Subgroups — C₃×#70659 #11784369

Order 1152.

521 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#4997	C₂	1	2	yes	maximal
5	C₃×#4532	C₂	1	2	yes	maximal
6	C₃×#1772	—	3	1	no	maximal
7	#70659	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃²×#117	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×#107	C₄	1	2	yes	complemented
14	C₃²×#106	C₄	1	2	yes	complemented
15	C₃×#819	C₄	1	4	yes	complemented
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	C₃×#803	—	2	2	no	—
18	C₃×C₄×C₃⋊C₈	—	2	2	no	—
19	C₂×C₃×#108	—	3	1	no	—
20	C₃×#427	—	2	1	no	—
21	C₃×#427	D₃	1	1	yes	characteristic
22	C₃×#427	C₂×C₃	1	1	yes	characteristic
23	C₃×#405	—	3	2	no	—
24	C₂×C₃×C₄×C₈	—	3	1	no	—
25	C₂×#819	C₂×C₃	1	1	yes	characteristic
26	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
27	#4997	C₂×C₃	1	2	yes	—
28	C₃×#339	—	3	2	no	—
29	#4532	C₂×C₃	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	1	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃²×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₃²×C₂²⋊C₄	—	2	2	no	—
38	C₃²×C₄²	—	2	2	no	—
39	C₃×C₄×Dic₃	—	2	1	no	—
40	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×#208	—	2	2	no	—
42	C₃×#208	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
48	#1772	—	3	1	no	Sylow
49	C₂×C₃×M₁₆	—	3	2	no	—
50	C₂²×C₃×C₈	—	2	1	no	—
51	C₂²×C₃×C₈	—	3	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₃×#117	—	2	2	no	—
55	C₃×#117	C₂×D₃	1	2	yes	—
56	C₃×#117	C₂²×C₃	1	2	yes	—
57	C₂×C₃×C₄²	—	3	1	no	—
58	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
59	C₂×#208	C₂²×C₃	1	2	yes	—
60	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₃×#108	—	3	4	no	—
62	C₃×#107	—	2	2	no	—
63	C₃×#107	Dic₃	1	2	yes	complemented
64	C₃×#107	C₃×C₄	1	2	yes	complemented
65	C₃×#106	—	2	2	no	—
66	C₃×#106	Dic₃	1	2	yes	complemented
67	C₃×#106	C₃×C₄	1	2	yes	complemented
68	C₃×#97	—	6	2	no	—
69	C₃×C₄×C₈	—	3	4	no	—
70	C₃×C₄×C₈	—	6	2	no	—
71	#819	C₃×C₄	1	4	yes	complemented
72	C₈×Dic₃	C₃×C₄	1	4	yes	complemented
73	#803	—	2	2	no	—
74	C₄×C₃⋊C₈	—	2	2	no	—
75	C₂³×C₃²	#41	1	1	yes	characteristic
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₃²×C₄	#41	1	1	yes	characteristic
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₂×C₃×Dic₃	—	2	2	no	—
85	C₃²×C₈	C₄²	1	2	yes	—
86	C₃²×C₈	C₄²	1	2	yes	—
87	C₃×C₃⋊C₈	—	2	4	no	—
88	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
89	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
90	C₂×#108	—	3	1	no	—
91	#427	C₃×D₃	1	1	yes	characteristic
92	#405	—	3	2	no	—
93	C₂×C₄×C₈	—	3	1	no	—
94	#339	—	3	2	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	6	1	no	—
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₂²×Dic₃	—	2	1	no	—
100	C₃×M₁₆	—	3	4	no	—
101	C₃×M₁₆	—	6	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	—	3	1	no	—
108	C₂×C₃×C₈	—	3	1	no	—
109	C₂×C₃×C₈	—	3	2	no	—
110	C₂×C₃×C₈	—	6	2	no	—
111	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
112	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
113	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
116	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
117	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	2	4	no	—
120	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
121	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
122	C₃×C₂²⋊C₄	—	2	2	no	—
123	C₃×C₂²⋊C₄	—	2	2	no	—
124	C₃×C₂²⋊C₄	—	4	2	no	—
125	C₃×C₄²	—	2	2	no	—
126	C₃×C₄²	—	2	2	no	—
127	C₃×C₄²	—	3	2	no	—
128	C₃×C₄²	—	4	2	no	—
129	C₃×C₄²	—	6	1	no	—
130	C₄×Dic₃	—	2	1	no	—
131	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
132	#208	—	2	2	no	—
133	#208	C₂×C₃×C₄	1	4	yes	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
136	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
137	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
138	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
139	C₂²×C₃²	—	2	2	no	—
140	C₂²×C₃²	#117	1	1	yes	characteristic
141	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
142	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
143	C₃²×C₄	—	4	4	no	—
144	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
145	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
147	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
148	C₃×Dic₃	—	2	4	no	—
149	C₂×M₁₆	—	3	2	no	—
150	C₂²×C₈	—	3	1	no	—
151	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
152	#117	C₂×C₃×D₃	1	2	yes	—
153	C₂×C₄²	—	3	1	no	—
154	#108	—	3	4	no	—
155	#107	C₃×Dic₃	1	2	yes	complemented
156	#106	C₃×Dic₃	1	2	yes	complemented
157	#97	—	6	2	no	—
158	C₄×C₈	—	3	4	no	—
159	C₄×C₈	—	6	2	no	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	#237	1	1	yes	characteristic
162	C₂³×C₃	C₃×#41	1	1	yes	characteristic
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	4	4	no	—
172	C₂×C₃×C₄	—	6	4	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
178	C₂×C₃×C₄	#239	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
185	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
186	C₂×Dic₃	—	2	4	no	—
187	C₂×Dic₃	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	3	2	no	—
191	C₃×C₈	—	3	2	no	—
192	C₃×C₈	—	6	4	no	—
193	C₃×C₈	C₄×Dic₃	1	2	yes	—
194	C₃×C₈	C₄×Dic₃	1	2	yes	—
195	C₃×C₈	C₃×C₄²	1	2	yes	—
196	C₃×C₈	C₃×C₄²	1	2	yes	—
197	C₃⋊C₈	—	2	4	no	—
198	C₃⋊C₈	C₃×C₄⋊C₄	1	2	yes	—
199	C₃⋊C₈	C₃×C₄⋊C₄	1	2	yes	—
200	C₂×C₃²	—	2	1	no	—
201	C₂×C₃²	—	2	1	no	—
202	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₃²	#443	1	2	yes	—
204	C₂²×C₄	—	6	1	no	—
205	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
206	M₁₆	—	3	4	no	—
207	M₁₆	—	6	2	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	—	3	1	no	—
210	C₂×C₈	—	3	1	no	—
211	C₂×C₈	—	3	2	no	—
212	C₂×C₈	—	6	2	no	—
213	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
214	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
215	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
217	C₄⋊C₄	C₂×C₃×Dic₃	1	4	yes	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	3	2	no	—
221	C₄²	—	6	1	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	#887	1	1	yes	characteristic
229	C₂²×C₃	#892	1	1	yes	characteristic
230	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
231	C₂²×C₃	C₃×#117	1	1	yes	characteristic
232	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
233	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	4	4	no	—
239	C₃×C₄	—	4	4	no	—
240	C₃×C₄	—	6	4	no	—
241	C₃×C₄	—	8	4	no	—
242	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
243	C₃×C₄	#893	1	1	yes	characteristic
244	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
245	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
246	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
247	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
248	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
249	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
250	Dic₃	—	2	4	no	—
251	C₃²	#1772	1	1	yes	characteristic, complemented, Sylow
252	C₂³	C₃×#237	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	6	4	no	—
255	C₂×C₄	—	6	2	no	—
256	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
258	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
259	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
260	C₂×C₄	C₃×#239	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
262	C₈	—	3	2	no	—
263	C₈	—	3	2	no	—
264	C₈	—	6	4	no	—
265	C₈	C₃×C₄×Dic₃	1	2	yes	—
266	C₈	C₃×C₄×Dic₃	1	2	yes	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	4	1	no	—
274	C₂×C₃	—	4	1	no	—
275	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
276	C₂×C₃	#5196	1	2	yes	—
277	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₃	C₃×#443	1	2	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	C₃×#887	1	1	yes	characteristic
281	C₂²	C₃×#892	1	1	yes	characteristic
282	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
283	C₄	—	4	4	no	—
284	C₄	—	6	4	no	—
285	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
286	C₄	C₃×#893	1	1	yes	characteristic
287	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
288	C₄	C₃×Q₈×Dic₃	1	1	yes	characteristic
289	C₃	—	2	1	no	—
290	C₃	#70659	1	1	yes	characteristic, direct product factor
291	C₃	C₃×#1772	1	1	yes	characteristic, complemented
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	C₃×Dic₃×C₄⋊C₄	1	1	yes	characteristic
295	C₂	C₃×#5196	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₃×C₄	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#427	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#427	D₃
hypocenter	C₃	C₃×#1772
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#427

Complements

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

N	H ≅ G/N	type
7. #70659	C₃	direct (N × H)
13. C₃²×#107	C₄	semidirect
14. C₃²×#106	C₄	semidirect
15. C₃×#819	C₄	semidirect
16. C₃×C₈×Dic₃	C₄	semidirect
63. C₃×#107	Dic₃	semidirect
64. C₃×#107	C₃×C₄	semidirect
66. C₃×#106	Dic₃	semidirect
67. C₃×#106	C₃×C₄	semidirect
71. #819	C₃×C₄	semidirect
72. C₈×Dic₃	C₃×C₄	semidirect
155. #107	C₃×Dic₃	semidirect
156. #106	C₃×Dic₃	semidirect
251. C₃²	#1772	semidirect
290. C₃	#70659	direct (N × H)
291. C₃	C₃×#1772	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#427	2
C₂×C₃×#819	2
C₂×C₃×C₈×Dic₃	2
C₃×#4997	2
C₃×#4532	2
C₃×#1772	3
#70659	3

Subgroups — C3×#70659 #11784369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70659 #11784369