Subgroups — #13143822

Order 1728.

610 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	C₄×#6233	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88523	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720542	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720537	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592428	—	3	1	no	maximal
9	#10588411	—	3	3	no	maximal
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×#6233	C₄	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	#88551	C₄	1	1	yes	characteristic
16	#88536	C₄	1	1	yes	characteristic
17	#88534	C₄	1	1	yes	characteristic
18	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
19	#88532	—	2	2	no	—
20	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
22	#88523	—	2	2	no	—
23	C₄×#638	—	3	1	no	—
24	C₂×#3748	—	3	1	no	—
25	C₂×#3743	—	3	1	no	—
26	C₃²×C₂²⋊C₈	D₃	1	1	yes	characteristic
27	#63647	—	3	1	no	—
28	#63641	—	3	1	no	—
29	C₈×C₃⋊Dic₃	—	3	1	no	—
30	C₄×#611	—	3	3	no	—
31	C₂×#3667	—	3	3	no	—
32	C₂×#3663	—	3	3	no	—
33	C₉×C₂²⋊C₈	D₃	1	3	yes	complemented
34	(C₂×D₉)⋊C₈	—	3	3	no	—
35	Dic₉⋊C₈	—	3	3	no	—
36	C₈×Dic₉	—	3	3	no	—
37	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
43	#6233	—	2	2	no	—
44	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
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	#4729	—	9	3	no	—
54	#4729	—	9	1	no	—
55	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂×#638	—	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	#3757	—	3	1	no	—
61	#3752	—	3	1	no	—
62	#3750	—	3	1	no	—
63	C₄×C₃⋊Dic₃	—	3	1	no	—
64	#3748	—	6	2	no	—
65	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
66	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
67	#3743	—	6	2	no	—
68	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
69	C₂×#611	—	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	#3676	—	3	3	no	—
74	(C₂×D₉)⋊C₄	—	3	3	no	—
75	Dic₉⋊C₄	—	3	3	no	—
76	C₄×Dic₉	—	3	3	no	—
77	#3667	—	6	6	no	—
78	C₂×C₉⋊C₈	—	3	3	no	—
79	C₂×C₉⋊C₈	—	3	3	no	—
80	#3663	—	6	6	no	—
81	C₂²×C₃×C₉	—	2	1	no	—
82	C₂²×C₃×C₉	—	2	1	no	—
83	C₂²×C₃×C₉	—	2	1	no	—
84	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃⋊D₉	—	2	1	no	—
86	C₂×C₃⋊D₉	—	2	1	no	—
87	C₂×C₃⋊D₉	—	2	1	no	—
88	C₃×C₄×C₉	—	2	1	no	—
89	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄×C₉	#41	1	1	yes	characteristic
91	C₃⋊Dic₉	—	2	1	no	—
92	C₃⋊Dic₉	—	2	1	no	—
93	C₃⋊Dic₉	M₁₆	1	2	yes	complemented
94	C₄×#67	—	9	3	no	—
95	C₄×#67	—	9	1	no	—
96	C₂×#208	—	9	3	no	—
97	C₂×#208	—	9	1	no	—
98	C₂×ΓL(1,𝔽₂₅)	—	9	3	no	—
99	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
100	C₃×C₂²⋊C₈	D₉	1	3	yes	complemented
101	C₃×C₂²⋊C₈	C₃⋊D₃	1	1	yes	characteristic
102	(C₂×D₃)⋊C₈	—	9	3	no	—
103	(C₂×D₃)⋊C₈	—	9	1	no	—
104	Dic₃⋊C₈	—	9	3	no	—
105	Dic₃⋊C₈	—	9	1	no	—
106	C₈×Dic₃	—	9	3	no	—
107	C₈×Dic₃	—	9	1	no	—
108	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₃⋊D₃	—	3	1	no	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	2	1	no	—
112	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
113	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
114	#638	—	6	2	no	—
115	C₂×C₃⋊Dic₃	—	3	1	no	—
116	C₂×C₃⋊Dic₃	—	3	1	no	—
117	C₂×C₃⋊Dic₃	—	3	1	no	—
118	C₄×C₃⋊D₃	—	6	1	no	—
119	C₄×C₃⋊D₃	—	6	1	no	—
120	C₂³×C₉	C₄×D₃	1	3	yes	—
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	—	2	3	no	—
127	C₂×C₄×C₉	—	2	3	no	—
128	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
129	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
130	#611	—	6	6	no	—
131	C₂×Dic₉	—	3	3	no	—
132	C₂×Dic₉	—	3	3	no	—
133	C₂×Dic₉	—	3	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	#436	—	27	1	no	Sylow
141	C₂×C₃×C₉	—	4	1	no	—
142	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₃×C₉	#131	1	1	yes	characteristic
145	C₃⋊D₉	—	4	1	no	—
146	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
147	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
148	C₂×#67	—	9	3	no	—
149	C₂×#67	—	9	1	no	—
150	C₂×C₄×D₃	—	9	3	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	#217	—	9	3	no	—
157	#217	—	9	1	no	—
158	(C₂×D₃)⋊C₄	—	9	3	no	—
159	(C₂×D₃)⋊C₄	—	9	1	no	—
160	Dic₃⋊C₄	—	9	3	no	—
161	Dic₃⋊C₄	—	9	1	no	—
162	C₄×Dic₃	—	9	3	no	—
163	C₄×Dic₃	—	9	1	no	—
164	#208	—	18	2	no	—
165	#208	—	18	6	no	—
166	C₂×C₃⋊C₈	—	9	3	no	—
167	C₂×C₃⋊C₈	—	9	1	no	—
168	C₂×C₃⋊C₈	—	9	3	no	—
169	C₂×C₃⋊C₈	—	9	1	no	—
170	ΓL(1,𝔽₂₅)	—	18	2	no	—
171	ΓL(1,𝔽₂₅)	—	18	6	no	—
172	C₂²×C₃²	—	2	1	no	—
173	C₂²×C₃²	—	2	1	no	—
174	C₂²×C₃²	—	2	1	no	—
175	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
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	D₃×D₄	1	1	yes	characteristic
181	C₃²×C₄	#237	1	1	yes	characteristic
182	C₃⋊Dic₃	—	3	2	no	—
183	C₃⋊Dic₃	—	6	1	no	—
184	C₃⋊Dic₃	—	6	1	no	—
185	C₂²×C₉	—	2	3	no	—
186	C₂²×C₉	—	2	3	no	—
187	C₂²×C₉	—	2	3	no	—
188	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
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	D₃×D₄	1	3	yes	—
194	C₄×C₉	#237	1	3	yes	—
195	Dic₉	—	3	6	no	—
196	Dic₉	—	6	3	no	—
197	Dic₉	—	6	3	no	—
198	C₂×M₁₆	—	27	1	no	—
199	C₂×M₁₆	—	27	1	no	—
200	C₄×D₄	—	27	1	no	—
201	C₄⋊C₈	—	27	1	no	—
202	C₂²⋊C₈	—	27	1	no	—
203	C₂²⋊C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
204	C₄×C₈	—	27	1	no	—
205	C₃×C₉	#436	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	C₄×D₉	1	3	yes	—
207	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
208	C₂²×D₃	—	9	3	no	—
209	C₂²×D₃	—	9	1	no	—
210	C₂×C₃×C₄	—	2	3	no	—
211	C₂×C₃×C₄	—	2	3	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
215	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
216	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
218	#67	—	18	2	no	—
219	#67	—	18	6	no	—
220	C₂×Dic₃	—	9	3	no	—
221	C₂×Dic₃	—	9	1	no	—
222	C₂×Dic₃	—	9	3	no	—
223	C₂×Dic₃	—	9	3	no	—
224	C₂×Dic₃	—	9	1	no	—
225	C₂×Dic₃	—	9	1	no	—
226	C₄×D₃	—	18	3	no	—
227	C₄×D₃	—	18	3	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	—	4	1	no	—
239	C₂×C₃²	#67⋊C₄	1	1	yes	characteristic
240	C₂×C₃²	#901	1	1	yes	characteristic
241	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
242	C₃⋊D₃	—	12	1	no	—
243	C₂×C₉	—	4	3	no	—
244	C₂×C₉	#67⋊C₄	1	3	yes	—
245	C₂×C₉	#901	1	3	yes	—
246	C₂×C₉	D₃×M₁₆	1	3	yes	—
247	D₉	—	12	3	no	—
248	C₂×D₄	—	27	1	no	—
249	C₂²×C₄	—	27	1	no	—
250	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
251	M₁₆	—	54	2	no	—
252	M₁₆	—	54	2	no	—
253	C₂×C₈	—	27	1	no	—
254	C₂×C₈	—	27	1	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	—	27	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	3	no	—
265	C₂²×C₃	—	2	3	no	—
266	C₂²×C₃	—	2	3	no	—
267	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
268	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
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	3	no	—
277	C₃×C₄	D₄×D₉	1	3	yes	—
278	C₃×C₄	#3699	1	3	yes	—
279	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
280	C₃×C₄	#3830	1	1	yes	characteristic
281	Dic₃	—	9	2	no	—
282	Dic₃	—	9	6	no	—
283	Dic₃	—	18	1	no	—
284	Dic₃	—	18	1	no	—
285	Dic₃	—	18	3	no	—
286	Dic₃	—	18	3	no	—
287	C₃²	#4729	1	1	yes	characteristic
288	C₉	#4729	1	3	yes	complemented
289	C₂³	—	27	1	no	—
290	C₂³	C₄×C₃⋊D₉	1	1	yes	characteristic
291	D₄	—	54	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	27	1	no	—
295	C₂×C₄	—	27	1	no	—
296	C₂×C₄	—	27	1	no	—
297	C₂×C₄	—	54	1	no	—
298	C₂×C₄	—	54	1	no	—
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	—	4	1	no	—
306	C₂×C₃	—	4	3	no	—
307	C₂×C₃	#611⋊C₄	1	3	yes	—
308	C₂×C₃	#63464	1	3	yes	—
309	C₂×C₃	M₁₆×D₉	1	3	yes	—
310	C₂×C₃	#64090	1	1	yes	characteristic
311	C₂×C₃	#64110	1	1	yes	characteristic
312	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
313	D₃	—	36	1	no	—
314	D₃	—	36	3	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	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	—	27	2	no	—
324	C₄	—	54	1	no	—
325	C₄	—	54	1	no	—
326	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
327	C₄	#88742	1	1	yes	characteristic
328	C₃	#10588411	1	3	yes	complemented
329	C₃	#10592428	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	108	1	no	—
332	C₂	#11721727	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×C₂²⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂×#6233	C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²⋊C₈	C₃⋊D₉
hypocenter	C₃×C₉	#436
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#88532	C₂	semidirect
3. C₂×#88523	C₂	semidirect
4. C₃×C₉×C₂²⋊C₈	C₂	semidirect
5. #11720542	C₂	semidirect
6. #11720537	C₂	semidirect
7. C₈×C₃⋊Dic₉	C₂	semidirect
33. C₉×C₂²⋊C₈	D₃	semidirect
52. C₃⋊(C₉⋊C₈)	D₄	semidirect
93. C₃⋊Dic₉	M₁₆	semidirect
100. C₃×C₂²⋊C₈	D₉	semidirect
203. C₂²⋊C₈	C₃⋊D₉	semidirect
205. C₃×C₉	#436	semidirect
288. C₉	#4729	semidirect
328. C₃	#10588411	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6233	2
C₂×#88532	2
C₂×#88523	2
C₃×C₉×C₂²⋊C₈	2
#11720542	2
#11720537	2
C₈×C₃⋊Dic₉	2
#10592428	3
#10588411	3