Subgroups — C₃×#67620 #11771202

Order 1152.

449 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#4924	C₂	1	1	yes	maximal, characteristic
4	C₃×#1303	—	3	1	no	maximal
5	#67620	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
7	C₂×C₃²×C₄²	—	2	1	no	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃×#208	—	2	1	no	—
10	C₃²×#95	C₄	1	2	yes	—
11	C₃×#830	—	2	1	no	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
14	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
15	C₃×C₄×C₃⋊C₈	—	2	1	no	—
16	C₃×#431	—	3	1	no	—
17	C₃×#423	—	3	1	no	—
18	C₃×C₄×C₄⋊C₄	—	2	1	no	—
19	C₃×C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
20	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
21	#4925	C₂×C₃	1	1	yes	characteristic
22	#4924	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	—	2	2	no	—
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₃²×C₄⋊C₄	—	2	4	no	—
28	C₃²×C₄²	—	4	1	no	—
29	C₃²×C₄²	—	4	1	no	—
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃²×C₄²	D₄	1	1	yes	characteristic
33	C₃²×C₄²	Q₈	1	1	yes	characteristic
34	C₃×#208	—	4	1	no	—
35	C₃×#208	—	4	1	no	—
36	C₂×C₃×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃×C₃⋊C₈	—	2	2	no	—
38	#1303	—	3	1	no	Sylow
39	C₂×C₃×M₁₆	—	6	1	no	—
40	C₂×C₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₃×C₄²	—	2	2	no	—
44	C₂×C₃×C₄²	—	2	1	no	—
45	C₂×C₃×C₄²	—	2	1	no	—
46	C₂×C₃×C₄²	—	2	1	no	—
47	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₂×#208	—	2	1	no	—
50	C₃×C₄⋊C₈	—	3	2	no	—
51	C₃×C₄⋊C₈	—	3	2	no	—
52	C₃×C₄⋊C₈	—	6	1	no	—
53	C₃×C₂²⋊C₈	—	6	1	no	—
54	C₃×C₄×C₈	—	6	1	no	—
55	C₃×#95	—	2	2	no	—
56	C₃×#95	Dic₃	1	2	yes	—
57	C₃×#95	C₃×C₄	1	2	yes	—
58	#830	—	2	1	no	—
59	C₄⋊(C₃⋊C₈)	—	2	1	no	—
60	C₄⋊(C₃⋊C₈)	C₃×C₄	1	2	yes	complemented
61	C₄⋊(C₃⋊C₈)	C₃×C₄	1	2	yes	complemented
62	C₄×C₃⋊C₈	—	2	1	no	—
63	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	2	4	no	—
66	C₂×C₃²×C₄	—	4	1	no	—
67	C₂×C₃²×C₄	—	4	2	no	—
68	C₂×C₃²×C₄	—	4	2	no	—
69	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
77	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
78	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
79	C₃×C₃⋊C₈	—	4	2	no	—
80	C₃×C₃⋊C₈	—	4	2	no	—
81	#431	—	3	1	no	—
82	#423	—	3	1	no	—
83	C₄×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	#67	1	1	yes	characteristic
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
96	C₃×M₁₆	—	12	1	no	—
97	C₃×M₁₆	—	12	1	no	—
98	C₂×C₃×C₈	—	6	2	no	—
99	C₂×C₃×C₈	—	6	2	no	—
100	C₃×C₄⋊C₄	—	2	8	no	—
101	C₃×C₄⋊C₄	—	2	4	no	—
102	C₃×C₄⋊C₄	—	2	4	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	—	2	1	no	—
106	C₃×C₄²	—	2	1	no	—
107	C₃×C₄²	—	4	2	no	—
108	C₃×C₄²	—	4	2	no	—
109	C₃×C₄²	—	4	1	no	—
110	C₃×C₄²	—	4	1	no	—
111	C₃×C₄²	—	4	1	no	—
112	C₃×C₄²	—	4	1	no	—
113	C₃×C₄²	Dic₆	1	1	yes	characteristic
114	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
115	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
116	C₃×C₄²	D₁₂	1	1	yes	characteristic
117	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
118	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
119	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
120	C₃×C₄²	C₃×Q₈	1	1	yes	characteristic
121	#208	—	4	1	no	—
122	#208	—	4	1	no	—
123	C₂×C₃⋊C₈	—	2	2	no	—
124	C₂×C₃⋊C₈	—	2	2	no	—
125	C₂²×C₃²	—	2	1	no	—
126	C₂²×C₃²	—	2	1	no	—
127	C₂²×C₃²	#95	1	1	yes	characteristic
128	C₂²×C₃²	#102	1	1	yes	characteristic
129	C₂²×C₃²	#103	1	1	yes	characteristic
130	C₃²×C₄	—	2	1	no	—
131	C₃²×C₄	—	2	1	no	—
132	C₃²×C₄	—	2	1	no	—
133	C₃²×C₄	—	4	4	no	—
134	C₃²×C₄	#102	1	1	yes	characteristic
135	C₃²×C₄	#103	1	1	yes	characteristic
136	C₃²×C₄	#104	1	2	yes	—
137	C₃²×C₄	#106	1	1	yes	characteristic
138	C₃²×C₄	#107	1	1	yes	characteristic
139	C₂×M₁₆	—	6	1	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
143	C₄⋊C₈	—	3	2	no	—
144	C₄⋊C₈	—	3	2	no	—
145	C₄⋊C₈	—	6	1	no	—
146	C₂²⋊C₈	—	6	1	no	—
147	C₄×C₈	—	6	1	no	—
148	#95	C₃×Dic₃	1	2	yes	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	#217	1	1	yes	characteristic
151	C₂³×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	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	—	2	1	no	—
168	C₂×C₃×C₄	—	4	1	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	—	4	4	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	4	4	no	—
173	C₂×C₃×C₄	—	4	2	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	—	4	2	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
178	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
181	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	#213	1	1	yes	characteristic
184	C₂×C₃×C₄	#214	1	1	yes	characteristic
185	C₂×C₃×C₄	#215	1	1	yes	characteristic
186	C₂×C₃×C₄	#216	1	1	yes	characteristic
187	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
188	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₃×D₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
195	C₂×C₃×C₄	C₃×SD₁₆	1	1	yes	characteristic
196	C₂×C₃×C₄	C₃×Q₁₆	1	1	yes	characteristic
197	C₃×C₈	—	12	2	no	—
198	C₃×C₈	—	12	2	no	—
199	C₃⋊C₈	—	4	2	no	—
200	C₃⋊C₈	—	4	2	no	—
201	C₂×C₃²	—	2	1	no	—
202	C₂×C₃²	—	2	1	no	—
203	C₂×C₃²	#339	1	1	yes	characteristic
204	C₂×C₃²	#340	1	1	yes	characteristic
205	C₂×C₃²	#344	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
208	C₂²×C₄	C₃×#67	1	1	yes	characteristic
209	C₂²×C₄	C₃×#67	1	1	yes	characteristic
210	M₁₆	—	12	1	no	—
211	M₁₆	—	12	1	no	—
212	C₂×C₈	—	6	2	no	—
213	C₂×C₈	—	6	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	1	no	—
217	C₄²	C₃×Dic₆	1	1	yes	characteristic
218	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
219	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
220	C₄²	C₃×D₁₂	1	1	yes	characteristic
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
231	C₂²×C₃	#832	1	1	yes	characteristic
232	C₂²×C₃	#835	1	1	yes	characteristic
233	C₂²×C₃	C₃×#95	1	1	yes	characteristic
234	C₂²×C₃	C₃×#102	1	1	yes	characteristic
235	C₂²×C₃	C₃×#103	1	1	yes	characteristic
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	4	1	no	—
249	C₃×C₄	—	4	1	no	—
250	C₃×C₄	—	4	8	no	—
251	C₃×C₄	—	4	4	no	—
252	C₃×C₄	—	4	4	no	—
253	C₃×C₄	#805	1	2	yes	—
254	C₃×C₄	#807	1	1	yes	characteristic
255	C₃×C₄	#808	1	1	yes	characteristic
256	C₃×C₄	#809	1	1	yes	characteristic
257	C₃×C₄	#810	1	1	yes	characteristic
258	C₃×C₄	C₃×#102	1	1	yes	characteristic
259	C₃×C₄	C₃×#103	1	1	yes	characteristic
260	C₃×C₄	C₃×#104	1	2	yes	—
261	C₃×C₄	C₃×#106	1	1	yes	characteristic
262	C₃×C₄	C₃×#107	1	1	yes	characteristic
263	C₃²	#1303	1	1	yes	characteristic, complemented, Sylow
264	C₂³	C₃×#217	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
271	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
274	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₃×#213	1	1	yes	characteristic
277	C₂×C₄	C₃×#214	1	1	yes	characteristic
278	C₂×C₄	C₃×#215	1	1	yes	characteristic
279	C₂×C₄	C₃×#216	1	1	yes	characteristic
280	C₈	—	12	2	no	—
281	C₈	—	12	2	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	2	2	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	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	#4523	1	1	yes	characteristic
292	C₂×C₃	#4529	1	1	yes	characteristic
293	C₂×C₃	#4533	1	1	yes	characteristic
294	C₂×C₃	C₃×#339	1	1	yes	characteristic
295	C₂×C₃	C₃×#340	1	1	yes	characteristic
296	C₂×C₃	C₃×#344	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	C₃×C₃⋊#95	1	1	yes	characteristic
300	C₂²	C₃×#832	1	1	yes	characteristic
301	C₂²	C₃×#835	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	4	no	—
306	C₄	C₃×#805	1	2	yes	—
307	C₄	C₃×#807	1	1	yes	characteristic
308	C₄	C₃×#808	1	1	yes	characteristic
309	C₄	C₃×#809	1	1	yes	characteristic
310	C₄	C₃×#810	1	1	yes	characteristic
311	C₃	—	2	1	no	—
312	C₃	#67620	1	1	yes	characteristic, direct product factor
313	C₃	C₃×#1303	1	1	yes	characteristic, complemented
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	C₃×#4523	1	1	yes	characteristic
317	C₂	C₃×#4529	1	1	yes	characteristic
318	C₂	C₃×#4533	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₃×C₄²
Frattini	C₂×C₄²	C₂×C₃×D₃
Fitting	C₃²×C₄×C₄⋊C₄	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂²×C₃
hypercenter	C₃×C₄×C₄⋊C₄	D₃
hypocenter	C₃	C₃×#1303
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
5. #67620	C₃	direct (N × H)
13. C₃×C₄⋊(C₃⋊C₈)	C₄	semidirect
14. C₃×C₄⋊(C₃⋊C₈)	C₄	semidirect
60. C₄⋊(C₃⋊C₈)	C₃×C₄	semidirect
61. C₄⋊(C₃⋊C₈)	C₃×C₄	semidirect
263. C₃²	#1303	semidirect
312. C₃	#67620	direct (N × H)
313. C₃	C₃×#1303	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₄⋊C₄	2
C₃×#4925	2
C₃×#4924	2
C₃×#1303	3
#67620	3

Subgroups — C3×#67620 #11771202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67620 #11771202