Subgroups — #13144051

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720542	C₂	1	2	yes	maximal
4	C₈×C₃⋊Dic₉	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592532	—	3	1	no	maximal
7	#10588517	—	3	3	no	maximal
8	C₂×C₄×C₃⋊D₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₃⋊D₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊Dic₉	C₄	1	1	yes	characteristic
14	C₄×C₃⋊Dic₉	C₄	1	1	yes	characteristic
15	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
17	C₄²×C₃⋊D₃	—	3	1	no	—
18	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
19	#63647	—	3	2	no	—
20	C₈×C₃⋊Dic₃	—	3	2	no	—
21	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
22	C₄²×D₉	—	3	3	no	—
23	C₉×C₄⋊C₈	D₃	1	3	yes	complemented
24	(C₂×D₉)⋊C₈	—	3	6	no	—
25	C₈×Dic₉	—	3	6	no	—
26	C₄⋊(C₉⋊C₈)	—	3	3	no	—
27	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₃⋊D₉	—	2	1	no	—
35	C₄×C₃⋊D₉	—	2	1	no	—
36	C₄×C₃⋊D₉	—	2	2	no	—
37	C₄×C₃⋊D₉	C₈	1	4	yes	complemented
38	C₃×C₈×C₉	—	2	2	no	—
39	C₃⋊(C₉⋊C₈)	—	2	2	no	—
40	#4833	—	9	3	no	—
41	#4833	—	9	1	no	—
42	C₂×C₄×C₃⋊D₃	—	3	1	no	—
43	C₂×C₄×C₃⋊D₃	—	3	1	no	—
44	C₂×C₄×C₃⋊D₃	—	3	1	no	—
45	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
46	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₄×C₃⋊Dic₃	—	3	1	no	—
48	C₄×C₃⋊Dic₃	—	3	1	no	—
49	C₄×C₃⋊Dic₃	—	3	1	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
51	C₂×C₄×D₉	—	3	3	no	—
52	C₂×C₄×D₉	—	3	3	no	—
53	C₂×C₄×D₉	—	3	3	no	—
54	C₂×C₈×C₉	C₂×D₃	1	6	yes	—
55	C₄²×C₉	C₂×D₃	1	3	yes	—
56	C₄×Dic₉	—	3	3	no	—
57	C₄×Dic₉	—	3	3	no	—
58	C₄×Dic₉	—	3	3	no	—
59	C₂×C₉⋊C₈	—	3	6	no	—
60	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃⋊D₉	—	2	1	no	—
62	C₂×C₃⋊D₉	—	2	1	no	—
63	C₂×C₃⋊D₉	C₂×C₈	1	2	yes	—
64	C₃×C₄×C₉	—	2	1	no	—
65	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
66	C₃×C₄×C₉	#41	1	1	yes	characteristic
67	C₃×C₄×C₉	#41	1	1	yes	characteristic
68	C₃⋊Dic₉	—	2	1	no	—
69	C₃⋊Dic₉	C₂×C₈	1	2	yes	—
70	C₃⋊Dic₉	M₁₆	1	2	yes	—
71	C₄²×D₃	—	9	3	no	—
72	C₄²×D₃	—	9	1	no	—
73	C₃×C₄⋊C₈	D₉	1	3	yes	complemented
74	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
75	(C₂×D₃)⋊C₈	—	9	6	no	—
76	(C₂×D₃)⋊C₈	—	9	2	no	—
77	C₈×Dic₃	—	9	6	no	—
78	C₈×Dic₃	—	9	2	no	—
79	C₄⋊(C₃⋊C₈)	—	9	3	no	—
80	C₄⋊(C₃⋊C₈)	—	9	1	no	—
81	C₂²×C₃⋊D₃	—	3	1	no	—
82	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃⋊Dic₃	—	3	1	no	—
86	C₂×C₃⋊Dic₃	—	3	1	no	—
87	C₂×C₃⋊Dic₃	—	3	1	no	—
88	C₄×C₃⋊D₃	—	3	4	no	—
89	C₄×C₃⋊D₃	—	6	2	no	—
90	C₄×C₃⋊D₃	—	6	1	no	—
91	C₄×C₃⋊D₃	—	6	1	no	—
92	C₂²×D₉	—	3	3	no	—
93	C₃²×C₈	—	2	2	no	—
94	C₃⋊(C₃⋊C₈)	—	6	2	no	—
95	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
96	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
97	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
98	C₂×Dic₉	—	3	3	no	—
99	C₂×Dic₉	—	3	3	no	—
100	C₂×Dic₉	—	3	3	no	—
101	C₄×D₉	—	3	12	no	—
102	C₄×D₉	—	6	6	no	—
103	C₄×D₉	—	6	3	no	—
104	C₄×D₉	—	6	3	no	—
105	C₈×C₉	—	2	6	no	—
106	C₉⋊C₈	—	6	6	no	—
107	#431	—	27	1	no	Sylow
108	C₂×C₃×C₉	#117	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
111	C₃⋊D₉	—	2	2	no	—
112	C₂×C₄×D₃	—	9	3	no	—
113	C₂×C₄×D₃	—	9	1	no	—
114	C₂×C₄×D₃	—	9	3	no	—
115	C₂×C₄×D₃	—	9	3	no	—
116	C₂×C₄×D₃	—	9	1	no	—
117	C₂×C₄×D₃	—	9	1	no	—
118	C₂×C₃×C₈	C₂×D₉	1	6	yes	—
119	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
120	C₃×C₄²	C₂×D₉	1	3	yes	—
121	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
122	C₄×Dic₃	—	9	3	no	—
123	C₄×Dic₃	—	9	3	no	—
124	C₄×Dic₃	—	9	3	no	—
125	C₄×Dic₃	—	9	1	no	—
126	C₄×Dic₃	—	9	1	no	—
127	C₄×Dic₃	—	9	1	no	—
128	C₂×C₃⋊C₈	—	9	6	no	—
129	C₂×C₃⋊C₈	—	9	2	no	—
130	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₃⋊D₃	—	3	2	no	—
132	C₂×C₃⋊D₃	—	6	1	no	—
133	C₂×C₃⋊D₃	—	6	1	no	—
134	C₃²×C₄	—	2	1	no	—
135	C₃²×C₄	D₃×C₈	1	2	yes	—
136	C₃²×C₄	#237	1	1	yes	characteristic
137	C₃²×C₄	#239	1	1	yes	characteristic
138	C₃⋊Dic₃	—	3	2	no	—
139	C₃⋊Dic₃	—	3	2	no	—
140	C₃⋊Dic₃	—	6	1	no	—
141	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
142	C₂×D₉	—	3	6	no	—
143	C₂×D₉	—	6	3	no	—
144	C₂×D₉	—	6	3	no	—
145	C₄×C₉	—	2	3	no	—
146	C₄×C₉	D₃×C₈	1	6	yes	—
147	C₄×C₉	#237	1	3	yes	—
148	C₄×C₉	#239	1	3	yes	—
149	Dic₉	—	3	6	no	—
150	Dic₉	—	3	6	no	—
151	Dic₉	—	6	3	no	—
152	C₂×C₄²	—	27	1	no	—
153	C₄⋊C₈	—	27	1	no	—
154	C₄⋊C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
155	C₂²⋊C₈	—	27	2	no	—
156	C₄×C₈	—	27	2	no	—
157	C₃×C₉	#431	1	1	yes	characteristic, complemented, Sylow
158	C₂²×D₃	—	9	3	no	—
159	C₂²×D₃	—	9	1	no	—
160	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
161	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
162	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
163	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
166	C₂×Dic₃	—	9	3	no	—
167	C₂×Dic₃	—	9	3	no	—
168	C₂×Dic₃	—	9	1	no	—
169	C₂×Dic₃	—	9	1	no	—
170	C₂×Dic₃	—	9	3	no	—
171	C₂×Dic₃	—	9	1	no	—
172	C₄×D₃	—	9	12	no	—
173	C₄×D₃	—	9	4	no	—
174	C₄×D₃	—	18	2	no	—
175	C₄×D₃	—	18	6	no	—
176	C₄×D₃	—	18	3	no	—
177	C₄×D₃	—	18	3	no	—
178	C₄×D₃	—	18	1	no	—
179	C₄×D₃	—	18	1	no	—
180	C₃×C₈	—	2	6	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃⋊C₈	—	18	6	no	—
183	C₃⋊C₈	—	18	2	no	—
184	C₂×C₃²	#892	1	1	yes	characteristic
185	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
186	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
187	C₃⋊D₃	—	6	2	no	—
188	C₂×C₉	#892	1	3	yes	—
189	C₂×C₉	C₂×D₃×C₈	1	3	yes	—
190	C₂×C₉	D₃×M₁₆	1	3	yes	—
191	D₉	—	6	6	no	—
192	C₂²×C₄	—	27	1	no	—
193	C₂²×C₄	—	27	1	no	—
194	C₂²×C₄	—	27	1	no	—
195	C₂×C₈	—	27	2	no	—
196	C₂×C₈	C₂×C₃⋊D₉	1	2	yes	—
197	C₄²	—	27	1	no	—
198	C₄²	—	27	1	no	—
199	C₄²	—	27	1	no	—
200	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
201	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
202	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
203	C₂×D₃	—	9	2	no	—
204	C₂×D₃	—	9	6	no	—
205	C₂×D₃	—	18	1	no	—
206	C₂×D₃	—	18	1	no	—
207	C₂×D₃	—	18	3	no	—
208	C₂×D₃	—	18	3	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	3	no	—
211	C₃×C₄	C₈×D₉	1	6	yes	—
212	C₃×C₄	#3699	1	3	yes	—
213	C₃×C₄	#3701	1	3	yes	—
214	C₃×C₄	C₈×C₃⋊D₃	1	2	yes	—
215	C₃×C₄	#3830	1	1	yes	characteristic
216	C₃×C₄	#3832	1	1	yes	characteristic
217	Dic₃	—	9	2	no	—
218	Dic₃	—	9	2	no	—
219	Dic₃	—	9	6	no	—
220	Dic₃	—	9	6	no	—
221	Dic₃	—	18	1	no	—
222	Dic₃	—	18	3	no	—
223	C₃²	#4833	1	1	yes	characteristic
224	C₉	#4833	1	3	yes	complemented
225	C₂³	—	27	1	no	—
226	C₂×C₄	—	27	1	no	—
227	C₂×C₄	—	27	1	no	—
228	C₂×C₄	—	27	4	no	—
229	C₂×C₄	—	27	1	no	—
230	C₂×C₄	—	54	2	no	—
231	C₂×C₄	—	54	1	no	—
232	C₂×C₄	—	54	1	no	—
233	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
234	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	54	2	no	—
238	C₂×C₃	#63454	1	3	yes	—
239	C₂×C₃	C₂×C₈×D₉	1	3	yes	—
240	C₂×C₃	M₁₆×D₉	1	3	yes	—
241	C₂×C₃	#64101	1	1	yes	characteristic
242	C₂×C₃	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
243	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
244	D₃	—	18	2	no	—
245	D₃	—	18	6	no	—
246	C₂²	—	27	2	no	—
247	C₂²	—	54	1	no	—
248	C₂²	—	54	1	no	—
249	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	27	2	no	—
252	C₄	—	27	2	no	—
253	C₄	—	54	1	no	—
254	C₄	C₈×C₃⋊D₉	1	2	yes	—
255	C₄	#88742	1	1	yes	characteristic
256	C₄	#88746	1	1	yes	characteristic
257	C₃	#10588517	1	3	yes	complemented
258	C₃	#10592532	1	1	yes	characteristic
259	C₂	—	54	2	no	—
260	C₂	#11721748	1	1	yes	characteristic
261	C₂	C₂×C₈×C₃⋊D₉	1	1	yes	characteristic
262	C₂	M₁₆×C₃⋊D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃⋊D₉	C₂×C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄⋊C₈	C₃⋊D₉
hypocenter	C₃×C₉	#431
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₄⋊C₈	C₂	semidirect
4. C₈×C₃⋊Dic₉	C₂	semidirect
5. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
23. C₉×C₄⋊C₈	D₃	semidirect
37. C₄×C₃⋊D₉	C₈	semidirect
73. C₃×C₄⋊C₈	D₉	semidirect
154. C₄⋊C₈	C₃⋊D₉	semidirect
157. C₃×C₉	#431	semidirect
224. C₉	#4833	semidirect
257. C₃	#10588517	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₃⋊D₉	2
C₃×C₉×C₄⋊C₈	2
#11720542	2
C₈×C₃⋊Dic₉	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592532	3
#10588517	3