Subgroups — #13160278

Order 1728.

1052 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₃²×#926	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	#10592696	—	3	1	no	maximal
7	#10591055	—	3	4	no	maximal
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	#88855	C₄	1	4	yes	complemented
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
11	C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
12	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₃²×#217	C₄	1	2	yes	—
15	C₃²×C₄⋊Dic₃	C₄	1	2	yes	—
16	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
17	C₃²×#117	—	3	1	no	—
18	C₂×#3748	—	3	1	no	—
19	C₂×#3748	—	3	4	no	—
20	C₂×#3748	—	6	4	no	—
21	C₂²×C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
22	C₃×#926	D₃	1	4	yes	—
23	C₂²×C₃×C₃⋊C₈	—	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	C₂×C₄	1	4	yes	—
34	C₃⋊(C₃⋊(C₃⋊C₈))	C₂×C₄	1	4	yes	—
35	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
36	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
37	#5101	—	9	1	no	—
38	#4997	—	9	4	no	—
39	C₂²×C₃²×C₄	—	2	4	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₃²×C₄⋊C₄	—	3	2	no	—
43	C₃²×C₂²⋊C₄	—	3	2	no	—
44	C₃²×C₄²	—	3	2	no	—
45	#3748	—	3	4	no	—
46	#3748	—	3	16	no	—
47	#3748	—	6	16	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
51	C₂×C₃⋊(C₃⋊C₈)	Dic₃	1	4	yes	complemented
52	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
54	C₃×#217	Dic₃	1	8	yes	—
55	C₃×C₄⋊Dic₃	Dic₃	1	8	yes	—
56	C₃×C₄×Dic₃	C₂×D₃	1	8	yes	—
57	C₂×C₃×C₃⋊C₈	—	3	16	no	—
58	C₂×C₃×C₃⋊C₈	—	3	4	no	—
59	C₂×C₃×C₃⋊C₈	—	3	4	no	—
60	C₂²×C₃³	—	2	2	no	—
61	C₂²×C₃³	C₄²	1	2	yes	—
62	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
63	C₃³×C₄	C₄²	1	2	yes	—
64	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃²×Dic₃	—	2	4	no	—
67	C₂²×C₃×C₈	—	9	1	no	—
68	C₃×#117	—	3	4	no	—
69	#926	C₃⋊D₃	1	1	yes	characteristic
70	C₂×#208	—	9	4	no	—
71	C₂×#208	—	9	1	no	—
72	C₂×#208	—	18	4	no	—
73	C₂²×C₃⋊C₈	—	3	4	no	—
74	#815	—	9	2	no	—
75	C₈×Dic₃	—	9	2	no	—
76	#803	—	9	8	no	—
77	C₄×C₃⋊C₈	—	9	8	no	—
78	C₂³×C₃²	—	2	4	no	—
79	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
80	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	8	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	3	4	no	—
86	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
87	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
88	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
90	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
91	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
93	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂×C₃×Dic₃	C₂×Dic₃	1	16	yes	—
95	C₃⋊(C₃⋊C₈)	—	3	4	no	—
96	C₃⋊(C₃⋊C₈)	—	3	16	no	—
97	C₃⋊(C₃⋊C₈)	—	6	16	no	—
98	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	2	yes	—
99	C₃⋊(C₃⋊C₈)	C₂×Dic₃	1	2	yes	—
100	C₃×C₃⋊C₈	—	3	8	no	—
101	C₃×C₃⋊C₈	—	3	8	no	—
102	#405	—	27	1	no	Sylow
103	C₂×C₃³	—	2	2	no	—
104	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₃³	#131	1	2	yes	—
106	C₂²×C₃×C₄	—	2	4	no	—
107	C₂²×C₃×C₄	D₃²	1	4	yes	—
108	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₂×C₃×C₈	—	9	4	no	—
110	C₂×C₃×C₈	—	9	1	no	—
111	C₂×C₃×C₈	—	9	1	no	—
112	C₃×C₄⋊C₄	—	3	8	no	—
113	C₃×C₂²⋊C₄	—	3	8	no	—
114	C₃×C₄²	—	3	8	no	—
115	#217	C₃⋊Dic₃	1	2	yes	—
116	C₄⋊Dic₃	C₃⋊Dic₃	1	2	yes	—
117	C₄×Dic₃	C₂×C₃⋊D₃	1	2	yes	—
118	#208	—	9	16	no	—
119	#208	—	9	4	no	—
120	#208	—	18	16	no	—
121	C₂×C₃⋊C₈	—	3	16	no	—
122	C₂×C₃⋊C₈	—	3	4	no	—
123	C₂×C₃⋊C₈	—	3	4	no	—
124	C₂×C₃⋊C₈	—	9	8	no	—
125	C₂×C₃⋊C₈	—	9	2	no	—
126	C₂×C₃⋊C₈	—	18	8	no	—
127	C₂²×C₃²	—	2	2	no	—
128	C₂²×C₃²	—	2	8	no	—
129	C₂²×C₃²	—	2	4	no	—
130	C₂²×C₃²	—	2	8	no	—
131	C₂²×C₃²	—	4	8	no	—
132	C₂²×C₃²	C₄×Dic₃	1	8	yes	—
133	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
134	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
136	C₃²×C₄	—	2	8	no	—
137	C₃²×C₄	—	2	4	no	—
138	C₃²×C₄	—	2	4	no	—
139	C₃²×C₄	—	6	4	no	—
140	C₃²×C₄	C₄×Dic₃	1	8	yes	—
141	C₃²×C₄	C₄×Dic₃	1	2	yes	—
142	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
143	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
144	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
145	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
146	C₃×Dic₃	—	2	16	no	—
147	C₂×M₁₆	—	27	1	no	—
148	C₂²×C₈	—	9	1	no	—
149	#117	—	3	1	no	—
150	#97	—	27	2	no	—
151	C₄×C₈	—	27	2	no	—
152	C₃³	#405	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	16	no	—
161	C₂×C₃×C₄	D₃×Dic₃	1	4	yes	—
162	C₂×C₃×C₄	D₃×Dic₃	1	8	yes	—
163	C₂×C₃×C₄	#624	1	8	yes	—
164	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
165	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
168	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
169	C₂×Dic₃	C₂×C₃⋊Dic₃	1	4	yes	—
170	C₃×C₈	—	9	2	no	—
171	C₃×C₈	—	9	2	no	—
172	C₃⋊C₈	—	3	8	no	—
173	C₃⋊C₈	—	3	8	no	—
174	C₃⋊C₈	—	9	16	no	—
175	C₃⋊C₈	—	9	4	no	—
176	C₃⋊C₈	—	18	16	no	—
177	C₂×C₃²	—	2	2	no	—
178	C₂×C₃²	—	2	8	no	—
179	C₂×C₃²	—	2	4	no	—
180	C₂×C₃²	—	2	8	no	—
181	C₂×C₃²	—	4	8	no	—
182	C₂×C₃²	#901	1	2	yes	—
183	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
184	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
185	C₂×C₃²	#948	1	8	yes	—
186	C₂²×C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
187	M₁₆	—	27	4	no	—
188	C₂×C₈	—	9	4	no	—
189	C₂×C₈	—	9	1	no	—
190	C₂×C₈	—	9	1	no	—
191	C₂×C₈	—	27	2	no	—
192	C₄⋊C₄	—	3	2	no	—
193	C₂²⋊C₄	—	3	2	no	—
194	C₄²	—	3	2	no	—
195	C₂²×C₃	—	2	8	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	2	8	no	—
199	C₂²×C₃	—	4	8	no	—
200	C₂²×C₃	Dic₃²	1	8	yes	—
201	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
202	C₂²×C₃	C₂×D₃×Dic₃	1	4	yes	—
203	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	2	4	no	—
206	C₃×C₄	—	2	8	no	—
207	C₃×C₄	—	6	16	no	—
208	C₃×C₄	Dic₃²	1	8	yes	—
209	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
210	C₃×C₄	C₂×D₃×Dic₃	1	4	yes	—
211	C₃×C₄	C₂×#624	1	4	yes	—
212	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
213	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
214	Dic₃	—	2	4	no	—
215	C₃²	—	2	4	no	—
216	C₃²	#4997	1	4	yes	complemented
217	C₃²	#5101	1	1	yes	characteristic, complemented
218	C₂³	D₃×C₃⋊Dic₃	1	1	yes	characteristic
219	C₂×C₄	—	3	4	no	—
220	C₂×C₄	D₃×C₃⋊Dic₃	1	1	yes	characteristic
221	C₂×C₄	Dic₃×C₃⋊D₃	1	2	yes	—
222	C₂×C₄	#6289	1	2	yes	—
223	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
224	C₈	—	9	2	no	—
225	C₈	—	9	2	no	—
226	C₈	—	27	4	no	—
227	C₂×C₃	—	2	8	no	—
228	C₂×C₃	—	2	2	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	2	8	no	—
231	C₂×C₃	—	4	8	no	—
232	C₂×C₃	#63814	1	8	yes	—
233	C₂×C₃	C₂×Dic₃²	1	4	yes	—
234	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
235	C₂×C₃	#64157	1	2	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	Dic₃×C₃⋊Dic₃	1	2	yes	—
238	C₂²	C₂×D₃×C₃⋊Dic₃	1	1	yes	characteristic
239	C₄	—	6	4	no	—
240	C₄	Dic₃×C₃⋊Dic₃	1	2	yes	—
241	C₄	C₂×Dic₃×C₃⋊D₃	1	1	yes	characteristic
242	C₄	C₂×#6289	1	1	yes	characteristic
243	C₃	—	2	4	no	—
244	C₃	#10591055	1	4	yes	complemented
245	C₃	#10592696	1	1	yes	characteristic, complemented
246	C₂	—	2	2	no	—
247	C₂	#11723349	1	2	yes	—
248	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	#405
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
9. #88855	C₄	semidirect
11. C₂×C₃×C₃⋊(C₃⋊C₈)	C₄	semidirect
51. C₂×C₃⋊(C₃⋊C₈)	Dic₃	semidirect
152. C₃³	#405	semidirect
216. C₃²	#4997	semidirect
217. C₃²	#5101	semidirect
244. C₃	#10591055	semidirect
245. C₃	#10592696	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
C₂²×C₃×C₃⋊(C₃⋊C₈)	2
C₃²×#926	2
#11722019	2
Dic₃×C₃⋊(C₃⋊C₈)	2
#10592696	3
#10591055	3