Subgroups — C₃×#70625 #11790483

Order 1152.

457 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
3	C₃×#5101	C₂	1	1	yes	maximal, characteristic
4	C₃×#4964	C₂	1	2	yes	maximal
5	C₃×#4530	C₂	1	2	yes	maximal
6	C₃×#1952	—	3	1	no	maximal
7	#70625	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	C₃²×#107	—	2	1	no	—
14	C₃²×#107	—	2	1	no	—
15	C₃×#819	C₄	1	4	yes	complemented
16	C₃×#815	—	2	1	no	—
17	C₃×C₈×Dic₃	—	2	1	no	—
18	C₃×#808	—	2	2	no	—
19	C₃×#807	—	2	2	no	—
20	C₂×C₃×#108	—	3	1	no	—
21	C₃×#428	—	3	2	no	—
22	C₂×C₃×#107	—	2	1	no	—
23	C₂×C₃×#107	D₃	1	1	yes	characteristic
24	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
25	C₃×#405	—	3	1	no	—
26	C₂×#819	C₂×C₃	1	1	yes	characteristic
27	#5101	C₂×C₃	1	1	yes	characteristic
28	#4964	C₂×C₃	1	2	yes	—
29	C₃×#343	—	3	2	no	—
30	#4530	C₂×C₃	1	2	yes	—
31	C₂²×C₃²×C₄	—	2	2	no	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	Q₈	1	2	yes	—
37	C₃²×C₄⋊C₄	—	2	2	no	—
38	C₃²×C₄⋊C₄	—	2	2	no	—
39	C₃×#217	—	2	1	no	—
40	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
41	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
42	C₃×C₄×Dic₃	—	2	1	no	—
43	C₃×#208	—	2	2	no	—
44	C₃×#208	—	2	1	no	—
45	C₃×#208	C₂×C₄	1	4	yes	—
46	C₃×#208	Q₈	1	2	yes	—
47	C₂×C₃×C₃⋊C₈	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	—	2	1	no	—
49	#1952	—	3	1	no	Sylow
50	C₂×C₃×M₁₆	—	3	1	no	—
51	C₂×C₃×M₁₆	—	3	2	no	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₃×#117	—	3	1	no	—
56	C₂×C₃×C₄⋊C₄	—	2	2	no	—
57	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	#926	C₂²×C₃	1	1	yes	characteristic
60	C₂×#208	C₂²×C₃	1	2	yes	—
61	C₂×#208	C₂²×C₃	1	1	yes	characteristic
62	C₃×#108	—	3	4	no	—
63	C₃×#107	—	2	2	no	—
64	C₃×#107	—	2	2	no	—
65	C₃×#107	—	2	1	no	—
66	C₃×#107	—	2	1	no	—
67	C₃×#107	—	2	1	no	—
68	C₃×#107	—	2	1	no	—
69	C₃×#107	—	6	2	no	—
70	C₃×#106	—	6	2	no	—
71	C₃×#97	—	6	1	no	—
72	C₃×C₄×C₈	—	6	1	no	—
73	#819	C₃×C₄	1	4	yes	complemented
74	#815	—	2	1	no	—
75	C₈×Dic₃	—	2	1	no	—
76	#808	—	2	2	no	—
77	#807	—	2	2	no	—
78	C₂³×C₃²	#41	1	1	yes	characteristic
79	C₂×C₃²×C₄	—	4	2	no	—
80	C₂×C₃²×C₄	—	8	2	no	—
81	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₃²×C₄	#41	1	1	yes	characteristic
87	C₂×C₃×Dic₃	—	2	1	no	—
88	C₂×C₃×Dic₃	—	2	1	no	—
89	C₃²×C₈	C₄⋊C₄	1	2	yes	—
90	C₃²×C₈	C₄⋊C₄	1	2	yes	—
91	C₃×C₃⋊C₈	—	2	4	no	—
92	C₃×C₃⋊C₈	—	2	2	no	—
93	C₂×#108	—	3	1	no	—
94	#428	—	3	2	no	—
95	C₂×#107	C₃×D₃	1	1	yes	characteristic
96	#405	—	3	1	no	—
97	#343	—	3	2	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	—	4	2	no	—
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₃×M₁₆	—	3	4	no	—
105	C₃×M₁₆	—	3	2	no	—
106	C₃×M₁₆	—	6	2	no	—
107	C₃×M₁₆	—	6	1	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	—	6	2	no	—
113	C₂×C₃×C₈	—	6	1	no	—
114	C₂×C₃×C₈	Dic₆	1	2	yes	—
115	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
116	C₂×C₃×C₈	#67	1	1	yes	characteristic
117	C₂×C₃×C₈	#67	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
119	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
122	C₃×C₄⋊C₄	—	2	4	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	3	1	no	—
128	C₃×C₄⋊C₄	—	3	1	no	—
129	C₃×C₄⋊C₄	—	4	2	no	—
130	C₃×C₂²⋊C₄	—	6	1	no	—
131	C₃×C₄²	—	6	1	no	—
132	#217	—	2	1	no	—
133	C₄⋊Dic₃	C₃×D₄	1	1	yes	characteristic
134	C₄⋊Dic₃	C₃×D₄	1	1	yes	characteristic
135	C₄×Dic₃	—	2	1	no	—
136	#208	—	2	2	no	—
137	#208	—	2	1	no	—
138	#208	C₂×C₃×C₄	1	4	yes	—
139	#208	C₃×Q₈	1	2	yes	—
140	C₂×C₃⋊C₈	—	2	2	no	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂²×C₃²	—	2	1	no	—
143	C₂²×C₃²	—	2	1	no	—
144	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₃²	#121	1	1	yes	characteristic
146	C₂²×C₃²	#125	1	1	yes	characteristic
147	C₃²×C₄	—	8	2	no	—
148	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
150	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
151	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
152	C₃×Dic₃	—	4	1	no	—
153	C₃×Dic₃	—	4	1	no	—
154	C₂×M₁₆	—	3	1	no	—
155	C₂×M₁₆	—	3	2	no	—
156	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
157	#117	—	3	1	no	—
158	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
159	#108	—	3	4	no	—
160	#107	—	2	1	no	—
161	#107	—	2	1	no	—
162	#107	—	6	2	no	—
163	#106	—	6	2	no	—
164	#97	—	6	1	no	—
165	C₄×C₈	—	6	1	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	#237	1	1	yes	characteristic
168	C₂³×C₃	C₃×#41	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	6	1	no	—
179	C₂×C₃×C₄	—	6	1	no	—
180	C₂×C₃×C₄	—	8	4	no	—
181	C₂×C₃×C₄	—	8	2	no	—
182	C₂×C₃×C₄	—	8	2	no	—
183	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
187	C₂×C₃×C₄	#239	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
195	C₂×Dic₃	—	2	1	no	—
196	C₂×Dic₃	—	2	1	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	—	6	4	no	—
200	C₃×C₈	—	6	2	no	—
201	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
202	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
203	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
204	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
205	C₃⋊C₈	—	2	4	no	—
206	C₃⋊C₈	—	2	2	no	—
207	C₂×C₃²	—	2	1	no	—
208	C₂×C₃²	—	2	1	no	—
209	C₂×C₃²	#389	1	1	yes	characteristic
210	C₂×C₃²	#472	1	1	yes	characteristic
211	C₂×C₃²	#473	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
214	M₁₆	—	3	4	no	—
215	M₁₆	—	3	2	no	—
216	M₁₆	—	6	2	no	—
217	M₁₆	—	6	1	no	—
218	C₂×C₈	—	6	2	no	—
219	C₂×C₈	—	6	1	no	—
220	C₂×C₈	C₃×Dic₆	1	2	yes	—
221	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
222	C₂×C₈	C₃×#67	1	1	yes	characteristic
223	C₂×C₈	C₃×#67	1	1	yes	characteristic
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	3	1	no	—
227	C₄⋊C₄	—	3	1	no	—
228	C₂²⋊C₄	—	6	1	no	—
229	C₄²	—	6	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	#887	1	1	yes	characteristic
240	C₂²×C₃	#890	1	1	yes	characteristic
241	C₂²×C₃	#938	1	1	yes	characteristic
242	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
243	C₂²×C₃	C₃×#121	1	1	yes	characteristic
244	C₂²×C₃	C₃×#125	1	1	yes	characteristic
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	—	8	4	no	—
250	C₃×C₄	—	8	2	no	—
251	C₃×C₄	—	8	2	no	—
252	C₃×C₄	—	12	1	no	—
253	C₃×C₄	—	12	1	no	—
254	C₃×C₄	#888	1	1	yes	characteristic
255	C₃×C₄	#893	1	1	yes	characteristic
256	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
257	C₃×C₄	#946	1	1	yes	characteristic
258	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
259	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
260	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
261	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
262	Dic₃	—	4	1	no	—
263	Dic₃	—	4	1	no	—
264	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
265	C₂³	C₃×#237	1	1	yes	characteristic
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	6	1	no	—
268	C₂×C₄	—	6	1	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
271	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
272	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
273	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
274	C₂×C₄	C₃×#239	1	1	yes	characteristic
275	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
276	C₈	—	6	4	no	—
277	C₈	—	6	2	no	—
278	C₈	C₃×Dic₃⋊C₄	1	2	yes	—
279	C₈	C₃×Dic₃⋊C₄	1	2	yes	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	#4972	1	1	yes	characteristic
290	C₂×C₃	#5160	1	1	yes	characteristic
291	C₂×C₃	#5192	1	1	yes	characteristic
292	C₂×C₃	C₃×#389	1	1	yes	characteristic
293	C₂×C₃	C₃×#472	1	1	yes	characteristic
294	C₂×C₃	C₃×#473	1	1	yes	characteristic
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	C₃×#887	1	1	yes	characteristic
298	C₂²	C₃×#890	1	1	yes	characteristic
299	C₂²	C₃×#938	1	1	yes	characteristic
300	C₄	—	8	2	no	—
301	C₄	—	12	1	no	—
302	C₄	—	12	1	no	—
303	C₄	C₃×#888	1	1	yes	characteristic
304	C₄	C₃×#893	1	1	yes	characteristic
305	C₄	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
306	C₄	C₃×#946	1	1	yes	characteristic
307	C₃	—	2	1	no	—
308	C₃	#70625	1	1	yes	characteristic, direct product factor
309	C₃	C₃×#1952	1	1	yes	characteristic, complemented
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₃×#4972	1	1	yes	characteristic
313	C₂	C₃×#5160	1	1	yes	characteristic
314	C₂	C₃×#5192	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#938
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#107	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#107	D₃
hypocenter	C₃	C₃×#1952
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#107

Complements

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

N	H ≅ G/N	type
7. #70625	C₃	direct (N × H)
15. C₃×#819	C₄	semidirect
73. #819	C₃×C₄	semidirect
264. C₃²	#1952	semidirect
308. C₃	#70625	direct (N × H)
309. C₃	C₃×#1952	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#107	2
C₂×C₃×#819	2
C₃×#5101	2
C₃×#4964	2
C₃×#4530	2
C₃×#1952	3
#70625	3

Subgroups — C3×#70625 #11790483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70625 #11790483