Subgroups — Dic₃×#3748 #13160280

Order 1728.

1124 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88855	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3748	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#11722019	C₂	1	2	yes	maximal, complemented
5	Dic₃×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
6	C₄×#3748	—	3	1	no	maximal
7	Dic₃×#208	—	3	4	no	maximal
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×Dic₃	C₄	1	2	yes	—
10	#88855	C₄	1	4	yes	complemented
11	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
12	C₃×#3748	C₄	1	4	yes	complemented
13	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₃²×C₄×Dic₃	C₄	1	2	yes	—
15	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃²×C₄²	—	3	1	no	—
18	C₂×#3748	—	3	1	no	—
19	C₂×#3748	—	3	4	no	—
20	C₂×#3748	—	6	4	no	—
21	C₂×#3748	D₃	1	1	yes	characteristic
22	C₂×C₃×C₄×Dic₃	D₃	1	4	yes	—
23	C₂×C₃×#208	—	3	4	no	—
24	#63630	—	3	2	no	—
25	C₄×C₃⋊(C₃⋊C₈)	—	3	2	no	—
26	#63554	—	3	8	no	—
27	Dic₃×C₃⋊C₈	—	3	8	no	—
28	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃²×Dic₃	—	2	4	no	—
34	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
35	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
36	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	4	yes	complemented
37	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	complemented
38	Dic₃×M₁₆	—	9	1	no	—
39	C₄×#208	—	9	4	no	—
40	C₂²×C₃²×C₄	—	2	4	no	—
41	C₂²×C₃²×C₄	—	3	2	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×Dic₃	Dic₃	1	8	yes	—
45	C₃²×C₄²	—	3	2	no	—
46	C₃²×C₄²	—	3	1	no	—
47	C₃²×C₄²	—	3	1	no	—
48	#3748	—	3	4	no	—
49	#3748	—	3	16	no	—
50	#3748	—	6	16	no	—
51	#3748	Dic₃	1	4	yes	direct product factor
52	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
53	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
54	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
55	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
56	C₃×C₄×Dic₃	Dic₃	1	8	yes	—
57	C₃×C₄×Dic₃	C₂×D₃	1	4	yes	—
58	C₃×C₄×Dic₃	C₂×D₃	1	4	yes	—
59	C₃×#208	—	3	16	no	—
60	C₂×C₃×C₃⋊C₈	—	3	8	no	—
61	C₂²×C₃³	—	2	2	no	—
62	C₂²×C₃³	C₄²	1	2	yes	—
63	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
64	C₃³×C₄	C₄²	1	2	yes	—
65	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₃²×Dic₃	—	2	2	no	—
68	C₃²×Dic₃	M₁₆	1	4	yes	complemented
69	C₂×C₃×M₁₆	—	9	1	no	—
70	C₂×C₃×C₄²	—	3	4	no	—
71	C₂×C₄×Dic₃	C₃⋊D₃	1	1	yes	characteristic
72	C₂×#208	—	3	4	no	—
73	C₂×#208	—	9	4	no	—
74	C₂×#208	—	9	1	no	—
75	C₂×#208	—	18	4	no	—
76	#815	—	9	2	no	—
77	C₈×Dic₃	—	9	2	no	—
78	#803	—	9	8	no	—
79	C₄×C₃⋊C₈	—	9	8	no	—
80	C₂³×C₃²	—	2	4	no	—
81	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
82	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	8	no	—
85	C₂×C₃²×C₄	—	2	8	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	—	3	2	no	—
88	C₂×C₃²×C₄	—	3	2	no	—
89	C₂×C₃²×C₄	—	6	4	no	—
90	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
91	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
92	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
94	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
95	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
96	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
97	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₃×Dic₃	—	2	16	no	—
99	C₂×C₃×Dic₃	C₂×Dic₃	1	8	yes	—
100	C₂×C₃×Dic₃	C₂×Dic₃	1	8	yes	—
101	C₃⋊(C₃⋊C₈)	—	3	4	no	—
102	C₃⋊(C₃⋊C₈)	—	3	16	no	—
103	C₃⋊(C₃⋊C₈)	—	6	16	no	—
104	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	4	yes	complemented
105	C₃×C₃⋊C₈	—	3	16	no	—
106	C₄×M₁₆	—	27	1	no	Sylow
107	C₂×C₃³	—	2	2	no	—
108	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₃³	C₂×M₁₆	1	2	yes	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	3	8	no	—
112	C₂²×C₃×C₄	D₃²	1	4	yes	—
113	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
114	C₂²×Dic₃	C₃⋊Dic₃	1	2	yes	—
115	C₃×M₁₆	—	9	4	no	—
116	C₂×C₃×C₈	—	9	2	no	—
117	C₃×C₄²	—	3	8	no	—
118	C₃×C₄²	—	3	4	no	—
119	C₃×C₄²	—	3	4	no	—
120	C₄×Dic₃	C₃⋊Dic₃	1	2	yes	—
121	C₄×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
122	C₄×Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
123	#208	—	3	16	no	—
124	#208	—	9	16	no	—
125	#208	—	9	4	no	—
126	#208	—	18	16	no	—
127	C₂×C₃⋊C₈	—	3	8	no	—
128	C₂×C₃⋊C₈	—	9	8	no	—
129	C₂×C₃⋊C₈	—	9	2	no	—
130	C₂×C₃⋊C₈	—	18	8	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	—	2	8	no	—
133	C₂²×C₃²	—	2	4	no	—
134	C₂²×C₃²	—	2	8	no	—
135	C₂²×C₃²	—	4	8	no	—
136	C₂²×C₃²	C₄×Dic₃	1	8	yes	—
137	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
138	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
140	C₃²×C₄	—	2	8	no	—
141	C₃²×C₄	—	2	4	no	—
142	C₃²×C₄	—	2	4	no	—
143	C₃²×C₄	—	3	4	no	—
144	C₃²×C₄	—	6	2	no	—
145	C₃²×C₄	C₄×Dic₃	1	8	yes	—
146	C₃²×C₄	C₄×Dic₃	1	2	yes	—
147	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
148	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
149	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
151	C₃×Dic₃	—	2	8	no	—
152	C₃×Dic₃	#208	1	16	yes	complemented
153	C₂×M₁₆	—	9	1	no	—
154	C₂×M₁₆	—	27	1	no	—
155	C₂×C₄²	—	3	1	no	—
156	#97	—	27	2	no	—
157	C₄×C₈	—	27	2	no	—
158	C₃³	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₃	—	2	4	no	—
160	C₂³×C₃	D₃×Dic₃	1	4	yes	—
161	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	3	8	no	—
167	C₂×C₃×C₄	—	3	8	no	—
168	C₂×C₃×C₄	—	6	16	no	—
169	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
170	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
171	C₂×C₃×C₄	#624	1	8	yes	—
172	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
173	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
176	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
177	C₂×Dic₃	—	2	4	no	—
178	C₂×Dic₃	C₂×C₃⋊Dic₃	1	2	yes	—
179	C₂×Dic₃	C₂×C₃⋊Dic₃	1	2	yes	—
180	C₃×C₈	—	9	4	no	—
181	C₃⋊C₈	—	3	16	no	—
182	C₃⋊C₈	—	9	16	no	—
183	C₃⋊C₈	—	9	4	no	—
184	C₃⋊C₈	—	18	16	no	—
185	C₂×C₃²	—	2	2	no	—
186	C₂×C₃²	—	2	8	no	—
187	C₂×C₃²	—	2	4	no	—
188	C₂×C₃²	—	2	8	no	—
189	C₂×C₃²	—	4	8	no	—
190	C₂×C₃²	D₃×M₁₆	1	2	yes	—
191	C₂×C₃²	C₂×#208	1	8	yes	—
192	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
193	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
194	C₂²×C₄	—	3	2	no	—
195	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
196	M₁₆	—	9	4	no	—
197	M₁₆	—	27	4	no	—
198	C₂×C₈	—	9	2	no	—
199	C₂×C₈	—	27	2	no	—
200	C₄²	—	3	2	no	—
201	C₄²	—	3	1	no	—
202	C₄²	—	3	1	no	—
203	C₂²×C₃	—	2	8	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	8	no	—
207	C₂²×C₃	—	4	8	no	—
208	C₂²×C₃	Dic₃²	1	8	yes	—
209	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
210	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
211	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	8	no	—
215	C₃×C₄	—	3	16	no	—
216	C₃×C₄	—	6	8	no	—
217	C₃×C₄	Dic₃²	1	8	yes	—
218	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
219	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
220	C₃×C₄	C₂×#624	1	4	yes	—
221	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
222	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
223	Dic₃	—	2	2	no	—
224	Dic₃	#3748	1	4	yes	direct product factor
225	C₃²	—	2	4	no	—
226	C₃²	C₄×#208	1	4	yes	complemented
227	C₃²	Dic₃×M₁₆	1	1	yes	characteristic, complemented
228	C₂³	D₃×C₃⋊Dic₃	1	1	yes	characteristic
229	C₂×C₄	—	3	2	no	—
230	C₂×C₄	—	3	2	no	—
231	C₂×C₄	—	6	4	no	—
232	C₂×C₄	D₃×C₃⋊Dic₃	1	1	yes	characteristic
233	C₂×C₄	Dic₃×C₃⋊D₃	1	2	yes	—
234	C₂×C₄	#6289	1	2	yes	—
235	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
236	C₈	—	9	4	no	—
237	C₈	—	27	4	no	—
238	C₂×C₃	—	2	8	no	—
239	C₂×C₃	—	2	2	no	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	—	2	8	no	—
242	C₂×C₃	—	4	8	no	—
243	C₂×C₃	D₃×#208	1	8	yes	—
244	C₂×C₃	C₂×Dic₃²	1	4	yes	—
245	C₂×C₃	C₂×#3748	1	2	yes	—
246	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
247	C₂²	—	2	2	no	—
248	C₂²	Dic₃×C₃⋊Dic₃	1	2	yes	—
249	C₂²	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
250	C₄	—	3	4	no	—
251	C₄	—	6	2	no	—
252	C₄	Dic₃×C₃⋊Dic₃	1	2	yes	—
253	C₄	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
254	C₄	C₂×#6289	1	1	yes	characteristic
255	C₃	—	2	4	no	—
256	C₃	Dic₃×#208	1	4	yes	complemented
257	C₃	C₄×#3748	1	1	yes	characteristic, complemented
258	C₂	—	2	2	no	—
259	C₂	D₃×#3748	1	2	yes	—
260	C₂	C₂×Dic₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator	C₂×C₃³	C₂×C₄²
Fitting	C₂²×C₃³×C₄	C₂²
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	C₄×M₁₆
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11722019	C₂	semidirect
5. Dic₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
10. #88855	C₄	semidirect
12. C₃×#3748	C₄	semidirect
36. C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	semidirect
37. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
51. #3748	Dic₃	direct (N × H)
68. C₃²×Dic₃	M₁₆	semidirect
104. C₃⋊(C₃⋊C₈)	C₂×Dic₃	semidirect
152. C₃×Dic₃	#208	semidirect
158. C₃³	C₄×M₁₆	semidirect
224. Dic₃	#3748	direct (N × H)
226. C₃²	C₄×#208	semidirect
227. C₃²	Dic₃×M₁₆	semidirect
256. C₃	Dic₃×#208	semidirect
257. C₃	C₄×#3748	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88855	2
C₂×C₃×#3748	2
C₂×C₃²×C₄×Dic₃	2
#11722019	2
Dic₃×C₃⋊(C₃⋊C₈)	2
C₄×#3748	3
Dic₃×#208	3

Subgroups — Dic3×#3748 #13160280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×#3748 #13160280