Subgroups — C₃×#67665 #11771143

Order 1152.

415 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#4932	C₂	1	1	yes	maximal, characteristic
3	C₃×#4926	C₂	1	1	yes	maximal, characteristic
4	C₃×#1300	—	3	1	no	maximal
5	#67665	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×M₁₆	—	2	1	no	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×Dic₃	—	2	1	no	—
9	C₃²×#97	C₄	1	2	yes	complemented
10	C₃²×C₄×C₈	—	2	1	no	—
11	C₃×C₃⋊#95	—	2	1	no	—
12	C₃×C₃⋊#95	C₄	1	2	yes	—
13	C₃×#830	—	2	1	no	—
14	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
15	C₃×#803	—	2	1	no	—
16	C₃×#432	—	3	1	no	—
17	C₃×C₄×M₁₆	—	2	1	no	—
18	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic
19	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
20	C₃×#376	—	3	1	no	—
21	#4932	C₂×C₃	1	1	yes	characteristic
22	#4926	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃×Dic₃	—	2	2	no	—
27	C₃²×M₁₆	—	2	4	no	—
28	C₂×C₃²×C₈	—	2	2	no	—
29	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃²×C₄²	D₄	1	1	yes	characteristic
32	C₃²×C₄²	Q₈	1	1	yes	characteristic
33	C₃×C₄×Dic₃	—	2	4	no	—
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	#1300	—	3	1	no	Sylow
36	C₂×C₃×M₁₆	—	2	2	no	—
37	C₂×C₃×M₁₆	—	2	1	no	—
38	C₂×C₃×M₁₆	—	2	1	no	—
39	C₂×C₃×C₄²	—	2	1	no	—
40	C₂×C₃×C₄²	—	6	1	no	—
41	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₄×Dic₃	—	2	1	no	—
44	C₃×C₄⋊C₈	—	3	2	no	—
45	C₃×C₂²⋊C₈	—	6	1	no	—
46	C₃×#97	—	2	2	no	—
47	C₃×#97	—	6	1	no	—
48	C₃×#97	Dic₃	1	2	yes	complemented
49	C₃×#97	C₃×C₄	1	2	yes	complemented
50	C₃×C₄×C₈	—	2	2	no	—
51	C₃×C₄×C₈	—	2	1	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×#95	—	3	2	no	—
54	C₃×#95	—	6	1	no	—
55	C₃⋊#95	—	2	1	no	—
56	C₃⋊#95	C₃×C₄	1	2	yes	—
57	#830	—	2	1	no	—
58	C₄⋊(C₃⋊C₈)	C₃×C₄	1	2	yes	complemented
59	#803	—	2	1	no	—
60	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	—	2	2	no	—
63	C₂×C₃²×C₄	—	4	1	no	—
64	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₄⋊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×C₃×Dic₃	—	2	4	no	—
71	C₂×C₃×Dic₃	—	4	2	no	—
72	C₂×C₃×Dic₃	—	4	2	no	—
73	C₃²×C₈	—	2	4	no	—
74	C₃×C₃⋊C₈	—	4	2	no	—
75	#432	—	3	1	no	—
76	C₄×M₁₆	C₃×D₃	1	1	yes	characteristic
77	#376	—	3	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	—	6	2	no	—
82	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
84	C₂²×C₃×C₄	#67	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
88	C₂²×Dic₃	—	2	2	no	—
89	C₃×M₁₆	—	2	8	no	—
90	C₃×M₁₆	—	2	4	no	—
91	C₃×M₁₆	—	2	4	no	—
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	—	6	2	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	—	2	1	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	—	6	4	no	—
101	C₃×C₄²	Dic₆	1	1	yes	characteristic
102	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
103	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
104	C₃×C₄²	#67	1	1	yes	characteristic
105	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
106	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
107	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
108	C₃×C₄²	C₃×Q₈	1	1	yes	characteristic
109	C₄×Dic₃	—	2	4	no	—
110	C₂×C₃⋊C₈	—	2	2	no	—
111	C₂²×C₃²	—	2	1	no	—
112	C₂²×C₃²	—	2	1	no	—
113	C₂²×C₃²	#95	1	1	yes	characteristic
114	C₂²×C₃²	#104	1	2	yes	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	—	2	1	no	—
117	C₃²×C₄	—	2	1	no	—
118	C₃²×C₄	—	2	1	no	—
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	#104	1	2	yes	—
121	C₃×Dic₃	—	4	4	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂×C₄²	—	6	1	no	—
124	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
125	C₄⋊C₈	—	3	2	no	—
126	C₂²⋊C₈	—	6	1	no	—
127	#97	—	6	1	no	—
128	#97	C₃×Dic₃	1	2	yes	complemented
129	C₄×C₈	—	2	1	no	—
130	#95	—	3	2	no	—
131	#95	—	6	1	no	—
132	C₂³×C₃	—	2	1	no	—
133	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
134	C₂³×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
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	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	—	4	1	no	—
148	C₂×C₃×C₄	—	4	1	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	—	6	4	no	—
151	C₂×C₃×C₄	—	12	2	no	—
152	C₂×C₃×C₄	—	12	2	no	—
153	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
157	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	C₂×C₃×C₄	#217	1	1	yes	characteristic
159	C₂×C₃×C₄	C₃×C₄²	1	1	yes	characteristic
160	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
165	C₂×Dic₃	—	2	4	no	—
166	C₂×Dic₃	—	4	2	no	—
167	C₂×Dic₃	—	4	2	no	—
168	C₃×C₈	—	2	8	no	—
169	C₃×C₈	—	2	4	no	—
170	C₃×C₈	—	2	4	no	—
171	C₃×C₈	—	12	2	no	—
172	C₃⋊C₈	—	4	2	no	—
173	C₂×C₃²	—	2	2	no	—
174	C₂×C₃²	#339	1	1	yes	characteristic
175	C₂×C₃²	#339	1	1	yes	characteristic
176	C₂×C₃²	#344	1	1	yes	characteristic
177	C₂²×C₄	—	6	2	no	—
178	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
179	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
180	C₂²×C₄	C₃×#67	1	1	yes	characteristic
181	M₁₆	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	6	2	no	—
184	C₄²	—	6	4	no	—
185	C₄²	C₃×Dic₆	1	1	yes	characteristic
186	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
187	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
188	C₄²	C₃×#67	1	1	yes	characteristic
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	#805	1	2	yes	—
198	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
199	C₂²×C₃	C₃×#95	1	1	yes	characteristic
200	C₂²×C₃	C₃×#104	1	2	yes	—
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	—	4	1	no	—
217	C₃×C₄	—	12	4	no	—
218	C₃×C₄	#825	1	2	yes	—
219	C₃×C₄	C₃×#104	1	2	yes	—
220	Dic₃	—	4	4	no	—
221	C₃²	#1300	1	1	yes	characteristic, complemented, Sylow
222	C₂³	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	6	4	no	—
227	C₂×C₄	—	12	2	no	—
228	C₂×C₄	—	12	2	no	—
229	C₂×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
230	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₃×C₄⋊Dic₃	1	1	yes	characteristic
233	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₃×#217	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	—	12	2	no	—
237	C₂×C₃	—	2	2	no	—
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	#4523	1	1	yes	characteristic
244	C₂×C₃	#4555	1	1	yes	characteristic
245	C₂×C₃	#4559	1	1	yes	characteristic
246	C₂×C₃	C₃×#339	1	1	yes	characteristic
247	C₂×C₃	C₃×#339	1	1	yes	characteristic
248	C₂×C₃	C₃×#344	1	1	yes	characteristic
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	C₃×#805	1	2	yes	—
252	C₂²	C₃×C₃⋊#95	1	1	yes	characteristic
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	12	4	no	—
259	C₄	C₃×#825	1	2	yes	—
260	C₃	—	2	1	no	—
261	C₃	#67665	1	1	yes	characteristic, direct product factor
262	C₃	C₃×#1300	1	1	yes	characteristic, complemented
263	C₂	—	2	2	no	—
264	C₂	C₃×#4523	1	1	yes	characteristic
265	C₂	C₃×#4555	1	1	yes	characteristic
266	C₂	C₃×#4559	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₃×C₄²
Frattini	C₂×C₄²	C₂×C₃×D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂²×C₃
hypercenter	C₃×C₄×M₁₆	D₃
hypocenter	C₃	C₃×#1300
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×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. #67665	C₃	direct (N × H)
9. C₃²×#97	C₄	semidirect
14. C₃×C₄⋊(C₃⋊C₈)	C₄	semidirect
48. C₃×#97	Dic₃	semidirect
49. C₃×#97	C₃×C₄	semidirect
58. C₄⋊(C₃⋊C₈)	C₃×C₄	semidirect
128. #97	C₃×Dic₃	semidirect
221. C₃²	#1300	semidirect
261. C₃	#67665	direct (N × H)
262. C₃	C₃×#1300	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×#4932	2
C₃×#4926	2
C₃×#1300	3
#67665	3

Subgroups — C3×#67665 #11771143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#67665 #11771143