Subgroups — C₃×#70560 #11790100

Order 1152.

561 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#4984	C₂	1	2	yes	maximal
5	C₃×#4529	C₂	1	2	yes	maximal
6	C₃×#1941	—	3	1	no	maximal
7	#70560	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	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃²×#106	C₄	1	4	yes	complemented
15	C₃×C₃⋊#95	—	2	2	no	—
16	C₃×#817	—	2	2	no	—
17	C₃×C₈×Dic₃	—	2	2	no	—
18	C₂×C₃×#106	—	2	1	no	—
19	C₂×C₃×#106	—	3	1	no	—
20	C₂×C₃×#106	D₃	1	1	yes	characteristic
21	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	—	3	1	no	—
23	C₃×#389	—	3	2	no	—
24	C₂×#817	C₂×C₃	1	1	yes	characteristic
25	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
26	#4984	C₂×C₃	1	2	yes	—
27	C₃×#340	—	3	2	no	—
28	#4529	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	C₂×C₄	1	4	yes	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	C₃×C₄⋊Dic₃	—	2	2	no	—
40	C₃×C₄⋊Dic₃	—	4	4	no	—
41	C₃×C₄×Dic₃	—	2	1	no	—
42	C₃×C₄×Dic₃	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
46	#1941	—	3	1	no	Sylow
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₃×C₄⋊C₄	—	3	2	no	—
53	C₂×C₃×C₄⋊C₄	—	6	2	no	—
54	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₂×C₄⋊Dic₃	—	2	2	no	—
58	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
59	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₃×#106	—	2	4	no	—
62	C₃×#106	—	6	2	no	—
63	C₃×#106	Dic₃	1	4	yes	complemented
64	C₃×#106	C₃×C₄	1	4	yes	complemented
65	C₃×C₄×C₈	—	6	2	no	—
66	C₃×#95	—	6	2	no	—
67	C₃⋊#95	—	2	2	no	—
68	#817	—	2	2	no	—
69	C₈×Dic₃	—	2	2	no	—
70	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	—	2	4	no	—
72	C₂×C₃²×C₄	—	4	2	no	—
73	C₂×C₃²×C₄	—	4	2	no	—
74	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×D₄	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×C₃²×C₄	#41	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	1	yes	characteristic
80	C₂×C₃×Dic₃	—	2	2	no	—
81	C₂×C₃×Dic₃	—	2	2	no	—
82	C₂×C₃×Dic₃	—	4	2	no	—
83	C₂×C₃×Dic₃	—	4	2	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	SD₁₆	1	4	yes	—
86	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
87	C₃×C₃⋊C₈	—	2	2	no	—
88	C₂×#106	—	3	1	no	—
89	C₂×#106	C₃×D₃	1	1	yes	characteristic
90	C₂×C₄×C₈	—	3	1	no	—
91	#389	—	3	2	no	—
92	#340	—	3	2	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	3	2	no	—
97	C₂²×C₃×C₄	—	4	2	no	—
98	C₂²×C₃×C₄	—	6	2	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	2	no	—
102	C₂²×Dic₃	C₃×D₄	1	2	yes	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	—	6	1	no	—
108	C₂×C₃×C₈	—	6	1	no	—
109	C₂×C₃×C₈	Dic₆	1	2	yes	—
110	C₂×C₃×C₈	D₁₂	1	2	yes	—
111	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
112	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
114	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	—	2	2	no	—
117	C₃×C₄⋊C₄	—	2	4	no	—
118	C₃×C₄⋊C₄	—	4	2	no	—
119	C₃×C₄⋊C₄	—	6	2	no	—
120	C₃×C₄⋊C₄	—	6	2	no	—
121	C₃×C₄⋊C₄	—	12	4	no	—
122	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
123	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
124	C₃×C₄²	—	6	1	no	—
125	C₃×C₄²	—	6	1	no	—
126	C₄⋊Dic₃	—	2	2	no	—
127	C₄⋊Dic₃	—	2	2	no	—
128	C₄⋊Dic₃	—	4	4	no	—
129	C₄×Dic₃	—	2	1	no	—
130	C₄×Dic₃	—	2	1	no	—
131	C₂×C₃⋊C₈	—	2	1	no	—
132	C₂×C₃⋊C₈	—	2	1	no	—
133	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
134	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
135	C₂²×C₃²	#121	1	1	yes	characteristic
136	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
138	C₂²×C₃²	#135	1	2	yes	—
139	C₃²×C₄	—	4	4	no	—
140	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
142	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
143	C₃²×C₄	#125	1	1	yes	characteristic
144	C₃×Dic₃	—	2	4	no	—
145	C₃×Dic₃	—	8	2	no	—
146	C₂²×C₈	—	3	1	no	—
147	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	—	3	2	no	—
149	C₂×C₄⋊C₄	—	6	2	no	—
150	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
151	C₂×C₄²	—	3	1	no	—
152	#106	—	6	2	no	—
153	#106	C₃×Dic₃	1	4	yes	complemented
154	C₄×C₈	—	6	2	no	—
155	#95	—	6	2	no	—
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
158	C₂³×C₃	C₂×C₃×D₄	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	—	3	4	no	—
168	C₂×C₃×C₄	—	4	4	no	—
169	C₂×C₃×C₄	—	4	2	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	4	2	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	—	6	2	no	—
175	C₂×C₃×C₄	—	8	2	no	—
176	C₂×C₃×C₄	—	8	2	no	—
177	C₂×C₃×C₄	—	12	2	no	—
178	C₂×C₃×C₄	—	12	2	no	—
179	C₂×C₃×C₄	—	12	2	no	—
180	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
182	C₂×C₃×C₄	#237	1	1	yes	characteristic
183	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₄	#239	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
190	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
191	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
192	C₂×Dic₃	—	2	2	no	—
193	C₂×Dic₃	—	2	2	no	—
194	C₂×Dic₃	—	4	2	no	—
195	C₂×Dic₃	—	4	2	no	—
196	C₂×Dic₃	—	4	2	no	—
197	C₂×Dic₃	C₃×SD₁₆	1	4	yes	—
198	C₃×C₈	—	2	4	no	—
199	C₃×C₈	—	6	2	no	—
200	C₃×C₈	C₄⋊Dic₃	1	4	yes	complemented
201	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	complemented
202	C₃⋊C₈	—	2	2	no	—
203	C₂×C₃²	#389	1	1	yes	characteristic
204	C₂×C₃²	C₄×SD₁₆	1	2	yes	—
205	C₂×C₃²	#465	1	2	yes	—
206	C₂×C₃²	#498	1	1	yes	characteristic
207	C₂×C₃²	#499	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	3	2	no	—
210	C₂²×C₄	—	6	2	no	—
211	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	C₂×C₈	—	3	2	no	—
213	C₂×C₈	—	6	1	no	—
214	C₂×C₈	—	6	1	no	—
215	C₂×C₈	C₃×Dic₆	1	2	yes	—
216	C₂×C₈	C₃×D₁₂	1	2	yes	—
217	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	6	2	no	—
220	C₄⋊C₄	—	6	2	no	—
221	C₄⋊C₄	—	12	4	no	—
222	C₄⋊C₄	C₂×C₃×Dic₃	1	4	yes	—
223	C₄²	—	6	1	no	—
224	C₄²	—	6	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	#888	1	1	yes	characteristic
231	C₂²×C₃	#895	1	1	yes	characteristic
232	C₂²×C₃	D₃×SD₁₆	1	2	yes	—
233	C₂²×C₃	#916	1	2	yes	—
234	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
235	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
236	C₂²×C₃	C₃×#121	1	1	yes	characteristic
237	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
239	C₂²×C₃	C₃×#135	1	2	yes	—
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	—	4	4	no	—
245	C₃×C₄	—	4	4	no	—
246	C₃×C₄	—	6	4	no	—
247	C₃×C₄	—	8	4	no	—
248	C₃×C₄	—	24	2	no	—
249	C₃×C₄	#890	1	1	yes	characteristic
250	C₃×C₄	#897	1	1	yes	characteristic
251	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
252	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
253	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
254	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
255	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
256	C₃×C₄	C₃×#125	1	1	yes	characteristic
257	Dic₃	—	2	4	no	—
258	Dic₃	—	8	2	no	—
259	C₃²	#1941	1	1	yes	characteristic, complemented, Sylow
260	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	3	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	6	2	no	—
267	C₂×C₄	—	12	2	no	—
268	C₂×C₄	—	12	2	no	—
269	C₂×C₄	—	12	2	no	—
270	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
271	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
272	C₂×C₄	C₃×#237	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	C₂²×C₃×Dic₃	1	1	yes	characteristic
276	C₈	—	6	2	no	—
277	C₈	C₃×C₄⋊Dic₃	1	4	yes	complemented
278	C₂×C₃	—	2	2	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	#4855	1	1	yes	characteristic
284	C₂×C₃	#4858	1	1	yes	characteristic
285	C₂×C₃	#4864	1	2	yes	—
286	C₂×C₃	#4984	1	1	yes	characteristic
287	C₂×C₃	Dic₃×SD₁₆	1	2	yes	—
288	C₂×C₃	C₃×#389	1	1	yes	characteristic
289	C₂×C₃	C₃×C₄×SD₁₆	1	2	yes	—
290	C₂×C₃	C₃×#465	1	2	yes	—
291	C₂×C₃	C₃×#498	1	1	yes	characteristic
292	C₂×C₃	C₃×#499	1	1	yes	characteristic
293	C₂²	C₃×#888	1	1	yes	characteristic
294	C₂²	C₃×#895	1	1	yes	characteristic
295	C₂²	C₃×D₃×SD₁₆	1	2	yes	—
296	C₂²	C₃×#916	1	2	yes	—
297	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
298	C₄	—	4	4	no	—
299	C₄	—	6	4	no	—
300	C₄	—	24	2	no	—
301	C₄	C₃×#890	1	1	yes	characteristic
302	C₄	C₃×#897	1	1	yes	characteristic
303	C₄	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
304	C₄	C₃×Q₈×Dic₃	1	1	yes	characteristic
305	C₃	—	2	1	no	—
306	C₃	#70560	1	1	yes	characteristic, direct product factor
307	C₃	C₃×#1941	1	1	yes	characteristic, complemented
308	C₂	C₃×#4855	1	1	yes	characteristic
309	C₂	C₃×#4858	1	1	yes	characteristic
310	C₂	C₃×#4864	1	2	yes	—
311	C₂	C₃×#4984	1	1	yes	characteristic
312	C₂	C₃×Dic₃×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#106	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#106	D₃
hypocenter	C₃	C₃×#1941
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#106

Complements

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

N	H ≅ G/N	type
7. #70560	C₃	direct (N × H)
14. C₃²×#106	C₄	semidirect
63. C₃×#106	Dic₃	semidirect
64. C₃×#106	C₃×C₄	semidirect
86. C₃²×C₈	C₄⋊C₄	semidirect
153. #106	C₃×Dic₃	semidirect
200. C₃×C₈	C₄⋊Dic₃	semidirect
201. C₃×C₈	C₃×C₄⋊C₄	semidirect
259. C₃²	#1941	semidirect
277. C₈	C₃×C₄⋊Dic₃	semidirect
306. C₃	#70560	direct (N × H)
307. C₃	C₃×#1941	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#106	2
C₂×C₃×#817	2
C₂×C₃×C₈×Dic₃	2
C₃×#4984	2
C₃×#4529	2
C₃×#1941	3
#70560	3

Subgroups — C3×#70560 #11790100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70560 #11790100