Subgroups — C₃×#10588517 #13142014

Order 1728.

392 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×D₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₂×D₉)⋊C₈	C₂	1	2	yes	maximal
4	C₃×C₈×Dic₉	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4833	—	3	1	no	maximal
7	#10588517	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×C₄×D₉	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×D₉	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
14	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
15	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₃×C₄²×D₃	—	3	1	no	—
18	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
19	C₃×(C₂×D₃)⋊C₈	—	3	2	no	—
20	C₃×C₈×Dic₃	—	3	2	no	—
21	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
22	C₄²×D₉	C₂×C₃	1	1	yes	characteristic
23	C₉×C₄⋊C₈	—	2	1	no	—
24	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	(C₂×D₉)⋊C₈	C₂×C₃	1	2	yes	—
26	C₈×Dic₉	C₂×C₃	1	2	yes	complemented
27	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₃×D₉	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×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×D₉	—	2	1	no	—
36	C₃×C₄×D₉	—	2	1	no	—
37	C₃×C₄×D₉	—	2	2	no	—
38	C₃×C₄×D₉	C₈	1	4	yes	complemented
39	C₃×C₈×C₉	—	2	2	no	—
40	C₃×C₉⋊C₈	—	2	2	no	—
41	C₃×#431	—	9	1	no	—
42	#4833	—	3	1	no	—
43	C₂×C₃×C₄×D₃	—	3	1	no	—
44	C₂×C₃×C₄×D₃	—	3	1	no	—
45	C₂×C₃×C₄×D₃	—	3	1	no	—
46	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
47	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₃×C₄×Dic₃	—	3	1	no	—
49	C₃×C₄×Dic₃	—	3	1	no	—
50	C₃×C₄×Dic₃	—	3	1	no	—
51	C₂×C₃×C₃⋊C₈	—	3	2	no	—
52	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
53	C₂×C₄×D₉	C₃×C₄	1	1	yes	characteristic
54	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₈×C₉	—	2	2	no	—
56	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
57	C₄²×C₉	—	2	1	no	—
58	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
59	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
60	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
61	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×D₉	—	2	1	no	—
65	C₂×C₃×D₉	—	2	1	no	—
66	C₂×C₃×D₉	C₂×C₈	1	2	yes	—
67	C₃×C₄×C₉	—	2	1	no	—
68	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
69	C₃×C₄×C₉	#41	1	1	yes	characteristic
70	C₃×C₄×C₉	#41	1	1	yes	characteristic
71	C₃×Dic₉	—	2	1	no	—
72	C₃×Dic₉	C₂×C₈	1	2	yes	—
73	C₃×Dic₉	M₁₆	1	2	yes	—
74	C₂×C₃×C₄²	—	9	1	no	—
75	C₄²×D₃	—	3	1	no	—
76	C₃×C₄⋊C₈	—	2	1	no	—
77	C₃×C₄⋊C₈	—	9	1	no	—
78	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
79	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
80	C₃×C₂²⋊C₈	—	9	2	no	—
81	C₃×C₄×C₈	—	9	2	no	—
82	(C₂×D₃)⋊C₈	—	3	2	no	—
83	C₈×Dic₃	—	3	2	no	—
84	C₄⋊(C₃⋊C₈)	—	3	1	no	—
85	C₂²×C₃×D₃	—	3	1	no	—
86	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₃×Dic₃	—	3	1	no	—
90	C₂×C₃×Dic₃	—	3	1	no	—
91	C₂×C₃×Dic₃	—	3	1	no	—
92	C₃×C₄×D₃	—	3	4	no	—
93	C₃×C₄×D₃	—	6	2	no	—
94	C₃×C₄×D₃	—	6	1	no	—
95	C₃×C₄×D₃	—	6	1	no	—
96	C₂²×D₉	C₂×C₃×C₄	1	1	yes	characteristic
97	C₃²×C₈	—	2	2	no	—
98	C₃×C₃⋊C₈	—	6	2	no	—
99	C₂×C₄×C₉	—	2	1	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
105	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
108	C₄×D₉	—	2	1	no	—
109	C₄×D₉	—	2	1	no	—
110	C₄×D₉	—	2	2	no	—
111	C₄×D₉	C₃×C₈	1	4	yes	complemented
112	C₈×C₉	—	2	2	no	—
113	C₈×C₉	—	4	2	no	—
114	C₉⋊C₈	—	2	2	no	—
115	#431	—	9	1	no	Sylow
116	C₂×C₃×C₉	#117	1	1	yes	characteristic
117	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
119	C₃×D₉	—	2	2	no	—
120	C₂²×C₃×C₄	—	9	1	no	—
121	C₂²×C₃×C₄	—	9	1	no	—
122	C₂²×C₃×C₄	—	9	1	no	—
123	C₂×C₄×D₃	—	3	1	no	—
124	C₂×C₄×D₃	—	3	1	no	—
125	C₂×C₄×D₃	—	3	1	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	9	2	no	—
128	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
129	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	9	1	no	—
132	C₃×C₄²	—	9	1	no	—
133	C₃×C₄²	—	9	1	no	—
134	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
135	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
136	C₄×Dic₃	—	3	1	no	—
137	C₄×Dic₃	—	3	1	no	—
138	C₄×Dic₃	—	3	1	no	—
139	C₂×C₃⋊C₈	—	3	2	no	—
140	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×D₃	—	3	2	no	—
142	C₂×C₃×D₃	—	6	1	no	—
143	C₂×C₃×D₃	—	6	1	no	—
144	C₃²×C₄	—	2	1	no	—
145	C₃²×C₄	D₃×C₈	1	2	yes	—
146	C₃²×C₄	#237	1	1	yes	characteristic
147	C₃²×C₄	#239	1	1	yes	characteristic
148	C₃×Dic₃	—	3	2	no	—
149	C₃×Dic₃	—	3	2	no	—
150	C₃×Dic₃	—	6	1	no	—
151	C₂²×C₉	—	2	1	no	—
152	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₂×D₉	—	2	1	no	—
154	C₂×D₉	—	2	1	no	—
155	C₂×D₉	C₂×C₃×C₈	1	2	yes	—
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	C₂×C₃×C₈	1	2	yes	—
162	C₄×C₉	C₃×#41	1	1	yes	characteristic
163	C₄×C₉	C₃×#41	1	1	yes	characteristic
164	Dic₉	—	2	1	no	—
165	Dic₉	C₂×C₃×C₈	1	2	yes	—
166	Dic₉	C₃×M₁₆	1	2	yes	—
167	C₂×C₄²	—	9	1	no	—
168	C₄⋊C₈	—	9	1	no	—
169	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic, complemented
170	C₂²⋊C₈	—	9	2	no	—
171	C₄×C₈	—	9	2	no	—
172	C₃×C₉	#431	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₃	—	9	1	no	—
174	C₂²×D₃	—	3	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	9	1	no	—
179	C₂×C₃×C₄	—	9	1	no	—
180	C₂×C₃×C₄	—	9	4	no	—
181	C₂×C₃×C₄	—	9	1	no	—
182	C₂×C₃×C₄	—	18	2	no	—
183	C₂×C₃×C₄	—	18	1	no	—
184	C₂×C₃×C₄	—	18	1	no	—
185	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
186	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
188	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
191	C₂×Dic₃	—	3	1	no	—
192	C₂×Dic₃	—	3	1	no	—
193	C₂×Dic₃	—	3	1	no	—
194	C₄×D₃	—	3	4	no	—
195	C₄×D₃	—	6	2	no	—
196	C₄×D₃	—	6	1	no	—
197	C₄×D₃	—	6	1	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	—	2	2	no	—
200	C₃×C₈	—	4	2	no	—
201	C₃×C₈	—	18	2	no	—
202	C₃⋊C₈	—	6	2	no	—
203	C₂×C₃²	#892	1	1	yes	characteristic
204	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
205	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
206	C₃×D₃	—	6	2	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	—	2	1	no	—
209	C₂×C₉	—	2	1	no	—
210	C₂×C₉	C₃×#117	1	1	yes	characteristic
211	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
212	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
213	D₉	—	2	2	no	—
214	C₂²×C₄	—	9	1	no	—
215	C₂²×C₄	—	9	1	no	—
216	C₂²×C₄	—	9	1	no	—
217	C₂×C₈	—	9	2	no	—
218	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
219	C₄²	—	9	1	no	—
220	C₄²	—	9	1	no	—
221	C₄²	—	9	1	no	—
222	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	9	2	no	—
225	C₂²×C₃	—	18	1	no	—
226	C₂²×C₃	—	18	1	no	—
227	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
228	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
229	C₂×D₃	—	3	2	no	—
230	C₂×D₃	—	6	1	no	—
231	C₂×D₃	—	6	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	4	1	no	—
238	C₃×C₄	—	9	2	no	—
239	C₃×C₄	—	9	2	no	—
240	C₃×C₄	—	18	1	no	—
241	C₃×C₄	C₈×D₉	1	2	yes	—
242	C₃×C₄	#3699	1	1	yes	characteristic
243	C₃×C₄	#3701	1	1	yes	characteristic
244	C₃×C₄	C₃×D₃×C₈	1	2	yes	—
245	C₃×C₄	C₃×#237	1	1	yes	characteristic
246	C₃×C₄	C₃×#239	1	1	yes	characteristic
247	Dic₃	—	3	2	no	—
248	Dic₃	—	3	2	no	—
249	Dic₃	—	6	1	no	—
250	C₃²	#4833	1	1	yes	characteristic
251	C₉	—	2	1	no	—
252	C₉	C₃×#431	1	1	yes	characteristic, complemented
253	C₂³	—	9	1	no	—
254	C₂×C₄	—	9	1	no	—
255	C₂×C₄	—	9	1	no	—
256	C₂×C₄	—	9	4	no	—
257	C₂×C₄	—	9	1	no	—
258	C₂×C₄	—	18	2	no	—
259	C₂×C₄	—	18	1	no	—
260	C₂×C₄	—	18	1	no	—
261	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
262	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	18	2	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	18	2	no	—
270	C₂×C₃	#63454	1	1	yes	characteristic
271	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
272	C₂×C₃	M₁₆×D₉	1	1	yes	characteristic
273	C₂×C₃	C₃×#892	1	1	yes	characteristic
274	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
275	C₂×C₃	C₃×D₃×M₁₆	1	1	yes	characteristic
276	D₃	—	6	2	no	—
277	C₂²	—	9	2	no	—
278	C₂²	—	18	1	no	—
279	C₂²	—	18	1	no	—
280	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	9	2	no	—
283	C₄	—	9	2	no	—
284	C₄	—	18	1	no	—
285	C₄	C₃×C₈×D₉	1	2	yes	—
286	C₄	C₃×#3699	1	1	yes	characteristic
287	C₄	C₃×#3701	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	#10588517	1	1	yes	characteristic, direct product factor
290	C₃	C₃×#4833	1	1	yes	characteristic
291	C₂	—	18	2	no	—
292	C₂	C₃×#63454	1	1	yes	characteristic
293	C₂	C₂×C₃×C₈×D₉	1	1	yes	characteristic
294	C₂	C₃×M₁₆×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₉	C₂×C₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#431
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₄⋊C₈	C₂	semidirect
4. C₃×C₈×Dic₉	C₂	semidirect
5. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
7. #10588517	C₃	direct (N × H)
24. C₉×C₄⋊C₈	C₂×C₃	semidirect
26. C₈×Dic₉	C₂×C₃	semidirect
27. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
38. C₃×C₄×D₉	C₈	semidirect
78. C₃×C₄⋊C₈	D₉	semidirect
111. C₄×D₉	C₃×C₈	semidirect
169. C₄⋊C₈	C₃×D₉	semidirect
172. C₃×C₉	#431	semidirect
252. C₉	C₃×#431	semidirect
289. C₃	#10588517	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄²×D₉	2
C₃×C₉×C₄⋊C₈	2
C₃×(C₂×D₉)⋊C₈	2
C₃×C₈×Dic₉	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#4833	3
#10588517	3

Subgroups — C3×#10588517 #13142014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588517 #13142014