Subgroups — C₉×#4717 #13141521

Order 1728.

352 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#904	C₂	1	1	yes	maximal, characteristic
2	C₉×#901	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#805	C₂	1	2	yes	maximal, complemented
6	C₃×#4717	C₃	1	1	yes	maximal, characteristic
7	C₉×#444	—	3	1	no	maximal
8	C₉×#235	C₂²	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₉×#208	C₂²	1	2	yes	—
13	C₉×Dic₁₂	C₄	1	1	yes	characteristic, complemented
14	C₉×D₂₄	C₄	1	1	yes	characteristic, complemented
15	C₉×#204	C₄	1	2	yes	complemented
16	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
17	D₃×C₈×C₉	—	2	2	no	—
18	C₃×#904	C₂×C₃	1	1	yes	characteristic
19	C₃×#901	C₂×C₃	1	2	yes	—
20	C₃²×#97	C₂×C₃	1	1	yes	characteristic
21	C₃×#819	C₂×C₃	1	1	yes	characteristic
22	C₃×#805	C₂×C₃	1	2	yes	—
23	C₉×#135	—	3	1	no	—
24	C₉×#131	—	3	2	no	—
25	C₉×#108	—	3	1	no	—
26	C₉×#104	—	3	2	no	—
27	C₉×#97	—	2	1	no	—
28	C₉×#97	D₃	1	1	yes	characteristic, complemented
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₉×#67	—	2	2	no	—
32	C₉×D₁₂	C₂×C₄	1	2	yes	—
33	C₄×D₃×C₉	—	2	2	no	—
34	C₉×Dic₆	C₂×C₄	1	2	yes	—
35	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×C₉	D₄	1	2	yes	—
38	C₉×C₃⋊C₈	—	2	2	no	—
39	C₃×#444	—	3	1	no	—
40	#4717	C₉	1	1	yes	characteristic, direct product factor
41	C₃×#235	C₂²×C₃	1	2	yes	—
42	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
45	C₃×#208	C₂²×C₃	1	2	yes	—
46	C₃×Dic₁₂	C₃×C₄	1	1	yes	characteristic
47	C₃×D₂₄	C₃×C₄	1	1	yes	characteristic
48	C₃×#204	C₃×C₄	1	2	yes	—
49	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
50	C₃×D₃×C₈	—	2	2	no	—
51	C₉×#41	—	3	2	no	—
52	C₉×Q₁₆	—	3	1	no	—
53	C₉×SD₁₆	—	3	2	no	—
54	C₉×D₈	—	3	1	no	—
55	C₉×M₁₆	—	3	2	no	—
56	C₉×M₁₆	—	6	2	no	—
57	C₂×C₈×C₉	—	2	1	no	—
58	C₂×C₈×C₉	—	2	1	no	—
59	C₂×C₈×C₉	—	6	2	no	—
60	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
61	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
62	C₄²×C₉	—	2	1	no	—
63	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₃×C₉	#41	1	1	yes	characteristic
65	C₂×D₃×C₉	—	2	2	no	—
66	C₃×C₄×C₉	—	4	1	no	—
67	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₉×Dic₃	—	2	2	no	—
70	C₃×#135	—	3	1	no	—
71	C₃×#131	—	3	2	no	—
72	#904	C₂×C₉	1	1	yes	characteristic
73	#901	C₂×C₉	1	2	yes	complemented
74	C₃×#108	—	3	1	no	—
75	C₃×#104	—	3	2	no	—
76	C₃×#97	—	2	1	no	—
77	C₃×#97	C₂×C₉	1	1	yes	characteristic, complemented
78	C₃×#97	C₃×D₃	1	1	yes	characteristic
79	#819	C₂×C₉	1	1	yes	characteristic, complemented
80	#805	C₂×C₉	1	2	yes	complemented
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
83	C₃×#67	—	2	2	no	—
84	C₃×D₁₂	C₂×C₃×C₄	1	2	yes	—
85	C₃×C₄×D₃	—	2	2	no	—
86	C₃×Dic₆	C₂×C₃×C₄	1	2	yes	—
87	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
88	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
89	C₃²×C₈	C₃×D₄	1	2	yes	—
90	C₃×C₃⋊C₈	—	2	2	no	—
91	Q₈×C₉	—	3	2	no	—
92	D₄×C₉	—	3	2	no	—
93	D₄×C₉	—	6	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	6	2	no	—
98	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₈×C₉	—	2	1	no	—
100	C₈×C₉	—	2	1	no	—
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	—	6	2	no	—
103	C₈×C₉	C₄×D₃	1	1	yes	characteristic
104	C₈×C₉	C₄×D₃	1	1	yes	characteristic
105	C₈×C₉	D₁₂	1	2	yes	—
106	#444	—	3	1	no	Sylow
107	C₂×C₃×C₉	—	2	1	no	—
108	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
109	D₃×C₉	—	4	2	no	—
110	C₃×#41	—	3	2	no	—
111	#235	C₂²×C₉	1	2	yes	—
112	C₃×Q₁₆	—	3	1	no	—
113	C₃×SD₁₆	—	3	2	no	—
114	C₃×D₈	—	3	1	no	—
115	C₃×M₁₆	—	3	2	no	—
116	C₃×M₁₆	—	6	2	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	6	2	no	—
120	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
126	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
127	#208	C₂²×C₉	1	2	yes	—
128	Dic₁₂	C₄×C₉	1	1	yes	characteristic, complemented
129	D₂₄	C₄×C₉	1	1	yes	characteristic, complemented
130	#204	C₄×C₉	1	2	yes	complemented
131	ΓL(1,𝔽₂₅)	—	2	2	no	—
132	D₃×C₈	—	2	2	no	—
133	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
134	C₂×C₃×D₃	—	2	2	no	—
135	C₃²×C₄	—	4	1	no	—
136	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
138	C₃×Dic₃	—	2	2	no	—
139	C₂²×C₉	—	2	1	no	—
140	C₂²×C₉	—	6	2	no	—
141	C₂²×C₉	#235	1	1	yes	characteristic
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	2	1	no	—
144	C₄×C₉	—	4	2	no	—
145	C₄×C₉	—	4	1	no	—
146	C₄×C₉	—	6	2	no	—
147	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
148	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
149	#135	—	3	1	no	—
150	#131	—	3	2	no	—
151	#108	—	3	1	no	—
152	#104	—	3	2	no	—
153	#97	D₃×C₉	1	1	yes	characteristic, complemented
154	C₃×C₉	#444	1	1	yes	characteristic, complemented, Sylow
155	C₃×Q₈	—	3	2	no	—
156	C₃×D₄	—	3	2	no	—
157	C₃×D₄	—	6	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	6	2	no	—
163	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
165	#67	—	2	2	no	—
166	D₁₂	C₂×C₄×C₉	1	2	yes	—
167	C₄×D₃	—	2	2	no	—
168	Dic₆	C₂×C₄×C₉	1	2	yes	—
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	2	1	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	6	2	no	—
173	C₃×C₈	C₂×C₄×C₉	1	1	yes	characteristic
174	C₃×C₈	C₂×C₄×C₉	1	1	yes	characteristic
175	C₃×C₈	D₄×C₉	1	2	yes	—
176	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
177	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
178	C₃×C₈	C₃×D₁₂	1	2	yes	—
179	C₃⋊C₈	—	2	2	no	—
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
182	C₃×D₃	—	4	2	no	—
183	C₂×C₉	—	2	2	no	—
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	12	2	no	—
187	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
188	#41	—	3	2	no	—
189	Q₁₆	—	3	1	no	—
190	SD₁₆	—	3	2	no	—
191	D₈	—	3	1	no	—
192	M₁₆	—	3	2	no	—
193	M₁₆	—	6	2	no	—
194	C₂×C₈	—	6	2	no	—
195	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
196	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
197	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	6	2	no	—
200	C₂²×C₃	C₉×#41	1	1	yes	characteristic
201	C₂²×C₃	C₃×#235	1	1	yes	characteristic
202	C₂×D₃	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	4	1	no	—
206	C₃×C₄	—	4	2	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	6	2	no	—
209	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
210	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
211	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
212	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
213	Dic₃	—	2	2	no	—
214	C₃²	C₃×#444	1	1	yes	characteristic
215	C₉	—	2	1	no	—
216	C₉	#4717	1	1	yes	characteristic, direct product factor
217	Q₈	—	3	2	no	—
218	D₄	—	3	2	no	—
219	D₄	—	6	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	6	2	no	—
222	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
223	C₈	—	6	2	no	—
224	C₈	C₄×D₃×C₉	1	1	yes	characteristic
225	C₈	C₄×D₃×C₉	1	1	yes	characteristic
226	C₈	C₉×D₁₂	1	2	yes	—
227	C₂×C₃	—	2	2	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	12	2	no	—
232	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
233	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
234	D₃	—	4	2	no	—
235	C₂²	—	6	2	no	—
236	C₂²	C₉×#235	1	1	yes	characteristic
237	C₄	—	4	1	no	—
238	C₄	—	6	2	no	—
239	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
240	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
241	C₃	—	2	1	no	—
242	C₃	C₉×#444	1	1	yes	characteristic, complemented
243	C₃	C₃×#4717	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	12	2	no	—
246	C₂	C₄×C₉×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#97	C₂
socle	C₂×C₃²	C₃×C₄×D₄
prime radical	C₃×#904	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#97	D₃
hypocenter	C₃	C₉×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₈×C₉ ◁ C₉×#97

Complements

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

N	H ≅ G/N	type
2. C₉×#901	C₂	semidirect
3. C₃×C₉×#97	C₂	semidirect
4. C₉×#819	C₂	semidirect
5. C₉×#805	C₂	semidirect
13. C₉×Dic₁₂	C₄	semidirect
14. C₉×D₂₄	C₄	semidirect
15. C₉×#204	C₄	semidirect
28. C₉×#97	D₃	semidirect
40. #4717	C₉	direct (N × H)
73. #901	C₂×C₉	semidirect
77. C₃×#97	C₂×C₉	semidirect
79. #819	C₂×C₉	semidirect
80. #805	C₂×C₉	semidirect
128. Dic₁₂	C₄×C₉	semidirect
129. D₂₄	C₄×C₉	semidirect
130. #204	C₄×C₉	semidirect
153. #97	D₃×C₉	semidirect
154. C₃×C₉	#444	semidirect
216. C₉	#4717	direct (N × H)
242. C₃	C₉×#444	semidirect

Maximal subgroups

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

subgroup	index
C₉×#904	2
C₉×#901	2
C₃×C₉×#97	2
C₉×#819	2
C₉×#805	2
C₃×#4717	3
C₉×#444	3

Subgroups — C9×#4717 #13141521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4717 #13141521