Subgroups — C₃×#10588606 #13142370

Order 1728.

438 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#63389	C₂	1	2	yes	maximal
3	C₃×C₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₃×#63362	C₂	1	2	yes	maximal, complemented
5	C₃×#4926	—	3	1	no	maximal
6	#10588606	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₉⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
16	C₃×#830	—	3	2	no	—
17	C₃×C₄⋊(C₃⋊C₈)	—	3	2	no	—
18	C₃×#803	—	3	2	no	—
19	C₂×C₄²×C₉	—	2	1	no	—
20	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
21	#63389	C₂×C₃	1	2	yes	—
22	C₄⋊(C₉⋊C₈)	C₂×C₃	1	2	yes	complemented
23	#63362	C₂×C₃	1	2	yes	complemented
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	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	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₃×C₉⋊C₈	—	2	4	no	—
35	C₃×#432	—	9	1	no	—
36	#4926	—	3	1	no	—
37	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
38	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₃²×C₄²	Dic₃	1	1	yes	characteristic
41	C₃²×C₄²	Dic₃	1	1	yes	characteristic
42	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	—	3	4	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
51	C₄²×C₉	—	2	1	no	—
52	C₄²×C₉	—	2	1	no	—
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	C₃×C₄	1	1	yes	characteristic
56	C₄²×C₉	C₃×C₄	1	1	yes	characteristic
57	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
58	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	—
60	C₂²×C₃×C₉	—	2	1	no	—
61	C₂²×C₃×C₉	—	2	1	no	—
62	C₂²×C₃×C₉	M₁₆	1	2	yes	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	—	2	1	no	—
65	C₃×C₄×C₉	—	2	1	no	—
66	C₃×C₄×C₉	—	2	1	no	—
67	C₃×C₄×C₉	—	2	1	no	—
68	C₃×C₄×C₉	M₁₆	1	2	yes	—
69	C₃×C₄×C₉	#41	1	2	yes	—
70	C₂×C₃×C₄²	—	2	1	no	—
71	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
72	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
73	C₃×C₄⋊C₈	—	9	2	no	—
74	C₃×C₂²⋊C₈	—	9	2	no	—
75	C₃×#97	—	9	2	no	—
76	#830	—	3	2	no	—
77	C₄⋊(C₃⋊C₈)	—	3	2	no	—
78	#803	—	3	2	no	—
79	C₂³×C₃²	C₂×Dic₃	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	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂³×C₉	—	2	1	no	—
90	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
91	C₃×C₃⋊C₈	—	6	4	no	—
92	C₂×C₄×C₉	—	2	4	no	—
93	C₂×C₄×C₉	—	2	4	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	1	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	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
110	C₉⋊C₈	—	2	4	no	—
111	#432	—	9	1	no	Sylow
112	C₂×C₃×C₉	—	2	2	no	—
113	C₂×C₃×C₉	#117	1	1	yes	characteristic
114	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
120	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂×C₃×C₈	—	9	4	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	Dic₉	1	1	yes	characteristic
131	C₃×C₄²	Dic₉	1	1	yes	characteristic
132	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
133	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
134	C₃×C₄²	C₃×Dic₃	1	1	yes	characteristic
135	C₃×C₄²	C₃×Dic₃	1	1	yes	characteristic
136	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
137	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂×C₃⋊C₈	—	3	4	no	—
139	C₂²×C₃²	—	2	1	no	—
140	C₂²×C₃²	—	2	1	no	—
141	C₂²×C₃²	#208	1	2	yes	—
142	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
143	C₃²×C₄	—	2	1	no	—
144	C₃²×C₄	—	2	1	no	—
145	C₃²×C₄	—	2	1	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	#208	1	2	yes	—
148	C₃²×C₄	#235	1	2	yes	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	2	no	—
151	C₂²×C₉	—	2	1	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	C₃×M₁₆	1	2	yes	—
156	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₄×C₉	—	2	2	no	—
158	C₄×C₉	—	2	2	no	—
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	2	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	2	no	—
166	C₄×C₉	—	2	2	no	—
167	C₄×C₉	C₃×M₁₆	1	2	yes	—
168	C₄×C₉	C₃×#41	1	2	yes	—
169	C₂×C₄²	C₃×D₉	1	1	yes	characteristic
170	C₄⋊C₈	—	9	2	no	—
171	C₂²⋊C₈	—	9	2	no	—
172	#97	—	9	2	no	—
173	C₃×C₉	#432	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₃	—	2	1	no	—
175	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
176	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	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	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
204	C₃×C₈	—	18	4	no	—
205	C₃⋊C₈	—	6	4	no	—
206	C₂×C₃²	—	2	2	no	—
207	C₂×C₃²	C₂×#208	1	1	yes	characteristic
208	C₂×C₃²	C₂×#208	1	1	yes	characteristic
209	C₂×C₃²	#926	1	1	yes	characteristic
210	C₂×C₉	—	2	4	no	—
211	C₂×C₉	—	2	2	no	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	C₃×#117	1	1	yes	characteristic
216	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
217	C₂×C₉	C₂×C₃×M₁₆	1	1	yes	characteristic
218	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
219	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
221	C₂×C₈	—	9	4	no	—
222	C₄²	C₃×Dic₉	1	1	yes	characteristic
223	C₄²	C₃×Dic₉	1	1	yes	characteristic
224	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
225	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	#3667	1	2	yes	—
235	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
236	C₂²×C₃	C₃×#208	1	2	yes	—
237	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	#3667	1	2	yes	—
253	C₃×C₄	#3697	1	2	yes	—
254	C₃×C₄	C₃×#208	1	2	yes	—
255	C₃×C₄	C₃×#235	1	2	yes	—
256	C₃²	#4926	1	1	yes	characteristic
257	C₉	—	2	1	no	—
258	C₉	C₃×#432	1	1	yes	characteristic, complemented
259	C₂³	C₂×C₃×Dic₉	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
264	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
265	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
269	C₈	—	18	4	no	—
270	C₂×C₃	—	2	4	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	C₂×#3667	1	1	yes	characteristic
277	C₂×C₃	C₂×#3667	1	1	yes	characteristic
278	C₂×C₃	#63488	1	1	yes	characteristic
279	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
280	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
281	C₂×C₃	C₃×#926	1	1	yes	characteristic
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	C₃×#3667	1	2	yes	—
285	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	C₃×#3667	1	2	yes	—
291	C₄	C₃×#3697	1	2	yes	—
292	C₃	—	2	1	no	—
293	C₃	#10588606	1	1	yes	characteristic, direct product factor
294	C₃	C₃×#4926	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×C₃×#3667	1	1	yes	characteristic
297	C₂	C₂×C₃×#3667	1	1	yes	characteristic
298	C₂	C₃×#63488	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	C₂×C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₃×#432
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
4. C₃×#63362	C₂	semidirect
6. #10588606	C₃	direct (N × H)
22. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
23. #63362	C₂×C₃	semidirect
173. C₃×C₉	#432	semidirect
258. C₉	C₃×#432	semidirect
293. C₃	#10588606	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×C₄²×C₉	2
C₃×#63389	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#63362	2
C₃×#4926	3
#10588606	3

Subgroups — C3×#10588606 #13142370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588606 #13142370