Subgroups — C₃×#10588665 #13142561

Order 1728.

494 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	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉⋊#95	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉⋊#95	C₂	1	2	yes	maximal
5	C₃×C₉⋊#95	C₂	1	2	yes	maximal
6	C₃×#4985	—	3	1	no	maximal
7	#10588665	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×C₉×C₄⋊C₄	—	2	2	no	—
14	C₃×Dic₉⋊C₄	—	2	2	no	—
15	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
16	C₂×C₃×Dic₃⋊C₄	—	3	1	no	—
17	C₃×C₃⋊#95	—	3	2	no	—
18	C₃×C₃⋊#95	—	3	2	no	—
19	C₃×C₃⋊#95	—	3	1	no	—
20	C₂×C₉×C₄⋊C₄	—	2	1	no	—
21	C₂×C₉×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₂×Dic₉⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₉⋊#95	C₂×C₃	1	1	yes	characteristic
24	C₉⋊#95	C₂×C₃	1	2	yes	—
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	4	no	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×Dic₉	—	2	1	no	—
32	C₂×C₃×Dic₉	—	2	1	no	—
33	C₂×C₃×Dic₉	—	2	1	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₂×C₃×Dic₉	—	2	2	no	—
36	C₂×C₃×Dic₉	—	2	2	no	—
37	C₂×C₃×Dic₉	—	2	2	no	—
38	C₂×C₃×Dic₉	Q₈	1	2	yes	—
39	C₃×#400	—	9	1	no	—
40	#4985	—	3	1	no	—
41	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃×Dic₃	—	3	1	no	—
44	C₂²×C₃×Dic₃	—	3	1	no	—
45	C₂²×C₃×Dic₃	—	3	2	no	—
46	C₃²×C₄⋊C₄	—	2	2	no	—
47	C₃×Dic₃⋊C₄	—	6	2	no	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
51	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
52	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
53	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
54	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
55	C₉×C₄⋊C₄	—	2	4	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	Dic₉⋊C₄	—	2	2	no	—
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₉	#41	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	1	no	—
67	C₃×Dic₉	—	4	2	no	—
68	C₂×C₃×C₄⋊C₄	—	2	1	no	—
69	C₂×C₃×C₄⋊C₄	—	9	1	no	—
70	C₂×C₃×C₄⋊C₄	D₉	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
72	C₂×Dic₃⋊C₄	—	3	1	no	—
73	C₃×#95	—	9	2	no	—
74	C₃×#95	—	9	1	no	—
75	C₃×#95	—	9	2	no	—
76	C₃⋊#95	—	3	2	no	—
77	C₃⋊#95	—	3	1	no	—
78	C₃⋊#95	—	3	2	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	4	no	—
83	C₂×C₃²×C₄	#67	1	2	yes	—
84	C₂×C₃×Dic₃	—	3	2	no	—
85	C₂×C₃×Dic₃	—	6	1	no	—
86	C₂×C₃×Dic₃	—	6	1	no	—
87	C₂×C₃×Dic₃	—	6	2	no	—
88	C₂×C₃×Dic₃	—	6	2	no	—
89	C₂×C₃×Dic₃	—	6	2	no	—
90	C₂×C₃×Dic₃	—	6	2	no	—
91	C₂×C₃×Dic₃	—	6	1	no	—
92	C₂³×C₉	—	2	1	no	—
93	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₄×C₉	—	2	4	no	—
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	4	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	4	4	no	—
101	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
102	C₂×Dic₉	—	2	1	no	—
103	C₂×Dic₉	—	2	1	no	—
104	C₂×Dic₉	—	2	1	no	—
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	C₃×Q₈	1	2	yes	—
110	#400	—	9	1	no	Sylow
111	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₃×C₉	#123	1	1	yes	characteristic
113	C₂×C₃×C₉	#124	1	1	yes	characteristic
114	C₂×C₃×C₉	#125	1	2	yes	—
115	C₂×C₃×C₉	#126	1	2	yes	—
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	—	2	2	no	—
118	C₂²×C₃×C₄	—	9	1	no	—
119	C₂²×C₃×C₄	—	9	2	no	—
120	C₂²×C₃×C₄	—	9	1	no	—
121	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
122	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
124	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂²×Dic₃	—	3	1	no	—
126	C₂²×Dic₃	—	3	2	no	—
127	C₂²×Dic₃	—	3	1	no	—
128	C₃×C₄⋊C₄	—	2	4	no	—
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	18	2	no	—
132	Dic₃⋊C₄	—	6	2	no	—
133	C₂²×C₃²	#235	1	2	yes	—
134	C₂²×C₃²	#237	1	2	yes	—
135	C₂²×C₃²	D₃×Q₈	1	1	yes	characteristic
136	C₂²×C₃²	#239	1	1	yes	characteristic
137	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
138	C₃²×C₄	—	2	2	no	—
139	C₃²×C₄	—	4	2	no	—
140	C₃×Dic₃	—	6	2	no	—
141	C₃×Dic₃	—	12	2	no	—
142	C₃×Dic₃	—	12	1	no	—
143	C₂²×C₉	—	2	2	no	—
144	C₂²×C₉	—	2	2	no	—
145	C₂²×C₉	—	2	1	no	—
146	C₂²×C₉	—	2	1	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
149	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
150	C₂²×C₉	C₃×#41	1	2	yes	—
151	C₂²×C₉	C₃×#41	1	2	yes	—
152	C₂²×C₉	C₃×#41	1	1	yes	characteristic
153	C₄×C₉	—	2	4	no	—
154	C₄×C₉	—	2	2	no	—
155	C₄×C₉	—	4	4	no	—
156	C₄×C₉	—	4	2	no	—
157	Dic₉	—	2	2	no	—
158	Dic₉	—	4	1	no	—
159	Dic₉	—	4	2	no	—
160	C₂×C₄⋊C₄	—	9	1	no	—
161	C₂×C₄⋊C₄	C₃×D₉	1	1	yes	characteristic
162	#95	—	9	2	no	—
163	#95	—	9	1	no	—
164	#95	—	9	2	no	—
165	C₃×C₉	#400	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
168	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	4	4	no	—
179	C₂×C₃×C₄	—	9	2	no	—
180	C₂×C₃×C₄	—	18	1	no	—
181	C₂×C₃×C₄	—	18	1	no	—
182	C₂×C₃×C₄	—	18	2	no	—
183	C₂×C₃×C₄	—	18	2	no	—
184	C₂×C₃×C₄	—	18	2	no	—
185	C₂×C₃×C₄	—	18	2	no	—
186	C₂×C₃×C₄	—	18	1	no	—
187	C₂×C₃×C₄	#611	1	2	yes	—
188	C₂×C₃×C₄	C₃×#67	1	2	yes	—
189	C₂×Dic₃	—	3	2	no	—
190	C₂×Dic₃	—	6	1	no	—
191	C₂×Dic₃	—	6	1	no	—
192	C₂×Dic₃	—	6	2	no	—
193	C₂×Dic₃	—	6	2	no	—
194	C₂×Dic₃	—	6	2	no	—
195	C₂×Dic₃	—	6	2	no	—
196	C₂×Dic₃	—	6	1	no	—
197	C₂×C₃²	#889	1	2	yes	—
198	C₂×C₃²	#898	1	2	yes	—
199	C₂×C₃²	#929	1	1	yes	characteristic
200	C₂×C₃²	#936	1	1	yes	characteristic
201	C₂×C₃²	#946	1	1	yes	characteristic
202	C₂×C₉	—	2	2	no	—
203	C₂×C₉	—	2	2	no	—
204	C₂×C₉	—	2	1	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	C₃×C₂²⋊Q₈	1	1	yes	characteristic
208	C₂×C₉	C₃×#123	1	1	yes	characteristic
209	C₂×C₉	C₃×#124	1	1	yes	characteristic
210	C₂×C₉	C₃×#125	1	2	yes	—
211	C₂×C₉	C₃×#126	1	2	yes	—
212	C₂²×C₄	—	9	1	no	—
213	C₂²×C₄	—	9	2	no	—
214	C₂²×C₄	—	9	1	no	—
215	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
216	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	18	2	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	#3697	1	2	yes	—
225	C₂²×C₃	#3699	1	2	yes	—
226	C₂²×C₃	Q₈×D₉	1	1	yes	characteristic
227	C₂²×C₃	#3701	1	1	yes	characteristic
228	C₂²×C₃	C₂×#611	1	1	yes	characteristic
229	C₂²×C₃	C₃×#235	1	2	yes	—
230	C₂²×C₃	C₃×#237	1	2	yes	—
231	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
232	C₂²×C₃	C₃×#239	1	1	yes	characteristic
233	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	4	4	no	—
238	C₃×C₄	—	4	2	no	—
239	C₃×C₄	—	4	2	no	—
240	C₃×C₄	—	18	2	no	—
241	C₃×C₄	—	36	1	no	—
242	C₃×C₄	—	36	2	no	—
243	Dic₃	—	6	2	no	—
244	Dic₃	—	12	2	no	—
245	Dic₃	—	12	1	no	—
246	C₃²	#4985	1	1	yes	characteristic
247	C₉	—	2	1	no	—
248	C₉	C₃×#400	1	1	yes	characteristic, complemented
249	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	9	2	no	—
254	C₂×C₄	—	18	1	no	—
255	C₂×C₄	—	18	1	no	—
256	C₂×C₄	—	18	2	no	—
257	C₂×C₄	—	18	2	no	—
258	C₂×C₄	—	18	2	no	—
259	C₂×C₄	—	18	2	no	—
260	C₂×C₄	—	18	1	no	—
261	C₂×C₄	C₃×#611	1	2	yes	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	#63451	1	2	yes	—
268	C₂×C₃	#63460	1	2	yes	—
269	C₂×C₃	#63491	1	1	yes	characteristic
270	C₂×C₃	#63498	1	1	yes	characteristic
271	C₂×C₃	#63508	1	1	yes	characteristic
272	C₂×C₃	C₃×#889	1	2	yes	—
273	C₂×C₃	C₃×#898	1	2	yes	—
274	C₂×C₃	C₃×#929	1	1	yes	characteristic
275	C₂×C₃	C₃×#936	1	1	yes	characteristic
276	C₂×C₃	C₃×#946	1	1	yes	characteristic
277	C₂²	C₃×#3697	1	2	yes	—
278	C₂²	C₃×#3699	1	2	yes	—
279	C₂²	C₃×Q₈×D₉	1	1	yes	characteristic
280	C₂²	C₃×#3701	1	1	yes	characteristic
281	C₂²	C₂×C₃×#611	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	18	2	no	—
285	C₄	—	36	1	no	—
286	C₄	—	36	2	no	—
287	C₃	—	2	1	no	—
288	C₃	#10588665	1	1	yes	characteristic, direct product factor
289	C₃	C₃×#4985	1	1	yes	characteristic
290	C₂	C₃×#63451	1	2	yes	—
291	C₂	C₃×#63460	1	2	yes	—
292	C₂	C₃×#63491	1	1	yes	characteristic
293	C₂	C₃×#63498	1	1	yes	characteristic
294	C₂	C₃×#63508	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	C₂×C₃×C₄⋊C₄	D₉
hypocenter	C₉	C₃×#400
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. #10588665	C₃	direct (N × H)
165. C₃×C₉	#400	semidirect
248. C₉	C₃×#400	semidirect
288. C₃	#10588665	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×C₉×C₄⋊C₄	2
C₂×C₃×Dic₉⋊C₄	2
C₃×C₉⋊#95	2
C₃×#4985	3
#10588665	3

Subgroups — C3×#10588665 #13142561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588665 #13142561