Subgroups — #13160394

Order 1728.

1028 conjugacy classes of subgroups.

The subgroup lattice has 256 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	Dic₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	4	yes	maximal, complemented
5	#10592623	—	3	1	no	maximal
6	#10591115	—	3	4	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×Dic₃	C₄	1	2	yes	—
9	#88855	—	2	2	no	—
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
11	C₃×#3748	—	2	2	no	—
12	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₃²×C₄×Dic₃	C₄	1	1	yes	characteristic
14	C₃²×C₄×Dic₃	C₄	1	1	yes	characteristic
15	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
16	C₂×C₃²×C₄²	—	3	1	no	—
17	C₂×#3748	—	3	1	no	—
18	C₂×#3748	—	3	4	no	—
19	C₂×#3748	—	6	4	no	—
20	C₂×#3748	D₃	1	1	yes	characteristic
21	C₂×C₃×C₄×Dic₃	D₃	1	4	yes	—
22	C₂×C₃×#208	—	3	4	no	—
23	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	4	no	—
24	Dic₃⋊(C₃⋊C₈)	—	3	16	no	—
25	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃³×C₄	D₄	1	2	yes	—
28	C₂×C₃³×C₄	Q₈	1	2	yes	—
29	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃²×Dic₃	—	2	4	no	—
31	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
32	C₂×C₃²×Dic₃	C₂×C₄	1	2	yes	—
33	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
34	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
35	#5118	—	9	1	no	—
36	#4924	—	9	4	no	—
37	C₂²×C₃²×C₄	—	2	4	no	—
38	C₂²×C₃²×C₄	—	3	2	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
41	C₂²×C₃×Dic₃	Dic₃	1	8	yes	—
42	C₃²×C₄²	—	3	1	no	—
43	C₃²×C₄²	—	3	1	no	—
44	C₃²×C₄²	—	3	2	no	—
45	#3748	—	2	2	no	—
46	#3748	—	6	16	no	—
47	#3748	—	6	8	no	—
48	#3748	—	6	2	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
52	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
53	C₃×C₄×Dic₃	Dic₃	1	4	yes	—
54	C₃×C₄×Dic₃	Dic₃	1	4	yes	—
55	C₃×C₄×Dic₃	C₂×D₃	1	8	yes	—
56	C₃×#208	—	6	8	no	—
57	C₂×C₃×C₃⋊C₈	—	3	8	no	—
58	C₂²×C₃³	—	2	2	no	—
59	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
60	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
61	C₃³×C₄	C₄⋊C₄	1	2	yes	—
62	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
63	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
64	C₃²×Dic₃	—	2	2	no	—
65	C₃²×Dic₃	M₁₆	1	4	yes	complemented
66	C₂×C₃×M₁₆	—	9	1	no	—
67	C₂×C₃×C₄²	—	3	4	no	—
68	C₂×C₄×Dic₃	C₃⋊D₃	1	1	yes	characteristic
69	C₂×#208	—	3	4	no	—
70	C₂×#208	—	9	4	no	—
71	C₂×#208	—	9	1	no	—
72	C₂×#208	—	18	4	no	—
73	Dic₃⋊C₈	—	9	4	no	—
74	C₄⋊(C₃⋊C₈)	—	9	16	no	—
75	C₂³×C₃²	—	2	4	no	—
76	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
77	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
78	C₂×C₃²×C₄	—	2	4	no	—
79	C₂×C₃²×C₄	—	2	8	no	—
80	C₂×C₃²×C₄	—	2	8	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	3	2	no	—
83	C₂×C₃²×C₄	—	3	2	no	—
84	C₂×C₃²×C₄	—	6	4	no	—
85	C₂×C₃²×C₄	Dic₆	1	8	yes	—
86	C₂×C₃²×C₄	Dic₆	1	2	yes	—
87	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
88	C₂×C₃²×C₄	D₁₂	1	8	yes	—
89	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
90	C₂×C₃²×C₄	#67	1	2	yes	—
91	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₃×Dic₃	—	2	16	no	—
94	C₂×C₃×Dic₃	C₂×Dic₃	1	8	yes	—
95	C₂×C₃×Dic₃	C₂×Dic₃	1	8	yes	—
96	C₃⋊(C₃⋊C₈)	—	2	2	no	—
97	C₃⋊(C₃⋊C₈)	—	6	16	no	—
98	C₃⋊(C₃⋊C₈)	—	6	8	no	—
99	C₃⋊(C₃⋊C₈)	—	6	2	no	—
100	C₃×C₃⋊C₈	—	6	8	no	—
101	#423	—	27	1	no	Sylow
102	C₂×C₃³	—	2	2	no	—
103	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₃³	C₂×M₁₆	1	2	yes	—
105	C₂²×C₃×C₄	—	2	4	no	—
106	C₂²×C₃×C₄	—	3	8	no	—
107	C₂²×C₃×C₄	D₃²	1	4	yes	—
108	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₂²×Dic₃	C₃⋊Dic₃	1	2	yes	—
110	C₃×M₁₆	—	18	2	no	—
111	C₂×C₃×C₈	—	9	2	no	—
112	C₃×C₄²	—	3	4	no	—
113	C₃×C₄²	—	3	4	no	—
114	C₃×C₄²	—	3	8	no	—
115	C₄×Dic₃	C₃⋊Dic₃	1	1	yes	characteristic
116	C₄×Dic₃	C₃⋊Dic₃	1	1	yes	characteristic
117	C₄×Dic₃	C₂×C₃⋊D₃	1	2	yes	—
118	#208	—	6	8	no	—
119	#208	—	18	16	no	—
120	#208	—	18	8	no	—
121	#208	—	18	2	no	—
122	C₂×C₃⋊C₈	—	3	8	no	—
123	C₂×C₃⋊C₈	—	9	8	no	—
124	C₂×C₃⋊C₈	—	9	2	no	—
125	C₂×C₃⋊C₈	—	18	8	no	—
126	C₂²×C₃²	—	2	2	no	—
127	C₂²×C₃²	—	2	8	no	—
128	C₂²×C₃²	—	2	4	no	—
129	C₂²×C₃²	—	2	8	no	—
130	C₂²×C₃²	—	4	8	no	—
131	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
132	C₂²×C₃²	C₄⋊Dic₃	1	8	yes	—
133	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
135	C₃²×C₄	—	2	8	no	—
136	C₃²×C₄	—	2	4	no	—
137	C₃²×C₄	—	2	4	no	—
138	C₃²×C₄	—	3	4	no	—
139	C₃²×C₄	—	6	2	no	—
140	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
141	C₃²×C₄	C₄⋊Dic₃	1	8	yes	—
142	C₃²×C₄	C₂×Dic₆	1	4	yes	—
143	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
144	C₃²×C₄	C₂×D₁₂	1	4	yes	—
145	C₃²×C₄	C₂×#67	1	1	yes	characteristic
146	C₃×Dic₃	—	2	8	no	—
147	C₃×Dic₃	#208	1	16	yes	complemented
148	C₂×M₁₆	—	9	1	no	—
149	C₂×M₁₆	—	27	1	no	—
150	C₂×C₄²	—	3	1	no	—
151	C₄⋊C₈	—	27	4	no	—
152	C₃³	#423	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₃	—	2	4	no	—
154	C₂³×C₃	D₃×Dic₃	1	4	yes	—
155	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	8	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	3	8	no	—
161	C₂×C₃×C₄	—	3	8	no	—
162	C₂×C₃×C₄	—	6	16	no	—
163	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
164	C₂×C₃×C₄	#626	1	8	yes	—
165	C₂×C₃×C₄	#627	1	8	yes	—
166	C₂×C₃×C₄	#634	1	2	yes	—
167	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
170	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
171	C₂×Dic₃	—	2	4	no	—
172	C₂×Dic₃	C₂×C₃⋊Dic₃	1	2	yes	—
173	C₂×Dic₃	C₂×C₃⋊Dic₃	1	2	yes	—
174	C₃×C₈	—	18	2	no	—
175	C₃⋊C₈	—	6	8	no	—
176	C₃⋊C₈	—	18	16	no	—
177	C₃⋊C₈	—	18	8	no	—
178	C₃⋊C₈	—	18	2	no	—
179	C₂×C₃²	—	2	2	no	—
180	C₂×C₃²	—	2	8	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	8	no	—
183	C₂×C₃²	—	4	8	no	—
184	C₂×C₃²	D₃×M₁₆	1	2	yes	—
185	C₂×C₃²	C₂×#208	1	8	yes	—
186	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
187	C₂×C₃²	C₂×C₄⋊Dic₃	1	4	yes	—
188	C₂²×C₄	—	3	2	no	—
189	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
190	M₁₆	—	18	2	no	—
191	M₁₆	—	54	2	no	—
192	C₂×C₈	—	9	2	no	—
193	C₂×C₈	—	27	2	no	—
194	C₄²	—	3	1	no	—
195	C₄²	—	3	1	no	—
196	C₄²	—	3	2	no	—
197	C₂²×C₃	—	2	8	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	8	no	—
201	C₂²×C₃	—	4	8	no	—
202	C₂²×C₃	Dic₃⋊Dic₃	1	8	yes	—
203	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
204	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
205	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	8	no	—
209	C₃×C₄	—	3	16	no	—
210	C₃×C₄	—	6	8	no	—
211	C₃×C₄	Dic₃⋊Dic₃	1	8	yes	—
212	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
213	C₃×C₄	C₂×#626	1	4	yes	—
214	C₃×C₄	C₂×#627	1	4	yes	—
215	C₃×C₄	C₂×#634	1	1	yes	characteristic
216	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
217	Dic₃	—	2	2	no	—
218	Dic₃	#3748	1	4	yes	complemented
219	C₃²	—	2	4	no	—
220	C₃²	#4924	1	4	yes	complemented
221	C₃²	#5118	1	1	yes	characteristic, complemented
222	C₂³	D₃×C₃⋊Dic₃	1	1	yes	characteristic
223	C₂×C₄	—	3	2	no	—
224	C₂×C₄	—	3	2	no	—
225	C₂×C₄	—	6	4	no	—
226	C₂×C₄	D₃×C₃⋊Dic₃	1	1	yes	characteristic
227	C₂×C₄	#6292	1	2	yes	—
228	C₂×C₄	#6293	1	2	yes	—
229	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
230	C₈	—	18	2	no	—
231	C₈	—	54	2	no	—
232	C₂×C₃	—	2	2	no	—
233	C₂×C₃	—	2	8	no	—
234	C₂×C₃	—	2	8	no	—
235	C₂×C₃	—	2	4	no	—
236	C₂×C₃	—	4	8	no	—
237	C₂×C₃	D₃×#208	1	8	yes	—
238	C₂×C₃	C₂×Dic₃⋊Dic₃	1	4	yes	—
239	C₂×C₃	C₂×#3748	1	2	yes	—
240	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	Dic₃⋊(C₃⋊Dic₃)	1	2	yes	—
243	C₂²	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
244	C₄	—	3	4	no	—
245	C₄	—	6	2	no	—
246	C₄	Dic₃⋊(C₃⋊Dic₃)	1	2	yes	—
247	C₄	C₂×#6292	1	1	yes	characteristic
248	C₄	C₂×#6293	1	1	yes	characteristic
249	C₃	—	2	4	no	—
250	C₃	#10591115	1	4	yes	complemented
251	C₃	#10592623	1	1	yes	characteristic, complemented
252	C₂	—	2	2	no	—
253	C₂	D₃×#3748	1	2	yes	—
254	C₂	C₂×Dic₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
commutator, socle	C₂²×C₃³	C₂²×C₄
Fitting	C₂²×C₃³×C₄	C₂²
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂²×C₄	D₃×C₃⋊D₃
hypocenter	C₃³	#423
LCS	G ▷ C₂²×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. Dic₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
65. C₃²×Dic₃	M₁₆	semidirect
147. C₃×Dic₃	#208	semidirect
152. C₃³	#423	semidirect
218. Dic₃	#3748	semidirect
220. C₃²	#4924	semidirect
221. C₃²	#5118	semidirect
250. C₃	#10591115	semidirect
251. C₃	#10592623	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
Dic₃⋊(C₃⋊(C₃⋊C₈))	2
#10592623	3
#10591115	3