Subgroups — C₃×#68532 #11776972

Order 1152.

363 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4837	C₂	1	1	yes	maximal, characteristic
2	C₃×D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#4685	C₂	1	1	yes	maximal, characteristic
4	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4457	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1570	—	3	1	no	maximal
9	#68532	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
11	C₂×C₃×D₃×C₈	—	2	1	no	—
12	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
13	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
17	C₃×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
18	C₃×Dic₃⋊C₈	—	2	1	no	—
19	C₃×Dic₃⋊C₈	—	2	1	no	—
20	C₃×Dic₃⋊C₈	C₄	1	1	yes	characteristic
21	C₃×Dic₃⋊C₈	C₄	1	1	yes	characteristic
22	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₃×#431	—	3	1	no	—
26	C₃×#423	—	3	1	no	—
27	C₂×C₃×C₄⋊C₈	—	3	1	no	—
28	#4837	C₂×C₃	1	1	yes	characteristic
29	D₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
30	#4685	C₂×C₃	1	1	yes	characteristic
31	C₃×#334	—	2	1	no	—
32	C₃×#334	—	3	1	no	—
33	C₃×#334	—	3	1	no	—
34	C₃×#334	—	3	1	no	—
35	C₃×#334	D₃	1	1	yes	characteristic, complemented
36	C₃×#334	C₂×C₃	1	1	yes	characteristic, complemented
37	#4480	C₂×C₃	1	1	yes	characteristic, complemented
38	#4480	C₂×C₃	1	1	yes	characteristic, complemented
39	#4457	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₂×C₃²×C₈	—	2	1	no	—
45	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃²×C₄²	C₂³	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
50	C₃×C₄×Dic₃	Q₈	1	1	yes	characteristic
51	C₂×C₃×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×ΓL(1,𝔽₂₅)	—	4	2	no	—
56	C₃×D₃×C₈	—	4	2	no	—
57	#1570	—	3	1	no	Sylow
58	C₂×C₃×M₁₆	—	6	1	no	—
59	C₂²×C₃×C₈	—	6	1	no	—
60	C₂×C₃×C₄²	—	3	1	no	—
61	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
62	C₂×D₃×C₈	—	2	1	no	—
63	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
64	C₃×C₄⋊C₈	—	2	1	no	—
65	C₃×C₄⋊C₈	—	2	1	no	—
66	C₃×C₄⋊C₈	—	3	1	no	—
67	C₃×C₄⋊C₈	—	3	1	no	—
68	C₃×C₄⋊C₈	—	3	1	no	—
69	C₃×C₄⋊C₈	—	3	1	no	—
70	C₃×C₄⋊C₈	—	6	1	no	—
71	C₃×C₄⋊C₈	—	6	1	no	—
72	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
73	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×C₂²⋊C₈	—	3	1	no	—
77	C₃×C₂²⋊C₈	—	3	1	no	—
78	C₃×C₄×C₈	—	2	1	no	—
79	C₃×C₄×C₈	—	3	1	no	—
80	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
82	(C₂×D₃)⋊C₈	C₃×C₄	1	1	yes	characteristic
83	(C₂×D₃)⋊C₈	C₃×C₄	1	1	yes	characteristic
84	Dic₃⋊C₈	—	2	1	no	—
85	Dic₃⋊C₈	—	2	1	no	—
86	Dic₃⋊C₈	C₃×C₄	1	1	yes	characteristic
87	Dic₃⋊C₈	C₃×C₄	1	1	yes	characteristic
88	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
89	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
90	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
91	C₂²×C₃×D₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₃×Dic₃	C₄⋊C₄	1	1	yes	characteristic
98	C₃×C₄×D₃	—	2	1	no	—
99	C₃×C₄×D₃	—	2	1	no	—
100	C₃×C₄×D₃	—	4	1	no	—
101	C₃×C₄×D₃	SD₁₆	1	2	yes	—
102	C₃×C₄×D₃	SD₁₆	1	2	yes	—
103	C₃²×C₈	—	2	1	no	—
104	C₃²×C₈	—	2	1	no	—
105	C₃²×C₈	—	4	1	no	—
106	C₃²×C₈	—	4	1	no	—
107	C₃×C₃⋊C₈	—	2	1	no	—
108	C₃×C₃⋊C₈	—	2	1	no	—
109	C₃×C₃⋊C₈	—	4	1	no	—
110	C₃×C₃⋊C₈	—	4	1	no	—
111	#431	—	3	1	no	—
112	#423	—	3	1	no	—
113	C₂×C₄⋊C₈	—	3	1	no	—
114	#334	—	3	1	no	—
115	#334	—	3	1	no	—
116	#334	—	3	1	no	—
117	#334	C₃×D₃	1	1	yes	characteristic, complemented
118	C₂²×C₃×C₄	—	3	1	no	—
119	C₂²×C₃×C₄	—	3	1	no	—
120	C₂²×C₃×C₄	—	3	1	no	—
121	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
122	C₂×C₄×D₃	C₃×D₄	1	1	yes	characteristic
123	C₂×C₄×D₃	C₃×Q₈	1	1	yes	characteristic
124	C₃×M₁₆	—	12	2	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	3	1	no	—
132	C₂×C₃×C₈	—	3	1	no	—
133	C₂×C₃×C₈	—	4	1	no	—
134	C₂×C₃×C₈	—	4	1	no	—
135	C₂×C₃×C₈	—	6	1	no	—
136	C₂×C₃×C₈	—	6	1	no	—
137	C₂×C₃×C₈	—	12	2	no	—
138	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
140	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
142	C₃×C₄²	—	2	1	no	—
143	C₃×C₄²	—	3	1	no	—
144	C₃×C₄²	—	3	1	no	—
145	C₃×C₄²	—	3	1	no	—
146	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
147	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
148	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
149	C₄×Dic₃	C₃×D₄	1	1	yes	characteristic
150	C₄×Dic₃	C₃×Q₈	1	1	yes	characteristic
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	—	2	1	no	—
153	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
154	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
155	ΓL(1,𝔽₂₅)	—	4	2	no	—
156	D₃×C₈	—	4	2	no	—
157	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₃×D₃	—	2	1	no	—
159	C₂×C₃×D₃	—	2	1	no	—
160	C₂×C₃×D₃	#106	1	2	yes	—
161	C₃²×C₄	—	2	1	no	—
162	C₃²×C₄	#131	1	1	yes	characteristic
163	C₃²×C₄	#131	1	1	yes	characteristic
164	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
165	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
166	C₃×Dic₃	—	2	1	no	—
167	C₃×Dic₃	—	2	1	no	—
168	C₃×Dic₃	#106	1	2	yes	—
169	C₂×M₁₆	—	6	1	no	—
170	C₂²×C₈	—	6	1	no	—
171	C₂×C₄²	—	3	1	no	—
172	C₄⋊C₈	—	3	1	no	—
173	C₄⋊C₈	—	3	1	no	—
174	C₄⋊C₈	—	3	1	no	—
175	C₄⋊C₈	—	3	1	no	—
176	C₄⋊C₈	—	6	1	no	—
177	C₄⋊C₈	—	6	1	no	—
178	C₄⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
179	C₄⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
180	C₂²⋊C₈	—	3	1	no	—
181	C₂²⋊C₈	—	3	1	no	—
182	C₄×C₈	—	3	1	no	—
183	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
184	C₂³×C₃	—	3	1	no	—
185	C₂²×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	3	1	no	—
190	C₂×C₃×C₄	—	3	2	no	—
191	C₂×C₃×C₄	—	3	2	no	—
192	C₂×C₃×C₄	—	3	1	no	—
193	C₂×C₃×C₄	—	3	1	no	—
194	C₂×C₃×C₄	—	6	1	no	—
195	C₂×C₃×C₄	—	6	1	no	—
196	C₂×C₃×C₄	—	12	1	no	—
197	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
203	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
204	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
205	C₂×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
206	C₄×D₃	—	2	1	no	—
207	C₄×D₃	—	2	1	no	—
208	C₄×D₃	—	4	1	no	—
209	C₄×D₃	C₃×SD₁₆	1	2	yes	—
210	C₄×D₃	C₃×SD₁₆	1	2	yes	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃×C₈	—	4	1	no	—
218	C₃×C₈	—	4	1	no	—
219	C₃×C₈	—	4	1	no	—
220	C₃×C₈	—	4	1	no	—
221	C₃×C₈	—	6	1	no	—
222	C₃×C₈	—	6	1	no	—
223	C₃×C₈	—	8	1	no	—
224	C₃×C₈	—	8	1	no	—
225	C₃×C₈	—	12	1	no	—
226	C₃×C₈	—	12	1	no	—
227	C₃⋊C₈	—	2	1	no	—
228	C₃⋊C₈	—	2	1	no	—
229	C₃⋊C₈	—	4	1	no	—
230	C₃⋊C₈	—	4	1	no	—
231	C₂×C₃²	#424	1	1	yes	characteristic
232	C₂×C₃²	C₂×#106	1	1	yes	characteristic
233	C₂×C₃²	#430	1	1	yes	characteristic
234	C₃×D₃	—	2	2	no	—
235	C₂²×C₄	—	3	1	no	—
236	C₂²×C₄	—	3	1	no	—
237	C₂²×C₄	—	3	1	no	—
238	M₁₆	—	12	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	—	3	1	no	—
243	C₂×C₈	—	6	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	12	2	no	—
246	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
247	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
248	C₄²	—	3	1	no	—
249	C₄²	—	3	1	no	—
250	C₄²	—	3	1	no	—
251	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	3	2	no	—
254	C₂²×C₃	—	6	1	no	—
255	C₂²×C₃	—	6	1	no	—
256	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
257	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
258	C₂×D₃	—	2	1	no	—
259	C₂×D₃	—	2	1	no	—
260	C₂×D₃	C₃×#106	1	2	yes	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	3	2	no	—
268	C₃×C₄	—	4	1	no	—
269	C₃×C₄	—	6	1	no	—
270	C₃×C₄	—	6	1	no	—
271	C₃×C₄	#901	1	1	yes	characteristic
272	C₃×C₄	#907	1	1	yes	characteristic
273	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
274	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
275	C₃×C₄	C₃×#131	1	1	yes	characteristic
276	C₃×C₄	C₃×#131	1	1	yes	characteristic
277	C₃×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
278	C₃×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
279	Dic₃	—	2	1	no	—
280	Dic₃	—	2	1	no	—
281	Dic₃	C₃×#106	1	2	yes	—
282	C₃²	#1570	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	3	1	no	—
284	C₂×C₄	—	3	1	no	—
285	C₂×C₄	—	3	2	no	—
286	C₂×C₄	—	3	2	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	6	1	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	12	1	no	—
292	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
293	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
294	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	4	1	no	—
298	C₈	—	4	1	no	—
299	C₈	—	6	1	no	—
300	C₈	—	6	1	no	—
301	C₈	—	12	1	no	—
302	C₈	—	12	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	6	2	no	—
307	C₂×C₃	#4839	1	1	yes	characteristic
308	C₂×C₃	D₃×#106	1	1	yes	characteristic
309	C₂×C₃	#4893	1	1	yes	characteristic
310	C₂×C₃	C₃×#424	1	1	yes	characteristic
311	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
312	C₂×C₃	C₃×#430	1	1	yes	characteristic
313	D₃	—	2	2	no	—
314	C₂²	—	3	2	no	—
315	C₂²	—	6	1	no	—
316	C₂²	—	6	1	no	—
317	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	3	2	no	—
320	C₄	—	6	1	no	—
321	C₄	—	6	1	no	—
322	C₄	C₃×#901	1	1	yes	characteristic
323	C₄	C₃×#907	1	1	yes	characteristic
324	C₄	C₃×D₃×SD₁₆	1	1	yes	characteristic
325	C₄	C₃×D₃×SD₁₆	1	1	yes	characteristic
326	C₃	—	2	1	no	—
327	C₃	#68532	1	1	yes	characteristic, direct product factor
328	C₃	C₃×#1570	1	1	yes	characteristic, complemented
329	C₂	—	6	2	no	—
330	C₂	C₃×#4839	1	1	yes	characteristic
331	C₂	C₃×D₃×#106	1	1	yes	characteristic
332	C₂	C₃×#4893	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#334	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#334	D₃
hypocenter	C₃	C₃×#1570
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#334

Complements

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

N	H ≅ G/N	type
4. C₃²×#334	C₂	semidirect
5. C₃×#4480	C₂	semidirect
6. C₃×#4480	C₂	semidirect
7. C₃×#4457	C₂	semidirect
9. #68532	C₃	direct (N × H)
35. C₃×#334	D₃	semidirect
36. C₃×#334	C₂×C₃	semidirect
37. #4480	C₂×C₃	semidirect
38. #4480	C₂×C₃	semidirect
39. #4457	C₂×C₃	semidirect
117. #334	C₃×D₃	semidirect
282. C₃²	#1570	semidirect
327. C₃	#68532	direct (N × H)
328. C₃	C₃×#1570	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4837	2
C₃×D₃×C₄⋊C₈	2
C₃×#4685	2
C₃²×#334	2
C₃×#4480	2
C₃×#4457	2
C₃×#1570	3
#68532	3

Subgroups — C3×#68532 #11776972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68532 #11776972