Subgroups — C₃×#10588689 #13142714

Order 1728.

460 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×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3667	C₂	1	2	yes	maximal, complemented
3	C₃×#63389	C₂	1	2	yes	maximal, complemented
4	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5012	—	3	1	no	maximal
7	#10588689	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×#3667	—	2	4	no	—
14	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
18	C₂×C₃×#208	—	3	2	no	—
19	C₃×#830	—	3	2	no	—
20	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₃×C₄×C₃⋊C₈	—	3	1	no	—
22	C₄×D₄×C₉	—	2	1	no	—
23	C₄×D₄×C₉	C₂×C₃	1	1	yes	characteristic
24	C₂×#3667	C₂×C₃	1	2	yes	complemented
25	#63389	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
27	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
29	C₃×D₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₃×C₉⋊C₈	—	2	2	no	—
37	C₃×C₉⋊C₈	—	2	1	no	—
38	C₃×C₉⋊C₈	D₄	1	2	yes	complemented
39	C₃×#436	—	9	1	no	—
40	#5012	—	3	1	no	—
41	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
43	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
44	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
45	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₃×#208	—	6	4	no	—
47	C₂×C₃×C₃⋊C₈	—	3	1	no	—
48	C₂×C₃×C₃⋊C₈	—	3	2	no	—
49	C₂×C₃×C₃⋊C₈	—	3	1	no	—
50	C₂×D₄×C₉	—	2	1	no	—
51	C₂×D₄×C₉	C₃×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
56	C₉×C₂²⋊C₄	—	2	2	no	—
57	C₉×C₂²⋊C₄	C₃×C₄	1	2	yes	—
58	C₄²×C₉	—	2	1	no	—
59	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
60	#3667	—	2	4	no	—
61	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
63	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₃×C₉	—	2	2	no	—
65	C₂²×C₃×C₉	—	2	2	no	—
66	C₂²×C₃×C₉	—	2	2	no	—
67	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	—	2	2	no	—
69	C₃×C₄×C₉	—	2	1	no	—
70	C₃×C₄×C₉	M₁₆	1	2	yes	complemented
71	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	#41	1	1	yes	characteristic
73	C₂×C₃×M₁₆	—	9	2	no	—
74	C₃×C₄×D₄	—	2	1	no	—
75	C₃×C₄×D₄	D₉	1	1	yes	characteristic
76	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
77	C₂×#208	—	3	2	no	—
78	C₃×C₄⋊C₈	—	9	1	no	—
79	C₃×C₂²⋊C₈	—	9	2	no	—
80	C₃×C₄×C₈	—	9	1	no	—
81	#830	—	3	2	no	—
82	C₄⋊(C₃⋊C₈)	—	3	1	no	—
83	C₄×C₃⋊C₈	—	3	1	no	—
84	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
85	C₃²×D₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
90	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
92	C₂³×C₉	—	2	2	no	—
93	C₂³×C₉	C₂×C₃×C₄	1	2	yes	—
94	C₃×C₃⋊C₈	—	3	2	no	—
95	C₃×C₃⋊C₈	—	6	2	no	—
96	C₃×C₃⋊C₈	—	6	1	no	—
97	D₄×C₉	—	2	4	no	—
98	D₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	4	2	no	—
106	C₂×C₄×C₉	—	4	2	no	—
107	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
111	C₉⋊C₈	—	2	2	no	—
112	C₉⋊C₈	—	2	1	no	—
113	C₉⋊C₈	C₃×D₄	1	2	yes	complemented
114	#436	—	9	1	no	Sylow
115	C₂×C₃×C₉	—	4	2	no	—
116	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₃×C₉	#131	1	1	yes	characteristic
119	C₂×C₃×D₄	—	2	1	no	—
120	C₂×C₃×D₄	Dic₉	1	1	yes	characteristic
121	C₂×C₃×D₄	C₃×Dic₃	1	1	yes	characteristic
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
124	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
125	C₃×M₁₆	—	18	4	no	—
126	C₂×C₃×C₈	—	9	1	no	—
127	C₂×C₃×C₈	—	9	1	no	—
128	C₂×C₃×C₈	—	9	2	no	—
129	C₃×C₄⋊C₄	—	2	1	no	—
130	C₃×C₄⋊C₄	Dic₉	1	1	yes	characteristic
131	C₃×C₄⋊C₄	C₃×Dic₃	1	1	yes	characteristic
132	C₃×C₂²⋊C₄	—	2	2	no	—
133	C₃×C₂²⋊C₄	Dic₉	1	2	yes	—
134	C₃×C₂²⋊C₄	C₃×Dic₃	1	2	yes	—
135	C₃×C₄²	—	2	1	no	—
136	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
137	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
138	#208	—	6	4	no	—
139	C₂×C₃⋊C₈	—	3	1	no	—
140	C₂×C₃⋊C₈	—	3	1	no	—
141	C₂×C₃⋊C₈	—	3	2	no	—
142	C₂²×C₃²	—	2	2	no	—
143	C₂²×C₃²	—	2	2	no	—
144	C₂²×C₃²	—	2	2	no	—
145	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
146	C₃²×C₄	—	2	2	no	—
147	C₃²×C₄	—	2	1	no	—
148	C₃²×C₄	#208	1	2	yes	—
149	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
150	C₃²×C₄	#237	1	1	yes	characteristic
151	C₂²×C₉	—	2	4	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	2	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	4	2	no	—
157	C₂²×C₉	—	4	2	no	—
158	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	4	1	no	—
165	C₄×C₉	—	4	2	no	—
166	C₄×C₉	C₃×M₁₆	1	2	yes	complemented
167	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
168	C₄×C₉	C₃×#41	1	1	yes	characteristic
169	C₂×M₁₆	—	9	2	no	—
170	C₄×D₄	C₃×D₉	1	1	yes	characteristic
171	C₄⋊C₈	—	9	1	no	—
172	C₂²⋊C₈	—	9	2	no	—
173	C₄×C₈	—	9	1	no	—
174	C₃×C₉	#436	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	C₂×Dic₉	1	2	yes	—
177	C₂³×C₃	C₂×C₃×Dic₃	1	2	yes	—
178	C₃×D₄	—	2	4	no	—
179	C₃×D₄	—	2	2	no	—
180	C₃×D₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	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	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
193	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
197	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
199	C₃×C₈	—	9	2	no	—
200	C₃×C₈	—	18	2	no	—
201	C₃×C₈	—	18	1	no	—
202	C₃⋊C₈	—	3	2	no	—
203	C₃⋊C₈	—	6	2	no	—
204	C₃⋊C₈	—	6	1	no	—
205	C₂×C₃²	—	4	2	no	—
206	C₂×C₃²	C₂×#208	1	1	yes	characteristic
207	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
208	C₂×C₃²	#948	1	1	yes	characteristic
209	C₂×C₉	—	2	1	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	4	4	no	—
213	C₂×C₉	—	4	2	no	—
214	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
215	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
216	C₂×C₉	C₃×#131	1	1	yes	characteristic
217	C₂×D₄	C₃×Dic₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
219	M₁₆	—	18	4	no	—
220	C₂×C₈	—	9	1	no	—
221	C₂×C₈	—	9	1	no	—
222	C₂×C₈	—	9	2	no	—
223	C₄⋊C₄	C₃×Dic₉	1	1	yes	characteristic
224	C₂²⋊C₄	C₃×Dic₉	1	2	yes	—
225	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
226	C₂²×C₃	—	2	4	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	—	4	2	no	—
236	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
237	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	—	4	1	no	—
247	C₃×C₄	#3667	1	2	yes	complemented
248	C₃×C₄	D₄×D₉	1	1	yes	characteristic
249	C₃×C₄	#3699	1	1	yes	characteristic
250	C₃×C₄	C₃×#208	1	2	yes	—
251	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
252	C₃×C₄	C₃×#237	1	1	yes	characteristic
253	C₃²	#5012	1	1	yes	characteristic
254	C₉	—	2	1	no	—
255	C₉	C₃×#436	1	1	yes	characteristic, complemented
256	C₂³	C₂×C₃×Dic₉	1	2	yes	—
257	D₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
261	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
262	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
264	C₈	—	9	2	no	—
265	C₈	—	18	2	no	—
266	C₈	—	18	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	4	4	no	—
271	C₂×C₃	—	4	2	no	—
272	C₂×C₃	—	4	2	no	—
273	C₂×C₃	C₂×#3667	1	1	yes	characteristic
274	C₂×C₃	D₄×Dic₉	1	1	yes	characteristic
275	C₂×C₃	#63510	1	1	yes	characteristic
276	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
277	C₂×C₃	C₃×D₄×Dic₃	1	1	yes	characteristic
278	C₂×C₃	C₃×#948	1	1	yes	characteristic
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	C₃×#3667	1	2	yes	complemented
286	C₄	C₃×D₄×D₉	1	1	yes	characteristic
287	C₄	C₃×#3699	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	#10588689	1	1	yes	characteristic, direct product factor
290	C₃	C₃×#5012	1	1	yes	characteristic
291	C₂	—	4	2	no	—
292	C₂	C₂×C₃×#3667	1	1	yes	characteristic
293	C₂	C₃×D₄×Dic₉	1	1	yes	characteristic
294	C₂	C₃×#63510	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂×C₃²×D₄	Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄×D₄	D₉
hypocenter	C₉	C₃×#436
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3667	C₂	semidirect
3. C₃×#63389	C₂	semidirect
4. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
5. C₃×C₄×C₉⋊C₈	C₂	semidirect
7. #10588689	C₃	direct (N × H)
24. C₂×#3667	C₂×C₃	semidirect
25. #63389	C₂×C₃	semidirect
26. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
27. C₄×C₉⋊C₈	C₂×C₃	semidirect
38. C₃×C₉⋊C₈	D₄	semidirect
70. C₃×C₄×C₉	M₁₆	semidirect
113. C₉⋊C₈	C₃×D₄	semidirect
166. C₄×C₉	C₃×M₁₆	semidirect
174. C₃×C₉	#436	semidirect
247. C₃×C₄	#3667	semidirect
255. C₉	C₃×#436	semidirect
285. C₄	C₃×#3667	semidirect
289. C₃	#10588689	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×C₄×D₄×C₉	2
C₂×C₃×#3667	2
C₃×#63389	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×C₄×C₉⋊C₈	2
C₃×#5012	3
#10588689	3

Subgroups — C3×#10588689 #13142714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588689 #13142714