Subgroups — C₃×#67668 #11770859

Order 1152.

391 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₃×#1283	—	3	1	no	maximal
5	#67668	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×M₁₆	C₄	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₃⋊C₈	C₄	1	2	yes	—
9	C₃²×#97	—	2	1	no	—
10	C₃²×C₄×C₈	—	2	1	no	—
11	C₃×#830	C₄	1	2	yes	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
14	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
15	C₃×C₄×C₃⋊C₈	—	2	1	no	—
16	C₃×#431	—	3	1	no	—
17	C₂×C₃×C₄⋊C₈	—	3	1	no	—
18	C₃×C₄×M₁₆	—	2	1	no	—
19	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic
20	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
21	#4925	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₃²×M₁₆	—	2	2	no	—
27	C₃²×M₁₆	C₈	1	4	yes	complemented
28	C₂×C₃²×C₈	—	2	2	no	—
29	C₃²×C₄²	D₄	1	1	yes	characteristic
30	C₃²×C₄²	D₄	1	1	yes	characteristic
31	C₃²×C₄²	D₄	1	1	yes	characteristic
32	C₃²×C₄²	Q₈	1	1	yes	characteristic
33	C₂×C₃×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	—	4	2	no	—
36	#1283	—	3	1	no	Sylow
37	C₂×C₃×M₁₆	—	2	2	no	—
38	C₂×C₃×M₁₆	Dic₃	1	2	yes	—
39	C₂×C₃×M₁₆	C₃×C₄	1	2	yes	—
40	C₂²×C₃×C₈	—	3	2	no	—
41	C₂×C₃×C₄²	—	2	1	no	—
42	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃⋊C₈	C₃×C₄	1	2	yes	—
45	C₃×C₄⋊C₈	—	6	1	no	—
46	C₃×C₄⋊C₈	—	6	1	no	—
47	C₃×C₄⋊C₈	—	6	1	no	—
48	C₃×C₂²⋊C₈	—	3	2	no	—
49	C₃×#97	—	2	2	no	—
50	C₃×#97	—	2	1	no	—
51	C₃×#97	—	2	1	no	—
52	C₃×C₄×C₈	—	2	2	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	—	6	1	no	—
56	#830	C₃×C₄	1	2	yes	—
57	C₄⋊(C₃⋊C₈)	—	2	1	no	—
58	C₄⋊(C₃⋊C₈)	—	2	1	no	—
59	C₄⋊(C₃⋊C₈)	—	2	1	no	—
60	C₄×C₃⋊C₈	—	2	1	no	—
61	C₂³×C₃²	C₄²	1	1	yes	characteristic
62	C₂×C₃²×C₄	—	2	1	no	—
63	C₂×C₃²×C₄	—	2	1	no	—
64	C₂×C₃²×C₄	—	4	1	no	—
65	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
72	C₂×C₃²×C₄	M₁₆	1	2	yes	—
73	C₃²×C₈	—	2	4	no	—
74	C₃×C₃⋊C₈	—	4	2	no	—
75	C₃×C₃⋊C₈	—	4	2	no	—
76	#431	—	3	1	no	—
77	C₂×C₄⋊C₈	—	3	1	no	—
78	C₄×M₁₆	C₃×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
88	C₃×M₁₆	—	2	4	no	—
89	C₃×M₁₆	—	2	2	no	—
90	C₃×M₁₆	—	2	2	no	—
91	C₃×M₁₆	—	2	4	no	—
92	C₃×M₁₆	C₃⋊C₈	1	4	yes	complemented
93	C₃×M₁₆	C₃×C₈	1	4	yes	complemented
94	C₂×C₃×C₈	—	2	4	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	6	2	no	—
98	C₂×C₃×C₈	—	6	2	no	—
99	C₂×C₃×C₈	—	12	2	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	Dic₆	1	1	yes	characteristic
105	C₃×C₄²	D₁₂	1	1	yes	characteristic
106	C₃×C₄²	#67	1	1	yes	characteristic
107	C₃×C₄²	#67	1	1	yes	characteristic
108	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
109	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
110	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
111	C₃×C₄²	C₃×Q₈	1	1	yes	characteristic
112	C₂×C₃⋊C₈	—	2	2	no	—
113	C₂×C₃⋊C₈	—	2	2	no	—
114	C₂×C₃⋊C₈	—	4	2	no	—
115	C₂²×C₃²	—	2	1	no	—
116	C₂²×C₃²	—	2	1	no	—
117	C₂²×C₃²	#95	1	1	yes	characteristic
118	C₂²×C₃²	C₄×C₈	1	1	yes	characteristic
119	C₂²×C₃²	#97	1	1	yes	characteristic
120	C₃²×C₄	—	2	1	no	—
121	C₃²×C₄	—	2	1	no	—
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
124	C₃²×C₄	#100	1	1	yes	characteristic
125	C₃²×C₄	#101	1	1	yes	characteristic
126	C₃²×C₄	C₄⋊C₈	1	2	yes	—
127	C₂×M₁₆	C₃×Dic₃	1	2	yes	—
128	C₂²×C₈	—	3	2	no	—
129	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
130	C₄⋊C₈	—	6	1	no	—
131	C₄⋊C₈	—	6	1	no	—
132	C₄⋊C₈	—	6	1	no	—
133	C₂²⋊C₈	—	3	2	no	—
134	#97	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	6	1	no	—
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
139	C₂³×C₃	C₃×C₄²	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	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	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	4	1	no	—
153	C₂×C₃×C₄	—	4	1	no	—
154	C₂×C₃×C₄	—	4	1	no	—
155	C₂×C₃×C₄	—	4	1	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
158	C₂×C₃×C₄	#208	1	2	yes	—
159	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
164	C₂×C₃×C₄	#217	1	1	yes	characteristic
165	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
172	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
173	C₃×C₈	—	2	8	no	—
174	C₃×C₈	—	2	4	no	—
175	C₃×C₈	—	2	4	no	—
176	C₃×C₈	—	12	2	no	—
177	C₃×C₈	—	12	2	no	—
178	C₃⋊C₈	—	4	2	no	—
179	C₃⋊C₈	—	4	2	no	—
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	—	2	1	no	—
182	C₂×C₃²	#336	1	1	yes	characteristic
183	C₂×C₃²	#343	1	1	yes	characteristic
184	C₂×C₃²	#344	1	1	yes	characteristic
185	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
186	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
188	M₁₆	—	2	2	no	—
189	M₁₆	C₃×C₃⋊C₈	1	4	yes	complemented
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	6	2	no	—
192	C₂×C₈	—	6	2	no	—
193	C₂×C₈	—	12	2	no	—
194	C₄²	C₃×Dic₆	1	1	yes	characteristic
195	C₄²	C₃×D₁₂	1	1	yes	characteristic
196	C₄²	C₃×#67	1	1	yes	characteristic
197	C₄²	C₃×#67	1	1	yes	characteristic
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	—	4	1	no	—
207	C₂²×C₃	C₄×C₃⋊C₈	1	1	yes	characteristic
208	C₂²×C₃	#803	1	1	yes	characteristic
209	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
210	C₂²×C₃	C₃×#95	1	1	yes	characteristic
211	C₂²×C₃	C₃×C₄×C₈	1	1	yes	characteristic
212	C₂²×C₃	C₃×#97	1	1	yes	characteristic
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
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	—	2	2	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	—	4	1	no	—
226	C₃×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
227	C₃×C₄	#823	1	1	yes	characteristic
228	C₃×C₄	#824	1	1	yes	characteristic
229	C₃×C₄	#830	1	2	yes	—
230	C₃×C₄	C₃×C₂²⋊C₈	1	2	yes	—
231	C₃×C₄	C₃×#100	1	1	yes	characteristic
232	C₃×C₄	C₃×#101	1	1	yes	characteristic
233	C₃×C₄	C₃×C₄⋊C₈	1	2	yes	—
234	C₃²	#1283	1	1	yes	characteristic, complemented, Sylow
235	C₂³	C₃×C₄×Dic₃	1	1	yes	characteristic
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
240	C₂×C₄	C₃×#208	1	2	yes	—
241	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
244	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
246	C₂×C₄	C₃×#217	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	—	12	2	no	—
249	C₈	—	12	2	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	4	1	no	—
258	C₂×C₃	—	4	1	no	—
259	C₂×C₃	#4524	1	1	yes	characteristic
260	C₂×C₃	#4554	1	1	yes	characteristic
261	C₂×C₃	#4559	1	1	yes	characteristic
262	C₂×C₃	C₃×#336	1	1	yes	characteristic
263	C₂×C₃	C₃×#343	1	1	yes	characteristic
264	C₂×C₃	C₃×#344	1	1	yes	characteristic
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	C₃×C₄×C₃⋊C₈	1	1	yes	characteristic
268	C₂²	C₃×#803	1	1	yes	characteristic
269	C₂²	C₃×C₃⋊#95	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	C₃×C₄⋊(C₃⋊C₈)	1	2	yes	—
274	C₄	C₃×#823	1	1	yes	characteristic
275	C₄	C₃×#824	1	1	yes	characteristic
276	C₄	C₃×#830	1	2	yes	—
277	C₃	—	2	1	no	—
278	C₃	#67668	1	1	yes	characteristic, direct product factor
279	C₃	C₃×#1283	1	1	yes	characteristic, complemented
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	C₃×#4524	1	1	yes	characteristic
283	C₂	C₃×#4554	1	1	yes	characteristic
284	C₂	C₃×#4559	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₃×C₄×C₈
Frattini	C₂×C₄²	C₂×C₃×D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄²
cosocle	C₂×C₃×C₄²	C₂²×C₃
hypercenter	C₃×C₄×M₁₆	D₃
hypocenter	C₃	C₃×#1283
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
5. #67668	C₃	direct (N × H)
27. C₃²×M₁₆	C₈	semidirect
92. C₃×M₁₆	C₃⋊C₈	semidirect
93. C₃×M₁₆	C₃×C₈	semidirect
189. M₁₆	C₃×C₃⋊C₈	semidirect
234. C₃²	#1283	semidirect
278. C₃	#67668	direct (N × H)
279. C₃	C₃×#1283	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×M₁₆	2
C₃×#4925	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₃×#1283	3
#67668	3

Subgroups — C3×#67668 #11770859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67668 #11770859