Subgroups — C₉×#5031 #13142787

Order 1728.

394 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	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#216	C₂	1	2	yes	maximal
3	C₉×#869	C₂	1	1	yes	maximal, characteristic
4	C₉×#810	C₂	1	2	yes	maximal
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₃×#5031	C₃	1	1	yes	maximal, characteristic
7	C₉×#462	—	3	1	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Dic₆	C₂²	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	—	2	1	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×#216	—	2	4	no	—
15	C₉×C₄⋊Dic₃	—	2	1	no	—
16	C₉×C₄⋊Dic₃	—	2	1	no	—
17	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
18	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#216	C₂×C₃	1	2	yes	—
20	C₃×#869	C₂×C₃	1	1	yes	characteristic
21	C₃×#810	C₂×C₃	1	2	yes	—
22	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	C₂×C₉×Q₁₆	—	3	2	no	—
24	C₉×C₄⋊Q₈	—	3	1	no	—
25	C₄×Q₈×C₉	—	2	1	no	—
26	C₄×Q₈×C₉	D₃	1	1	yes	characteristic
27	C₉×C₄⋊C₈	—	3	1	no	—
28	C₉×#103	—	3	2	no	—
29	C₃×Q₈×C₉	—	2	1	no	—
30	C₃×Q₈×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂×C₉×Dic₃	—	2	2	no	—
37	C₉×Dic₆	—	2	2	no	—
38	C₉×Dic₆	—	2	2	no	—
39	C₉×C₃⋊C₈	—	2	2	no	—
40	C₃×#462	—	3	1	no	—
41	#5031	C₉	1	1	yes	characteristic, direct product factor
42	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
44	C₃²×C₄⋊C₄	—	2	1	no	—
45	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₄²	—	2	1	no	—
47	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₃×#216	—	2	4	no	—
49	C₃×C₄⋊Dic₃	—	2	1	no	—
50	C₃×C₄⋊Dic₃	—	2	1	no	—
51	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
52	C₂×Q₈×C₉	—	2	1	no	—
53	C₂×Q₈×C₉	—	3	2	no	—
54	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
55	C₉×Q₁₆	—	6	4	no	—
56	C₂×C₈×C₉	—	3	2	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	1	no	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	—	6	1	no	—
61	C₉×C₄⋊C₄	—	6	1	no	—
62	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
63	C₄²×C₉	—	2	2	no	—
64	C₄²×C₉	—	2	1	no	—
65	C₄²×C₉	—	2	1	no	—
66	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×C₉	—	2	2	no	—
69	C₃×C₄×C₉	—	2	1	no	—
70	C₃×C₄×C₉	—	4	1	no	—
71	C₃×C₄×C₉	Q₁₆	1	2	yes	complemented
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×C₉	#41	1	1	yes	characteristic
74	C₉×Dic₃	—	4	2	no	—
75	C₂×C₃×Q₁₆	—	3	2	no	—
76	C₃×C₄⋊Q₈	—	3	1	no	—
77	C₃×C₄×Q₈	—	2	1	no	—
78	C₃×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
79	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
80	C₂×#216	C₂×C₉	1	2	yes	—
81	#869	C₂×C₉	1	1	yes	characteristic
82	C₃×C₄⋊C₈	—	3	1	no	—
83	C₃×#103	—	3	2	no	—
84	#810	C₂×C₉	1	2	yes	—
85	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic
86	C₃²×Q₈	—	2	1	no	—
87	C₃²×Q₈	C₃×D₄	1	2	yes	—
88	C₂×C₃²×C₄	—	2	1	no	—
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₃×Dic₃	—	2	2	no	—
94	C₃×Dic₆	—	2	2	no	—
95	C₃×Dic₆	—	2	2	no	—
96	C₃×C₃⋊C₈	—	2	2	no	—
97	Q₈×C₉	—	2	2	no	—
98	Q₈×C₉	—	2	1	no	—
99	Q₈×C₉	—	2	2	no	—
100	Q₈×C₉	—	6	2	no	—
101	Q₈×C₉	—	6	2	no	—
102	Q₈×C₉	D₁₂	1	2	yes	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	6	2	no	—
111	C₂×C₄×C₉	#67	1	1	yes	characteristic
112	C₂×C₄×C₉	#67	1	1	yes	characteristic
113	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
114	C₈×C₉	—	6	2	no	—
115	#462	—	3	1	no	Sylow
116	C₂×C₃×C₉	#121	1	1	yes	characteristic
117	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₃×C₉	#137	1	1	yes	characteristic
119	C₂×C₃×Q₈	—	2	1	no	—
120	C₂×C₃×Q₈	—	3	2	no	—
121	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
122	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂×Dic₆	C₂²×C₉	1	2	yes	—
124	C₃×Q₁₆	—	6	4	no	—
125	C₂×C₃×C₈	—	3	2	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₄⋊C₄	—	2	1	no	—
129	C₃×C₄⋊C₄	—	2	1	no	—
130	C₃×C₄⋊C₄	—	6	1	no	—
131	C₃×C₄⋊C₄	—	6	1	no	—
132	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
133	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	—	2	1	no	—
136	C₃×C₄²	—	2	1	no	—
137	C₃×C₄²	—	2	1	no	—
138	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
139	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
140	#216	—	2	4	no	—
141	C₄⋊Dic₃	—	2	1	no	—
142	C₄⋊Dic₃	—	2	1	no	—
143	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
144	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
145	C₃²×C₄	—	2	2	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	—	4	1	no	—
148	C₃²×C₄	C₃×Q₁₆	1	2	yes	—
149	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	C₃²×C₄	C₃×#41	1	1	yes	characteristic
151	C₃×Dic₃	—	4	2	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	C₂×#67	1	1	yes	characteristic
154	C₄×C₉	—	2	4	no	—
155	C₄×C₉	—	2	2	no	—
156	C₄×C₉	—	2	1	no	—
157	C₄×C₉	—	2	2	no	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	4	1	no	—
161	C₄×C₉	—	4	2	no	—
162	C₄×C₉	—	4	1	no	—
163	C₄×C₉	—	12	2	no	—
164	C₄×C₉	#216	1	2	yes	complemented
165	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
166	C₄×C₉	#235	1	1	yes	characteristic
167	C₂×Q₁₆	—	3	2	no	—
168	C₄⋊Q₈	—	3	1	no	—
169	C₄×Q₈	D₃×C₉	1	1	yes	characteristic
170	C₄⋊C₈	—	3	1	no	—
171	#103	—	3	2	no	—
172	C₃×C₉	#462	1	1	yes	characteristic, complemented, Sylow
173	C₃×Q₈	—	2	2	no	—
174	C₃×Q₈	—	2	1	no	—
175	C₃×Q₈	—	2	1	no	—
176	C₃×Q₈	—	2	2	no	—
177	C₃×Q₈	—	6	2	no	—
178	C₃×Q₈	—	6	2	no	—
179	C₃×Q₈	D₄×C₉	1	2	yes	—
180	C₃×Q₈	C₃×D₁₂	1	2	yes	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	6	2	no	—
191	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
192	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
195	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
197	C₂×Dic₃	—	2	2	no	—
198	Dic₆	—	2	2	no	—
199	Dic₆	—	2	2	no	—
200	C₃×C₈	—	6	2	no	—
201	C₃⋊C₈	—	2	2	no	—
202	C₂×C₃²	C₃×#121	1	1	yes	characteristic
203	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
204	C₂×C₃²	C₃×#137	1	1	yes	characteristic
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	#930	1	1	yes	characteristic
209	C₂×C₉	C₂×#216	1	1	yes	characteristic
210	C₂×C₉	#951	1	1	yes	characteristic
211	C₂×Q₈	—	3	2	no	—
212	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
213	Q₁₆	—	6	4	no	—
214	C₂×C₈	—	3	2	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	6	1	no	—
217	C₄⋊C₄	—	6	1	no	—
218	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
223	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	4	1	no	—
233	C₃×C₄	—	4	2	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	—	4	1	no	—
236	C₃×C₄	—	12	2	no	—
237	C₃×C₄	C₉×Q₁₆	1	2	yes	complemented
238	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
239	C₃×C₄	C₉×#41	1	1	yes	characteristic
240	C₃×C₄	C₃×#216	1	2	yes	—
241	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
242	C₃×C₄	C₃×#235	1	1	yes	characteristic
243	Dic₃	—	4	2	no	—
244	C₃²	C₃×#462	1	1	yes	characteristic
245	C₉	—	2	1	no	—
246	C₉	#5031	1	1	yes	characteristic, direct product factor
247	Q₈	—	2	1	no	—
248	Q₈	—	6	2	no	—
249	Q₈	—	6	2	no	—
250	Q₈	C₉×D₁₂	1	2	yes	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	C₉×#67	1	1	yes	characteristic
255	C₂×C₄	C₉×#67	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
257	C₈	—	6	2	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	C₉×#121	1	1	yes	characteristic
262	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
263	C₂×C₃	C₉×#137	1	1	yes	characteristic
264	C₂×C₃	C₃×#930	1	1	yes	characteristic
265	C₂×C₃	C₂×C₃×#216	1	1	yes	characteristic
266	C₂×C₃	C₃×#951	1	1	yes	characteristic
267	C₂²	C₂×C₉×#67	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	12	2	no	—
272	C₄	C₉×#216	1	2	yes	complemented
273	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
274	C₄	C₉×#235	1	1	yes	characteristic
275	C₃	—	2	1	no	—
276	C₃	C₉×#462	1	1	yes	characteristic, complemented
277	C₃	C₃×#5031	1	1	yes	characteristic
278	C₂	C₉×#930	1	1	yes	characteristic
279	C₂	C₂×C₉×#216	1	1	yes	characteristic
280	C₂	C₉×#951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×Q₈×C₉	D₃
hypocenter	C₃	C₉×#462
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×Q₈×C₉

Complements

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

N	H ≅ G/N	type
41. #5031	C₉	direct (N × H)
71. C₃×C₄×C₉	Q₁₆	semidirect
164. C₄×C₉	#216	semidirect
172. C₃×C₉	#462	semidirect
237. C₃×C₄	C₉×Q₁₆	semidirect
246. C₉	#5031	direct (N × H)
272. C₄	C₉×#216	semidirect
276. C₃	C₉×#462	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₂×C₉×#216	2
C₉×#869	2
C₉×#810	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#5031	3
C₉×#462	3

Subgroups — C9×#5031 #13142787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5031 #13142787