Subgroups — C₉×#4685 #13141426

Order 1728.

398 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
4	C₉×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4685	C₃	1	1	yes	maximal, characteristic
7	C₉×#431	—	3	1	no	maximal
8	C₂×C₄×D₃×C₉	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃×C₉	C₄	1	1	yes	characteristic
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₄×C₉×Dic₃	C₄	1	1	yes	characteristic
14	C₄×C₉×Dic₃	C₄	1	1	yes	characteristic
15	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃×C₄²×D₃	C₂×C₃	1	1	yes	characteristic
18	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	2	yes	—
20	C₃×Dic₃⋊C₈	C₂×C₃	1	2	yes	—
21	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄²×C₉	—	3	1	no	—
23	C₉×C₄⋊C₈	—	3	2	no	—
24	C₉×C₂²⋊C₈	—	3	2	no	—
25	C₄×C₈×C₉	—	2	1	no	—
26	C₄×C₈×C₉	—	3	1	no	—
27	C₄×C₈×C₉	D₃	1	1	yes	characteristic, complemented
28	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₄×D₃×C₉	—	2	2	no	—
36	C₄×D₃×C₉	—	2	2	no	—
37	C₄×D₃×C₉	C₈	1	4	yes	complemented
38	C₃×C₈×C₉	—	2	2	no	—
39	C₉×C₃⋊C₈	—	2	2	no	—
40	C₃×#431	—	3	1	no	—
41	#4685	C₉	1	1	yes	characteristic, direct product factor
42	C₂×C₃×C₄×D₃	C₃×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×D₃	C₃×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
46	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
47	C₃×C₄×Dic₃	C₃×C₄	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₃×C₄	1	1	yes	characteristic
49	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
51	C₂²×C₄×C₉	—	3	1	no	—
52	C₂²×C₄×C₉	—	3	1	no	—
53	C₂²×C₄×C₉	—	3	1	no	—
54	C₂×C₈×C₉	—	2	2	no	—
55	C₂×C₈×C₉	—	3	2	no	—
56	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
57	C₄²×C₉	—	2	1	no	—
58	C₄²×C₉	—	3	1	no	—
59	C₄²×C₉	—	3	1	no	—
60	C₄²×C₉	—	3	1	no	—
61	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×D₃×C₉	—	2	1	no	—
64	C₂×D₃×C₉	—	2	1	no	—
65	C₂×D₃×C₉	C₂×C₈	1	2	yes	—
66	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
67	C₃×C₄×C₉	M₁₆	1	2	yes	—
68	C₃×C₄×C₉	#41	1	2	yes	—
69	C₉×Dic₃	—	2	1	no	—
70	C₉×Dic₃	—	2	1	no	—
71	C₉×Dic₃	C₂×C₈	1	2	yes	—
72	C₂×C₃×C₄²	—	3	1	no	—
73	C₄²×D₃	C₂×C₉	1	1	yes	characteristic
74	C₃×C₄⋊C₈	—	3	2	no	—
75	C₃×C₂²⋊C₈	—	3	2	no	—
76	C₃×C₄×C₈	—	2	1	no	—
77	C₃×C₄×C₈	—	3	1	no	—
78	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
79	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
80	(C₂×D₃)⋊C₈	C₂×C₉	1	2	yes	—
81	Dic₃⋊C₈	C₂×C₉	1	2	yes	complemented
82	C₄×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
83	C₂²×C₃×D₃	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
90	C₃×C₄×D₃	—	2	2	no	—
91	C₃×C₄×D₃	—	2	2	no	—
92	C₃×C₄×D₃	C₃×C₈	1	4	yes	—
93	C₂³×C₉	—	3	1	no	—
94	C₃²×C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	—	2	2	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	3	1	no	—
100	C₂×C₄×C₉	—	3	4	no	—
101	C₂×C₄×C₉	—	3	1	no	—
102	C₂×C₄×C₉	—	3	1	no	—
103	C₂×C₄×C₉	—	6	2	no	—
104	C₂×C₄×C₉	—	6	2	no	—
105	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
106	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
108	C₈×C₉	—	2	4	no	—
109	C₈×C₉	—	2	2	no	—
110	C₈×C₉	—	6	2	no	—
111	#431	—	3	1	no	Sylow
112	C₂×C₃×C₉	#117	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
115	D₃×C₉	—	2	2	no	—
116	C₂²×C₃×C₄	—	3	1	no	—
117	C₂²×C₃×C₄	—	3	1	no	—
118	C₂²×C₃×C₄	—	3	1	no	—
119	C₂×C₄×D₃	C₄×C₉	1	1	yes	characteristic
120	C₂×C₄×D₃	C₄×C₉	1	1	yes	characteristic
121	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	3	2	no	—
124	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
125	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	3	1	no	—
128	C₃×C₄²	—	3	1	no	—
129	C₃×C₄²	—	3	1	no	—
130	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
131	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
132	C₄×Dic₃	C₄×C₉	1	1	yes	characteristic
133	C₄×Dic₃	C₄×C₉	1	1	yes	characteristic
134	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
135	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
136	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×C₃×D₃	—	2	1	no	—
138	C₂×C₃×D₃	—	2	1	no	—
139	C₂×C₃×D₃	C₂×C₃×C₈	1	2	yes	—
140	C₃²×C₄	C₂×C₃×C₈	1	2	yes	—
141	C₃²×C₄	C₃×M₁₆	1	2	yes	—
142	C₃²×C₄	C₃×#41	1	2	yes	—
143	C₃×Dic₃	—	2	1	no	—
144	C₃×Dic₃	—	2	1	no	—
145	C₃×Dic₃	C₂×C₃×C₈	1	2	yes	—
146	C₂²×C₉	—	2	1	no	—
147	C₂²×C₉	—	3	2	no	—
148	C₂²×C₉	—	6	1	no	—
149	C₂²×C₉	—	6	1	no	—
150	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
151	C₄×C₉	—	2	2	no	—
152	C₄×C₉	—	2	2	no	—
153	C₄×C₉	—	2	2	no	—
154	C₄×C₉	—	3	2	no	—
155	C₄×C₉	—	6	1	no	—
156	C₄×C₉	—	6	1	no	—
157	C₄×C₉	D₃×C₈	1	2	yes	—
158	C₄×C₉	ΓL(1,𝔽₂₅)	1	2	yes	—
159	C₄×C₉	#235	1	2	yes	—
160	C₂×C₄²	—	3	1	no	—
161	C₄⋊C₈	—	3	2	no	—
162	C₂²⋊C₈	—	3	2	no	—
163	C₄×C₈	—	3	1	no	—
164	C₄×C₈	D₃×C₉	1	1	yes	characteristic, complemented
165	C₃×C₉	#431	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₃	—	3	1	no	—
167	C₂²×D₃	C₂×C₄×C₉	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	3	4	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	6	2	no	—
176	C₂×C₃×C₄	—	6	2	no	—
177	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
180	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
183	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
184	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
185	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
186	C₄×D₃	—	2	2	no	—
187	C₄×D₃	—	2	2	no	—
188	C₄×D₃	C₈×C₉	1	4	yes	complemented
189	C₃×C₈	—	2	4	no	—
190	C₃×C₈	—	2	2	no	—
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	—	6	2	no	—
193	C₃⋊C₈	—	2	2	no	—
194	C₂×C₃²	C₃×#117	1	1	yes	characteristic
195	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
196	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
197	C₃×D₃	—	2	2	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	2	1	no	—
201	C₂×C₉	—	6	2	no	—
202	C₂×C₉	#872	1	1	yes	characteristic
203	C₂×C₉	C₂×D₃×C₈	1	1	yes	characteristic
204	C₂×C₉	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₂²×C₄	—	3	1	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
210	C₄²	—	3	1	no	—
211	C₄²	—	3	1	no	—
212	C₄²	—	3	1	no	—
213	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	3	2	no	—
216	C₂²×C₃	—	6	1	no	—
217	C₂²×C₃	—	6	1	no	—
218	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
220	C₂×D₃	—	2	1	no	—
221	C₂×D₃	—	2	1	no	—
222	C₂×D₃	C₂×C₈×C₉	1	2	yes	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	3	2	no	—
227	C₃×C₄	—	6	1	no	—
228	C₃×C₄	—	6	1	no	—
229	C₃×C₄	C₂×C₈×C₉	1	2	yes	—
230	C₃×C₄	C₉×M₁₆	1	2	yes	—
231	C₃×C₄	C₉×#41	1	2	yes	—
232	C₃×C₄	C₃×D₃×C₈	1	2	yes	—
233	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
234	C₃×C₄	C₃×#235	1	2	yes	—
235	Dic₃	—	2	1	no	—
236	Dic₃	—	2	1	no	—
237	Dic₃	C₂×C₈×C₉	1	2	yes	—
238	C₃²	C₃×#431	1	1	yes	characteristic
239	C₉	—	2	1	no	—
240	C₉	#4685	1	1	yes	characteristic, direct product factor
241	C₂³	—	3	1	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	4	no	—
244	C₂×C₄	—	3	1	no	—
245	C₂×C₄	—	3	1	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	—	6	2	no	—
248	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
249	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	6	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	6	2	no	—
257	C₂×C₃	C₉×#117	1	1	yes	characteristic
258	C₂×C₃	C₂²×C₈×C₉	1	1	yes	characteristic
259	C₂×C₃	C₂×C₉×M₁₆	1	1	yes	characteristic
260	C₂×C₃	C₃×#872	1	1	yes	characteristic
261	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
262	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
263	D₃	—	2	2	no	—
264	C₂²	—	3	2	no	—
265	C₂²	—	6	1	no	—
266	C₂²	—	6	1	no	—
267	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
268	C₄	—	3	2	no	—
269	C₄	—	6	1	no	—
270	C₄	—	6	1	no	—
271	C₄	D₃×C₈×C₉	1	2	yes	—
272	C₄	C₉×ΓL(1,𝔽₂₅)	1	2	yes	—
273	C₄	C₉×#235	1	2	yes	—
274	C₃	—	2	1	no	—
275	C₃	C₉×#431	1	1	yes	characteristic, complemented
276	C₃	C₃×#4685	1	1	yes	characteristic
277	C₂	—	6	2	no	—
278	C₂	C₉×#872	1	1	yes	characteristic
279	C₂	C₂×D₃×C₈×C₉	1	1	yes	characteristic
280	C₂	C₂×C₉×ΓL(1,𝔽₂₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₉	C₂×D₃
commutator	C₂×C₃	C₂²×C₈×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂²×C₃×D₃	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₉×#431
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₉ ◁ C₄×C₈×C₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₉×Dic₃⋊C₈	C₂	semidirect
5. C₄×C₉×C₃⋊C₈	C₂	semidirect
27. C₄×C₈×C₉	D₃	semidirect
37. C₄×D₃×C₉	C₈	semidirect
41. #4685	C₉	direct (N × H)
78. C₃×C₄×C₈	C₂×C₉	semidirect
81. Dic₃⋊C₈	C₂×C₉	semidirect
82. C₄×C₃⋊C₈	C₂×C₉	semidirect
164. C₄×C₈	D₃×C₉	semidirect
165. C₃×C₉	#431	semidirect
188. C₄×D₃	C₈×C₉	semidirect
240. C₉	#4685	direct (N × H)
275. C₃	C₉×#431	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₃×C₉	2
C₃×C₄×C₈×C₉	2
C₉×(C₂×D₃)⋊C₈	2
C₉×Dic₃⋊C₈	2
C₄×C₉×C₃⋊C₈	2
C₃×#4685	3
C₉×#431	3

Subgroups — C9×#4685 #13141426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4685 #13141426