Subgroups — C₉×#4894 #13142261

Order 1728.

342 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#907	C₂	1	2	yes	maximal, complemented
2	C₉×#904	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#825	C₂	1	2	yes	maximal, complemented
5	C₉×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4894	C₃	1	1	yes	maximal, characteristic
7	C₉×#444	—	3	1	no	maximal
8	C₉×#235	C₂²	1	2	yes	—
9	C₃×C₉×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
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×#907	C₂×C₃	1	2	yes	—
19	C₃×#904	C₂×C₃	1	1	yes	characteristic
20	C₃²×#108	C₂×C₃	1	1	yes	characteristic
21	C₃×#825	C₂×C₃	1	2	yes	—
22	C₃×#815	C₂×C₃	1	1	yes	characteristic
23	C₉×#135	—	3	1	no	—
24	C₉×#131	—	3	2	no	—
25	C₉×#108	—	2	1	no	—
26	C₉×#108	D₃	1	1	yes	characteristic, complemented
27	C₉×#104	—	3	2	no	—
28	C₉×#97	—	3	1	no	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₉×#67	—	2	2	no	—
31	C₂×C₉×Dic₃	—	2	1	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	—	2	2	no	—
36	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
38	C₉×C₃⋊C₈	D₄	1	2	yes	—
39	C₃×#444	—	3	1	no	—
40	#4894	C₉	1	1	yes	characteristic, direct product factor
41	C₃×#235	C₂²×C₃	1	2	yes	—
42	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
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	—	2	2	no	—
56	C₉×M₁₆	—	6	2	no	—
57	C₉×M₁₆	C₂×D₃	1	2	yes	—
58	C₂×C₈×C₉	—	2	1	no	—
59	C₂×C₈×C₉	—	3	1	no	—
60	C₂×C₈×C₉	—	6	2	no	—
61	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
62	C₄²×C₉	—	3	1	no	—
63	C₂²×C₃×C₉	#41	1	1	yes	characteristic
64	C₂×D₃×C₉	—	2	2	no	—
65	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₉×Dic₃	—	2	2	no	—
68	C₉×Dic₃	—	4	1	no	—
69	C₃×#135	—	3	1	no	—
70	C₃×#131	—	3	2	no	—
71	#907	C₂×C₉	1	2	yes	complemented
72	#904	C₂×C₉	1	1	yes	characteristic
73	C₃×#108	—	2	1	no	—
74	C₃×#108	C₂×C₉	1	1	yes	characteristic, complemented
75	C₃×#108	C₃×D₃	1	1	yes	characteristic
76	C₃×#104	—	3	2	no	—
77	C₃×#97	—	3	1	no	—
78	#825	C₂×C₉	1	2	yes	complemented
79	#815	C₂×C₉	1	1	yes	characteristic, complemented
80	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
81	C₃×#67	—	2	2	no	—
82	C₂×C₃×Dic₃	—	2	1	no	—
83	C₃×D₁₂	C₂×C₃×C₄	1	2	yes	—
84	C₃×C₄×D₃	—	2	2	no	—
85	C₃×Dic₆	C₂×C₃×C₄	1	2	yes	—
86	C₃²×C₈	—	2	2	no	—
87	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
88	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
89	C₃×C₃⋊C₈	C₃×D₄	1	2	yes	—
90	Q₈×C₉	—	3	2	no	—
91	D₄×C₉	—	3	2	no	—
92	D₄×C₉	—	6	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	6	2	no	—
95	C₂×C₄×C₉	—	6	1	no	—
96	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
97	C₈×C₉	—	2	2	no	—
98	C₈×C₉	—	2	1	no	—
99	C₈×C₉	—	2	1	no	—
100	C₈×C₉	—	3	2	no	—
101	C₈×C₉	—	4	2	no	—
102	C₈×C₉	C₄×D₃	1	1	yes	characteristic
103	C₈×C₉	C₄×D₃	1	1	yes	characteristic
104	#444	—	3	1	no	Sylow
105	C₂×C₃×C₉	—	2	1	no	—
106	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
107	D₃×C₉	—	4	2	no	—
108	C₃×#41	—	3	2	no	—
109	#235	C₂²×C₉	1	2	yes	—
110	C₃×Q₁₆	—	3	1	no	—
111	C₃×SD₁₆	—	3	2	no	—
112	C₃×D₈	—	3	1	no	—
113	C₃×M₁₆	—	2	2	no	—
114	C₃×M₁₆	—	6	2	no	—
115	C₃×M₁₆	C₂²×C₉	1	2	yes	—
116	C₃×M₁₆	C₂×C₃×D₃	1	2	yes	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	3	1	no	—
119	C₂×C₃×C₈	—	6	2	no	—
120	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
122	C₃×C₄²	—	3	1	no	—
123	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
124	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
125	Dic₁₂	C₄×C₉	1	1	yes	characteristic, complemented
126	D₂₄	C₄×C₉	1	1	yes	characteristic, complemented
127	#204	C₄×C₉	1	2	yes	complemented
128	ΓL(1,𝔽₂₅)	—	2	2	no	—
129	D₃×C₈	—	2	2	no	—
130	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
131	C₂×C₃×D₃	—	2	2	no	—
132	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
134	C₃×Dic₃	—	2	2	no	—
135	C₃×Dic₃	—	4	1	no	—
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	—	6	2	no	—
138	C₂²×C₉	#239	1	1	yes	characteristic
139	C₄×C₉	—	2	1	no	—
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	6	2	no	—
142	C₄×C₉	—	12	1	no	—
143	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
144	C₄×C₉	D₃×D₄	1	1	yes	characteristic
145	#135	—	3	1	no	—
146	#131	—	3	2	no	—
147	#108	D₃×C₉	1	1	yes	characteristic, complemented
148	#104	—	3	2	no	—
149	#97	—	3	1	no	—
150	C₃×C₉	#444	1	1	yes	characteristic, complemented, Sylow
151	C₃×Q₈	—	3	2	no	—
152	C₃×D₄	—	3	2	no	—
153	C₃×D₄	—	6	2	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	6	2	no	—
156	C₂×C₃×C₄	—	6	1	no	—
157	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
159	#67	—	2	2	no	—
160	C₂×Dic₃	—	2	1	no	—
161	D₁₂	C₂×C₄×C₉	1	2	yes	—
162	C₄×D₃	—	2	2	no	—
163	Dic₆	C₂×C₄×C₉	1	2	yes	—
164	C₃×C₈	—	2	2	no	—
165	C₃×C₈	—	2	2	no	—
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	3	2	no	—
169	C₃×C₈	—	4	2	no	—
170	C₃×C₈	C₂×C₄×C₉	1	1	yes	characteristic
171	C₃×C₈	C₂×C₄×C₉	1	1	yes	characteristic
172	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
173	C₃×C₈	C₃×C₄×D₃	1	1	yes	characteristic
174	C₃⋊C₈	D₄×C₉	1	2	yes	—
175	C₂×C₃²	—	2	1	no	—
176	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
177	C₃×D₃	—	4	2	no	—
178	C₂×C₉	—	2	1	no	—
179	C₂×C₉	—	2	1	no	—
180	C₂×C₉	—	4	1	no	—
181	C₂×C₉	—	12	2	no	—
182	C₂×C₉	#893	1	1	yes	characteristic
183	#41	—	3	2	no	—
184	Q₁₆	—	3	1	no	—
185	SD₁₆	—	3	2	no	—
186	D₈	—	3	1	no	—
187	M₁₆	—	6	2	no	—
188	M₁₆	C₂×D₃×C₉	1	2	yes	—
189	C₂×C₈	—	3	1	no	—
190	C₂×C₈	—	6	2	no	—
191	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
192	C₄²	—	3	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	6	2	no	—
195	C₂²×C₃	C₉×#41	1	1	yes	characteristic
196	C₂²×C₃	C₃×#239	1	1	yes	characteristic
197	C₂×D₃	—	2	2	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	6	2	no	—
201	C₃×C₄	—	12	1	no	—
202	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
203	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
204	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
205	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
206	Dic₃	—	2	2	no	—
207	Dic₃	—	4	1	no	—
208	C₃²	C₃×#444	1	1	yes	characteristic
209	C₉	—	2	1	no	—
210	C₉	#4894	1	1	yes	characteristic, direct product factor
211	Q₈	—	3	2	no	—
212	D₄	—	3	2	no	—
213	D₄	—	6	2	no	—
214	C₂×C₄	—	6	2	no	—
215	C₂×C₄	—	6	1	no	—
216	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	3	2	no	—
219	C₈	C₄×D₃×C₉	1	1	yes	characteristic
220	C₈	C₄×D₃×C₉	1	1	yes	characteristic
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	4	1	no	—
225	C₂×C₃	—	12	2	no	—
226	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
227	C₂×C₃	C₃×#893	1	1	yes	characteristic
228	D₃	—	4	2	no	—
229	C₂²	—	6	2	no	—
230	C₂²	C₉×#239	1	1	yes	characteristic
231	C₄	—	6	2	no	—
232	C₄	—	12	1	no	—
233	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
234	C₄	D₃×D₄×C₉	1	1	yes	characteristic
235	C₃	—	2	1	no	—
236	C₃	C₉×#444	1	1	yes	characteristic, complemented
237	C₃	C₃×#4894	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	12	2	no	—
240	C₂	C₉×#893	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#108	C₂
socle	C₂×C₃²	C₃×C₄×D₄
prime radical	C₃×#904	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#108	D₃
hypocenter	C₃	C₉×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#108

Complements

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

N	H ≅ G/N	type
1. C₉×#907	C₂	semidirect
3. C₃×C₉×#108	C₂	semidirect
4. C₉×#825	C₂	semidirect
5. C₉×#815	C₂	semidirect
13. C₉×Dic₁₂	C₄	semidirect
14. C₉×D₂₄	C₄	semidirect
15. C₉×#204	C₄	semidirect
26. C₉×#108	D₃	semidirect
40. #4894	C₉	direct (N × H)
71. #907	C₂×C₉	semidirect
74. C₃×#108	C₂×C₉	semidirect
78. #825	C₂×C₉	semidirect
79. #815	C₂×C₉	semidirect
125. Dic₁₂	C₄×C₉	semidirect
126. D₂₄	C₄×C₉	semidirect
127. #204	C₄×C₉	semidirect
147. #108	D₃×C₉	semidirect
150. C₃×C₉	#444	semidirect
210. C₉	#4894	direct (N × H)
236. 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×#907	2
C₉×#904	2
C₃×C₉×#108	2
C₉×#825	2
C₉×#815	2
C₃×#4894	3
C₉×#444	3

Subgroups — C9×#4894 #13142261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4894 #13142261