Subgroups — C₉×#4975 #13142566

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×Dic₃⋊C₄	C₂	1	4	yes	maximal
4	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₃×#4975	C₃	1	1	yes	maximal, characteristic
6	C₉×#384	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	—	2	2	no	—
12	C₉×C₄⋊Dic₃	—	2	2	no	—
13	C₉×Dic₃⋊C₄	—	2	8	no	—
14	C₄×C₉×Dic₃	C₄	1	4	yes	complemented
15	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	4	yes	—
18	C₂×C₃×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	—	2	1	no	—
20	C₂×C₉×C₄⋊C₄	—	3	4	no	—
21	C₂×C₉×C₄⋊C₄	—	3	1	no	—
22	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic
23	C₂×C₄²×C₉	—	3	1	no	—
24	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	—	2	4	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
30	C₂×C₉×Dic₃	—	2	4	no	—
31	C₂×C₉×Dic₃	—	2	2	no	—
32	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
33	C₂×C₉×Dic₃	D₄	1	4	yes	—
34	C₂×C₉×Dic₃	Q₈	1	4	yes	—
35	C₃×#384	—	3	1	no	—
36	#4975	C₉	1	1	yes	characteristic, direct product factor
37	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
38	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
40	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
41	C₃²×C₄⋊C₄	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×Dic₃⋊C₄	—	2	8	no	—
44	C₃×C₄×Dic₃	C₃×C₄	1	4	yes	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	—	2	2	no	—
47	C₂²×C₄×C₉	—	3	2	no	—
48	C₂²×C₄×C₉	—	3	2	no	—
49	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
50	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
51	C₉×C₄⋊C₄	—	2	4	no	—
52	C₉×C₄⋊C₄	—	2	2	no	—
53	C₉×C₄⋊C₄	—	6	8	no	—
54	C₉×C₄⋊C₄	—	6	2	no	—
55	C₄²×C₉	—	3	4	no	—
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
61	C₃×C₄×C₉	—	4	2	no	—
62	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	complemented
63	C₉×Dic₃	—	4	2	no	—
64	C₉×Dic₃	C₄⋊C₄	1	8	yes	complemented
65	C₂×C₃×C₄⋊C₄	—	2	1	no	—
66	C₂×C₃×C₄⋊C₄	—	3	4	no	—
67	C₂×C₃×C₄⋊C₄	—	3	1	no	—
68	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
69	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
70	C₂×C₃×C₄²	—	3	1	no	—
71	C₂×C₄⋊Dic₃	C₂×C₉	1	1	yes	characteristic
72	C₂×Dic₃⋊C₄	C₂×C₉	1	4	yes	—
73	C₂×C₄×Dic₃	C₂×C₉	1	1	yes	characteristic
74	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
79	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
80	C₂×C₃×Dic₃	—	2	4	no	—
81	C₂×C₃×Dic₃	—	2	2	no	—
82	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
83	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
84	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	3	4	no	—
94	C₂×C₄×C₉	—	3	4	no	—
95	C₂×C₄×C₉	—	3	4	no	—
96	C₂×C₄×C₉	—	4	4	no	—
97	C₂×C₄×C₉	—	6	4	no	—
98	C₂×C₄×C₉	—	6	2	no	—
99	C₂×C₄×C₉	Dic₆	1	2	yes	—
100	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
101	C₂×C₄×C₉	#67	1	2	yes	—
102	#384	—	3	1	no	Sylow
103	C₂×C₃×C₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₉	#127	1	1	yes	characteristic
106	C₂×C₃×C₉	C₄⋊Q₈	1	2	yes	—
107	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	—	3	2	no	—
111	C₂²×C₃×C₄	—	3	2	no	—
112	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
113	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
116	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
117	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
118	C₃×C₄⋊C₄	—	2	4	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	—	6	8	no	—
122	C₃×C₄⋊C₄	—	6	2	no	—
123	C₃×C₄²	—	3	4	no	—
124	C₄⋊Dic₃	—	2	2	no	—
125	Dic₃⋊C₄	—	2	8	no	—
126	C₄×Dic₃	C₄×C₉	1	4	yes	complemented
127	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
129	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
131	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
132	C₃²×C₄	—	4	2	no	—
133	C₃²×C₄	C₃×C₄⋊C₄	1	4	yes	—
134	C₃×Dic₃	—	4	2	no	—
135	C₃×Dic₃	C₃×C₄⋊C₄	1	8	yes	—
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	—	2	1	no	—
138	C₂²×C₉	—	2	2	no	—
139	C₂²×C₉	—	2	2	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₂×Dic₆	1	1	yes	characteristic
142	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂²×C₉	D₃×D₄	1	2	yes	—
144	C₂²×C₉	D₃×Q₈	1	2	yes	—
145	C₂²×C₉	C₂×#67	1	1	yes	characteristic
146	C₄×C₉	—	2	4	no	—
147	C₄×C₉	—	3	8	no	—
148	C₄×C₉	—	4	4	no	—
149	C₄×C₉	—	4	2	no	—
150	C₄×C₉	—	12	2	no	—
151	C₄×C₉	Dic₃⋊C₄	1	4	yes	complemented
152	C₂×C₄⋊C₄	—	3	4	no	—
153	C₂×C₄⋊C₄	—	3	1	no	—
154	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic
155	C₂×C₄²	—	3	1	no	—
156	C₃×C₉	#384	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
159	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	3	4	no	—
169	C₂×C₃×C₄	—	3	4	no	—
170	C₂×C₃×C₄	—	3	4	no	—
171	C₂×C₃×C₄	—	4	4	no	—
172	C₂×C₃×C₄	—	6	4	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
175	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
176	C₂×C₃×C₄	Q₈×C₉	1	2	yes	—
177	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
178	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
179	C₂×C₃×C₄	C₃×#67	1	2	yes	—
180	C₂×Dic₃	—	2	4	no	—
181	C₂×Dic₃	—	2	2	no	—
182	C₂×Dic₃	C₂×C₄×C₉	1	4	yes	—
183	C₂×Dic₃	D₄×C₉	1	4	yes	—
184	C₂×Dic₃	Q₈×C₉	1	4	yes	—
185	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	2	yes	—
186	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₃²	C₃×#127	1	1	yes	characteristic
188	C₂×C₃²	C₃×C₄⋊Q₈	1	2	yes	—
189	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
190	C₂×C₉	—	2	2	no	—
191	C₂×C₉	—	2	2	no	—
192	C₂×C₉	—	2	1	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	#888	1	2	yes	—
196	C₂×C₉	D₃×C₄⋊C₄	1	2	yes	—
197	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
198	C₂×C₉	#940	1	1	yes	characteristic
199	C₂×C₉	#944	1	1	yes	characteristic
200	C₂²×C₄	—	3	2	no	—
201	C₂²×C₄	—	3	2	no	—
202	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
203	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	6	8	no	—
206	C₄⋊C₄	—	6	2	no	—
207	C₄²	—	3	4	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
214	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
215	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
216	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
217	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
218	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
220	C₂²×C₃	C₃×D₃×D₄	1	2	yes	—
221	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
222	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	3	8	no	—
225	C₃×C₄	—	4	4	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	—	12	2	no	—
229	C₃×C₄	C₉×C₄⋊C₄	1	4	yes	complemented
230	C₃×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
231	Dic₃	—	4	2	no	—
232	Dic₃	C₉×C₄⋊C₄	1	8	yes	complemented
233	C₃²	C₃×#384	1	1	yes	characteristic
234	C₉	—	2	1	no	—
235	C₉	#4975	1	1	yes	characteristic, direct product factor
236	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	3	4	no	—
240	C₂×C₄	—	3	4	no	—
241	C₂×C₄	—	3	4	no	—
242	C₂×C₄	—	6	4	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	C₉×Dic₆	1	2	yes	—
245	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
246	C₂×C₄	C₉×#67	1	2	yes	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₃	C₂×C₉×C₄⋊C₄	1	2	yes	—
254	C₂×C₃	C₉×#127	1	1	yes	characteristic
255	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₃	C₉×C₄⋊Q₈	1	2	yes	—
257	C₂×C₃	C₃×#888	1	2	yes	—
258	C₂×C₃	C₃×D₃×C₄⋊C₄	1	2	yes	—
259	C₂×C₃	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
260	C₂×C₃	C₃×#940	1	1	yes	characteristic
261	C₂×C₃	C₃×#944	1	1	yes	characteristic
262	C₂²	C₂×C₉×Dic₆	1	1	yes	characteristic
263	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
264	C₂²	D₃×D₄×C₉	1	2	yes	—
265	C₂²	D₃×Q₈×C₉	1	2	yes	—
266	C₂²	C₂×C₉×#67	1	1	yes	characteristic
267	C₄	—	3	8	no	—
268	C₄	—	4	2	no	—
269	C₄	—	12	2	no	—
270	C₄	C₉×Dic₃⋊C₄	1	4	yes	complemented
271	C₃	—	2	1	no	—
272	C₃	C₉×#384	1	1	yes	characteristic, complemented
273	C₃	C₃×#4975	1	1	yes	characteristic
274	C₂	C₉×#888	1	2	yes	—
275	C₂	D₃×C₉×C₄⋊C₄	1	2	yes	—
276	C₂	C₂×C₉×Dic₃⋊C₄	1	1	yes	characteristic
277	C₂	C₉×#940	1	1	yes	characteristic
278	C₂	C₉×#944	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	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#384
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₄×C₉×Dic₃	C₄	semidirect
36. #4975	C₉	direct (N × H)
62. C₃×C₄×C₉	C₄⋊C₄	semidirect
64. C₉×Dic₃	C₄⋊C₄	semidirect
126. C₄×Dic₃	C₄×C₉	semidirect
151. C₄×C₉	Dic₃⋊C₄	semidirect
156. C₃×C₉	#384	semidirect
229. C₃×C₄	C₉×C₄⋊C₄	semidirect
232. Dic₃	C₉×C₄⋊C₄	semidirect
235. C₉	#4975	direct (N × H)
270. C₄	C₉×Dic₃⋊C₄	semidirect
272. C₃	C₉×#384	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₉×C₄⋊Dic₃	2
C₂×C₉×Dic₃⋊C₄	2
C₂×C₄×C₉×Dic₃	2
C₃×#4975	3
C₉×#384	3

Subgroups — C9×#4975 #13142566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4975 #13142566