Subgroups — C₃×#67671 #11772348

Order 1152.

337 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×#1379	—	3	1	no	maximal
5	#67671	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₃²×#117	C₄	1	1	yes	characteristic
9	C₃²×#97	—	2	1	no	—
10	C₃²×C₄×C₈	—	2	1	no	—
11	C₃×#812	—	2	1	no	—
12	C₃×#812	C₄	1	2	yes	complemented
13	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
14	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
15	C₂×C₃×C₃⋊C₁₆	—	2	1	no	—
16	C₂×C₃×C₃⋊C₁₆	C₄	1	2	yes	complemented
17	C₂×C₃×M₃₂	—	3	1	no	—
18	C₂²×C₃×C₁₆	—	3	1	no	—
19	C₃×#405	—	2	1	no	—
20	C₃×#405	D₃	1	1	yes	characteristic
21	C₃×#405	C₂×C₃	1	1	yes	characteristic
22	C₂×#812	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₃²×M₁₆	—	2	1	no	—
26	C₃²×M₁₆	C₈	1	2	yes	—
27	C₂×C₃²×C₈	—	2	1	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
34	C₃²×C₄⋊C₄	C₈	1	2	yes	—
35	C₃²×C₂²⋊C₄	—	2	1	no	—
36	C₃²×C₄²	—	2	1	no	—
37	C₃×C₃⋊C₁₆	—	2	2	no	—
38	C₃×C₃⋊C₁₆	—	2	2	no	—
39	#1379	—	3	1	no	Sylow
40	C₂×C₃×M₁₆	—	2	1	no	—
41	C₂×C₃×M₁₆	Dic₃	1	1	yes	characteristic
42	C₂×C₃×M₁₆	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₈	—	2	1	no	—
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃×#117	—	2	1	no	—
47	C₃×#117	Dic₃	1	1	yes	characteristic
48	C₃×#117	C₃×C₄	1	1	yes	characteristic
49	C₃×M₃₂	—	3	2	no	—
50	C₃×M₃₂	—	6	1	no	—
51	C₂×C₃×C₁₆	—	3	2	no	—
52	C₂×C₃×C₁₆	—	6	1	no	—
53	C₂×C₃×C₁₆	—	6	1	no	—
54	C₂×C₃×C₁₆	—	6	1	no	—
55	C₃×#97	—	2	2	no	—
56	C₃×#97	—	2	1	no	—
57	C₃×#97	—	2	1	no	—
58	C₃×C₄×C₈	—	2	2	no	—
59	C₃×C₄×C₈	—	2	1	no	—
60	C₃×C₄×C₈	—	2	1	no	—
61	#812	—	2	1	no	—
62	#812	C₃×C₄	1	2	yes	complemented
63	C₂×C₃⋊C₁₆	—	2	1	no	—
64	C₂×C₃⋊C₁₆	—	2	1	no	—
65	C₂×C₃⋊C₁₆	—	2	1	no	—
66	C₂×C₃⋊C₁₆	C₃×C₄	1	2	yes	complemented
67	C₂³×C₃²	M₁₆	1	1	yes	characteristic
68	C₂×C₃²×C₄	—	2	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	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
74	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
75	C₃²×C₈	—	2	2	no	—
76	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
80	C₂×M₃₂	—	3	1	no	—
81	C₂²×C₁₆	—	3	1	no	—
82	#405	C₃×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
86	C₃×M₁₆	—	2	2	no	—
87	C₃×M₁₆	—	2	1	no	—
88	C₃×M₁₆	—	2	1	no	—
89	C₃×M₁₆	—	2	2	no	—
90	C₃×M₁₆	C₃⋊C₈	1	2	yes	—
91	C₃×M₁₆	C₃×C₈	1	2	yes	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
102	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
103	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
104	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
105	C₂×C₃×C₈	#67	1	1	yes	characteristic
106	C₂×C₃×C₈	#67	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
111	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
112	C₂×C₃×C₈	C₃×Q₈	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	2	2	no	—
114	C₃×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
115	C₃×C₄⋊C₄	C₃×C₈	1	2	yes	—
116	C₃×C₂²⋊C₄	—	2	2	no	—
117	C₃×C₂²⋊C₄	—	2	1	no	—
118	C₃×C₂²⋊C₄	—	2	1	no	—
119	C₃×C₄²	—	2	2	no	—
120	C₃×C₄²	—	2	1	no	—
121	C₃×C₄²	—	2	1	no	—
122	C₃×C₁₆	—	6	2	no	—
123	C₃×C₁₆	—	6	2	no	—
124	C₃⋊C₁₆	—	2	2	no	—
125	C₃⋊C₁₆	—	2	2	no	—
126	C₂²×C₃²	—	2	2	no	—
127	C₂²×C₃²	#97	1	1	yes	characteristic
128	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
130	C₃²×C₄	—	4	2	no	—
131	C₃²×C₄	#95	1	1	yes	characteristic
132	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
133	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
134	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
135	C₂×M₁₆	C₃×Dic₃	1	1	yes	characteristic
136	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
137	#117	C₃×Dic₃	1	1	yes	characteristic
138	M₃₂	—	3	2	no	—
139	M₃₂	—	6	1	no	—
140	C₂×C₁₆	—	3	2	no	—
141	C₂×C₁₆	—	6	1	no	—
142	C₂×C₁₆	—	6	1	no	—
143	C₂×C₁₆	—	6	1	no	—
144	#97	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	#208	1	1	yes	characteristic
148	C₂³×C₃	C₃×M₁₆	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
160	C₂×C₃×C₄	#208	1	1	yes	characteristic
161	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
163	C₂×C₃×C₄	#217	1	1	yes	characteristic
164	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
165	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×M₁₆	1	1	yes	characteristic
170	C₃×C₈	—	2	4	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	2	1	no	—
174	C₃×C₈	—	2	1	no	—
175	C₃×C₈	—	2	1	no	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
178	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
179	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
182	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
183	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
184	C₃×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₃²	—	2	1	no	—
186	C₂×C₃²	—	2	1	no	—
187	C₂×C₃²	#336	1	1	yes	characteristic
188	C₂×C₃²	#350	1	2	yes	—
189	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
190	M₁₆	—	2	1	no	—
191	M₁₆	C₃×C₃⋊C₈	1	2	yes	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	C₃×Dic₆	1	1	yes	characteristic
194	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
195	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
196	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
197	C₂×C₈	C₃×#67	1	1	yes	characteristic
198	C₂×C₈	C₃×#67	1	1	yes	characteristic
199	C₄⋊C₄	C₃×C₃⋊C₈	1	2	yes	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₁₆	—	6	2	no	—
203	C₁₆	—	6	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	#803	1	1	yes	characteristic
211	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
212	C₂²×C₃	#830	1	1	yes	characteristic
213	C₂²×C₃	C₃×#97	1	1	yes	characteristic
214	C₂²×C₃	C₃×C₂²⋊C₈	1	1	yes	characteristic
215	C₂²×C₃	C₃×C₄⋊C₈	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	4	4	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	—	4	2	no	—
223	C₃×C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
224	C₃×C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
225	C₃×C₄	#830	1	1	yes	characteristic
226	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
227	C₃×C₄	C₃×#95	1	1	yes	characteristic
228	C₃×C₄	C₃×C₄×C₈	1	1	yes	characteristic
229	C₃×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
230	C₃×C₄	C₃×C₄⋊C₈	1	1	yes	characteristic
231	C₃²	#1379	1	1	yes	characteristic, complemented, Sylow
232	C₂³	C₃×#208	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₃×#208	1	1	yes	characteristic
237	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
238	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
239	C₂×C₄	C₃×#217	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	C₃×Dic₃⋊C₄	1	1	yes	characteristic
242	C₈	C₃×Dic₃⋊C₄	1	1	yes	characteristic
243	C₈	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
244	C₈	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	—	4	1	no	—
253	C₂×C₃	#4524	1	1	yes	characteristic
254	C₂×C₃	#4561	1	2	yes	—
255	C₂×C₃	C₃×#336	1	1	yes	characteristic
256	C₂×C₃	C₃×#350	1	2	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	C₃×#803	1	1	yes	characteristic
259	C₂²	C₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
260	C₂²	C₃×#830	1	1	yes	characteristic
261	C₄	—	4	2	no	—
262	C₄	C₃×C₄×C₃⋊C₈	1	1	yes	characteristic
263	C₄	C₃×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
264	C₄	C₃×#830	1	1	yes	characteristic
265	C₄	C₃×C₃⋊#95	1	1	yes	characteristic
266	C₃	—	2	1	no	—
267	C₃	#67671	1	1	yes	characteristic, direct product factor
268	C₃	C₃×#1379	1	1	yes	characteristic, complemented
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	C₃×#4524	1	1	yes	characteristic
272	C₂	C₃×#4561	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₈	#67
commutator	C₃×C₄	C₃×C₄×C₈
Frattini	C₂²×C₈	C₂×C₃×D₃
Fitting	C₃²×#405	C₂
socle	C₂²×C₃²	C₂²⋊C₈
prime radical	C₂³×C₃²	M₁₆
cosocle	C₂²×C₃×C₈	C₂²×C₃
hypercenter	C₃×#405	D₃
hypocenter	C₃	C₃×#1379
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈ ◁ C₃×#405

Complements

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

N	H ≅ G/N	type
5. #67671	C₃	direct (N × H)
12. C₃×#812	C₄	semidirect
16. C₂×C₃×C₃⋊C₁₆	C₄	semidirect
62. #812	C₃×C₄	semidirect
66. C₂×C₃⋊C₁₆	C₃×C₄	semidirect
231. C₃²	#1379	semidirect
267. C₃	#67671	direct (N × H)
268. C₃	C₃×#1379	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#405	2
C₂×C₃×#812	2
C₂²×C₃×C₃⋊C₁₆	2
C₃×#1379	3
#67671	3

Subgroups — C3×#67671 #11772348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67671 #11772348