Subgroups — C₃×#68596 #11783467

Order 1152.

651 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Dic₃	C₂	1	2	yes	maximal
2	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃²×#336	C₂	1	1	yes	maximal, characteristic
4	C₃×#4550	C₂	1	2	yes	maximal
5	C₃×#4524	C₂	1	1	yes	maximal, characteristic
6	C₃×#1744	—	3	1	no	maximal
7	#68596	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃×C₈×Dic₃	C₄	1	8	yes	complemented
16	C₂×C₃×C₄×C₈	—	3	2	no	—
17	C₃×C₄³	—	3	1	no	—
18	C₂×C₈×Dic₃	C₂×C₃	1	2	yes	—
19	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic
20	C₃×#336	—	2	1	no	—
21	C₃×#336	—	3	2	no	—
22	C₃×#336	—	3	1	no	—
23	C₃×#336	D₃	1	1	yes	characteristic
24	C₃×#336	C₂×C₃	1	1	yes	characteristic
25	#4550	C₂×C₃	1	2	yes	—
26	#4524	C₂×C₃	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
30	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
35	C₃²×C₄²	—	2	2	no	—
36	C₃×C₄×Dic₃	—	2	1	no	—
37	C₃×C₄×Dic₃	—	2	1	no	—
38	C₃×C₄×Dic₃	—	2	4	no	—
39	C₃×C₄×Dic₃	C₈	1	8	yes	complemented
40	C₃×C₄×Dic₃	C₂×C₄	1	4	yes	—
41	C₂×C₃×C₃⋊C₈	—	2	2	no	—
42	C₂×C₃×C₃⋊C₈	—	2	2	no	—
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
44	#1744	—	3	1	no	Sylow
45	C₂²×C₃×C₈	—	2	2	no	—
46	C₂²×C₃×C₈	—	3	2	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
48	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
49	C₂×C₃×C₄²	—	2	1	no	—
50	C₂×C₃×C₄²	—	3	2	no	—
51	C₂×C₃×C₄²	—	3	1	no	—
52	C₂×C₃×C₄²	—	3	2	no	—
53	C₂×C₃×C₄²	—	3	1	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄×Dic₃	C₃×C₄	1	2	yes	—
57	C₂×C₄×Dic₃	C₃×C₄	1	2	yes	—
58	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
61	C₃×C₄×C₈	—	3	8	no	—
62	C₈×Dic₃	C₃×C₄	1	8	yes	complemented
63	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	2	2	no	—
66	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
67	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
68	C₂×C₃²×C₄	#41	1	2	yes	—
69	C₂×C₃²×C₄	#41	1	1	yes	characteristic
70	C₂×C₃²×C₄	#41	1	1	yes	characteristic
71	C₂×C₃×Dic₃	—	2	2	no	—
72	C₂×C₃×Dic₃	—	2	2	no	—
73	C₂×C₃×Dic₃	C₄²	1	2	yes	—
74	C₂×C₃×Dic₃	C₄²	1	2	yes	—
75	C₂×C₃×Dic₃	C₂×C₈	1	4	yes	—
76	C₂×C₃×Dic₃	C₂×C₈	1	4	yes	—
77	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
78	C₃²×C₈	—	2	4	no	—
79	C₃×C₃⋊C₈	—	2	4	no	—
80	C₂×C₄×C₈	—	3	2	no	—
81	C₄³	—	3	1	no	—
82	#336	—	3	2	no	—
83	#336	—	3	1	no	—
84	#336	C₃×D₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂²×C₃×C₄	—	3	1	no	—
89	C₂²×C₃×C₄	—	3	2	no	—
90	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
91	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
92	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
95	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	4	no	—
102	C₂×C₃×C₈	—	3	4	no	—
103	C₂×C₃×C₈	—	4	2	no	—
104	C₂×C₃×C₈	—	4	2	no	—
105	C₂×C₃×C₈	—	6	2	no	—
106	C₂×C₃×C₈	—	6	2	no	—
107	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
108	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
109	C₃×C₄²	—	2	2	no	—
110	C₃×C₄²	—	2	2	no	—
111	C₃×C₄²	—	3	8	no	—
112	C₃×C₄²	—	3	4	no	—
113	C₃×C₄²	—	4	2	no	—
114	C₃×C₄²	—	6	4	no	—
115	C₃×C₄²	—	6	1	no	—
116	C₃×C₄²	—	6	1	no	—
117	C₄×Dic₃	—	2	1	no	—
118	C₄×Dic₃	—	2	1	no	—
119	C₄×Dic₃	—	2	4	no	—
120	C₄×Dic₃	C₃×C₈	1	8	yes	complemented
121	C₄×Dic₃	C₂×C₃×C₄	1	4	yes	—
122	C₂×C₃⋊C₈	—	2	2	no	—
123	C₂×C₃⋊C₈	—	2	2	no	—
124	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
125	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
126	C₂²×C₃²	#117	1	1	yes	characteristic
127	C₂²×C₃²	#117	1	1	yes	characteristic
128	C₂²×C₃²	C₂²×C₈	1	2	yes	—
129	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
130	C₃²×C₄	—	2	4	no	—
131	C₃²×C₄	#117	1	2	yes	—
132	C₃²×C₄	#117	1	2	yes	—
133	C₃×Dic₃	—	2	4	no	—
134	C₃×Dic₃	C₄×C₈	1	4	yes	—
135	C₃×Dic₃	#97	1	4	yes	—
136	C₂²×C₈	—	3	2	no	—
137	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
138	C₂×C₄²	—	3	2	no	—
139	C₂×C₄²	—	3	1	no	—
140	C₂×C₄²	—	3	2	no	—
141	C₂×C₄²	—	3	1	no	—
142	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
143	C₄×C₈	—	3	8	no	—
144	C₂³×C₃	—	2	1	no	—
145	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
146	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	3	2	no	—
157	C₂×C₃×C₄	—	3	2	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	3	4	no	—
160	C₂×C₃×C₄	—	3	4	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	6	2	no	—
164	C₂×C₃×C₄	—	6	2	no	—
165	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
166	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
167	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
168	C₂×C₃×C₄	#237	1	1	yes	characteristic
169	C₂×C₃×C₄	#237	1	2	yes	—
170	C₂×C₃×C₄	#239	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
172	C₂×C₃×C₄	C₃×#41	1	2	yes	—
173	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
174	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
175	C₂×Dic₃	—	2	2	no	—
176	C₂×Dic₃	—	2	2	no	—
177	C₂×Dic₃	C₃×C₄²	1	2	yes	—
178	C₂×Dic₃	C₃×C₄²	1	2	yes	—
179	C₂×Dic₃	C₂×C₃×C₈	1	4	yes	—
180	C₂×Dic₃	C₂×C₃×C₈	1	4	yes	—
181	C₂×Dic₃	C₃×M₁₆	1	4	yes	—
182	C₃×C₈	—	2	4	no	—
183	C₃×C₈	—	2	4	no	—
184	C₃×C₈	—	4	4	no	—
185	C₃×C₈	—	6	4	no	—
186	C₃⋊C₈	—	2	4	no	—
187	C₂×C₃²	#376	1	1	yes	characteristic
188	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
189	C₂×C₃²	C₂×#97	1	1	yes	characteristic
190	C₂×C₃²	#431	1	2	yes	—
191	C₂×C₃²	#431	1	2	yes	—
192	C₂²×C₄	—	3	1	no	—
193	C₂²×C₄	—	3	1	no	—
194	C₂²×C₄	—	3	2	no	—
195	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
196	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	3	4	no	—
200	C₂×C₈	—	6	2	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
203	C₄²	—	2	2	no	—
204	C₄²	—	3	8	no	—
205	C₄²	—	3	4	no	—
206	C₄²	—	6	4	no	—
207	C₄²	—	6	1	no	—
208	C₄²	—	6	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
215	C₂²×C₃	#877	1	1	yes	characteristic
216	C₂²×C₃	#892	1	1	yes	characteristic
217	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
218	C₂²×C₃	D₃×M₁₆	1	2	yes	—
219	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
220	C₂²×C₃	C₃×#117	1	1	yes	characteristic
221	C₂²×C₃	C₃×#117	1	1	yes	characteristic
222	C₂²×C₃	C₂²×C₃×C₈	1	2	yes	—
223	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	3	4	no	—
229	C₃×C₄	—	3	4	no	—
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	—	6	4	no	—
232	C₃×C₄	#877	1	2	yes	—
233	C₃×C₄	#892	1	2	yes	—
234	C₃×C₄	C₃×#117	1	2	yes	—
235	C₃×C₄	C₃×#117	1	2	yes	—
236	Dic₃	—	2	4	no	—
237	Dic₃	C₃×C₄×C₈	1	4	yes	—
238	Dic₃	C₃×#97	1	4	yes	—
239	C₃²	#1744	1	1	yes	characteristic, complemented, Sylow
240	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	3	4	no	—
244	C₂×C₄	—	3	4	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	4	no	—
248	C₂×C₄	—	6	2	no	—
249	C₂×C₄	—	6	2	no	—
250	C₂×C₄	C₃×D₃×C₈	1	4	yes	—
251	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
252	C₂×C₄	C₃×#237	1	1	yes	characteristic
253	C₂×C₄	C₃×#237	1	2	yes	—
254	C₂×C₄	C₃×#239	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	6	4	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	#4648	1	1	yes	characteristic
263	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
264	C₂×C₃	D₃×#97	1	1	yes	characteristic
265	C₂×C₃	#4718	1	2	yes	—
266	C₂×C₃	#4833	1	2	yes	—
267	C₂×C₃	C₃×#376	1	1	yes	characteristic
268	C₂×C₃	C₂×C₃×C₄×C₈	1	1	yes	characteristic
269	C₂×C₃	C₂×C₃×#97	1	1	yes	characteristic
270	C₂×C₃	C₃×#431	1	2	yes	—
271	C₂×C₃	C₃×#431	1	2	yes	—
272	C₂²	C₃×C₄²×D₃	1	1	yes	characteristic
273	C₂²	C₃×#877	1	1	yes	characteristic
274	C₂²	C₃×#892	1	1	yes	characteristic
275	C₂²	C₂×C₃×D₃×C₈	1	2	yes	—
276	C₂²	C₃×D₃×M₁₆	1	2	yes	—
277	C₄	—	2	4	no	—
278	C₄	—	3	4	no	—
279	C₄	—	3	4	no	—
280	C₄	—	6	4	no	—
281	C₄	C₃×#877	1	2	yes	—
282	C₄	C₃×#892	1	2	yes	—
283	C₃	—	2	1	no	—
284	C₃	#68596	1	1	yes	characteristic, direct product factor
285	C₃	C₃×#1744	1	1	yes	characteristic, complemented
286	C₂	C₃×#4648	1	1	yes	characteristic
287	C₂	C₃×C₄×D₃×C₈	1	1	yes	characteristic
288	C₂	C₃×D₃×#97	1	1	yes	characteristic
289	C₂	C₃×#4718	1	2	yes	—
290	C₂	C₃×#4833	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₃×C₄×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#336	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#336	D₃
hypocenter	C₃	C₃×#1744
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×#336

Complements

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

N	H ≅ G/N	type
7. #68596	C₃	direct (N × H)
15. C₃×C₈×Dic₃	C₄	semidirect
39. C₃×C₄×Dic₃	C₈	semidirect
62. C₈×Dic₃	C₃×C₄	semidirect
120. C₄×Dic₃	C₃×C₈	semidirect
239. C₃²	#1744	semidirect
284. C₃	#68596	direct (N × H)
285. C₃	C₃×#1744	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Dic₃	2
C₃×C₄²×Dic₃	2
C₃²×#336	2
C₃×#4550	2
C₃×#4524	2
C₃×#1744	3
#68596	3

Subgroups — C3×#68596 #11783467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68596 #11783467