Subgroups — #13135137

Order 1728.

462 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589156	—	3	1	no	maximal
5	#10588228	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
8	C₄×C₃⋊Dic₉	—	2	1	no	—
9	C₄×C₃⋊Dic₉	—	2	1	no	—
10	C₂×C₉×C₃⋊C₈	—	2	1	no	—
11	C₂×C₉×C₃⋊C₈	—	2	1	no	—
12	C₂×C₉×C₃⋊C₈	C₄	1	2	yes	complemented
13	C₂×C₃×C₉⋊C₈	—	2	1	no	—
14	C₂×C₃×C₉⋊C₈	—	2	1	no	—
15	C₂×C₃×C₉⋊C₈	C₄	1	2	yes	complemented
16	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
17	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
18	C₂²×C₃×C₃⋊C₈	D₃	1	1	yes	characteristic
19	C₂²×C₈×C₉	—	3	1	no	—
20	C₂×C₄×Dic₉	—	3	1	no	—
21	C₂×C₄×Dic₉	—	6	1	no	—
22	C₂²×C₉⋊C₈	D₃	1	1	yes	characteristic
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₂×C₃⋊Dic₉	—	2	2	no	—
32	C₂×C₃⋊Dic₉	C₈	1	4	yes	complemented
33	C₉×C₃⋊C₈	—	2	2	no	—
34	C₃×C₉⋊C₈	—	2	2	no	—
35	#4550	—	9	1	no	—
36	#4550	—	9	1	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	—	3	2	no	—
39	C₄×C₃⋊Dic₃	—	6	1	no	—
40	C₄×C₃⋊Dic₃	—	6	1	no	—
41	C₂×C₃×C₃⋊C₈	—	2	1	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	—	3	2	no	—
44	C₂×C₃×C₃⋊C₈	—	6	1	no	—
45	C₂×C₃×C₃⋊C₈	—	6	1	no	—
46	C₂×C₃×C₃⋊C₈	Dic₃	1	2	yes	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
49	C₂²×Dic₉	—	3	2	no	—
50	C₂²×Dic₉	—	6	2	no	—
51	C₂×C₈×C₉	—	3	2	no	—
52	C₂×C₈×C₉	—	6	1	no	—
53	C₂×C₈×C₉	—	6	1	no	—
54	C₄×Dic₉	—	6	2	no	—
55	C₄×Dic₉	—	6	2	no	—
56	C₄×Dic₉	—	6	1	no	—
57	C₄×Dic₉	—	6	1	no	—
58	C₂×C₉⋊C₈	—	2	1	no	—
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂×C₉⋊C₈	Dic₃	1	2	yes	complemented
61	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
65	C₂²×C₃×C₉	M₁₆	1	2	yes	—
66	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
70	C₃⋊Dic₉	—	2	4	no	—
71	C₂²×C₃×C₈	—	3	1	no	—
72	C₂²×C₃×C₈	—	9	1	no	—
73	C₂×C₄×Dic₃	—	9	1	no	—
74	C₂×C₄×Dic₃	—	9	1	no	—
75	C₂×C₄×Dic₃	—	18	1	no	—
76	C₂²×C₃⋊C₈	—	3	1	no	—
77	C₂²×C₃⋊C₈	D₉	1	1	yes	characteristic
78	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	#67	1	1	yes	characteristic
84	C₂×C₃²×C₄	#67	1	1	yes	characteristic
85	C₂×C₃⋊Dic₃	—	3	4	no	—
86	C₂×C₃⋊Dic₃	—	6	2	no	—
87	C₂×C₃⋊Dic₃	—	6	2	no	—
88	C₂³×C₉	—	2	1	no	—
89	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
90	C₃×C₃⋊C₈	—	2	2	no	—
91	C₃×C₃⋊C₈	—	6	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
99	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₄×C₉	#67	1	1	yes	characteristic
103	C₂×C₄×C₉	#67	1	1	yes	characteristic
104	C₂×Dic₉	—	3	4	no	—
105	C₂×Dic₉	—	6	4	no	—
106	C₂×Dic₉	—	6	4	no	—
107	C₂×Dic₉	—	6	4	no	—
108	C₂×Dic₉	—	6	2	no	—
109	C₂×Dic₉	—	6	2	no	—
110	C₈×C₉	—	6	2	no	—
111	C₉⋊C₈	—	2	2	no	—
112	#336	—	27	1	no	Sylow
113	C₂×C₃×C₉	#95	1	1	yes	characteristic
114	C₂×C₃×C₉	C₄×C₈	1	1	yes	characteristic
115	C₂×C₃×C₉	#97	1	1	yes	characteristic
116	C₂×C₃×C₉	C₂²⋊C₈	1	2	yes	—
117	C₂×C₃×C₉	C₄⋊C₈	1	2	yes	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
120	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
121	C₂²×Dic₃	—	9	2	no	—
122	C₂²×Dic₃	—	9	2	no	—
123	C₂²×Dic₃	—	18	2	no	—
124	C₂×C₃×C₈	—	3	2	no	—
125	C₂×C₃×C₈	—	6	1	no	—
126	C₂×C₃×C₈	—	6	1	no	—
127	C₂×C₃×C₈	—	9	2	no	—
128	C₂×C₃×C₈	—	18	1	no	—
129	C₂×C₃×C₈	—	18	1	no	—
130	C₄×Dic₃	—	18	2	no	—
131	C₄×Dic₃	—	18	2	no	—
132	C₄×Dic₃	—	18	1	no	—
133	C₄×Dic₃	—	18	1	no	—
134	C₄×Dic₃	—	18	1	no	—
135	C₄×Dic₃	—	18	1	no	—
136	C₂×C₃⋊C₈	—	2	1	no	—
137	C₂×C₃⋊C₈	—	2	1	no	—
138	C₂×C₃⋊C₈	—	3	2	no	—
139	C₂×C₃⋊C₈	—	6	1	no	—
140	C₂×C₃⋊C₈	—	6	1	no	—
141	C₂×C₃⋊C₈	Dic₉	1	2	yes	complemented
142	C₂²×C₃²	D₃×C₈	1	2	yes	—
143	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	2	yes	—
144	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
145	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
146	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
147	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
148	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
149	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
150	C₃²×C₄	#217	1	1	yes	characteristic
151	C₃⋊Dic₃	—	6	4	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	D₃×C₈	1	2	yes	—
158	C₂²×C₉	ΓL(1,𝔽₂₅)	1	2	yes	—
159	C₂²×C₉	C₄×Dic₃	1	1	yes	characteristic
160	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
161	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
167	C₄×C₉	C₄⋊Dic₃	1	1	yes	characteristic
168	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₄×C₉	#217	1	1	yes	characteristic
170	Dic₉	—	6	8	no	—
171	Dic₉	—	6	4	no	—
172	C₂²×C₈	—	3	1	no	—
173	C₂²×C₈	—	9	1	no	—
174	C₂×C₄²	—	27	1	no	—
175	C₃×C₉	#336	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₃	—	2	1	no	—
177	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
178	C₂³×C₃	#624	1	1	yes	characteristic
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
187	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
189	C₂×C₃×C₄	#611	1	1	yes	characteristic
190	C₂×C₃×C₄	#611	1	1	yes	characteristic
191	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
192	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
193	C₂×C₃×C₄	#625	1	1	yes	characteristic
194	C₂×C₃×C₄	#626	1	1	yes	characteristic
195	C₂×C₃×C₄	#626	1	1	yes	characteristic
196	C₂×C₃×C₄	#627	1	1	yes	characteristic
197	C₂×Dic₃	—	9	4	no	—
198	C₂×Dic₃	—	9	4	no	—
199	C₂×Dic₃	—	18	4	no	—
200	C₂×Dic₃	—	18	4	no	—
201	C₂×Dic₃	—	18	4	no	—
202	C₂×Dic₃	—	18	2	no	—
203	C₂×Dic₃	—	18	2	no	—
204	C₂×Dic₃	—	18	2	no	—
205	C₂×Dic₃	—	18	2	no	—
206	C₃×C₈	—	6	2	no	—
207	C₃×C₈	—	18	2	no	—
208	C₃⋊C₈	—	2	2	no	—
209	C₃⋊C₈	—	6	2	no	—
210	C₂×C₃²	C₈×Dic₃	1	1	yes	characteristic
211	C₂×C₃²	Dic₃⋊C₈	1	2	yes	—
212	C₂×C₃²	#815	1	1	yes	characteristic
213	C₂×C₃²	(C₂×D₃)⋊C₈	1	2	yes	—
214	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	—	2	2	no	—
218	C₂×C₉	—	2	2	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	C₈×Dic₃	1	1	yes	characteristic
221	C₂×C₉	Dic₃⋊C₈	1	2	yes	—
222	C₂×C₉	#815	1	1	yes	characteristic
223	C₂×C₉	(C₂×D₃)⋊C₈	1	2	yes	—
224	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
225	C₂²×C₄	—	27	2	no	—
226	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
227	C₂×C₈	—	3	2	no	—
228	C₂×C₈	—	6	1	no	—
229	C₂×C₈	—	6	1	no	—
230	C₂×C₈	—	9	2	no	—
231	C₂×C₈	—	18	1	no	—
232	C₂×C₈	—	18	1	no	—
233	C₄²	—	54	1	no	—
234	C₄²	—	54	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	C₈×D₉	1	2	yes	—
241	C₂²×C₃	#3663	1	2	yes	—
242	C₂²×C₃	C₄×Dic₉	1	1	yes	characteristic
243	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
244	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
245	C₂²×C₃	#3710	1	2	yes	—
246	C₂²×C₃	#3712	1	2	yes	—
247	C₂²×C₃	Dic₃²	1	1	yes	characteristic
248	C₂²×C₃	#3722	1	1	yes	characteristic
249	C₂²×C₃	#3724	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
255	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
256	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
257	C₃×C₄	#3676	1	1	yes	characteristic
258	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
259	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
260	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
261	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
262	Dic₃	—	18	8	no	—
263	Dic₃	—	18	4	no	—
264	Dic₃	—	18	4	no	—
265	C₃²	#4550	1	1	yes	characteristic
266	C₉	—	2	1	no	—
267	C₉	#4550	1	1	yes	characteristic, complemented
268	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	—	27	4	no	—
270	C₂×C₄	—	54	2	no	—
271	C₂×C₄	—	54	2	no	—
272	C₂×C₄	#6189	1	1	yes	characteristic
273	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
274	C₂×C₄	#6192	1	1	yes	characteristic
275	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
276	C₂×C₄	#6194	1	1	yes	characteristic
277	C₂×C₄	#6195	1	1	yes	characteristic
278	C₈	—	6	2	no	—
279	C₈	—	18	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	C₈×Dic₉	1	1	yes	characteristic
286	C₂×C₃	Dic₉⋊C₈	1	2	yes	—
287	C₂×C₃	#63375	1	1	yes	characteristic
288	C₂×C₃	(C₂×D₉)⋊C₈	1	2	yes	—
289	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
290	C₂×C₃	#63553	1	1	yes	characteristic
291	C₂×C₃	#63555	1	1	yes	characteristic
292	C₂×C₃	#63557	1	2	yes	—
293	C₂×C₃	#63572	1	2	yes	—
294	C₂×C₃	#63579	1	1	yes	characteristic
295	C₂²	(C₃⋊D₉)⋊C₈	1	2	yes	—
296	C₂²	#88414	1	2	yes	—
297	C₂²	Dic₃×Dic₉	1	1	yes	characteristic
298	C₂²	#88442	1	1	yes	characteristic
299	C₂²	#88445	1	1	yes	characteristic
300	C₄	—	54	4	no	—
301	C₄	Dic₉⋊Dic₃	1	1	yes	characteristic
302	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
303	C₄	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
304	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
305	C₃	—	2	1	no	—
306	C₃	#10588228	1	1	yes	characteristic, complemented
307	C₃	#10589156	1	1	yes	characteristic
308	C₂	#11720271	1	1	yes	characteristic
309	C₂	#11720273	1	2	yes	—
310	C₂	#11720275	1	1	yes	characteristic
311	C₂	#11720282	1	2	yes	—
312	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₉
commutator	C₂×C₃×C₉	C₄×C₈
Frattini	C₂²×C₃×C₄	D₃²
Fitting, cosocle	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₄×D₃
hypocenter	C₃×C₉	#336
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
12. C₂×C₉×C₃⋊C₈	C₄	semidirect
15. C₂×C₃×C₉⋊C₈	C₄	semidirect
32. C₂×C₃⋊Dic₉	C₈	semidirect
60. C₂×C₉⋊C₈	Dic₃	semidirect
141. C₂×C₃⋊C₈	Dic₉	semidirect
175. C₃×C₉	#336	semidirect
267. C₉	#4550	semidirect
306. C₃	#10588228	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊Dic₉	2
C₂²×C₉×C₃⋊C₈	2
C₂²×C₃×C₉⋊C₈	2
#10589156	3
#10588228	3