Subgroups — #13144056

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88523	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
3	#11721709	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720537	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720537	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592538	—	3	1	no	maximal
9	#10588523	—	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	#88523	—	2	2	no	—
22	C₈×C₃⋊D₉	—	2	2	no	—
23	C₂×#3743	—	3	1	no	—
24	C₂×C₈×C₃⋊D₃	—	3	1	no	—
25	#64081	—	3	1	no	—
26	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
27	#63641	—	3	1	no	—
28	#63641	—	3	1	no	—
29	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
30	C₂×#3663	—	3	3	no	—
31	C₂×C₈×D₉	—	3	3	no	—
32	#63434	—	3	3	no	—
33	C₉×C₄⋊C₈	D₃	1	3	yes	complemented
34	Dic₉⋊C₈	—	3	3	no	—
35	Dic₉⋊C₈	—	3	3	no	—
36	C₄⋊(C₉⋊C₈)	—	3	3	no	—
37	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₃⋊D₉	D₄	1	2	yes	—
45	C₄×C₃⋊D₉	Q₈	1	2	yes	—
46	C₃×C₈×C₉	—	2	1	no	—
47	C₃×C₈×C₉	—	2	1	no	—
48	C₃⋊(C₉⋊C₈)	—	2	1	no	—
49	C₃⋊(C₉⋊C₈)	—	2	1	no	—
50	#4839	—	9	3	no	—
51	#4839	—	9	1	no	—
52	C₂×C₄×C₃⋊D₃	—	3	1	no	—
53	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
55	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
56	#3752	—	3	1	no	—
57	#3752	—	3	1	no	—
58	#3750	—	3	1	no	—
59	#3750	—	3	1	no	—
60	C₄×C₃⋊Dic₃	—	3	1	no	—
61	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
62	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
63	#3743	—	6	2	no	—
64	C₈×C₃⋊D₃	—	6	2	no	—
65	C₂×C₄×D₉	—	3	3	no	—
66	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
67	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
68	C₄²×C₉	C₂×D₃	1	3	yes	—
69	(C₂×D₉)⋊C₄	—	3	3	no	—
70	(C₂×D₉)⋊C₄	—	3	3	no	—
71	Dic₉⋊C₄	—	3	3	no	—
72	Dic₉⋊C₄	—	3	3	no	—
73	C₄×Dic₉	—	3	3	no	—
74	C₂×C₉⋊C₈	—	3	3	no	—
75	C₂×C₉⋊C₈	—	3	3	no	—
76	#3663	—	6	6	no	—
77	C₈×D₉	—	6	6	no	—
78	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃⋊D₉	—	2	1	no	—
80	C₂×C₃⋊D₉	—	2	1	no	—
81	C₂×C₃⋊D₉	C₄⋊C₄	1	2	yes	—
82	C₃×C₄×C₉	—	2	1	no	—
83	C₃×C₄×C₉	—	2	1	no	—
84	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
86	C₃⋊Dic₉	—	2	1	no	—
87	C₃⋊Dic₉	—	2	1	no	—
88	C₃⋊Dic₉	C₄⋊C₄	1	2	yes	—
89	C₂×ΓL(1,𝔽₂₅)	—	9	3	no	—
90	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
91	C₂×D₃×C₈	—	9	3	no	—
92	C₂×D₃×C₈	—	9	1	no	—
93	#872	—	9	3	no	—
94	#872	—	9	1	no	—
95	C₃×C₄⋊C₈	D₉	1	3	yes	complemented
96	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
97	Dic₃⋊C₈	—	9	3	no	—
98	Dic₃⋊C₈	—	9	3	no	—
99	Dic₃⋊C₈	—	9	1	no	—
100	Dic₃⋊C₈	—	9	1	no	—
101	C₄⋊(C₃⋊C₈)	—	9	3	no	—
102	C₄⋊(C₃⋊C₈)	—	9	1	no	—
103	C₂²×C₃⋊D₃	—	3	1	no	—
104	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃⋊Dic₃	—	3	1	no	—
108	C₂×C₃⋊Dic₃	—	3	1	no	—
109	C₂×C₃⋊Dic₃	—	3	1	no	—
110	C₄×C₃⋊D₃	—	3	2	no	—
111	C₄×C₃⋊D₃	—	3	2	no	—
112	C₂²×D₉	—	3	3	no	—
113	C₃²×C₈	—	2	1	no	—
114	C₃²×C₈	—	2	1	no	—
115	C₃⋊(C₃⋊C₈)	—	6	1	no	—
116	C₃⋊(C₃⋊C₈)	—	6	1	no	—
117	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
118	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
119	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
120	C₂×Dic₉	—	3	3	no	—
121	C₂×Dic₉	—	3	3	no	—
122	C₂×Dic₉	—	3	3	no	—
123	C₄×D₉	—	3	6	no	—
124	C₄×D₉	—	3	6	no	—
125	C₈×C₉	—	2	3	no	—
126	C₈×C₉	—	2	3	no	—
127	C₉⋊C₈	—	6	3	no	—
128	C₉⋊C₈	—	6	3	no	—
129	#424	—	27	1	no	Sylow
130	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₉	#131	1	1	yes	characteristic
132	C₂×C₃×C₉	#131	1	1	yes	characteristic
133	C₃⋊D₉	—	2	2	no	—
134	C₂×C₄×D₃	—	9	3	no	—
135	C₂×C₄×D₃	—	9	1	no	—
136	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
137	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
138	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
140	C₃×C₄²	C₂×D₉	1	3	yes	—
141	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
142	(C₂×D₃)⋊C₄	—	9	3	no	—
143	(C₂×D₃)⋊C₄	—	9	3	no	—
144	(C₂×D₃)⋊C₄	—	9	1	no	—
145	(C₂×D₃)⋊C₄	—	9	1	no	—
146	Dic₃⋊C₄	—	9	3	no	—
147	Dic₃⋊C₄	—	9	3	no	—
148	Dic₃⋊C₄	—	9	1	no	—
149	Dic₃⋊C₄	—	9	1	no	—
150	C₄×Dic₃	—	9	3	no	—
151	C₄×Dic₃	—	9	1	no	—
152	C₂×C₃⋊C₈	—	9	3	no	—
153	C₂×C₃⋊C₈	—	9	3	no	—
154	C₂×C₃⋊C₈	—	9	1	no	—
155	C₂×C₃⋊C₈	—	9	1	no	—
156	ΓL(1,𝔽₂₅)	—	18	2	no	—
157	ΓL(1,𝔽₂₅)	—	18	6	no	—
158	D₃×C₈	—	18	2	no	—
159	D₃×C₈	—	18	6	no	—
160	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₃⋊D₃	—	3	2	no	—
162	C₂×C₃⋊D₃	—	6	1	no	—
163	C₂×C₃⋊D₃	—	6	1	no	—
164	C₃²×C₄	—	2	1	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
167	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
168	C₃⋊Dic₃	—	3	2	no	—
169	C₃⋊Dic₃	—	6	1	no	—
170	C₃⋊Dic₃	—	6	1	no	—
171	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
172	C₂×D₉	—	3	6	no	—
173	C₂×D₉	—	6	3	no	—
174	C₂×D₉	—	6	3	no	—
175	C₄×C₉	—	2	3	no	—
176	C₄×C₉	—	2	3	no	—
177	C₄×C₉	D₃×D₄	1	3	yes	—
178	C₄×C₉	D₃×Q₈	1	3	yes	—
179	Dic₉	—	3	6	no	—
180	Dic₉	—	6	3	no	—
181	Dic₉	—	6	3	no	—
182	C₂×M₁₆	—	27	1	no	—
183	C₂²×C₈	—	27	1	no	—
184	#117	—	27	1	no	—
185	C₄⋊C₈	—	27	1	no	—
186	C₄⋊C₈	—	27	1	no	—
187	C₄⋊C₈	—	27	1	no	—
188	C₄⋊C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
189	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₃	—	9	3	no	—
191	C₂²×D₃	—	9	1	no	—
192	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
193	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
194	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
195	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
196	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
198	C₂×Dic₃	—	9	3	no	—
199	C₂×Dic₃	—	9	1	no	—
200	C₂×Dic₃	—	9	3	no	—
201	C₂×Dic₃	—	9	3	no	—
202	C₂×Dic₃	—	9	1	no	—
203	C₂×Dic₃	—	9	1	no	—
204	C₄×D₃	—	9	6	no	—
205	C₄×D₃	—	9	6	no	—
206	C₄×D₃	—	9	2	no	—
207	C₄×D₃	—	9	2	no	—
208	C₃×C₈	—	2	3	no	—
209	C₃×C₈	—	2	3	no	—
210	C₃×C₈	—	2	1	no	—
211	C₃×C₈	—	2	1	no	—
212	C₃⋊C₈	—	18	3	no	—
213	C₃⋊C₈	—	18	3	no	—
214	C₃⋊C₈	—	18	1	no	—
215	C₃⋊C₈	—	18	1	no	—
216	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₃²	#901	1	1	yes	characteristic
218	C₂×C₃²	#907	1	1	yes	characteristic
219	C₃⋊D₃	—	6	2	no	—
220	C₂×C₉	D₃×C₄⋊C₄	1	3	yes	—
221	C₂×C₉	#901	1	3	yes	—
222	C₂×C₉	#907	1	3	yes	—
223	D₉	—	6	6	no	—
224	C₂²×C₄	—	27	1	no	—
225	M₁₆	—	54	2	no	—
226	C₂×C₈	—	27	1	no	—
227	C₂×C₈	—	27	1	no	—
228	C₂×C₈	—	54	2	no	—
229	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
231	C₄⋊C₄	—	27	1	no	—
232	C₄⋊C₄	—	27	1	no	—
233	C₂²⋊C₄	—	27	1	no	—
234	C₂²⋊C₄	—	27	1	no	—
235	C₄²	—	27	1	no	—
236	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
238	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
239	C₂×D₃	—	9	2	no	—
240	C₂×D₃	—	9	6	no	—
241	C₂×D₃	—	18	1	no	—
242	C₂×D₃	—	18	1	no	—
243	C₂×D₃	—	18	3	no	—
244	C₂×D₃	—	18	3	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	3	no	—
248	C₃×C₄	—	2	3	no	—
249	C₃×C₄	D₄×D₉	1	3	yes	—
250	C₃×C₄	Q₈×D₉	1	3	yes	—
251	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
252	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
253	Dic₃	—	9	2	no	—
254	Dic₃	—	9	6	no	—
255	Dic₃	—	18	1	no	—
256	Dic₃	—	18	1	no	—
257	Dic₃	—	18	3	no	—
258	Dic₃	—	18	3	no	—
259	C₃²	#4839	1	1	yes	characteristic
260	C₉	#4839	1	3	yes	complemented
261	C₂³	—	27	1	no	—
262	C₂×C₄	—	27	1	no	—
263	C₂×C₄	—	27	2	no	—
264	C₂×C₄	—	27	2	no	—
265	C₂×C₄	—	27	1	no	—
266	C₂×C₄	—	27	1	no	—
267	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
268	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	54	1	no	—
273	C₈	—	54	1	no	—
274	C₂×C₃	C₄⋊C₄×D₉	1	3	yes	—
275	C₂×C₃	#63464	1	3	yes	—
276	C₂×C₃	#63469	1	3	yes	—
277	C₂×C₃	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
278	C₂×C₃	#64110	1	1	yes	characteristic
279	C₂×C₃	#64116	1	1	yes	characteristic
280	D₃	—	18	2	no	—
281	D₃	—	18	6	no	—
282	C₂²	—	27	2	no	—
283	C₂²	—	54	1	no	—
284	C₂²	—	54	1	no	—
285	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	27	2	no	—
289	C₄	—	54	1	no	—
290	C₄	—	54	1	no	—
291	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
292	C₄	Q₈×C₃⋊D₉	1	1	yes	characteristic
293	C₃	#10588523	1	3	yes	complemented
294	C₃	#10592538	1	1	yes	characteristic
295	C₂	—	54	2	no	—
296	C₂	C₄⋊C₄×C₃⋊D₉	1	1	yes	characteristic
297	C₂	#11721770	1	1	yes	characteristic
298	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	#424
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
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. #11720537	C₂	semidirect
6. #11720537	C₂	semidirect
7. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
33. C₉×C₄⋊C₈	D₃	semidirect
95. C₃×C₄⋊C₈	D₉	semidirect
188. C₄⋊C₈	C₃⋊D₉	semidirect
189. C₃×C₉	#424	semidirect
260. C₉	#4839	semidirect
293. C₃	#10588523	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88523	2
C₂×C₈×C₃⋊D₉	2
#11721709	2
C₃×C₉×C₄⋊C₈	2
#11720537	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592538	3
#10588523	3