Subgroups — C₃×#70744 #11773674

Order 1152.

415 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#431	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#4925	C₂	1	1	yes	maximal, characteristic
4	C₃×#4455	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1454	—	3	1	no	maximal
9	#70744	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₃²×C₄⋊C₈	C₄	1	2	yes	—
12	C₃²×C₂²⋊C₈	C₄	1	2	yes	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×#830	C₄	1	2	yes	—
16	C₃×#830	C₄	1	2	yes	—
17	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
18	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
19	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×#431	—	2	1	no	—
24	C₃×#431	—	3	1	no	—
25	C₃×#431	—	3	1	no	—
26	C₃×#431	D₃	1	1	yes	characteristic
27	C₃×#431	C₂×C₃	1	1	yes	characteristic
28	#4925	C₂×C₃	1	1	yes	characteristic
29	#4925	C₂×C₃	1	1	yes	characteristic
30	C₃×#322	—	3	1	no	—
31	C₃×#322	—	3	1	no	—
32	C₃×#322	—	3	1	no	—
33	C₃×C₈²	—	3	1	no	—
34	#4455	C₂×C₃	1	1	yes	characteristic, complemented
35	#4454	C₂×C₃	1	1	yes	characteristic, complemented
36	#4454	C₂×C₃	1	1	yes	characteristic, complemented
37	C₈×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
43	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
45	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
46	C₃²×C₄²	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	#1454	—	3	1	no	Sylow
52	C₂×C₃×C₄²	—	2	1	no	—
53	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	2	no	—
56	C₃×C₄⋊C₈	—	3	2	no	—
57	C₃×C₄⋊C₈	—	3	2	no	—
58	C₃×C₄⋊C₈	Dic₃	1	2	yes	—
59	C₃×C₄⋊C₈	C₃×C₄	1	2	yes	—
60	C₃×C₂²⋊C₈	—	2	2	no	—
61	C₃×C₂²⋊C₈	—	3	2	no	—
62	C₃×C₂²⋊C₈	—	3	2	no	—
63	C₃×C₂²⋊C₈	Dic₃	1	2	yes	—
64	C₃×C₂²⋊C₈	C₃×C₄	1	2	yes	—
65	C₃×C₄×C₈	—	2	1	no	—
66	C₃×C₄×C₈	—	2	1	no	—
67	C₃×C₄×C₈	—	3	1	no	—
68	C₃×C₄×C₈	—	3	1	no	—
69	C₃×C₄×C₈	—	3	1	no	—
70	C₃×C₄×C₈	—	3	1	no	—
71	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	#830	C₃×C₄	1	2	yes	—
76	#830	C₃×C₄	1	2	yes	—
77	C₄⋊(C₃⋊C₈)	C₃×C₄	1	2	yes	—
78	C₄⋊(C₃⋊C₈)	C₃×C₄	1	2	yes	—
79	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
80	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
83	C₂³×C₃²	C₄²	1	1	yes	characteristic
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₃²×C₈	—	2	2	no	—
94	C₃²×C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	—	2	2	no	—
96	C₃×C₃⋊C₈	—	2	2	no	—
97	C₃×C₃⋊C₈	—	2	2	no	—
98	C₃×C₃⋊C₈	—	2	2	no	—
99	#431	—	3	1	no	—
100	#431	—	3	1	no	—
101	#431	C₃×D₃	1	1	yes	characteristic
102	#322	—	3	1	no	—
103	#322	—	3	1	no	—
104	#322	—	3	1	no	—
105	C₈²	—	3	1	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₂×C₃×C₈	—	3	2	no	—
120	C₂×C₃×C₈	—	3	2	no	—
121	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
122	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
123	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
125	C₃×C₄²	—	2	1	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
130	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
131	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
132	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
133	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
134	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
135	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
136	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
137	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
138	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
139	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
140	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
141	C₂²×C₃²	—	2	1	no	—
142	C₂²×C₃²	—	2	1	no	—
143	C₂²×C₃²	—	2	1	no	—
144	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
145	C₃²×C₄	—	2	1	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	—	2	1	no	—
148	C₃²×C₄	#131	1	2	yes	—
149	C₃²×C₄	#131	1	2	yes	—
150	C₃²×C₄	#131	1	2	yes	—
151	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
152	C₄⋊C₈	—	3	2	no	—
153	C₄⋊C₈	—	3	2	no	—
154	C₄⋊C₈	C₃×Dic₃	1	2	yes	—
155	C₂²⋊C₈	—	3	2	no	—
156	C₂²⋊C₈	—	3	2	no	—
157	C₂²⋊C₈	C₃×Dic₃	1	2	yes	—
158	C₄×C₈	—	3	1	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	3	1	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
163	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
166	C₂³×C₃	C₃×C₄²	1	1	yes	characteristic
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	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	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
187	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	4	2	no	—
199	C₃×C₈	—	4	2	no	—
200	C₃×C₈	—	6	2	no	—
201	C₃×C₈	—	6	2	no	—
202	C₃×C₈	—	6	2	no	—
203	C₃×C₈	—	6	2	no	—
204	C₃⋊C₈	—	2	2	no	—
205	C₃⋊C₈	—	2	2	no	—
206	C₃⋊C₈	—	2	2	no	—
207	C₃⋊C₈	—	2	2	no	—
208	C₂×C₃²	—	4	1	no	—
209	C₂×C₃²	#405	1	1	yes	characteristic
210	C₂×C₃²	#405	1	1	yes	characteristic
211	C₂×C₃²	#405	1	1	yes	characteristic
212	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
213	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
215	C₂×C₈	—	3	2	no	—
216	C₂×C₈	—	3	2	no	—
217	C₂×C₈	—	3	2	no	—
218	C₂×C₈	—	3	2	no	—
219	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
220	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
221	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
222	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
223	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
224	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	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	—	4	1	no	—
234	C₂²×C₃	—	4	1	no	—
235	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
236	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	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	—	2	2	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	4	1	no	—
248	C₃×C₄	—	4	1	no	—
249	C₃×C₄	#901	1	2	yes	—
250	C₃×C₄	#907	1	2	yes	—
251	C₃×C₄	#948	1	2	yes	—
252	C₃×C₄	C₃×#131	1	2	yes	—
253	C₃×C₄	C₃×#131	1	2	yes	—
254	C₃×C₄	C₃×#131	1	2	yes	—
255	C₃²	#1454	1	1	yes	characteristic, complemented, Sylow
256	C₂³	C₃×C₄×Dic₃	1	1	yes	characteristic
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
261	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
262	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	6	2	no	—
269	C₈	—	6	2	no	—
270	C₈	—	6	2	no	—
271	C₈	—	6	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	4	2	no	—
276	C₂×C₃	—	4	1	no	—
277	C₂×C₃	—	4	1	no	—
278	C₂×C₃	#4997	1	1	yes	characteristic
279	C₂×C₃	#5101	1	1	yes	characteristic
280	C₂×C₃	#5122	1	1	yes	characteristic
281	C₂×C₃	C₃×#405	1	1	yes	characteristic
282	C₂×C₃	C₃×#405	1	1	yes	characteristic
283	C₂×C₃	C₃×#405	1	1	yes	characteristic
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	C₃×#901	1	2	yes	—
292	C₄	C₃×#907	1	2	yes	—
293	C₄	C₃×#948	1	2	yes	—
294	C₃	—	2	1	no	—
295	C₃	#70744	1	1	yes	characteristic, direct product factor
296	C₃	C₃×#1454	1	1	yes	characteristic, complemented
297	C₂	—	4	1	no	—
298	C₂	C₃×#4997	1	1	yes	characteristic
299	C₂	C₃×#5101	1	1	yes	characteristic
300	C₂	C₃×#5122	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄²	C₂²×D₃
commutator	C₂²×C₃	C₂×C₃×C₄²
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#431	C₂
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂³×C₃²	C₄²
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#431	D₃
hypocenter	C₃	C₃×#1454
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄² ◁ C₃×#431

Complements

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

N	H ≅ G/N	type
4. C₃×#4455	C₂	semidirect
5. C₃×#4454	C₂	semidirect
6. C₃×#4454	C₂	semidirect
7. C₃×C₈×C₃⋊C₈	C₂	semidirect
9. #70744	C₃	direct (N × H)
34. #4455	C₂×C₃	semidirect
35. #4454	C₂×C₃	semidirect
36. #4454	C₂×C₃	semidirect
37. C₈×C₃⋊C₈	C₂×C₃	semidirect
255. C₃²	#1454	semidirect
295. C₃	#70744	direct (N × H)
296. C₃	C₃×#1454	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#431	2
C₃×#4925	2
C₃×#4455	2
C₃×#4454	2
C₃×C₈×C₃⋊C₈	2
C₃×#1454	3
#70744	3

Subgroups — C3×#70744 #11773674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70744 #11773674