Subgroups — C₃×#68608 #11786136

Order 1152.

451 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4932	C₂	1	1	yes	maximal, characteristic
2	C₃²×#336	C₂	1	1	yes	maximal, characteristic
3	C₃×#4550	C₂	1	2	yes	maximal
4	C₃×#4550	C₂	1	2	yes	maximal
5	C₃×#4524	C₂	1	1	yes	maximal, characteristic
6	C₃×#1840	—	3	1	no	maximal
7	#68608	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	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃×C₃⋊#95	C₄	1	2	yes	—
14	C₃×C₃⋊#95	C₄	1	2	yes	—
15	C₃×#376	—	3	1	no	—
16	#4932	C₂×C₃	1	1	yes	characteristic
17	C₃×#336	—	2	1	no	—
18	C₃×#336	—	3	2	no	—
19	C₃×#336	—	3	2	no	—
20	C₃×#336	—	3	1	no	—
21	C₃×#336	D₃	1	1	yes	characteristic
22	C₃×#336	C₂×C₃	1	1	yes	characteristic
23	#4550	C₂×C₃	1	2	yes	—
24	#4550	C₂×C₃	1	2	yes	—
25	#4524	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₃²×C₄²	—	2	2	no	—
35	C₃×C₄×Dic₃	—	2	2	no	—
36	C₃×C₄×Dic₃	—	2	1	no	—
37	C₃×C₄×Dic₃	—	2	1	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
41	#1840	—	3	1	no	Sylow
42	C₂²×C₃×C₈	—	2	2	no	—
43	C₂²×C₃×C₈	—	3	2	no	—
44	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
45	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
46	C₂×C₃×C₄²	—	2	1	no	—
47	C₂×C₃×C₄²	—	3	1	no	—
48	C₂×C₃×C₄²	—	3	1	no	—
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
54	C₃×#95	—	3	2	no	—
55	C₃×#95	—	3	2	no	—
56	C₃⋊#95	C₃×C₄	1	2	yes	—
57	C₃⋊#95	C₃×C₄	1	2	yes	—
58	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	2	no	—
60	C₂×C₃²×C₄	—	2	2	no	—
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	#41	1	2	yes	—
63	C₂×C₃²×C₄	#41	1	2	yes	—
64	C₂×C₃²×C₄	#41	1	1	yes	characteristic
65	C₂×C₃²×C₄	#41	1	1	yes	characteristic
66	C₂×C₃×Dic₃	—	2	1	no	—
67	C₂×C₃×Dic₃	—	2	1	no	—
68	C₂×C₃×Dic₃	—	2	2	no	—
69	C₂×C₃×Dic₃	—	2	2	no	—
70	C₂×C₃×Dic₃	—	2	2	no	—
71	C₂×C₃×Dic₃	—	2	2	no	—
72	C₂×C₃×Dic₃	—	2	2	no	—
73	C₃²×C₈	—	4	2	no	—
74	C₃×C₃⋊C₈	—	4	2	no	—
75	#376	—	3	1	no	—
76	#336	—	3	2	no	—
77	#336	—	3	2	no	—
78	#336	—	3	1	no	—
79	#336	C₃×D₃	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	2	2	no	—
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	—	3	1	no	—
83	C₂²×C₃×C₄	—	3	1	no	—
84	C₂²×C₃×C₄	—	3	2	no	—
85	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
86	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
87	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
89	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
90	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	4	2	no	—
100	C₂×C₃×C₈	—	4	2	no	—
101	C₂×C₃×C₈	—	6	2	no	—
102	C₂×C₃×C₈	—	6	2	no	—
103	C₂×C₃×C₈	—	6	2	no	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	—	4	2	no	—
107	C₃×C₄²	—	6	2	no	—
108	C₃×C₄²	—	6	1	no	—
109	C₃×C₄²	—	6	1	no	—
110	C₄×Dic₃	—	2	2	no	—
111	C₄×Dic₃	—	2	1	no	—
112	C₄×Dic₃	—	2	1	no	—
113	C₂×C₃⋊C₈	—	2	2	no	—
114	C₂×C₃⋊C₈	—	2	2	no	—
115	C₂×C₃⋊C₈	—	2	2	no	—
116	C₂²×C₃²	#117	1	1	yes	characteristic
117	C₂²×C₃²	#117	1	1	yes	characteristic
118	C₂²×C₃²	#117	1	1	yes	characteristic
119	C₂²×C₃²	#131	1	2	yes	—
120	C₂²×C₃²	#131	1	2	yes	—
121	C₃²×C₄	—	4	2	no	—
122	C₃²×C₄	#126	1	2	yes	—
123	C₃²×C₄	#126	1	2	yes	—
124	C₃×Dic₃	—	4	1	no	—
125	C₃×Dic₃	—	4	1	no	—
126	C₃×Dic₃	—	4	2	no	—
127	C₂²×C₈	—	3	2	no	—
128	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
129	C₂×C₄²	—	3	1	no	—
130	C₂×C₄²	—	3	1	no	—
131	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
132	#95	—	3	2	no	—
133	#95	—	3	2	no	—
134	C₂³×C₃	—	2	1	no	—
135	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	—	6	1	no	—
151	C₂×C₃×C₄	—	6	1	no	—
152	C₂×C₃×C₄	—	6	2	no	—
153	C₂×C₃×C₄	—	6	2	no	—
154	C₂×C₃×C₄	—	6	2	no	—
155	C₂×C₃×C₄	—	6	2	no	—
156	C₂×C₃×C₄	—	6	2	no	—
157	C₂×C₃×C₄	#235	1	2	yes	—
158	C₂×C₃×C₄	#237	1	1	yes	characteristic
159	C₂×C₃×C₄	#237	1	2	yes	—
160	C₂×C₃×C₄	#239	1	1	yes	characteristic
161	C₂×C₃×C₄	C₃×#41	1	2	yes	—
162	C₂×C₃×C₄	C₃×#41	1	2	yes	—
163	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
164	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
165	C₂×Dic₃	—	2	2	no	—
166	C₂×Dic₃	—	2	1	no	—
167	C₂×Dic₃	—	2	1	no	—
168	C₂×Dic₃	—	2	2	no	—
169	C₂×Dic₃	—	2	2	no	—
170	C₂×Dic₃	—	2	2	no	—
171	C₂×Dic₃	—	2	2	no	—
172	C₃×C₈	—	4	4	no	—
173	C₃×C₈	—	4	2	no	—
174	C₃×C₈	—	4	2	no	—
175	C₃×C₈	—	12	2	no	—
176	C₃⋊C₈	—	4	2	no	—
177	C₂×C₃²	#383	1	1	yes	characteristic
178	C₂×C₃²	#433	1	1	yes	characteristic
179	C₂×C₃²	#433	1	1	yes	characteristic
180	C₂×C₃²	#433	1	2	yes	—
181	C₂×C₃²	#433	1	2	yes	—
182	C₂²×C₄	—	3	2	no	—
183	C₂²×C₄	—	3	1	no	—
184	C₂²×C₄	—	3	1	no	—
185	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
186	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	6	2	no	—
191	C₂×C₈	—	6	2	no	—
192	C₂×C₈	—	6	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	6	2	no	—
195	C₄²	—	6	1	no	—
196	C₄²	—	6	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	#872	1	1	yes	characteristic
203	C₂²×C₃	#877	1	1	yes	characteristic
204	C₂²×C₃	#892	1	1	yes	characteristic
205	C₂²×C₃	#901	1	2	yes	—
206	C₂²×C₃	#907	1	2	yes	—
207	C₂²×C₃	C₃×#117	1	1	yes	characteristic
208	C₂²×C₃	C₃×#117	1	1	yes	characteristic
209	C₂²×C₃	C₃×#117	1	1	yes	characteristic
210	C₂²×C₃	C₃×#131	1	2	yes	—
211	C₂²×C₃	C₃×#131	1	2	yes	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	4	no	—
215	C₃×C₄	—	4	2	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	—	12	1	no	—
218	C₃×C₄	—	12	1	no	—
219	C₃×C₄	—	12	2	no	—
220	C₃×C₄	#879	1	2	yes	—
221	C₃×C₄	#898	1	2	yes	—
222	C₃×C₄	C₃×#126	1	2	yes	—
223	C₃×C₄	C₃×#126	1	2	yes	—
224	Dic₃	—	4	1	no	—
225	Dic₃	—	4	1	no	—
226	Dic₃	—	4	2	no	—
227	C₃²	#1840	1	1	yes	characteristic, complemented, Sylow
228	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	—	6	2	no	—
234	C₂×C₄	—	6	2	no	—
235	C₂×C₄	—	6	2	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	6	1	no	—
238	C₂×C₄	—	6	1	no	—
239	C₂×C₄	C₃×#235	1	2	yes	—
240	C₂×C₄	C₃×#237	1	1	yes	characteristic
241	C₂×C₄	C₃×#237	1	2	yes	—
242	C₂×C₄	C₃×#239	1	1	yes	characteristic
243	C₈	—	4	2	no	—
244	C₈	—	12	2	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	2	no	—
249	C₂×C₃	—	2	2	no	—
250	C₂×C₃	#4651	1	1	yes	characteristic
251	C₂×C₃	#4690	1	1	yes	characteristic
252	C₂×C₃	#4709	1	1	yes	characteristic
253	C₂×C₃	#4720	1	2	yes	—
254	C₂×C₃	#4840	1	2	yes	—
255	C₂×C₃	C₃×#383	1	1	yes	characteristic
256	C₂×C₃	C₃×#433	1	1	yes	characteristic
257	C₂×C₃	C₃×#433	1	1	yes	characteristic
258	C₂×C₃	C₃×#433	1	2	yes	—
259	C₂×C₃	C₃×#433	1	2	yes	—
260	C₂²	C₃×#872	1	1	yes	characteristic
261	C₂²	C₃×#877	1	1	yes	characteristic
262	C₂²	C₃×#892	1	1	yes	characteristic
263	C₂²	C₃×#901	1	2	yes	—
264	C₂²	C₃×#907	1	2	yes	—
265	C₄	—	4	2	no	—
266	C₄	—	12	1	no	—
267	C₄	—	12	1	no	—
268	C₄	—	12	2	no	—
269	C₄	C₃×#879	1	2	yes	—
270	C₄	C₃×#898	1	2	yes	—
271	C₃	—	2	1	no	—
272	C₃	#68608	1	1	yes	characteristic, direct product factor
273	C₃	C₃×#1840	1	1	yes	characteristic, complemented
274	C₂	C₃×#4651	1	1	yes	characteristic
275	C₂	C₃×#4690	1	1	yes	characteristic
276	C₂	C₃×#4709	1	1	yes	characteristic
277	C₂	C₃×#4720	1	2	yes	—
278	C₂	C₃×#4840	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₃×C₄
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×#1840
LCS	G ▷ C₂³×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. #68608	C₃	direct (N × H)
227. C₃²	#1840	semidirect
272. C₃	#68608	direct (N × H)
273. C₃	C₃×#1840	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4932	2
C₃²×#336	2
C₃×#4550	2
C₃×#4524	2
C₃×#1840	3
#68608	3

Subgroups — C3×#68608 #11786136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68608 #11786136