Subgroups — C₃×#10588695 #13142726

Order 1728.

454 conjugacy classes of subgroups.

The subgroup lattice has 314 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×#3672	C₂	1	2	yes	maximal
3	C₃×#63431	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63368	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5019	—	3	1	no	maximal
7	#10588695	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	—	2	1	no	—
10	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₃×#3672	—	2	4	no	—
15	C₃×C₄⋊Dic₉	—	2	1	no	—
16	C₃×C₄⋊Dic₉	—	2	1	no	—
17	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
18	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
19	C₂×C₃×#214	—	3	2	no	—
20	C₃×#869	—	3	1	no	—
21	C₃×#810	—	3	2	no	—
22	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
23	C₄×D₄×C₉	—	2	1	no	—
24	C₄×D₄×C₉	C₂×C₃	1	1	yes	characteristic
25	C₂×#3672	C₂×C₃	1	2	yes	—
26	#63431	C₂×C₃	1	1	yes	characteristic, complemented
27	#63368	C₂×C₃	1	2	yes	complemented
28	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃×C₉	—	2	1	no	—
30	C₃×D₄×C₉	—	2	1	no	—
31	C₃×D₄×C₉	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	—	4	2	no	—
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₉	—	2	2	no	—
38	C₃×Dic₁₈	—	2	2	no	—
39	C₃×Dic₁₈	—	2	2	no	—
40	C₃×C₉⋊C₈	—	2	2	no	—
41	C₃×#461	—	9	1	no	—
42	#5019	—	3	1	no	—
43	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃²×C₄	—	2	1	no	—
45	C₂×C₃×Dic₆	—	3	2	no	—
46	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
47	C₃²×C₂²⋊C₄	—	2	1	no	—
48	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₃×#214	—	6	4	no	—
50	C₃×C₄⋊Dic₃	—	6	1	no	—
51	C₃×C₄⋊Dic₃	—	6	1	no	—
52	C₂×C₃×C₃⋊C₈	—	3	2	no	—
53	C₂×D₄×C₉	—	2	1	no	—
54	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
55	C₂²×C₄×C₉	—	2	2	no	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
58	C₉×C₄⋊C₄	—	2	1	no	—
59	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₉×C₂²⋊C₄	—	2	2	no	—
61	C₉×C₂²⋊C₄	—	2	1	no	—
62	C₄²×C₉	—	2	1	no	—
63	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
64	#3672	—	2	4	no	—
65	C₄⋊Dic₉	—	2	1	no	—
66	C₄⋊Dic₉	—	2	1	no	—
67	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
68	C₂²×C₃×C₉	—	2	2	no	—
69	C₂²×C₃×C₉	—	4	1	no	—
70	C₂²×C₃×C₉	—	4	1	no	—
71	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	—	2	1	no	—
73	C₃×C₄×C₉	—	4	1	no	—
74	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
75	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×C₉	#41	1	1	yes	characteristic
77	C₃×Dic₉	—	4	2	no	—
78	C₂×C₃×SD₁₆	—	9	2	no	—
79	C₃×C₄⋊Q₈	—	9	1	no	—
80	C₃×C₄×D₄	—	2	1	no	—
81	C₃×C₄×D₄	D₉	1	1	yes	characteristic
82	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
83	C₂×#214	—	3	2	no	—
84	#869	—	3	1	no	—
85	C₃×C₄⋊C₈	—	9	1	no	—
86	C₃×#103	—	9	2	no	—
87	#810	—	3	2	no	—
88	C₄⋊(C₃⋊C₈)	—	3	1	no	—
89	C₂³×C₃²	—	2	1	no	—
90	C₃²×D₄	—	2	1	no	—
91	C₃²×D₄	D₁₂	1	2	yes	—
92	C₂×C₃²×C₄	—	2	1	no	—
93	C₂×C₃²×C₄	—	4	2	no	—
94	C₂×C₃²×C₄	#67	1	1	yes	characteristic
95	C₂×C₃²×C₄	#67	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃×Dic₃	—	6	2	no	—
98	C₃×Dic₆	—	6	2	no	—
99	C₃×Dic₆	—	6	2	no	—
100	C₂³×C₉	—	2	2	no	—
101	C₂³×C₉	—	2	1	no	—
102	C₃×C₃⋊C₈	—	6	2	no	—
103	D₄×C₉	—	2	2	no	—
104	D₄×C₉	—	2	1	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	C₃×D₄	1	2	yes	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	4	4	no	—
113	C₂×C₄×C₉	—	4	2	no	—
114	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
117	C₂×Dic₉	—	2	2	no	—
118	Dic₁₈	—	2	2	no	—
119	Dic₁₈	—	2	2	no	—
120	C₉⋊C₈	—	2	2	no	—
121	#461	—	9	1	no	Sylow
122	C₂×C₃×C₉	—	4	2	no	—
123	C₂×C₃×C₉	#121	1	1	yes	characteristic
124	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₃×C₉	#137	1	1	yes	characteristic
126	C₂×C₃×Q₈	—	9	2	no	—
127	C₂×C₃×D₄	—	2	1	no	—
128	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
129	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	1	no	—
132	C₂²×C₃×C₄	—	2	1	no	—
133	C₂×Dic₆	—	3	2	no	—
134	C₃×SD₁₆	—	18	4	no	—
135	C₂×C₃×C₈	—	9	2	no	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	18	1	no	—
138	C₃×C₄⋊C₄	—	18	1	no	—
139	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
141	C₃×C₂²⋊C₄	—	2	2	no	—
142	C₃×C₂²⋊C₄	—	2	1	no	—
143	C₃×C₂²⋊C₄	—	2	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
146	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
147	#214	—	6	4	no	—
148	C₄⋊Dic₃	—	6	1	no	—
149	C₄⋊Dic₃	—	6	1	no	—
150	C₂×C₃⋊C₈	—	3	2	no	—
151	C₂²×C₃²	—	2	2	no	—
152	C₂²×C₃²	—	4	1	no	—
153	C₂²×C₃²	—	4	1	no	—
154	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
155	C₃²×C₄	—	2	1	no	—
156	C₃²×C₄	—	4	1	no	—
157	C₃²×C₄	#214	1	2	yes	—
158	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
159	C₃²×C₄	#235	1	1	yes	characteristic
160	C₃×Dic₃	—	12	2	no	—
161	C₂²×C₉	—	2	4	no	—
162	C₂²×C₉	—	2	2	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	—	4	2	no	—
166	C₂²×C₉	—	4	1	no	—
167	C₂²×C₉	—	4	1	no	—
168	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	—	2	2	no	—
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	—	2	1	no	—
173	C₄×C₉	—	4	1	no	—
174	C₄×C₉	—	4	2	no	—
175	C₄×C₉	—	4	1	no	—
176	C₄×C₉	C₃×SD₁₆	1	2	yes	complemented
177	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
178	C₄×C₉	C₃×#41	1	1	yes	characteristic
179	Dic₉	—	4	2	no	—
180	C₂×SD₁₆	—	9	2	no	—
181	C₄⋊Q₈	—	9	1	no	—
182	C₄×D₄	C₃×D₉	1	1	yes	characteristic
183	C₄⋊C₈	—	9	1	no	—
184	#103	—	9	2	no	—
185	C₃×C₉	#461	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₃	—	2	2	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	—	2	1	no	—
189	C₃×Q₈	—	18	2	no	—
190	C₃×Q₈	—	18	2	no	—
191	C₃×D₄	—	2	2	no	—
192	C₃×D₄	—	2	1	no	—
193	C₃×D₄	—	2	1	no	—
194	C₃×D₄	—	2	2	no	—
195	C₃×D₄	D₃₆	1	2	yes	—
196	C₃×D₄	C₃×D₁₂	1	2	yes	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	4	4	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	18	2	no	—
207	C₂×C₃×C₄	#611	1	1	yes	characteristic
208	C₂×C₃×C₄	#611	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
210	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
211	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
213	C₂×Dic₃	—	6	2	no	—
214	Dic₆	—	6	2	no	—
215	Dic₆	—	6	2	no	—
216	C₃×C₈	—	18	2	no	—
217	C₃⋊C₈	—	6	2	no	—
218	C₂×C₃²	—	4	2	no	—
219	C₂×C₃²	#930	1	1	yes	characteristic
220	C₂×C₃²	C₂×#214	1	1	yes	characteristic
221	C₂×C₃²	#951	1	1	yes	characteristic
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	4	4	no	—
226	C₂×C₉	—	4	2	no	—
227	C₂×C₉	C₃×#121	1	1	yes	characteristic
228	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
229	C₂×C₉	C₃×#137	1	1	yes	characteristic
230	C₂×Q₈	—	9	2	no	—
231	C₂×D₄	C₂×C₃×D₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	SD₁₆	—	18	4	no	—
234	C₂×C₈	—	9	2	no	—
235	C₄⋊C₄	—	18	1	no	—
236	C₄⋊C₄	—	18	1	no	—
237	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
238	C₂²⋊C₄	—	2	1	no	—
239	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	4	2	no	—
245	C₂²×C₃	—	4	2	no	—
246	C₂²×C₃	—	4	1	no	—
247	C₂²×C₃	—	4	1	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	1	no	—
250	C₂²×C₃	C₂×#611	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	36	2	no	—
262	C₃×C₄	#3672	1	2	yes	complemented
263	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
264	C₃×C₄	#3697	1	1	yes	characteristic
265	C₃×C₄	C₃×#214	1	2	yes	—
266	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
267	C₃×C₄	C₃×#235	1	1	yes	characteristic
268	Dic₃	—	12	2	no	—
269	C₃²	#5019	1	1	yes	characteristic
270	C₉	—	2	1	no	—
271	C₉	C₃×#461	1	1	yes	characteristic, complemented
272	C₂³	—	2	1	no	—
273	Q₈	—	18	2	no	—
274	Q₈	—	18	2	no	—
275	D₄	—	2	1	no	—
276	D₄	C₃×D₃₆	1	2	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	18	2	no	—
280	C₂×C₄	C₃×#611	1	1	yes	characteristic
281	C₂×C₄	C₃×#611	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
283	C₈	—	18	2	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	4	4	no	—
288	C₂×C₃	—	4	2	no	—
289	C₂×C₃	—	4	2	no	—
290	C₂×C₃	#63492	1	1	yes	characteristic
291	C₂×C₃	C₂×#3672	1	1	yes	characteristic
292	C₂×C₃	#63511	1	1	yes	characteristic
293	C₂×C₃	C₃×#930	1	1	yes	characteristic
294	C₂×C₃	C₂×C₃×#214	1	1	yes	characteristic
295	C₂×C₃	C₃×#951	1	1	yes	characteristic
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂×C₃×#611	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	36	2	no	—
303	C₄	C₃×#3672	1	2	yes	complemented
304	C₄	C₂×C₃×D₃₆	1	1	yes	characteristic
305	C₄	C₃×#3697	1	1	yes	characteristic
306	C₃	—	2	1	no	—
307	C₃	#10588695	1	1	yes	characteristic, direct product factor
308	C₃	C₃×#5019	1	1	yes	characteristic
309	C₂	—	4	2	no	—
310	C₂	C₃×#63492	1	1	yes	characteristic
311	C₂	C₂×C₃×#3672	1	1	yes	characteristic
312	C₂	C₃×#63511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₃²×D₄	C₂×D₃
hypercenter	C₃×C₄×D₄	D₉
hypocenter	C₉	C₃×#461
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₃×#63431	C₂	semidirect
4. C₃×#63368	C₂	semidirect
5. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
7. #10588695	C₃	direct (N × H)
26. #63431	C₂×C₃	semidirect
27. #63368	C₂×C₃	semidirect
28. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
74. C₃×C₄×C₉	SD₁₆	semidirect
176. C₄×C₉	C₃×SD₁₆	semidirect
185. C₃×C₉	#461	semidirect
262. C₃×C₄	#3672	semidirect
271. C₉	C₃×#461	semidirect
303. C₄	C₃×#3672	semidirect
307. C₃	#10588695	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×#3672	2
C₃×#63431	2
C₃×#63368	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#5019	3
#10588695	3

Subgroups — C3×#10588695 #13142726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588695 #13142726