Subgroups — C₃×#10588330 #13141273

Order 1728.

538 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Dic₉⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₃×C₉⋊#95	C₂	1	2	yes	maximal
6	C₃×#4650	—	3	1	no	maximal
7	#10588330	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
11	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
13	C₃×Dic₉⋊C₄	—	2	4	no	—
14	C₃×C₄×Dic₉	C₄	1	4	yes	complemented
15	C₂×C₃×Dic₃⋊C₄	—	3	2	no	—
16	C₂×C₃×C₄×Dic₃	—	3	1	no	—
17	C₃²×#95	D₃	1	1	yes	characteristic
18	C₃×C₃⋊#95	—	3	2	no	—
19	C₃×C₃⋊#95	—	3	1	no	—
20	C₂×Dic₉⋊C₄	C₂×C₃	1	2	yes	—
21	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
22	C₉×#95	—	2	1	no	—
23	C₉×#95	C₂×C₃	1	1	yes	characteristic
24	C₉⋊#95	C₂×C₃	1	1	yes	characteristic
25	C₉⋊#95	C₂×C₃	1	2	yes	—
26	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×Dic₉	—	2	2	no	—
33	C₂×C₃×Dic₉	—	2	2	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₂×C₃×Dic₉	—	2	2	no	—
36	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
37	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
38	C₂×C₃×Dic₉	D₄	1	2	yes	—
39	C₂×C₃×Dic₉	Q₈	1	2	yes	—
40	C₃×#382	—	9	1	no	—
41	#4650	—	3	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×Dic₃	—	3	1	no	—
45	C₂²×C₃×Dic₃	—	3	1	no	—
46	C₂²×C₃×Dic₃	—	3	2	no	—
47	C₃×Dic₃⋊C₄	—	6	4	no	—
48	C₃×C₄×Dic₃	—	3	4	no	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
52	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
53	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
54	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
55	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
56	Dic₉⋊C₄	—	2	4	no	—
57	C₄×Dic₉	C₃×C₄	1	4	yes	complemented
58	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₃×C₉	#41	1	2	yes	—
62	C₂²×C₃×C₉	#41	1	2	yes	—
63	C₃×C₄×C₉	—	2	2	no	—
64	C₃×C₄×C₉	—	4	2	no	—
65	C₃×Dic₉	—	2	2	no	—
66	C₃×Dic₉	—	4	2	no	—
67	C₃×Dic₉	C₄⋊C₄	1	4	yes	—
68	C₂×C₃×C₄⋊C₄	—	9	2	no	—
69	C₂×C₃×C₄²	—	9	1	no	—
70	C₂×Dic₃⋊C₄	—	3	2	no	—
71	C₂×C₄×Dic₃	—	3	1	no	—
72	C₃×#95	—	2	1	no	—
73	C₃×#95	—	9	2	no	—
74	C₃×#95	—	9	1	no	—
75	C₃×#95	D₉	1	1	yes	characteristic
76	C₃×#95	C₃×D₃	1	1	yes	characteristic
77	C₃⋊#95	—	3	2	no	—
78	C₃⋊#95	—	3	1	no	—
79	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
85	C₂×C₃×Dic₃	—	3	2	no	—
86	C₂×C₃×Dic₃	—	3	2	no	—
87	C₂×C₃×Dic₃	—	3	2	no	—
88	C₂×C₃×Dic₃	—	3	2	no	—
89	C₂×C₃×Dic₃	—	6	2	no	—
90	C₂×C₃×Dic₃	—	6	2	no	—
91	C₂×C₃×Dic₃	—	6	2	no	—
92	C₂×C₃×Dic₃	—	6	2	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	4	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	4	2	no	—
102	C₂×C₄×C₉	—	4	2	no	—
103	C₂×C₄×C₉	—	4	2	no	—
104	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
105	C₂×Dic₉	—	2	2	no	—
106	C₂×Dic₉	—	2	2	no	—
107	C₂×Dic₉	—	2	2	no	—
108	C₂×Dic₉	—	2	2	no	—
109	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
110	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
111	C₂×Dic₉	C₃×D₄	1	2	yes	—
112	C₂×Dic₉	C₃×Q₈	1	2	yes	—
113	#382	—	9	1	no	Sylow
114	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₃×C₉	#117	1	1	yes	characteristic
116	C₂×C₃×C₉	#117	1	1	yes	characteristic
117	C₂×C₃×C₉	#124	1	2	yes	—
118	C₂×C₃×C₉	#125	1	2	yes	—
119	C₂²×C₃×C₄	—	2	2	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	9	1	no	—
122	C₂²×C₃×C₄	—	9	1	no	—
123	C₂²×C₃×C₄	—	9	2	no	—
124	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
125	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
127	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
128	C₂²×Dic₃	—	3	1	no	—
129	C₂²×Dic₃	—	3	1	no	—
130	C₂²×Dic₃	—	3	2	no	—
131	C₃×C₄⋊C₄	—	18	4	no	—
132	C₃×C₄²	—	9	4	no	—
133	Dic₃⋊C₄	—	6	4	no	—
134	C₄×Dic₃	—	3	4	no	—
135	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂²×C₃²	#235	1	2	yes	—
137	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
138	C₂²×C₃²	#237	1	2	yes	—
139	C₂²×C₃²	D₃×Q₈	1	1	yes	characteristic
140	C₃²×C₄	—	2	2	no	—
141	C₃²×C₄	—	4	2	no	—
142	C₃×Dic₃	—	3	4	no	—
143	C₃×Dic₃	—	6	2	no	—
144	C₃×Dic₃	—	12	2	no	—
145	C₂²×C₉	—	2	1	no	—
146	C₂²×C₉	—	2	2	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	—	2	2	no	—
149	C₂²×C₉	—	2	1	no	—
150	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
151	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
153	C₂²×C₉	C₃×#41	1	2	yes	—
154	C₂²×C₉	C₃×#41	1	2	yes	—
155	C₄×C₉	—	2	2	no	—
156	C₄×C₉	—	4	2	no	—
157	C₄×C₉	—	4	4	no	—
158	C₄×C₉	—	4	2	no	—
159	Dic₉	—	2	2	no	—
160	Dic₉	—	4	2	no	—
161	Dic₉	C₃×C₄⋊C₄	1	4	yes	—
162	C₂×C₄⋊C₄	—	9	2	no	—
163	C₂×C₄²	—	9	1	no	—
164	#95	—	9	2	no	—
165	#95	—	9	1	no	—
166	#95	C₃×D₉	1	1	yes	characteristic
167	C₃×C₉	#382	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₃	—	2	1	no	—
169	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
170	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	4	2	no	—
184	C₂×C₃×C₄	—	9	2	no	—
185	C₂×C₃×C₄	—	9	2	no	—
186	C₂×C₃×C₄	—	9	2	no	—
187	C₂×C₃×C₄	—	9	2	no	—
188	C₂×C₃×C₄	—	18	2	no	—
189	C₂×C₃×C₄	—	18	2	no	—
190	C₂×C₃×C₄	—	18	2	no	—
191	C₂×C₃×C₄	—	18	2	no	—
192	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
193	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
194	C₂×Dic₃	—	3	2	no	—
195	C₂×Dic₃	—	3	2	no	—
196	C₂×Dic₃	—	3	2	no	—
197	C₂×Dic₃	—	3	2	no	—
198	C₂×Dic₃	—	6	2	no	—
199	C₂×Dic₃	—	6	2	no	—
200	C₂×Dic₃	—	6	2	no	—
201	C₂×Dic₃	—	6	2	no	—
202	C₂×C₃²	#872	1	1	yes	characteristic
203	C₂×C₃²	#877	1	1	yes	characteristic
204	C₂×C₃²	#885	1	2	yes	—
205	C₂×C₃²	#889	1	2	yes	—
206	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	—	2	2	no	—
209	C₂×C₉	—	2	2	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₉	C₃×#117	1	1	yes	characteristic
214	C₂×C₉	C₃×#117	1	1	yes	characteristic
215	C₂×C₉	C₃×#124	1	2	yes	—
216	C₂×C₉	C₃×#125	1	2	yes	—
217	C₂²×C₄	—	9	1	no	—
218	C₂²×C₄	—	9	1	no	—
219	C₂²×C₄	—	9	2	no	—
220	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
221	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
222	C₄⋊C₄	—	18	4	no	—
223	C₄²	—	9	4	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
230	C₂²×C₃	#3697	1	2	yes	—
231	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
232	C₂²×C₃	#3699	1	2	yes	—
233	C₂²×C₃	Q₈×D₉	1	1	yes	characteristic
234	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
235	C₂²×C₃	C₃×#235	1	2	yes	—
236	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
237	C₂²×C₃	C₃×#237	1	2	yes	—
238	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	—	4	4	no	—
243	C₃×C₄	—	4	2	no	—
244	C₃×C₄	—	4	2	no	—
245	C₃×C₄	—	9	4	no	—
246	C₃×C₄	—	18	2	no	—
247	C₃×C₄	—	36	2	no	—
248	Dic₃	—	3	4	no	—
249	Dic₃	—	6	2	no	—
250	Dic₃	—	12	2	no	—
251	C₃²	#4650	1	1	yes	characteristic
252	C₉	—	2	1	no	—
253	C₉	C₃×#382	1	1	yes	characteristic, complemented
254	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	9	2	no	—
260	C₂×C₄	—	9	2	no	—
261	C₂×C₄	—	9	2	no	—
262	C₂×C₄	—	9	2	no	—
263	C₂×C₄	—	18	2	no	—
264	C₂×C₄	—	18	2	no	—
265	C₂×C₄	—	18	2	no	—
266	C₂×C₄	—	18	2	no	—
267	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	2	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	#63434	1	1	yes	characteristic
274	C₂×C₃	#63439	1	1	yes	characteristic
275	C₂×C₃	#63447	1	2	yes	—
276	C₂×C₃	#63451	1	2	yes	—
277	C₂×C₃	C₄⋊C₄×D₉	1	1	yes	characteristic
278	C₂×C₃	C₃×#872	1	1	yes	characteristic
279	C₂×C₃	C₃×#877	1	1	yes	characteristic
280	C₂×C₃	C₃×#885	1	2	yes	—
281	C₂×C₃	C₃×#889	1	2	yes	—
282	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
283	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
284	C₂²	C₃×#3697	1	2	yes	—
285	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
286	C₂²	C₃×#3699	1	2	yes	—
287	C₂²	C₃×Q₈×D₉	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	9	4	no	—
291	C₄	—	18	2	no	—
292	C₄	—	36	2	no	—
293	C₃	—	2	1	no	—
294	C₃	#10588330	1	1	yes	characteristic, direct product factor
295	C₃	C₃×#4650	1	1	yes	characteristic
296	C₂	C₃×#63434	1	1	yes	characteristic
297	C₂	C₃×#63439	1	1	yes	characteristic
298	C₂	C₃×#63447	1	2	yes	—
299	C₂	C₃×#63451	1	2	yes	—
300	C₂	C₃×C₄⋊C₄×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₃×C₉×#95	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₃×#95	D₉
hypocenter	C₉	C₃×#382
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

Complements

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

N	H ≅ G/N	type
7. #10588330	C₃	direct (N × H)
14. C₃×C₄×Dic₉	C₄	semidirect
57. C₄×Dic₉	C₃×C₄	semidirect
167. C₃×C₉	#382	semidirect
253. C₉	C₃×#382	semidirect
294. C₃	#10588330	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×Dic₉⋊C₄	2
C₂×C₃×C₄×Dic₉	2
C₃×C₉×#95	2
C₃×C₉⋊#95	2
C₃×#4650	3
#10588330	3

Subgroups — C3×#10588330 #13141273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588330 #13141273