Subgroups — #13144054

Order 1728.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 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, complemented
2	C₂×C₈×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720542	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720542	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720516	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592535	—	3	1	no	maximal
9	#10588520	—	3	3	no	maximal
10	C₂×C₄⋊(C₃⋊D₉)	C₄	1	1	yes	characteristic
11	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	#88536	C₄	1	1	yes	characteristic
17	#88536	C₄	1	1	yes	characteristic
18	C₄⋊(C₃⋊Dic₉)	C₄	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	C₄×C₄⋊(C₃⋊D₃)	—	3	1	no	—
26	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
27	#63647	—	3	1	no	—
28	#63647	—	3	1	no	—
29	#63630	—	3	1	no	—
30	C₂×#3663	—	3	3	no	—
31	C₂×C₈×D₉	—	3	3	no	—
32	C₄×D₃₆	—	3	3	no	—
33	C₉×C₄⋊C₈	D₃	1	3	yes	complemented
34	(C₂×D₉)⋊C₈	—	3	3	no	—
35	(C₂×D₉)⋊C₈	—	3	3	no	—
36	#63362	—	3	3	no	—
37	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₉	C₂³	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)	—	2	2	no	—
45	C₄×C₃⋊D₉	—	2	1	no	—
46	C₄×C₃⋊D₉	—	2	1	no	—
47	C₄×C₃⋊D₉	—	2	1	no	—
48	C₄×C₃⋊D₉	—	2	1	no	—
49	C₃×C₈×C₉	—	2	1	no	—
50	C₃×C₈×C₉	—	2	1	no	—
51	C₃⋊(C₉⋊C₈)	—	2	1	no	—
52	C₃⋊(C₉⋊C₈)	D₄	1	2	yes	complemented
53	#4836	—	9	3	no	—
54	#4836	—	9	1	no	—
55	C₂×C₄⋊(C₃⋊D₃)	—	3	1	no	—
56	C₂×C₄×C₃⋊D₃	—	3	1	no	—
57	C₂×C₄×C₃⋊D₃	—	3	1	no	—
58	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
60	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
61	#3752	—	3	1	no	—
62	#3752	—	3	1	no	—
63	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
64	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
65	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
66	#3743	—	6	2	no	—
67	C₈×C₃⋊D₃	—	6	2	no	—
68	C₂×D₃₆	—	3	3	no	—
69	C₂×C₄×D₉	—	3	3	no	—
70	C₂×C₄×D₉	—	3	3	no	—
71	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
72	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
73	C₄²×C₉	C₂×D₃	1	3	yes	—
74	(C₂×D₉)⋊C₄	—	3	3	no	—
75	(C₂×D₉)⋊C₄	—	3	3	no	—
76	C₄⋊Dic₉	—	3	3	no	—
77	C₂×C₉⋊C₈	—	3	3	no	—
78	C₂×C₉⋊C₈	—	3	3	no	—
79	#3663	—	6	6	no	—
80	C₈×D₉	—	6	6	no	—
81	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃⋊D₉	—	2	1	no	—
83	C₂×C₃⋊D₉	—	2	1	no	—
84	C₂×C₃⋊D₉	—	2	1	no	—
85	C₂×C₃⋊D₉	—	2	1	no	—
86	C₂×C₃⋊D₉	—	2	1	no	—
87	C₂×C₃⋊D₉	M₁₆	1	2	yes	complemented
88	C₃×C₄×C₉	—	2	1	no	—
89	C₃×C₄×C₉	—	2	1	no	—
90	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄×C₉	#41	1	1	yes	characteristic
92	C₃⋊Dic₉	—	2	1	no	—
93	C₃⋊Dic₉	—	2	1	no	—
94	C₂×ΓL(1,𝔽₂₅)	—	9	3	no	—
95	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
96	C₂×D₃×C₈	—	9	3	no	—
97	C₂×D₃×C₈	—	9	1	no	—
98	C₄×D₁₂	—	9	3	no	—
99	C₄×D₁₂	—	9	1	no	—
100	C₃×C₄⋊C₈	D₉	1	3	yes	complemented
101	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic
102	(C₂×D₃)⋊C₈	—	9	3	no	—
103	(C₂×D₃)⋊C₈	—	9	3	no	—
104	(C₂×D₃)⋊C₈	—	9	1	no	—
105	(C₂×D₃)⋊C₈	—	9	1	no	—
106	#803	—	9	3	no	—
107	#803	—	9	1	no	—
108	C₂²×C₃⋊D₃	—	3	1	no	—
109	C₂²×C₃⋊D₃	—	3	1	no	—
110	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₃⋊Dic₃	—	3	1	no	—
114	C₂×C₃⋊Dic₃	—	3	1	no	—
115	C₄⋊(C₃⋊D₃)	—	6	2	no	—
116	C₄×C₃⋊D₃	—	6	1	no	—
117	C₄×C₃⋊D₃	—	6	1	no	—
118	C₄×C₃⋊D₃	—	6	1	no	—
119	C₄×C₃⋊D₃	—	6	1	no	—
120	C₂²×D₉	—	3	3	no	—
121	C₂²×D₉	—	3	3	no	—
122	C₃²×C₈	—	2	1	no	—
123	C₃²×C₈	—	2	1	no	—
124	C₃⋊(C₃⋊C₈)	—	3	2	no	—
125	C₃⋊(C₃⋊C₈)	—	6	1	no	—
126	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
127	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
128	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
129	C₂×Dic₉	—	3	3	no	—
130	C₂×Dic₉	—	3	3	no	—
131	D₃₆	—	6	6	no	—
132	C₄×D₉	—	6	3	no	—
133	C₄×D₉	—	6	3	no	—
134	C₄×D₉	—	6	3	no	—
135	C₄×D₉	—	6	3	no	—
136	C₈×C₉	—	2	3	no	—
137	C₈×C₉	—	2	3	no	—
138	C₉⋊C₈	—	3	6	no	—
139	C₉⋊C₈	—	6	3	no	—
140	#435	—	27	1	no	Sylow
141	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₃×C₉	#131	1	1	yes	characteristic
144	C₃⋊D₉	—	2	2	no	—
145	C₃⋊D₉	—	4	1	no	—
146	C₂×D₁₂	—	9	3	no	—
147	C₂×D₁₂	—	9	1	no	—
148	C₂×C₄×D₃	—	9	3	no	—
149	C₂×C₄×D₃	—	9	3	no	—
150	C₂×C₄×D₃	—	9	1	no	—
151	C₂×C₄×D₃	—	9	1	no	—
152	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
153	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
154	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
156	C₃×C₄²	C₂×D₉	1	3	yes	—
157	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
158	(C₂×D₃)⋊C₄	—	9	3	no	—
159	(C₂×D₃)⋊C₄	—	9	3	no	—
160	(C₂×D₃)⋊C₄	—	9	1	no	—
161	(C₂×D₃)⋊C₄	—	9	1	no	—
162	C₄⋊Dic₃	—	9	3	no	—
163	C₄⋊Dic₃	—	9	1	no	—
164	C₂×C₃⋊C₈	—	9	3	no	—
165	C₂×C₃⋊C₈	—	9	1	no	—
166	C₂×C₃⋊C₈	—	9	3	no	—
167	C₂×C₃⋊C₈	—	9	1	no	—
168	ΓL(1,𝔽₂₅)	—	18	2	no	—
169	ΓL(1,𝔽₂₅)	—	18	6	no	—
170	D₃×C₈	—	18	2	no	—
171	D₃×C₈	—	18	6	no	—
172	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₃⋊D₃	—	3	2	no	—
174	C₂×C₃⋊D₃	—	6	1	no	—
175	C₂×C₃⋊D₃	—	6	1	no	—
176	C₂×C₃⋊D₃	—	6	1	no	—
177	C₂×C₃⋊D₃	—	6	1	no	—
178	C₂×C₃⋊D₃	—	6	1	no	—
179	C₃²×C₄	—	2	1	no	—
180	C₃²×C₄	—	2	1	no	—
181	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
182	C₃²×C₄	#239	1	1	yes	characteristic
183	C₃⋊Dic₃	—	6	1	no	—
184	C₃⋊Dic₃	—	6	1	no	—
185	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
186	C₂×D₉	—	3	6	no	—
187	C₂×D₉	—	6	3	no	—
188	C₂×D₉	—	6	3	no	—
189	C₂×D₉	—	6	3	no	—
190	C₂×D₉	—	6	3	no	—
191	C₂×D₉	—	6	3	no	—
192	C₄×C₉	—	2	3	no	—
193	C₄×C₉	—	2	3	no	—
194	C₄×C₉	D₃×D₄	1	3	yes	—
195	C₄×C₉	#239	1	3	yes	—
196	Dic₉	—	6	3	no	—
197	Dic₉	—	6	3	no	—
198	C₂×M₁₆	—	27	1	no	—
199	C₂²×C₈	—	27	1	no	—
200	C₄×D₄	—	27	1	no	—
201	C₄⋊C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
202	C₂²⋊C₈	—	27	1	no	—
203	C₂²⋊C₈	—	27	1	no	—
204	#97	—	27	1	no	—
205	C₃×C₉	#435	1	1	yes	characteristic, complemented, Sylow
206	C₂²×D₃	—	9	3	no	—
207	C₂²×D₃	—	9	1	no	—
208	C₂²×D₃	—	9	3	no	—
209	C₂²×D₃	—	9	1	no	—
210	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
211	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
212	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
213	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
214	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
216	C₂×Dic₃	—	9	3	no	—
217	C₂×Dic₃	—	9	3	no	—
218	C₂×Dic₃	—	9	1	no	—
219	C₂×Dic₃	—	9	1	no	—
220	D₁₂	—	18	2	no	—
221	D₁₂	—	18	6	no	—
222	C₄×D₃	—	18	3	no	—
223	C₄×D₃	—	18	3	no	—
224	C₄×D₃	—	18	3	no	—
225	C₄×D₃	—	18	3	no	—
226	C₄×D₃	—	18	1	no	—
227	C₄×D₃	—	18	1	no	—
228	C₄×D₃	—	18	1	no	—
229	C₄×D₃	—	18	1	no	—
230	C₃×C₈	—	2	3	no	—
231	C₃×C₈	—	2	3	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	2	1	no	—
234	C₃⋊C₈	—	9	6	no	—
235	C₃⋊C₈	—	9	2	no	—
236	C₃⋊C₈	—	18	3	no	—
237	C₃⋊C₈	—	18	1	no	—
238	C₂×C₃²	#893	1	1	yes	characteristic
239	C₂×C₃²	#901	1	1	yes	characteristic
240	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
241	C₃⋊D₃	—	6	2	no	—
242	C₃⋊D₃	—	12	1	no	—
243	C₂×C₉	#893	1	3	yes	—
244	C₂×C₉	#901	1	3	yes	—
245	C₂×C₉	D₃×M₁₆	1	3	yes	—
246	D₉	—	6	6	no	—
247	D₉	—	12	3	no	—
248	C₂×D₄	—	27	1	no	—
249	C₂²×C₄	—	27	1	no	—
250	C₂²×C₄	—	27	1	no	—
251	M₁₆	—	54	2	no	—
252	C₂×C₈	—	27	1	no	—
253	C₂×C₈	—	27	1	no	—
254	C₂×C₈	—	54	2	no	—
255	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
256	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
257	C₄⋊C₄	—	27	1	no	—
258	C₂²⋊C₄	—	27	1	no	—
259	C₂²⋊C₄	—	27	1	no	—
260	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
261	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
262	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
263	C₂×D₃	—	9	2	no	—
264	C₂×D₃	—	9	6	no	—
265	C₂×D₃	—	18	1	no	—
266	C₂×D₃	—	18	1	no	—
267	C₂×D₃	—	18	3	no	—
268	C₂×D₃	—	18	3	no	—
269	C₂×D₃	—	18	1	no	—
270	C₂×D₃	—	18	1	no	—
271	C₂×D₃	—	18	1	no	—
272	C₂×D₃	—	18	3	no	—
273	C₂×D₃	—	18	3	no	—
274	C₂×D₃	—	18	3	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	3	no	—
278	C₃×C₄	—	2	3	no	—
279	C₃×C₄	D₄×D₉	1	3	yes	—
280	C₃×C₄	#3701	1	3	yes	—
281	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
282	C₃×C₄	#3832	1	1	yes	characteristic
283	Dic₃	—	18	1	no	—
284	Dic₃	—	18	1	no	—
285	Dic₃	—	18	3	no	—
286	Dic₃	—	18	3	no	—
287	C₃²	#4836	1	1	yes	characteristic
288	C₉	#4836	1	3	yes	complemented
289	C₂³	—	27	1	no	—
290	C₂³	—	27	1	no	—
291	D₄	—	54	2	no	—
292	C₂×C₄	—	27	1	no	—
293	C₂×C₄	—	27	1	no	—
294	C₂×C₄	—	54	1	no	—
295	C₂×C₄	—	54	1	no	—
296	C₂×C₄	—	54	1	no	—
297	C₂×C₄	—	54	1	no	—
298	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
299	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	27	2	no	—
304	C₈	—	54	1	no	—
305	C₂×C₃	#63455	1	3	yes	—
306	C₂×C₃	#63464	1	3	yes	—
307	C₂×C₃	M₁₆×D₉	1	3	yes	—
308	C₂×C₃	#64102	1	1	yes	characteristic
309	C₂×C₃	#64110	1	1	yes	characteristic
310	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
311	D₃	—	18	2	no	—
312	D₃	—	18	6	no	—
313	D₃	—	36	1	no	—
314	D₃	—	36	3	no	—
315	C₂²	—	27	2	no	—
316	C₂²	—	54	1	no	—
317	C₂²	—	54	1	no	—
318	C₂²	—	54	1	no	—
319	C₂²	—	54	1	no	—
320	C₂²	—	54	1	no	—
321	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	54	1	no	—
325	C₄	—	54	1	no	—
326	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
327	C₄	#88746	1	1	yes	characteristic
328	C₃	#10588520	1	3	yes	complemented
329	C₃	#10592535	1	1	yes	characteristic
330	C₂	—	54	2	no	—
331	C₂	—	108	1	no	—
332	C₂	#11721749	1	1	yes	characteristic
333	C₂	#11721770	1	1	yes	characteristic
334	C₂	M₁₆×C₃⋊D₉	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	C₂×C₄⋊(C₃⋊D₉)	C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄⋊C₈	C₃⋊D₉
hypocenter	C₃×C₉	#435
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
1. C₂×#88523	C₂	semidirect
2. C₂×C₈×C₃⋊D₉	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. #11720542	C₂	semidirect
6. #11720542	C₂	semidirect
7. #11720516	C₂	semidirect
33. C₉×C₄⋊C₈	D₃	semidirect
52. C₃⋊(C₉⋊C₈)	D₄	semidirect
87. C₂×C₃⋊D₉	M₁₆	semidirect
100. C₃×C₄⋊C₈	D₉	semidirect
201. C₄⋊C₈	C₃⋊D₉	semidirect
205. C₃×C₉	#435	semidirect
288. C₉	#4836	semidirect
328. C₃	#10588520	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
C₄×C₄⋊(C₃⋊D₉)	2
C₃×C₉×C₄⋊C₈	2
#11720542	2
#11720516	2
#10592535	3
#10588520	3