Subgroups — C₃×#68606 #11785973

Order 1152.

491 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Dic₃⋊C₈	C₂	1	2	yes	maximal
2	C₃×#4932	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×#1832	—	3	1	no	maximal
7	#68606	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×Dic₃⋊C₈	—	2	4	no	—
16	C₂×C₃×C₄⋊C₈	—	3	2	no	—
17	C₃×#376	—	3	1	no	—
18	C₂×Dic₃⋊C₈	C₂×C₃	1	2	yes	—
19	#4932	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	1	yes	characteristic
30	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	—	2	2	no	—
35	C₃²×C₄²	—	2	2	no	—
36	C₃×C₄×Dic₃	—	2	2	no	—
37	C₃×C₄×Dic₃	D₄	1	2	yes	—
38	C₃×C₄×Dic₃	Q₈	1	2	yes	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	—	2	2	no	—
42	#1832	—	3	1	no	Sylow
43	C₂²×C₃×C₈	—	2	2	no	—
44	C₂²×C₃×C₈	—	3	2	no	—
45	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
46	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
47	C₂×C₃×C₄²	—	2	1	no	—
48	C₂×C₃×C₄²	—	3	1	no	—
49	C₂×C₃×C₄²	—	3	1	no	—
50	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
55	C₃×C₄⋊C₈	—	6	4	no	—
56	C₃×#95	—	3	2	no	—
57	C₃×#95	—	3	2	no	—
58	C₃⋊#95	C₃×C₄	1	2	yes	—
59	C₃⋊#95	C₃×C₄	1	2	yes	—
60	Dic₃⋊C₈	—	2	4	no	—
61	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	—	2	2	no	—
63	C₂×C₃²×C₄	—	2	2	no	—
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	#41	1	2	yes	—
68	C₂×C₃²×C₄	#41	1	2	yes	—
69	C₂×C₃×Dic₃	—	2	2	no	—
70	C₂×C₃×Dic₃	—	2	2	no	—
71	C₂×C₃×Dic₃	—	2	1	no	—
72	C₂×C₃×Dic₃	—	2	1	no	—
73	C₂×C₃×Dic₃	—	2	2	no	—
74	C₂×C₃×Dic₃	—	2	2	no	—
75	C₂×C₃×Dic₃	C₄⋊C₄	1	4	yes	—
76	C₃²×C₈	—	4	2	no	—
77	C₃×C₃⋊C₈	—	4	2	no	—
78	C₂×C₄⋊C₈	—	3	2	no	—
79	#376	—	3	1	no	—
80	#336	—	3	2	no	—
81	#336	—	3	1	no	—
82	#336	C₃×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	2	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	3	2	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
89	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂²×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₃×C₈	—	2	4	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	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	4	2	no	—
103	C₂×C₃×C₈	—	4	2	no	—
104	C₂×C₃×C₈	—	6	2	no	—
105	C₂×C₃×C₈	—	6	2	no	—
106	C₂×C₃×C₈	—	6	2	no	—
107	C₃×C₄²	—	2	2	no	—
108	C₃×C₄²	—	2	2	no	—
109	C₃×C₄²	—	3	2	no	—
110	C₃×C₄²	—	3	2	no	—
111	C₃×C₄²	—	4	2	no	—
112	C₃×C₄²	—	6	2	no	—
113	C₄×Dic₃	—	2	2	no	—
114	C₄×Dic₃	C₃×D₄	1	2	yes	—
115	C₄×Dic₃	C₃×Q₈	1	2	yes	—
116	C₂×C₃⋊C₈	—	2	2	no	—
117	C₂×C₃⋊C₈	—	2	2	no	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₃²	#117	1	1	yes	characteristic
121	C₂²×C₃²	#117	1	1	yes	characteristic
122	C₂²×C₃²	#131	1	2	yes	—
123	C₂²×C₃²	#131	1	2	yes	—
124	C₃²×C₄	—	4	2	no	—
125	C₃²×C₄	#124	1	2	yes	—
126	C₃²×C₄	#125	1	2	yes	—
127	C₃×Dic₃	—	2	4	no	—
128	C₃×Dic₃	—	4	1	no	—
129	C₃×Dic₃	—	4	1	no	—
130	C₂²×C₈	—	3	2	no	—
131	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
132	C₂×C₄²	—	3	1	no	—
133	C₂×C₄²	—	3	1	no	—
134	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
135	C₄⋊C₈	—	6	4	no	—
136	#95	—	3	2	no	—
137	#95	—	3	2	no	—
138	C₂³×C₃	—	2	1	no	—
139	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
140	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₂×C₃×C₄	—	2	4	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	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	3	4	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	6	2	no	—
156	C₂×C₃×C₄	—	6	2	no	—
157	C₂×C₃×C₄	—	6	1	no	—
158	C₂×C₃×C₄	—	6	1	no	—
159	C₂×C₃×C₄	—	6	2	no	—
160	C₂×C₃×C₄	—	6	2	no	—
161	C₂×C₃×C₄	#235	1	2	yes	—
162	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	#237	1	2	yes	—
164	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃×#41	1	2	yes	—
168	C₂×C₃×C₄	C₃×#41	1	2	yes	—
169	C₂×Dic₃	—	2	2	no	—
170	C₂×Dic₃	—	2	2	no	—
171	C₂×Dic₃	—	2	1	no	—
172	C₂×Dic₃	—	2	1	no	—
173	C₂×Dic₃	—	2	2	no	—
174	C₂×Dic₃	—	2	2	no	—
175	C₂×Dic₃	C₃×C₄⋊C₄	1	4	yes	—
176	C₃×C₈	—	4	4	no	—
177	C₃×C₈	—	4	2	no	—
178	C₃×C₈	—	4	2	no	—
179	C₃×C₈	—	12	2	no	—
180	C₃⋊C₈	—	4	2	no	—
181	C₂×C₃²	#382	1	1	yes	characteristic
182	C₂×C₃²	#424	1	2	yes	—
183	C₂×C₃²	#433	1	1	yes	characteristic
184	C₂×C₃²	#433	1	1	yes	characteristic
185	C₂×C₃²	#433	1	2	yes	—
186	C₂²×C₄	—	3	2	no	—
187	C₂²×C₄	—	3	1	no	—
188	C₂²×C₄	—	3	1	no	—
189	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
190	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	6	2	no	—
195	C₂×C₈	—	6	2	no	—
196	C₂×C₈	—	6	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	3	2	no	—
199	C₄²	—	3	2	no	—
200	C₄²	—	6	2	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	#872	1	1	yes	characteristic
207	C₂²×C₃	#877	1	1	yes	characteristic
208	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₃	#901	1	2	yes	—
210	C₂²×C₃	#907	1	2	yes	—
211	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₃	C₃×#117	1	1	yes	characteristic
213	C₂²×C₃	C₃×#117	1	1	yes	characteristic
214	C₂²×C₃	C₃×#131	1	2	yes	—
215	C₂²×C₃	C₃×#131	1	2	yes	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	4	4	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	6	4	no	—
222	C₃×C₄	—	12	1	no	—
223	C₃×C₄	—	12	1	no	—
224	C₃×C₄	#885	1	2	yes	—
225	C₃×C₄	#889	1	2	yes	—
226	C₃×C₄	C₃×#124	1	2	yes	—
227	C₃×C₄	C₃×#125	1	2	yes	—
228	Dic₃	—	2	4	no	—
229	Dic₃	—	4	1	no	—
230	Dic₃	—	4	1	no	—
231	C₃²	#1832	1	1	yes	characteristic, complemented, Sylow
232	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	3	4	no	—
237	C₂×C₄	—	6	2	no	—
238	C₂×C₄	—	6	2	no	—
239	C₂×C₄	—	6	2	no	—
240	C₂×C₄	—	6	2	no	—
241	C₂×C₄	—	6	1	no	—
242	C₂×C₄	—	6	1	no	—
243	C₂×C₄	C₃×#235	1	2	yes	—
244	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
245	C₂×C₄	C₃×#237	1	2	yes	—
246	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
247	C₈	—	4	2	no	—
248	C₈	—	12	2	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	—	2	2	no	—
254	C₂×C₃	#4650	1	1	yes	characteristic
255	C₂×C₃	#4690	1	1	yes	characteristic
256	C₂×C₃	#4709	1	1	yes	characteristic
257	C₂×C₃	#4720	1	2	yes	—
258	C₂×C₃	#4839	1	2	yes	—
259	C₂×C₃	C₃×#382	1	1	yes	characteristic
260	C₂×C₃	C₃×#424	1	2	yes	—
261	C₂×C₃	C₃×#433	1	1	yes	characteristic
262	C₂×C₃	C₃×#433	1	1	yes	characteristic
263	C₂×C₃	C₃×#433	1	2	yes	—
264	C₂²	C₃×#872	1	1	yes	characteristic
265	C₂²	C₃×#877	1	1	yes	characteristic
266	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
267	C₂²	C₃×#901	1	2	yes	—
268	C₂²	C₃×#907	1	2	yes	—
269	C₄	—	4	2	no	—
270	C₄	—	6	4	no	—
271	C₄	—	12	1	no	—
272	C₄	—	12	1	no	—
273	C₄	C₃×#885	1	2	yes	—
274	C₄	C₃×#889	1	2	yes	—
275	C₃	—	2	1	no	—
276	C₃	#68606	1	1	yes	characteristic, direct product factor
277	C₃	C₃×#1832	1	1	yes	characteristic, complemented
278	C₂	C₃×#4650	1	1	yes	characteristic
279	C₂	C₃×#4690	1	1	yes	characteristic
280	C₂	C₃×#4709	1	1	yes	characteristic
281	C₂	C₃×#4720	1	2	yes	—
282	C₂	C₃×#4839	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×#1832
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. #68606	C₃	direct (N × H)
231. C₃²	#1832	semidirect
276. C₃	#68606	direct (N × H)
277. C₃	C₃×#1832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Dic₃⋊C₈	2
C₃×#4932	2
C₃²×#336	2
C₃×#4550	2
C₃×#4524	2
C₃×#1832	3
#68606	3

Subgroups — C3×#68606 #11785973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68606 #11785973