—
67	C₂³×C₉	—	4	4	no	—
68	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂³×C₉	C₂×D₄	1	2	yes	—
70	D₄×C₉	—	2	8	no	—
71	D₄×C₉	—	4	4	no	—
72	D₄×C₉	—	4	8	no	—
73	D₄×C₉	—	4	8	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	4	2	no	—
78	C₂×C₄×C₉	—	4	1	no	—
79	C₂×C₄×C₉	—	4	4	no	—
80	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
83	C₈×C₉	—	2	4	no	—
84	C₂×#121	C₂×C₉	1	2	yes	complemented
85	#481	C₂×C₉	1	4	yes	complemented
86	#469	C₂×C₉	1	4	yes	complemented
87	#447	C₂×C₉	1	4	yes	complemented
88	#407	C₂×C₉	1	1	yes	characteristic, complemented
89	C₂⁴×C₃	—	2	2	no	—
90	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
91	C₂×C₃×D₄	—	2	4	no	—
92	C₂×C₃×D₄	—	2	4	no	—
93	C₂×C₃×D₄	—	2	2	no	—
94	C₂×C₃×D₄	—	4	4	no	—
95	C₂×C₃×D₄	—	4	8	no	—
96	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	4	2	no	—
99	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
100	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
102	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₃×D₈	—	4	4	no	—
104	C₃×M₁₆	—	4	2	no	—
105	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
108	C₃×C₂²⋊C₄	—	2	4	no	—
109	C₃×C₂²⋊C₄	—	2	4	no	—
110	C₂²×C₉	—	2	2	no	—
111	C₂²×C₉	—	2	2	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	—	2	2	no	—
114	C₂²×C₉	—	4	4	no	—
115	C₂²×C₉	—	4	4	no	—
116	C₂²×C₉	—	4	4	no	—
117	C₂²×C₉	—	4	4	no	—
118	C₂²×C₉	—	8	8	no	—
119	C₂²×C₉	AGL(1,ℤ/8)	1	4	yes	—
120	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₉	—	2	1	no	—
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	—	4	4	no	—
124	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
125	C₂²×D₄	—	2	2	no	—
126	C₂²×D₄	C₂²×C₉	1	2	yes	complemented
127	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
128	C₂×D₈	C₂²×C₉	1	4	yes	complemented
129	C₂×M₁₆	C₂²×C₉	1	2	yes	complemented
130	#121	—	2	2	no	—
131	#121	—	2	2	no	—
132	#121	C₂²×C₉	1	4	yes	—
133	#121	C₂²×C₉	1	4	yes	complemented
134	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	complemented
135	C₂×C₂²⋊C₄	—	2	2	no	—
136	#106	C₂²×C₉	1	4	yes	complemented
137	#102	C₂²×C₉	1	8	yes	complemented
138	C₂²⋊C₈	C₂²×C₉	1	4	yes	complemented
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	4	1	no	—
142	C₂³×C₃	—	4	4	no	—
143	C₂³×C₃	—	4	4	no	—
144	C₂³×C₃	—	4	4	no	—
145	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
147	C₃×D₄	—	2	8	no	—
148	C₃×D₄	—	4	4	no	—
149	C₃×D₄	—	4	8	no	—
150	C₃×D₄	—	4	8	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	4	1	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	4	no	—
157	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
159	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
160	C₃×C₈	—	2	4	no	—
161	C₂×C₉	—	2	4	no	—
162	C₂×C₉	—	4	1	no	—
163	C₂×C₉	—	8	4	no	—
164	C₂×C₉	#534	1	1	yes	characteristic
165	C₂×C₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
166	C₂⁴	—	2	2	no	—
167	C₂⁴	D₄×C₉	1	1	yes	characteristic
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	—	4	8	no	—
173	C₂×D₄	C₂³×C₉	1	4	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	D₄×C₉	1	2	yes	—
177	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₉	1	2	yes	—
179	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
180	D₈	—	4	4	no	—
181	M₁₆	—	4	2	no	—
182	C₂×C₈	C₂³×C₉	1	4	yes	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	C₂³×C₉	1	4	yes	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	4	4	no	—
192	C₂²×C₃	—	4	4	no	—
193	C₂²×C₃	—	4	4	no	—
194	C₂²×C₃	—	4	4	no	—
195	C₂²×C₃	—	8	8	no	—
196	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	4	yes	—
197	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	4	4	no	—
201	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
202	C₉	#3192	1	1	yes	characteristic, direct product factor, Sylow
203	C₂³	—	2	4	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	4	1	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	4	4	no	—
208	C₂³	—	4	4	no	—
209	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
210	C₂³	C₂×D₄×C₉	1	2	yes	—
211	D₄	—	2	8	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	8	no	—
214	D₄	—	4	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
223	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₂×C₃	—	2	4	no	—
226	C₂×C₃	—	4	1	no	—
227	C₂×C₃	—	8	4	no	—
228	C₂×C₃	C₃×#534	1	1	yes	characteristic
229	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	8	no	—
239	C₂²	C₉×AGL(1,ℤ/8)	1	4	yes	—
240	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	4	no	—
244	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
245	C₃	C₃×#3192	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	4	1	no	—
248	C₂	—	8	4	no	—
249	C₂	C₉×#534	1	1	yes	characteristic
250	C₂	C₂×C₉×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#3192	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#121	C₂	semidirect
2. C₉×#481	C₂	semidirect
3. C₉×#469	C₂	semidirect
4. C₉×#447	C₂	semidirect
5. C₉×#407	C₂	semidirect
8. C₂²×D₄×C₉	C₂²	semidirect
10. C₂×C₉×D₈	C₂²	semidirect
11. C₂×C₉×M₁₆	C₂²	semidirect
15. C₉×#121	C₂²	semidirect
16. C₂×C₉×C₄⋊C₄	C₂²	semidirect
18. C₉×#106	C₂²	semidirect
19. C₉×#102	C₂²	semidirect
20. C₉×C₂²⋊C₈	C₂²	semidirect
47. #3192	C₉	direct (N × H)
84. C₂×#121	C₂×C₉	semidirect
85. #481	C₂×C₉	semidirect
86. #469	C₂×C₉	semidirect
87. #447	C₂×C₉	semidirect
88. #407	C₂×C₉	semidirect
126. C₂²×D₄	C₂²×C₉	semidirect
128. C₂×D₈	C₂²×C₉	semidirect
129. C₂×M₁₆	C₂²×C₉	semidirect
133. #121	C₂²×C₉	semidirect
134. C₂×C₄⋊C₄	C₂²×C₉	semidirect
136. #106	C₂²×C₉	semidirect
137. #102	C₂²×C₉	semidirect
138. C₂²⋊C₈	C₂²×C₉	semidirect
202. C₉	#3192	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×#121	2
C₉×#481	2
C₉×#469	2
C₉×#447	2
C₉×#407	2
C₃×#3192	3

Subgroups — C9×#3192 #11767913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3192 #11767913