Subgroups — C₉×#5211 #13143520

Order 1728.

890 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#217	C₂	1	2	yes	maximal
3	C₉×C₃⋊#95	C₂	1	4	yes	maximal, complemented
4	C₃×#5211	C₃	1	1	yes	maximal, characteristic
5	C₉×#381	—	3	1	no	maximal
6	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₉×Dic₃	C₂²	1	4	yes	—
11	C₉×#217	—	2	4	no	—
12	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃×#217	C₂×C₃	1	2	yes	—
14	C₂³×C₄×C₉	—	2	1	no	—
15	C₂³×C₄×C₉	D₃	1	1	yes	characteristic
16	C₃×C₃⋊#95	C₂×C₃	1	4	yes	—
17	C₂×C₉×C₂²⋊C₄	—	3	2	no	—
18	C₉×#95	—	3	4	no	—
19	C₂³×C₃×C₉	—	2	4	no	—
20	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
21	C₂³×C₃×C₉	D₄	1	4	yes	—
22	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	—	2	8	no	—
24	C₂×C₃×C₄×C₉	—	2	4	no	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
26	C₂×C₉×Dic₃	—	2	4	no	—
27	C₂×C₉×Dic₃	—	2	8	no	—
28	C₃×#381	—	3	1	no	—
29	#5211	C₉	1	1	yes	characteristic, direct product factor
30	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
31	C₂²×C₃²×C₄	—	2	4	no	—
32	C₂²×C₃²×C₄	C₃×C₄	1	4	yes	—
33	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
34	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	—
35	C₂⁴×C₉	—	2	1	no	—
36	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
37	C₃×#217	—	2	4	no	—
38	C₂²×C₄×C₉	—	2	8	no	—
39	C₂²×C₄×C₉	—	2	4	no	—
40	C₂²×C₄×C₉	—	2	2	no	—
41	C₂²×C₄×C₉	—	2	4	no	—
42	C₂²×C₄×C₉	—	3	4	no	—
43	C₂²×C₄×C₉	Dic₃	1	4	yes	complemented
44	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
45	C₉×C₂²⋊C₄	—	6	4	no	—
46	C₂²×C₃×C₉	—	2	4	no	—
47	C₂²×C₃×C₉	—	2	8	no	—
48	C₂²×C₃×C₉	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
51	C₂²×C₃×C₉	#41	1	4	yes	—
52	C₃×C₄×C₉	—	2	4	no	—
53	C₉×Dic₃	—	4	4	no	—
54	C₂³×C₃×C₄	—	2	1	no	—
55	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic
56	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
57	C₂×C₃×C₂²⋊C₄	—	3	2	no	—
58	C₂×#217	C₂×C₉	1	2	yes	—
59	C₃×#95	—	3	4	no	—
60	C₃⋊#95	C₂×C₉	1	4	yes	complemented
61	C₂³×C₃²	—	2	4	no	—
62	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
63	C₂³×C₃²	C₃×D₄	1	4	yes	—
64	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
65	C₂×C₃²×C₄	—	2	8	no	—
66	C₂×C₃²×C₄	—	2	4	no	—
67	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
68	C₂×C₃×Dic₃	—	2	4	no	—
69	C₂×C₃×Dic₃	—	2	8	no	—
70	C₂³×C₉	—	2	8	no	—
71	C₂³×C₉	—	2	4	no	—
72	C₂³×C₉	—	2	2	no	—
73	C₂³×C₉	—	2	4	no	—
74	C₂³×C₉	—	2	1	no	—
75	C₂³×C₉	C₂×Dic₃	1	2	yes	—
76	C₂³×C₉	#67	1	4	yes	—
77	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
78	C₂×C₄×C₉	—	2	16	no	—
79	C₂×C₄×C₉	—	2	8	no	—
80	C₂×C₄×C₉	—	2	8	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	4	no	—
83	C₂×C₄×C₉	—	6	8	no	—
84	C₂×C₄×C₉	—	6	4	no	—
85	C₂×C₄×C₉	C₂×Dic₃	1	4	yes	—
86	#381	—	3	1	no	Sylow
87	C₂×C₃×C₉	—	2	4	no	—
88	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃×C₉	#117	1	2	yes	—
90	C₂×C₃×C₉	#123	1	4	yes	—
91	C₂⁴×C₃	—	2	1	no	—
92	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
93	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	—	2	8	no	—
95	C₂²×C₃×C₄	—	2	4	no	—
96	C₂²×C₃×C₄	—	2	4	no	—
97	C₂²×C₃×C₄	—	2	4	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	3	4	no	—
100	C₂²×C₃×C₄	C₄×C₉	1	4	yes	complemented
101	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
102	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
103	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
104	C₂²×Dic₃	C₂²×C₉	1	4	yes	—
105	C₃×C₂²⋊C₄	—	6	4	no	—
106	#217	—	2	4	no	—
107	C₂²×C₃²	—	2	4	no	—
108	C₂²×C₃²	—	2	8	no	—
109	C₂²×C₃²	C₃×C₂²⋊C₄	1	4	yes	—
110	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
112	C₂²×C₃²	C₃×#41	1	4	yes	—
113	C₃²×C₄	—	2	4	no	—
114	C₃×Dic₃	—	4	4	no	—
115	C₂²×C₉	—	2	8	no	—
116	C₂²×C₉	—	2	16	no	—
117	C₂²×C₉	—	2	4	no	—
118	C₂²×C₉	—	2	8	no	—
119	C₂²×C₉	—	2	4	no	—
120	C₂²×C₉	—	2	4	no	—
121	C₂²×C₉	—	2	1	no	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	#217	1	4	yes	—
124	C₂²×C₉	#235	1	4	yes	—
125	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
126	C₂²×C₉	C₂×#67	1	2	yes	—
127	C₄×C₉	—	2	8	no	—
128	C₄×C₉	—	2	4	no	—
129	C₄×C₉	—	12	4	no	—
130	C₂³×C₄	D₃×C₉	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	—	3	2	no	—
132	#95	—	3	4	no	—
133	C₃×C₉	#381	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₃	—	2	8	no	—
135	C₂³×C₃	—	2	4	no	—
136	C₂³×C₃	—	2	4	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	—	2	4	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	C₂×C₄×C₉	1	2	yes	—
141	C₂³×C₃	D₄×C₉	1	4	yes	—
142	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
143	C₂³×C₃	C₂×C₃×Dic₃	1	2	yes	—
144	C₂³×C₃	C₃×#67	1	4	yes	—
145	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
146	C₂×C₃×C₄	—	2	8	no	—
147	C₂×C₃×C₄	—	2	16	no	—
148	C₂×C₃×C₄	—	2	8	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	8	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	6	8	no	—
154	C₂×C₃×C₄	—	6	4	no	—
155	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
156	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
157	C₂×Dic₃	—	2	4	no	—
158	C₂×Dic₃	—	2	8	no	—
159	C₂×C₃²	—	2	4	no	—
160	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₃²	C₃×#117	1	2	yes	—
162	C₂×C₃²	C₃×#123	1	4	yes	—
163	C₂×C₉	—	2	8	no	—
164	C₂×C₉	—	2	4	no	—
165	C₂×C₉	—	2	2	no	—
166	C₂×C₉	—	2	4	no	—
167	C₂×C₉	—	2	1	no	—
168	C₂×C₉	#926	1	2	yes	—
169	C₂×C₉	#929	1	4	yes	—
170	C₂×C₉	C₂×#217	1	1	yes	characteristic
171	C₂⁴	C₂×D₃×C₉	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	3	4	no	—
174	C₂²×C₄	C₉×Dic₃	1	4	yes	complemented
175	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
176	C₂²⋊C₄	—	6	4	no	—
177	C₂²×C₃	—	2	8	no	—
178	C₂²×C₃	—	2	16	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	8	no	—
181	C₂²×C₃	—	2	4	no	—
182	C₂²×C₃	—	2	8	no	—
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	4	no	—
187	C₂²×C₃	C₉×C₂²⋊C₄	1	4	yes	—
188	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
189	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
190	C₂²×C₃	C₉×#41	1	4	yes	—
191	C₂²×C₃	C₃×#217	1	4	yes	—
192	C₂²×C₃	C₃×#235	1	4	yes	—
193	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
194	C₂²×C₃	C₂×C₃×#67	1	2	yes	—
195	C₃×C₄	—	2	8	no	—
196	C₃×C₄	—	2	4	no	—
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	—	12	4	no	—
199	Dic₃	—	4	4	no	—
200	C₃²	C₃×#381	1	1	yes	characteristic
201	C₉	—	2	1	no	—
202	C₉	#5211	1	1	yes	characteristic, direct product factor
203	C₂³	—	2	4	no	—
204	C₂³	C₂×C₉×Dic₃	1	2	yes	—
205	C₂³	C₉×#67	1	4	yes	—
206	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	6	8	no	—
210	C₂×C₄	—	6	4	no	—
211	C₂×C₄	C₂×C₉×Dic₃	1	4	yes	—
212	C₂×C₃	—	2	8	no	—
213	C₂×C₃	—	2	4	no	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₃	C₉×#117	1	2	yes	—
220	C₂×C₃	C₉×#123	1	4	yes	—
221	C₂×C₃	C₃×#926	1	2	yes	—
222	C₂×C₃	C₃×#929	1	4	yes	—
223	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
224	C₂²	—	2	4	no	—
225	C₂²	—	2	8	no	—
226	C₂²	C₉×#217	1	4	yes	—
227	C₂²	C₉×#235	1	4	yes	—
228	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
229	C₂²	C₂×C₉×#67	1	2	yes	—
230	C₄	—	2	4	no	—
231	C₄	—	12	4	no	—
232	C₃	—	2	1	no	—
233	C₃	C₉×#381	1	1	yes	characteristic, complemented
234	C₃	C₃×#5211	1	1	yes	characteristic
235	C₂	—	2	4	no	—
236	C₂	C₉×#926	1	2	yes	—
237	C₂	C₉×#929	1	4	yes	—
238	C₂	C₂×C₉×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle, cosocle	C₂³×C₃²	C₂³×C₃
prime radical	C₂⁴×C₃²	C₂²×C₃
hypercenter	C₂³×C₄×C₉	D₃
hypocenter	C₃	C₉×#381
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
3. C₉×C₃⋊#95	C₂	semidirect
8. C₂²×C₃×C₄×C₉	C₄	semidirect
29. #5211	C₉	direct (N × H)
43. C₂²×C₄×C₉	Dic₃	semidirect
60. C₃⋊#95	C₂×C₉	semidirect
100. C₂²×C₃×C₄	C₄×C₉	semidirect
133. C₃×C₉	#381	semidirect
174. C₂²×C₄	C₉×Dic₃	semidirect
202. C₉	#5211	direct (N × H)
233. C₃	C₉×#381	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×C₉×#217	2
C₉×C₃⋊#95	2
C₃×#5211	3
C₉×#381	3

Subgroups — C9×#5211 #13143520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5211 #13143520