Subgroups — C₉×#5752 #13153407

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#1009	C₂	1	2	yes	maximal, complemented
2	C₃×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#908	C₂	1	6	yes	maximal, complemented
4	C₉×#904	C₂	1	3	yes	maximal, complemented
5	C₂×C₉×D₂₄	C₂	1	3	yes	maximal, complemented
6	C₃×#5752	C₃	1	1	yes	maximal, characteristic
7	C₉×#576	—	3	1	no	maximal
8	C₃×C₉×#41	C₂²	1	1	yes	characteristic
9	C₉×#239	—	2	2	no	—
10	D₃×D₄×C₉	—	2	6	no	—
11	C₉×#235	C₂²	1	6	yes	complemented
12	C₂×C₉×D₁₂	C₂²	1	6	yes	complemented
13	C₃×C₉×M₁₆	C₂²	1	3	yes	complemented
14	C₂×C₃×C₈×C₉	C₂²	1	3	yes	complemented
15	C₉×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
16	C₉×D₂₄	C₂²	1	3	yes	complemented
17	C₉×D₂₄	C₂²	1	6	yes	complemented
18	C₉×#204	C₂²	1	6	yes	complemented
19	C₃×#1009	C₂×C₃	1	2	yes	—
20	C₃²×#131	C₂×C₃	1	1	yes	characteristic
21	C₃×#908	C₂×C₃	1	6	yes	—
22	C₃×#904	C₂×C₃	1	3	yes	—
23	C₂×C₃×D₂₄	C₂×C₃	1	3	yes	—
24	C₉×H(5,𝔽₂)	—	3	2	no	—
25	C₉×AGL(1,ℤ/8)	—	3	6	no	—
26	C₉×#135	—	3	3	no	—
27	C₂×C₉×D₈	—	3	3	no	—
28	C₉×#131	—	2	1	no	—
29	C₉×#131	D₃	1	1	yes	characteristic, complemented
30	C₂²×D₃×C₉	—	2	6	no	—
31	C₃×Q₈×C₉	D₄	1	1	yes	characteristic
32	C₃×D₄×C₉	D₄	1	3	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
34	C₉×#67	—	2	6	no	—
35	C₉×D₁₂	—	2	6	no	—
36	C₉×D₁₂	C₂³	1	6	yes	—
37	C₄×D₃×C₉	—	2	6	no	—
38	C₉×Dic₆	C₂³	1	2	yes	—
39	C₃×C₈×C₉	C₂³	1	3	yes	—
40	C₃×C₈×C₉	C₂³	1	1	yes	characteristic
41	C₃×#576	—	3	1	no	—
42	#5752	C₉	1	1	yes	characteristic, direct product factor
43	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
44	C₃×#239	—	2	2	no	—
45	C₃×D₃×D₄	—	2	6	no	—
46	C₃×#235	C₂²×C₃	1	6	yes	—
47	C₂×C₃×D₁₂	C₂²×C₃	1	6	yes	—
48	C₃²×M₁₆	C₂²×C₃	1	3	yes	—
49	C₂×C₃²×C₈	C₂²×C₃	1	3	yes	—
50	C₃×Dic₁₂	C₂²×C₃	1	1	yes	characteristic
51	C₃×D₂₄	C₂²×C₃	1	3	yes	—
52	C₃×D₂₄	C₂²×C₃	1	6	yes	—
53	C₃×#204	C₂²×C₃	1	6	yes	—
54	C₉×#41	—	2	1	no	—
55	C₉×#41	—	3	6	no	—
56	C₉×#41	—	6	2	no	—
57	C₉×#41	C₂×D₃	1	1	yes	characteristic
58	C₂×D₄×C₉	—	3	6	no	—
59	C₂×D₄×C₉	—	6	6	no	—
60	C₉×Q₁₆	—	3	1	no	—
61	C₉×SD₁₆	—	3	6	no	—
62	C₉×D₈	—	3	3	no	—
63	C₉×D₈	—	3	6	no	—
64	C₉×M₁₆	—	2	3	no	—
65	C₉×M₁₆	C₂×D₃	1	3	yes	complemented
66	C₂×C₈×C₉	—	2	3	no	—
67	C₂×C₈×C₉	C₂×D₃	1	3	yes	complemented
68	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
69	C₂×D₃×C₉	—	2	6	no	—
70	C₂×D₃×C₉	—	8	6	no	—
71	C₃×C₄×C₉	C₂×D₄	1	3	yes	—
72	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
73	C₉×Dic₃	—	2	2	no	—
74	C₃×H(5,𝔽₂)	—	3	2	no	—
75	#1009	C₂×C₉	1	2	yes	complemented
76	C₃×AGL(1,ℤ/8)	—	3	6	no	—
77	C₃×#135	—	3	3	no	—
78	C₂×C₃×D₈	—	3	3	no	—
79	C₃×#131	—	2	1	no	—
80	C₃×#131	C₂×C₉	1	1	yes	characteristic, complemented
81	C₃×#131	C₃×D₃	1	1	yes	characteristic
82	#908	C₂×C₉	1	6	yes	complemented
83	#904	C₂×C₉	1	3	yes	complemented
84	C₂×D₂₄	C₂×C₉	1	3	yes	complemented
85	C₂²×C₃×D₃	—	2	6	no	—
86	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
87	C₃²×D₄	C₃×D₄	1	3	yes	—
88	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
89	C₃×#67	—	2	6	no	—
90	C₃×D₁₂	—	2	6	no	—
91	C₃×D₁₂	C₂³×C₃	1	6	yes	—
92	C₃×C₄×D₃	—	2	6	no	—
93	C₃×Dic₆	C₂³×C₃	1	2	yes	—
94	C₂³×C₉	—	6	6	no	—
95	C₃²×C₈	C₂³×C₃	1	3	yes	—
96	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
97	Q₈×C₉	—	2	1	no	—
98	Q₈×C₉	—	3	2	no	—
99	Q₈×C₉	D₁₂	1	1	yes	characteristic
100	D₄×C₉	—	2	3	no	—
101	D₄×C₉	—	3	6	no	—
102	D₄×C₉	—	6	6	no	—
103	D₄×C₉	—	6	6	no	—
104	D₄×C₉	D₁₂	1	3	yes	—
105	C₂×C₄×C₉	—	2	3	no	—
106	C₂×C₄×C₉	—	6	6	no	—
107	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
108	C₈×C₉	—	2	1	no	—
109	C₈×C₉	—	2	3	no	—
110	C₈×C₉	C₂²×D₃	1	1	yes	characteristic
111	C₈×C₉	C₂²×D₃	1	3	yes	—
112	#576	—	3	1	no	Sylow
113	C₂×C₃×C₉	—	2	3	no	—
114	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
115	D₃×C₉	—	4	6	no	—
116	C₃×#41	—	2	1	no	—
117	C₃×#41	—	3	6	no	—
118	C₃×#41	—	6	2	no	—
119	C₃×#41	C₂²×C₉	1	1	yes	characteristic
120	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
121	C₂×C₃×D₄	—	3	6	no	—
122	C₂×C₃×D₄	—	6	6	no	—
123	#239	—	2	2	no	—
124	D₃×D₄	—	2	6	no	—
125	#235	C₂²×C₉	1	6	yes	complemented
126	C₂×D₁₂	C₂²×C₉	1	6	yes	complemented
127	C₃×Q₁₆	—	3	1	no	—
128	C₃×SD₁₆	—	3	6	no	—
129	C₃×D₈	—	3	3	no	—
130	C₃×D₈	—	3	6	no	—
131	C₃×M₁₆	—	2	3	no	—
132	C₃×M₁₆	C₂²×C₉	1	3	yes	complemented
133	C₃×M₁₆	C₂×C₃×D₃	1	3	yes	—
134	C₂×C₃×C₈	—	2	3	no	—
135	C₂×C₃×C₈	C₂²×C₉	1	3	yes	complemented
136	C₂×C₃×C₈	C₂×C₃×D₃	1	3	yes	—
137	Dic₁₂	C₂²×C₉	1	1	yes	characteristic, complemented
138	D₂₄	C₂²×C₉	1	3	yes	complemented
139	D₂₄	C₂²×C₉	1	6	yes	complemented
140	#204	C₂²×C₉	1	6	yes	complemented
141	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
142	C₂×C₃×D₃	—	2	6	no	—
143	C₂×C₃×D₃	—	8	6	no	—
144	C₃²×C₄	C₂×C₃×D₄	1	3	yes	—
145	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
146	C₃×Dic₃	—	2	2	no	—
147	C₂²×C₉	—	2	3	no	—
148	C₂²×C₉	—	6	6	no	—
149	C₂²×C₉	—	24	6	no	—
150	C₂²×C₉	C₂×D₁₂	1	3	yes	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	2	3	no	—
153	C₄×C₉	—	6	2	no	—
154	C₄×C₉	C₂×D₁₂	1	3	yes	—
155	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
156	H(5,𝔽₂)	—	3	2	no	—
157	AGL(1,ℤ/8)	—	3	6	no	—
158	#135	—	3	3	no	—
159	C₂×D₈	—	3	3	no	—
160	#131	D₃×C₉	1	1	yes	characteristic, complemented
161	C₃×C₉	#576	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₃	—	6	6	no	—
163	C₂²×D₃	—	2	6	no	—
164	C₃×Q₈	—	2	1	no	—
165	C₃×Q₈	—	3	2	no	—
166	C₃×Q₈	D₄×C₉	1	1	yes	characteristic
167	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
168	C₃×D₄	—	2	3	no	—
169	C₃×D₄	—	3	6	no	—
170	C₃×D₄	—	6	6	no	—
171	C₃×D₄	—	6	6	no	—
172	C₃×D₄	D₄×C₉	1	3	yes	—
173	C₃×D₄	C₃×D₁₂	1	3	yes	—
174	C₂×C₃×C₄	—	2	3	no	—
175	C₂×C₃×C₄	—	6	6	no	—
176	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
177	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
178	#67	—	2	6	no	—
179	D₁₂	—	2	6	no	—
180	D₁₂	C₂³×C₉	1	6	yes	—
181	C₄×D₃	—	2	6	no	—
182	Dic₆	C₂³×C₉	1	2	yes	—
183	C₃×C₈	—	2	1	no	—
184	C₃×C₈	—	2	3	no	—
185	C₃×C₈	C₂³×C₉	1	3	yes	—
186	C₃×C₈	C₂³×C₉	1	1	yes	characteristic
187	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
188	C₃×C₈	C₂²×C₃×D₃	1	3	yes	—
189	C₂×C₃²	—	2	3	no	—
190	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
191	C₃×D₃	—	4	6	no	—
192	C₂×C₉	—	2	3	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	4	3	no	—
195	C₂×C₉	—	12	6	no	—
196	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
197	#41	—	3	6	no	—
198	#41	—	6	2	no	—
199	#41	C₂×D₃×C₉	1	1	yes	characteristic
200	C₂×D₄	—	3	6	no	—
201	C₂×D₄	—	6	6	no	—
202	Q₁₆	—	3	1	no	—
203	SD₁₆	—	3	6	no	—
204	D₈	—	3	3	no	—
205	D₈	—	3	6	no	—
206	M₁₆	C₂×D₃×C₉	1	3	yes	complemented
207	C₂×C₈	C₂×D₃×C₉	1	3	yes	complemented
208	C₂²×C₃	—	2	3	no	—
209	C₂²×C₃	—	6	6	no	—
210	C₂²×C₃	—	24	6	no	—
211	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
212	C₂²×C₃	C₂×C₃×D₁₂	1	3	yes	—
213	C₂×D₃	—	2	6	no	—
214	C₂×D₃	—	8	6	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	3	no	—
217	C₃×C₄	—	6	2	no	—
218	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
219	C₃×C₄	C₂⁴×C₉	1	1	yes	characteristic
220	C₃×C₄	C₂×C₃×D₁₂	1	3	yes	—
221	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
222	Dic₃	—	2	2	no	—
223	C₃²	C₃×#576	1	1	yes	characteristic
224	C₉	—	2	1	no	—
225	C₉	#5752	1	1	yes	characteristic, direct product factor
226	C₂³	—	6	6	no	—
227	Q₈	—	3	2	no	—
228	Q₈	C₉×D₁₂	1	1	yes	characteristic
229	D₄	—	3	6	no	—
230	D₄	—	6	6	no	—
231	D₄	—	6	6	no	—
232	D₄	C₉×D₁₂	1	3	yes	—
233	C₂×C₄	—	6	6	no	—
234	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
235	C₈	C₂²×D₃×C₉	1	1	yes	characteristic
236	C₈	C₂²×D₃×C₉	1	3	yes	—
237	C₂×C₃	—	2	3	no	—
238	C₂×C₃	—	2	3	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	4	3	no	—
241	C₂×C₃	—	12	6	no	—
242	C₂×C₃	C₂²×D₄×C₉	1	1	yes	characteristic
243	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
244	D₃	—	4	6	no	—
245	C₂²	—	6	6	no	—
246	C₂²	—	24	6	no	—
247	C₂²	C₂×C₉×D₁₂	1	3	yes	—
248	C₄	—	6	2	no	—
249	C₄	C₂×C₉×D₁₂	1	3	yes	—
250	C₄	C₂³×D₃×C₉	1	1	yes	characteristic
251	C₃	—	2	1	no	—
252	C₃	C₉×#576	1	1	yes	characteristic, complemented
253	C₃	C₃×#5752	1	1	yes	characteristic
254	C₂	—	2	3	no	—
255	C₂	—	12	6	no	—
256	C₂	C₂²×C₉×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²×D₁₂
commutator	C₃×C₄	C₂⁴×C₉
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₃×C₉×#131	C₂
socle	C₂×C₃²	C₂²×C₃×D₄
prime radical	C₃×#5752	C₃
cosocle	C₃²×C₄	C₂⁴×C₃
hypercenter	C₉×#131	D₃
hypocenter	C₃	C₉×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₉×#41 ◁ C₉×#131

Complements

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

N	H ≅ G/N	type
1. C₉×#1009	C₂	semidirect
2. C₃×C₉×#131	C₂	semidirect
3. C₉×#908	C₂	semidirect
4. C₉×#904	C₂	semidirect
5. C₂×C₉×D₂₄	C₂	semidirect
11. C₉×#235	C₂²	semidirect
12. C₂×C₉×D₁₂	C₂²	semidirect
13. C₃×C₉×M₁₆	C₂²	semidirect
14. C₂×C₃×C₈×C₉	C₂²	semidirect
15. C₉×Dic₁₂	C₂²	semidirect
16. C₉×D₂₄	C₂²	semidirect
17. C₉×D₂₄	C₂²	semidirect
18. C₉×#204	C₂²	semidirect
29. C₉×#131	D₃	semidirect
42. #5752	C₉	direct (N × H)
65. C₉×M₁₆	C₂×D₃	semidirect
67. C₂×C₈×C₉	C₂×D₃	semidirect
75. #1009	C₂×C₉	semidirect
80. C₃×#131	C₂×C₉	semidirect
82. #908	C₂×C₉	semidirect
83. #904	C₂×C₉	semidirect
84. C₂×D₂₄	C₂×C₉	semidirect
125. #235	C₂²×C₉	semidirect
126. C₂×D₁₂	C₂²×C₉	semidirect
132. C₃×M₁₆	C₂²×C₉	semidirect
135. C₂×C₃×C₈	C₂²×C₉	semidirect
137. Dic₁₂	C₂²×C₉	semidirect
138. D₂₄	C₂²×C₉	semidirect
139. D₂₄	C₂²×C₉	semidirect
140. #204	C₂²×C₉	semidirect
160. #131	D₃×C₉	semidirect
161. C₃×C₉	#576	semidirect
206. M₁₆	C₂×D₃×C₉	semidirect
207. C₂×C₈	C₂×D₃×C₉	semidirect
225. C₉	#5752	direct (N × H)
252. C₃	C₉×#576	semidirect

Maximal subgroups

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

subgroup	index
C₉×#1009	2
C₃×C₉×#131	2
C₉×#908	2
C₉×#904	2
C₂×C₉×D₂₄	2
C₃×#5752	3
C₉×#576	3

Subgroups — C9×#5752 #13153407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5752 #13153407