Subgroups — #13144057

Order 1728.

480 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11721709	C₂	1	1	yes	maximal, characteristic
2	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720542	C₂	1	1	yes	maximal, characteristic
4	#11720542	C₂	1	1	yes	maximal, characteristic
5	#11720538	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592539	—	3	1	no	maximal
9	#10588524	—	3	3	no	maximal
10	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	#88536	C₄	1	1	yes	characteristic
15	#88536	C₄	1	1	yes	characteristic
16	#88534	C₄	1	1	yes	characteristic
17	#88534	C₄	1	1	yes	characteristic
18	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
21	#64081	—	3	1	no	—
22	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
23	#63647	—	3	1	no	—
24	#63647	—	3	1	no	—
25	#63642	—	3	1	no	—
26	C₈×C₃⋊Dic₃	—	3	1	no	—
27	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
28	#63434	—	3	3	no	—
29	C₉×C₄⋊C₈	D₃	1	3	yes	complemented
30	(C₂×D₉)⋊C₈	—	3	3	no	—
31	(C₂×D₉)⋊C₈	—	3	3	no	—
32	#63375	—	3	3	no	—
33	C₈×Dic₉	—	3	3	no	—
34	C₄⋊(C₉⋊C₈)	—	3	3	no	—
35	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₃⋊D₉	—	2	1	no	—
43	C₄×C₃⋊D₉	—	2	1	no	—
44	C₃×C₈×C₉	—	2	1	no	—
45	C₃×C₈×C₉	—	2	1	no	—
46	C₃⋊(C₉⋊C₈)	—	2	1	no	—
47	C₃⋊(C₉⋊C₈)	—	2	1	no	—
48	#4840	—	9	3	no	—
49	#4840	—	9	1	no	—
50	C₂×C₄×C₃⋊D₃	—	3	1	no	—
51	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
54	#3752	—	3	1	no	—
55	#3752	—	3	1	no	—
56	#3750	—	3	1	no	—
57	#3750	—	3	1	no	—
58	C₄×C₃⋊Dic₃	—	3	1	no	—
59	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
60	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
61	C₂×C₄×D₉	—	3	3	no	—
62	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
63	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
64	C₄²×C₉	C₂×D₃	1	3	yes	—
65	(C₂×D₉)⋊C₄	—	3	3	no	—
66	(C₂×D₉)⋊C₄	—	3	3	no	—
67	Dic₉⋊C₄	—	3	3	no	—
68	Dic₉⋊C₄	—	3	3	no	—
69	C₄×Dic₉	—	3	3	no	—
70	C₂×C₉⋊C₈	—	3	3	no	—
71	C₂×C₉⋊C₈	—	3	3	no	—
72	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃⋊D₉	—	2	1	no	—
74	C₂×C₃⋊D₉	—	2	1	no	—
75	C₂×C₃⋊D₉	—	2	1	no	—
76	C₃×C₄×C₉	—	2	1	no	—
77	C₃×C₄×C₉	—	2	1	no	—
78	C₃×C₄×C₉	#41	1	1	yes	characteristic
79	C₃×C₄×C₉	#41	1	1	yes	characteristic
80	C₃⋊Dic₉	—	2	1	no	—
81	C₃⋊Dic₉	—	2	1	no	—
82	C₃⋊Dic₉	—	2	1	no	—
83	#872	—	9	3	no	—
84	#872	—	9	1	no	—
85	C₃×C₄⋊C₈	D₉	1	3	yes	complemented
86	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
87	(C₂×D₃)⋊C₈	—	9	3	no	—
88	(C₂×D₃)⋊C₈	—	9	3	no	—
89	(C₂×D₃)⋊C₈	—	9	1	no	—
90	(C₂×D₃)⋊C₈	—	9	1	no	—
91	#815	—	9	3	no	—
92	#815	—	9	1	no	—
93	C₈×Dic₃	—	9	3	no	—
94	C₈×Dic₃	—	9	1	no	—
95	C₄⋊(C₃⋊C₈)	—	9	3	no	—
96	C₄⋊(C₃⋊C₈)	—	9	1	no	—
97	C₂²×C₃⋊D₃	—	3	1	no	—
98	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂×C₃⋊Dic₃	—	3	1	no	—
102	C₂×C₃⋊Dic₃	—	3	1	no	—
103	C₂×C₃⋊Dic₃	—	3	1	no	—
104	C₄×C₃⋊D₃	—	6	1	no	—
105	C₄×C₃⋊D₃	—	6	1	no	—
106	C₂²×D₉	—	3	3	no	—
107	C₃²×C₈	—	2	1	no	—
108	C₃²×C₈	—	2	1	no	—
109	C₃⋊(C₃⋊C₈)	—	6	1	no	—
110	C₃⋊(C₃⋊C₈)	—	6	1	no	—
111	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
112	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
113	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
114	C₂×Dic₉	—	3	3	no	—
115	C₂×Dic₉	—	3	3	no	—
116	C₂×Dic₉	—	3	3	no	—
117	C₄×D₉	—	6	3	no	—
118	C₄×D₉	—	6	3	no	—
119	C₈×C₉	—	2	3	no	—
120	C₈×C₉	—	2	3	no	—
121	C₉⋊C₈	—	6	3	no	—
122	C₉⋊C₈	—	6	3	no	—
123	#433	—	27	1	no	Sylow
124	C₂×C₃×C₉	#117	1	1	yes	characteristic
125	C₂×C₃×C₉	#131	1	1	yes	characteristic
126	C₂×C₃×C₉	#131	1	1	yes	characteristic
127	C₃⋊D₉	—	4	1	no	—
128	C₂×C₄×D₃	—	9	3	no	—
129	C₂×C₄×D₃	—	9	1	no	—
130	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
131	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
132	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
134	C₃×C₄²	C₂×D₉	1	3	yes	—
135	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	—	9	3	no	—
137	(C₂×D₃)⋊C₄	—	9	3	no	—
138	(C₂×D₃)⋊C₄	—	9	1	no	—
139	(C₂×D₃)⋊C₄	—	9	1	no	—
140	Dic₃⋊C₄	—	9	3	no	—
141	Dic₃⋊C₄	—	9	3	no	—
142	Dic₃⋊C₄	—	9	1	no	—
143	Dic₃⋊C₄	—	9	1	no	—
144	C₄×Dic₃	—	9	3	no	—
145	C₄×Dic₃	—	9	1	no	—
146	C₂×C₃⋊C₈	—	9	3	no	—
147	C₂×C₃⋊C₈	—	9	3	no	—
148	C₂×C₃⋊C₈	—	9	1	no	—
149	C₂×C₃⋊C₈	—	9	1	no	—
150	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×C₃⋊D₃	—	6	1	no	—
152	C₂×C₃⋊D₃	—	6	1	no	—
153	C₂×C₃⋊D₃	—	6	1	no	—
154	C₃²×C₄	—	2	1	no	—
155	C₃²×C₄	—	2	1	no	—
156	C₃²×C₄	#237	1	1	yes	characteristic
157	C₃²×C₄	#239	1	1	yes	characteristic
158	C₃⋊Dic₃	—	6	1	no	—
159	C₃⋊Dic₃	—	6	1	no	—
160	C₃⋊Dic₃	—	6	1	no	—
161	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
162	C₂×D₉	—	6	3	no	—
163	C₂×D₉	—	6	3	no	—
164	C₂×D₉	—	6	3	no	—
165	C₄×C₉	—	2	3	no	—
166	C₄×C₉	—	2	3	no	—
167	C₄×C₉	#237	1	3	yes	—
168	C₄×C₉	#239	1	3	yes	—
169	Dic₉	—	6	3	no	—
170	Dic₉	—	6	3	no	—
171	Dic₉	—	6	3	no	—
172	#117	—	27	1	no	—
173	C₄⋊C₈	—	27	1	no	—
174	C₄⋊C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
175	C₂²⋊C₈	—	27	1	no	—
176	C₂²⋊C₈	—	27	1	no	—
177	#97	—	27	1	no	—
178	C₄×C₈	—	27	1	no	—
179	C₃×C₉	#433	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₃	—	9	3	no	—
181	C₂²×D₃	—	9	1	no	—
182	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
183	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
184	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
185	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
188	C₂×Dic₃	—	9	3	no	—
189	C₂×Dic₃	—	9	3	no	—
190	C₂×Dic₃	—	9	3	no	—
191	C₂×Dic₃	—	9	1	no	—
192	C₂×Dic₃	—	9	1	no	—
193	C₂×Dic₃	—	9	1	no	—
194	C₄×D₃	—	18	3	no	—
195	C₄×D₃	—	18	3	no	—
196	C₄×D₃	—	18	1	no	—
197	C₄×D₃	—	18	1	no	—
198	C₃×C₈	—	2	3	no	—
199	C₃×C₈	—	2	3	no	—
200	C₃×C₈	—	2	1	no	—
201	C₃×C₈	—	2	1	no	—
202	C₃⋊C₈	—	18	3	no	—
203	C₃⋊C₈	—	18	3	no	—
204	C₃⋊C₈	—	18	1	no	—
205	C₃⋊C₈	—	18	1	no	—
206	C₂×C₃²	#892	1	1	yes	characteristic
207	C₂×C₃²	#901	1	1	yes	characteristic
208	C₂×C₃²	#907	1	1	yes	characteristic
209	C₃⋊D₃	—	12	1	no	—
210	C₂×C₉	#892	1	3	yes	—
211	C₂×C₉	#901	1	3	yes	—
212	C₂×C₉	#907	1	3	yes	—
213	D₉	—	12	3	no	—
214	C₂²×C₄	—	27	1	no	—
215	C₂×C₈	—	27	1	no	—
216	C₂×C₈	—	27	1	no	—
217	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
218	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
219	C₄⋊C₄	—	27	1	no	—
220	C₄⋊C₄	—	27	1	no	—
221	C₂²⋊C₄	—	27	1	no	—
222	C₂²⋊C₄	—	27	1	no	—
223	C₄²	—	27	1	no	—
224	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
225	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
226	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
227	C₂×D₃	—	18	1	no	—
228	C₂×D₃	—	18	1	no	—
229	C₂×D₃	—	18	1	no	—
230	C₂×D₃	—	18	3	no	—
231	C₂×D₃	—	18	3	no	—
232	C₂×D₃	—	18	3	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	3	no	—
236	C₃×C₄	—	2	3	no	—
237	C₃×C₄	#3699	1	3	yes	—
238	C₃×C₄	#3701	1	3	yes	—
239	C₃×C₄	#3830	1	1	yes	characteristic
240	C₃×C₄	#3832	1	1	yes	characteristic
241	Dic₃	—	18	1	no	—
242	Dic₃	—	18	1	no	—
243	Dic₃	—	18	1	no	—
244	Dic₃	—	18	3	no	—
245	Dic₃	—	18	3	no	—
246	Dic₃	—	18	3	no	—
247	C₃²	#4840	1	1	yes	characteristic
248	C₉	#4840	1	3	yes	complemented
249	C₂³	—	27	1	no	—
250	C₂×C₄	—	27	1	no	—
251	C₂×C₄	—	27	1	no	—
252	C₂×C₄	—	27	1	no	—
253	C₂×C₄	—	54	1	no	—
254	C₂×C₄	—	54	1	no	—
255	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
256	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	54	1	no	—
261	C₈	—	54	1	no	—
262	C₂×C₃	#63454	1	3	yes	—
263	C₂×C₃	#63464	1	3	yes	—
264	C₂×C₃	#63469	1	3	yes	—
265	C₂×C₃	#64101	1	1	yes	characteristic
266	C₂×C₃	#64110	1	1	yes	characteristic
267	C₂×C₃	#64116	1	1	yes	characteristic
268	D₃	—	36	1	no	—
269	D₃	—	36	3	no	—
270	C₂²	—	54	1	no	—
271	C₂²	—	54	1	no	—
272	C₂²	—	54	1	no	—
273	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	54	1	no	—
277	C₄	—	54	1	no	—
278	C₄	—	54	1	no	—
279	C₄	#88742	1	1	yes	characteristic
280	C₄	#88746	1	1	yes	characteristic
281	C₃	#10588524	1	3	yes	complemented
282	C₃	#10592539	1	1	yes	characteristic
283	C₂	—	108	1	no	—
284	C₂	#11721748	1	1	yes	characteristic
285	C₂	#11721770	1	1	yes	characteristic
286	C₂	#11721782	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
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	#433
LCS	G ▷ C₂²×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
5. #11720538	C₂	semidirect
6. C₈×C₃⋊Dic₉	C₂	semidirect
7. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
29. C₉×C₄⋊C₈	D₃	semidirect
85. C₃×C₄⋊C₈	D₉	semidirect
174. C₄⋊C₈	C₃⋊D₉	semidirect
179. C₃×C₉	#433	semidirect
248. C₉	#4840	semidirect
281. C₃	#10588524	semidirect

Maximal subgroups

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

subgroup	index
#11721709	2
C₃×C₉×C₄⋊C₈	2
#11720542	2
#11720538	2
C₈×C₃⋊Dic₉	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592539	3
#10588524	3