Subgroups — C₃×#70607 #11786249

Order 1152.

561 conjugacy classes of subgroups.

The subgroup lattice has 301 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×#818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#4977	C₂	1	2	yes	maximal
5	C₂×C₃×#808	C₂	1	2	yes	maximal
6	C₃×#1846	—	3	1	no	maximal
7	#70607	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	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×#107	—	2	2	no	—
14	C₃×C₃⋊#95	—	2	4	no	—
15	C₃×#818	—	2	2	no	—
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	C₃×#808	—	2	4	no	—
18	C₂×C₃×#107	—	2	1	no	—
19	C₂×C₃×#107	—	3	1	no	—
20	C₂×C₃×#107	D₃	1	1	yes	characteristic
21	C₂×C₃×#107	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×#106	—	3	2	no	—
23	C₂×C₃×C₄×C₈	—	3	1	no	—
24	C₃×#382	—	3	2	no	—
25	C₂×#818	C₂×C₃	1	1	yes	characteristic
26	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
27	#4977	C₂×C₃	1	2	yes	—
28	C₂×#808	C₂×C₃	1	2	yes	—
29	C₂²×C₃²×C₄	—	2	2	no	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	D₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₂×C₃²×C₈	Q₈	1	2	yes	—
36	C₃²×C₄⋊C₄	—	2	2	no	—
37	C₃²×C₄⋊C₄	—	2	2	no	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	C₃×C₄⋊Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×C₄×Dic₃	Q₈	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
45	#1846	—	3	1	no	Sylow
46	C₂²×C₃×C₈	—	2	1	no	—
47	C₂²×C₃×C₈	—	3	1	no	—
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
53	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
56	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃×#107	—	2	4	no	—
59	C₃×#107	—	2	2	no	—
60	C₃×#107	—	2	2	no	—
61	C₃×#107	—	6	2	no	—
62	C₃×#106	—	6	4	no	—
63	C₃×C₄×C₈	—	3	4	no	—
64	C₃×#95	—	6	4	no	—
65	C₃⋊#95	—	2	4	no	—
66	#818	—	2	2	no	—
67	C₈×Dic₃	C₃×C₄	1	4	yes	complemented
68	#808	—	2	4	no	—
69	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	—	4	2	no	—
71	C₂×C₃²×C₄	—	4	4	no	—
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃×Dic₃	—	2	2	no	—
79	C₂×C₃×Dic₃	—	2	2	no	—
80	C₂×C₃×Dic₃	—	4	2	no	—
81	C₂×C₃×Dic₃	—	4	4	no	—
82	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
83	C₃²×C₈	C₄⋊C₄	1	4	yes	—
84	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	—
85	C₂×#107	—	3	1	no	—
86	C₂×#107	C₃×D₃	1	1	yes	characteristic
87	C₂×#106	—	3	2	no	—
88	C₂×C₄×C₈	—	3	1	no	—
89	#382	—	3	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	—	3	2	no	—
94	C₂²×C₃×C₄	—	4	2	no	—
95	C₂²×C₃×C₄	—	6	2	no	—
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂²×Dic₃	—	2	2	no	—
99	C₂²×Dic₃	C₃×D₄	1	2	yes	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	3	2	no	—
104	C₂×C₃×C₈	—	3	2	no	—
105	C₂×C₃×C₈	—	3	2	no	—
106	C₂×C₃×C₈	Dic₆	1	2	yes	—
107	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
108	C₂×C₃×C₈	#67	1	2	yes	—
109	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
110	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
111	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	2	2	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	—	2	2	no	—
117	C₃×C₄⋊C₄	—	4	2	no	—
118	C₃×C₄⋊C₄	—	6	2	no	—
119	C₃×C₄⋊C₄	—	6	2	no	—
120	C₃×C₄²	—	3	2	no	—
121	C₃×C₄²	—	3	2	no	—
122	C₄⋊Dic₃	—	2	2	no	—
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
125	C₄×Dic₃	C₃×Q₈	1	2	yes	—
126	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
127	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
128	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
129	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₃²	#127	1	1	yes	characteristic
131	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₃²	#135	1	2	yes	—
133	C₂²×C₃²	#135	1	2	yes	—
134	C₃²×C₄	—	8	2	no	—
135	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
138	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
139	C₃×Dic₃	—	2	4	no	—
140	C₃×Dic₃	—	8	2	no	—
141	C₂²×C₈	—	3	1	no	—
142	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	3	2	no	—
144	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
145	C₂×C₄²	—	3	1	no	—
146	#107	—	2	2	no	—
147	#107	—	6	2	no	—
148	#106	—	6	4	no	—
149	C₄×C₈	—	3	4	no	—
150	#95	—	6	4	no	—
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
153	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	3	4	no	—
161	C₂×C₃×C₄	—	4	4	no	—
162	C₂×C₃×C₄	—	4	4	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	4	no	—
166	C₂×C₃×C₄	—	6	2	no	—
167	C₂×C₃×C₄	—	6	2	no	—
168	C₂×C₃×C₄	—	8	4	no	—
169	C₂×C₃×C₄	—	12	4	no	—
170	C₂×C₃×C₄	—	12	2	no	—
171	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₃×D₄	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	C₂×C₃×Q₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
183	C₂×Dic₃	—	2	2	no	—
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	4	2	no	—
186	C₂×Dic₃	—	4	4	no	—
187	C₂×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
188	C₃×C₈	—	2	4	no	—
189	C₃×C₈	—	3	4	no	—
190	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
191	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
192	C₃⋊C₈	C₃×C₄⋊C₄	1	4	yes	—
193	C₂×C₃²	#384	1	1	yes	characteristic
194	C₂×C₃²	#427	1	2	yes	—
195	C₂×C₃²	#495	1	1	yes	characteristic
196	C₂×C₃²	#495	1	1	yes	characteristic
197	C₂×C₃²	#499	1	2	yes	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₂²×C₄	—	6	2	no	—
201	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	3	2	no	—
204	C₂×C₈	—	3	2	no	—
205	C₂×C₈	C₃×Dic₆	1	2	yes	—
206	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
207	C₂×C₈	C₃×#67	1	2	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	6	2	no	—
211	C₄⋊C₄	—	6	2	no	—
212	C₄²	—	3	2	no	—
213	C₄²	—	3	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	#888	1	1	yes	characteristic
220	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	#912	1	2	yes	—
222	C₂²×C₃	#919	1	2	yes	—
223	C₂²×C₃	#940	1	1	yes	characteristic
224	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
225	C₂²×C₃	C₃×#127	1	1	yes	characteristic
226	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
227	C₂²×C₃	C₃×#135	1	2	yes	—
228	C₂²×C₃	C₃×#135	1	2	yes	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	6	4	no	—
234	C₃×C₄	—	8	4	no	—
235	C₃×C₄	—	8	2	no	—
236	C₃×C₄	—	8	2	no	—
237	C₃×C₄	—	24	2	no	—
238	C₃×C₄	#888	1	1	yes	characteristic
239	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
240	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
241	C₃×C₄	#944	1	1	yes	characteristic
242	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
243	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
244	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
245	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
246	Dic₃	—	2	4	no	—
247	Dic₃	—	8	2	no	—
248	C₃²	#1846	1	1	yes	characteristic, complemented, Sylow
249	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
250	C₂×C₄	—	3	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	—	12	4	no	—
256	C₂×C₄	—	12	2	no	—
257	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
259	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
260	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
261	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
262	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
263	C₈	—	3	4	no	—
264	C₈	C₃×Dic₃⋊C₄	1	4	yes	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	2	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	#4878	1	2	yes	—
271	C₂×C₃	#4880	1	2	yes	—
272	C₂×C₃	#4975	1	1	yes	characteristic
273	C₂×C₃	#5155	1	1	yes	characteristic
274	C₂×C₃	#5191	1	1	yes	characteristic
275	C₂×C₃	C₃×#384	1	1	yes	characteristic
276	C₂×C₃	C₃×#427	1	2	yes	—
277	C₂×C₃	C₃×#495	1	1	yes	characteristic
278	C₂×C₃	C₃×#495	1	1	yes	characteristic
279	C₂×C₃	C₃×#499	1	2	yes	—
280	C₂²	C₃×#888	1	1	yes	characteristic
281	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
282	C₂²	C₃×#912	1	2	yes	—
283	C₂²	C₃×#919	1	2	yes	—
284	C₂²	C₃×#940	1	1	yes	characteristic
285	C₄	—	6	4	no	—
286	C₄	—	8	2	no	—
287	C₄	—	24	2	no	—
288	C₄	C₃×#888	1	1	yes	characteristic
289	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
290	C₄	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
291	C₄	C₃×#944	1	1	yes	characteristic
292	C₃	—	2	1	no	—
293	C₃	#70607	1	1	yes	characteristic, direct product factor
294	C₃	C₃×#1846	1	1	yes	characteristic, complemented
295	C₂	C₃×#4878	1	2	yes	—
296	C₂	C₃×#4880	1	2	yes	—
297	C₂	C₃×#4975	1	1	yes	characteristic
298	C₂	C₃×#5155	1	1	yes	characteristic
299	C₂	C₃×#5191	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#107	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#107	D₃
hypocenter	C₃	C₃×#1846
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#107

Complements

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

N	H ≅ G/N	type
7. #70607	C₃	direct (N × H)
16. C₃×C₈×Dic₃	C₄	semidirect
67. C₈×Dic₃	C₃×C₄	semidirect
248. C₃²	#1846	semidirect
293. C₃	#70607	direct (N × H)
294. C₃	C₃×#1846	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×#818	2
C₂×C₃×C₈×Dic₃	2
C₃×#4977	2
C₂×C₃×#808	2
C₃×#1846	3
#70607	3

Subgroups — C3×#70607 #11786249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70607 #11786249