Subgroups — C₃×#70660 #11784404

Order 1152.

505 conjugacy classes of subgroups.

The subgroup lattice has 289 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×#815	C₂	1	1	yes	maximal, characteristic
4	C₃×#4997	C₂	1	2	yes	maximal
5	C₃×#4532	C₂	1	2	yes	maximal
6	C₃×#1773	—	3	1	no	maximal
7	#70660	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×#815	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×#97	—	3	1	no	—
25	C₂×#819	C₂×C₃	1	1	yes	characteristic
26	C₂×#815	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	2	yes	—
46	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
47	#1773	—	3	1	no	Sylow
48	C₂×C₃×M₁₆	—	3	2	no	—
49	C₂²×C₃×C₈	—	2	1	no	—
50	C₂²×C₃×C₈	—	3	1	no	—
51	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₃×#117	—	2	2	no	—
54	C₃×#117	C₂×D₃	1	2	yes	—
55	C₃×#117	C₂²×C₃	1	2	yes	—
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
58	C₂×#208	C₂²×C₃	1	2	yes	—
59	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×#108	—	3	4	no	—
61	C₃×#107	—	2	2	no	—
62	C₃×#107	Dic₃	1	2	yes	complemented
63	C₃×#107	C₃×C₄	1	2	yes	complemented
64	C₃×#106	—	2	2	no	—
65	C₃×#106	Dic₃	1	2	yes	complemented
66	C₃×#106	C₃×C₄	1	2	yes	complemented
67	C₃×#97	—	3	4	no	—
68	C₃×#97	—	6	2	no	—
69	C₃×C₄×C₈	—	6	2	no	—
70	#819	C₃×C₄	1	4	yes	complemented
71	#815	C₃×C₄	1	4	yes	complemented
72	#803	—	2	2	no	—
73	C₄×C₃⋊C₈	—	2	2	no	—
74	C₂³×C₃²	#41	1	1	yes	characteristic
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	2	2	no	—
83	C₂×C₃×Dic₃	—	4	2	no	—
84	C₃²×C₈	C₄²	1	2	yes	—
85	C₃²×C₈	C₄²	1	2	yes	—
86	C₃×C₃⋊C₈	—	2	4	no	—
87	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	—
88	C₂×#108	—	3	1	no	—
89	#427	C₃×D₃	1	1	yes	characteristic
90	#405	—	3	2	no	—
91	C₂×#97	—	3	1	no	—
92	#339	—	3	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	6	1	no	—
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₂²×Dic₃	—	2	1	no	—
98	C₃×M₁₆	—	3	4	no	—
99	C₃×M₁₆	—	6	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	3	2	no	—
105	C₂×C₃×C₈	—	3	2	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	—	6	2	no	—
108	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
109	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
110	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
116	C₃×C₄⋊C₄	—	2	4	no	—
117	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
118	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
119	C₃×C₂²⋊C₄	—	2	2	no	—
120	C₃×C₂²⋊C₄	—	2	2	no	—
121	C₃×C₂²⋊C₄	—	4	2	no	—
122	C₃×C₄²	—	2	2	no	—
123	C₃×C₄²	—	2	2	no	—
124	C₃×C₄²	—	3	2	no	—
125	C₃×C₄²	—	4	2	no	—
126	C₃×C₄²	—	6	1	no	—
127	C₄×Dic₃	—	2	1	no	—
128	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
129	#208	—	2	2	no	—
130	#208	C₂×C₃×C₄	1	4	yes	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
133	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
134	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
135	C₂²×C₃²	—	2	2	no	—
136	C₂²×C₃²	#117	1	1	yes	characteristic
137	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
138	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
139	C₃²×C₄	—	4	4	no	—
140	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
141	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
143	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
144	C₃×Dic₃	—	4	2	no	—
145	C₂×M₁₆	—	3	2	no	—
146	C₂²×C₈	—	3	1	no	—
147	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
148	#117	C₂×C₃×D₃	1	2	yes	—
149	C₂×C₄²	—	3	1	no	—
150	#108	—	3	4	no	—
151	#107	C₃×Dic₃	1	2	yes	complemented
152	#106	C₃×Dic₃	1	2	yes	complemented
153	#97	—	3	4	no	—
154	#97	—	6	2	no	—
155	C₄×C₈	—	6	2	no	—
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	#237	1	1	yes	characteristic
158	C₂³×C₃	C₃×#41	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	4	4	no	—
168	C₂×C₃×C₄	—	6	2	no	—
169	C₂×C₃×C₄	—	12	2	no	—
170	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
174	C₂×C₃×C₄	#239	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
181	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
182	C₂×Dic₃	—	2	2	no	—
183	C₂×Dic₃	—	4	2	no	—
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	3	4	no	—
187	C₃×C₈	—	6	4	no	—
188	C₃×C₈	C₄×Dic₃	1	2	yes	—
189	C₃×C₈	C₄×Dic₃	1	2	yes	—
190	C₃×C₈	C₃×C₄²	1	2	yes	—
191	C₃×C₈	C₃×C₄²	1	2	yes	—
192	C₃⋊C₈	—	2	4	no	—
193	C₃⋊C₈	C₃×C₄⋊C₄	1	4	yes	—
194	C₂×C₃²	—	2	1	no	—
195	C₂×C₃²	—	2	1	no	—
196	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₃²	#444	1	2	yes	—
198	C₂²×C₄	—	6	1	no	—
199	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
200	M₁₆	—	3	4	no	—
201	M₁₆	—	6	2	no	—
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	3	2	no	—
204	C₂×C₈	—	3	2	no	—
205	C₂×C₈	—	6	2	no	—
206	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
207	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
208	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
209	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂×C₃×Dic₃	1	4	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	3	2	no	—
214	C₄²	—	6	1	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	#887	1	1	yes	characteristic
222	C₂²×C₃	#892	1	1	yes	characteristic
223	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
224	C₂²×C₃	C₃×#117	1	1	yes	characteristic
225	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
226	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	4	4	no	—
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	—	8	4	no	—
234	C₃×C₄	—	12	2	no	—
235	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
236	C₃×C₄	#893	1	1	yes	characteristic
237	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
238	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
239	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
240	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
241	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
242	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
243	Dic₃	—	4	2	no	—
244	C₃²	#1773	1	1	yes	characteristic, complemented, Sylow
245	C₂³	C₃×#237	1	1	yes	characteristic
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	6	2	no	—
248	C₂×C₄	—	12	2	no	—
249	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
251	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
252	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
253	C₂×C₄	C₃×#239	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
255	C₈	—	3	4	no	—
256	C₈	—	6	4	no	—
257	C₈	C₃×C₄×Dic₃	1	2	yes	—
258	C₈	C₃×C₄×Dic₃	1	2	yes	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	4	1	no	—
266	C₂×C₃	—	4	1	no	—
267	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
268	C₂×C₃	#5197	1	2	yes	—
269	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
270	C₂×C₃	C₃×#444	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	C₃×#887	1	1	yes	characteristic
273	C₂²	C₃×#892	1	1	yes	characteristic
274	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
275	C₄	—	4	4	no	—
276	C₄	—	12	2	no	—
277	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
278	C₄	C₃×#893	1	1	yes	characteristic
279	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
280	C₄	C₃×Q₈×Dic₃	1	1	yes	characteristic
281	C₃	—	2	1	no	—
282	C₃	#70660	1	1	yes	characteristic, direct product factor
283	C₃	C₃×#1773	1	1	yes	characteristic, complemented
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	C₃×Dic₃×C₄⋊C₄	1	1	yes	characteristic
287	C₂	C₃×#5197	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×#1773
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. #70660	C₃	direct (N × H)
13. C₃²×#107	C₄	semidirect
14. C₃²×#106	C₄	semidirect
15. C₃×#819	C₄	semidirect
16. C₃×#815	C₄	semidirect
62. C₃×#107	Dic₃	semidirect
63. C₃×#107	C₃×C₄	semidirect
65. C₃×#106	Dic₃	semidirect
66. C₃×#106	C₃×C₄	semidirect
70. #819	C₃×C₄	semidirect
71. #815	C₃×C₄	semidirect
151. #107	C₃×Dic₃	semidirect
152. #106	C₃×Dic₃	semidirect
244. C₃²	#1773	semidirect
282. C₃	#70660	direct (N × H)
283. C₃	C₃×#1773	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×#815	2
C₃×#4997	2
C₃×#4532	2
C₃×#1773	3
#70660	3

Subgroups — C3×#70660 #11784404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70660 #11784404