Subgroups — #13139344

Order 1728.

404 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
2	#11720571	C₂	1	1	yes	maximal, characteristic
3	C₉×#830	C₂	1	1	yes	maximal, characteristic
4	Dic₉⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₃⋊C₈×Dic₉	C₂	1	2	yes	maximal, complemented
6	#10591053	—	3	1	no	maximal
7	#10588398	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₉	C₄	1	1	yes	characteristic
10	C₂²×C₃×Dic₉	C₄	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
12	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
13	C₃×C₄×Dic₉	C₄	1	1	yes	characteristic
14	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃×C₄×Dic₃	—	3	1	no	—
18	#63657	—	3	1	no	—
19	C₃×#830	D₃	1	1	yes	characteristic
20	Dic₃⋊(C₃⋊C₈)	—	3	2	no	—
21	Dic₃×C₃⋊C₈	—	3	2	no	—
22	C₂×C₄×Dic₉	D₃	1	1	yes	characteristic
23	C₉×C₂²⋊C₈	—	3	1	no	—
24	#63389	—	3	1	no	—
25	#63389	—	6	1	no	—
26	Dic₉⋊C₈	—	3	2	no	—
27	C₈×Dic₉	—	3	2	no	—
28	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₃×Dic₉	—	2	2	no	—
33	C₂×C₃×Dic₉	C₈	1	4	yes	complemented
34	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
38	C₃⋊(C₉⋊C₈)	—	2	2	no	—
39	C₉×C₃⋊C₈	—	2	2	no	—
40	#4925	—	9	1	no	—
41	#4718	—	9	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃×Dic₃	—	3	1	no	—
44	C₂²×C₃×Dic₃	—	3	1	no	—
45	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
46	C₃×C₄×Dic₃	—	3	1	no	—
47	C₃×C₄×Dic₃	—	3	1	no	—
48	C₃×C₄×Dic₃	—	3	1	no	—
49	C₃×C₄×Dic₃	—	3	1	no	—
50	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
53	C₂²×Dic₉	Dic₃	1	1	yes	characteristic
54	C₂²×Dic₉	Dic₃	1	1	yes	characteristic
55	C₂×C₈×C₉	—	3	2	no	—
56	C₄×Dic₉	Dic₃	1	1	yes	characteristic
57	C₄×Dic₉	Dic₃	1	1	yes	characteristic
58	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
59	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
60	C₂×C₉⋊C₈	—	3	2	no	—
61	C₂×C₉⋊C₈	—	6	2	no	—
62	C₂²×C₃×C₉	—	2	1	no	—
63	C₂²×C₃×C₉	—	2	1	no	—
64	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
65	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	—	2	1	no	—
67	C₃×C₄×C₉	#41	1	2	yes	—
68	C₃×Dic₉	—	2	1	no	—
69	C₃×Dic₉	C₂×C₈	1	2	yes	—
70	C₃×Dic₉	C₂×C₈	1	2	yes	—
71	C₃×Dic₉	M₁₆	1	2	yes	—
72	C₂×C₃×C₄²	—	9	1	no	—
73	C₂×C₄×Dic₃	—	3	1	no	—
74	C₃×C₂²⋊C₈	—	3	1	no	—
75	#830	—	9	1	no	—
76	#830	—	9	1	no	—
77	#830	—	18	1	no	—
78	#830	D₉	1	1	yes	characteristic
79	Dic₃⋊C₈	—	9	2	no	—
80	C₈×Dic₃	—	9	2	no	—
81	C₄⋊(C₃⋊C₈)	—	9	2	no	—
82	C₄×C₃⋊C₈	—	9	2	no	—
83	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃×Dic₃	—	3	4	no	—
88	C₂×C₃×Dic₃	—	3	1	no	—
89	C₂×C₃×Dic₃	—	3	1	no	—
90	C₂×C₃×Dic₃	—	3	1	no	—
91	C₂×C₃×Dic₃	—	3	1	no	—
92	C₂×C₃×Dic₃	—	6	2	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
95	C₃⋊(C₃⋊C₈)	—	6	2	no	—
96	C₃×C₃⋊C₈	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	1	no	—
100	C₂×C₄×C₉	—	4	2	no	—
101	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
103	C₂×Dic₉	—	2	2	no	—
104	C₂×Dic₉	C₃⋊C₈	1	4	yes	complemented
105	C₂×Dic₉	C₂×Dic₃	1	1	yes	characteristic
106	C₂×Dic₉	C₂×Dic₃	1	1	yes	characteristic
107	C₂×Dic₉	C₂×Dic₃	1	1	yes	characteristic
108	C₂×Dic₉	C₂×Dic₃	1	1	yes	characteristic
109	C₈×C₉	—	6	2	no	—
110	C₉⋊C₈	—	6	2	no	—
111	C₉⋊C₈	—	12	2	no	—
112	#431	—	27	1	no	Sylow
113	C₂×C₃×C₉	—	2	2	no	—
114	C₂×C₃×C₉	#117	1	1	yes	characteristic
115	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	9	1	no	—
119	C₂²×C₃×C₄	—	9	1	no	—
120	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
121	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
122	C₂²×Dic₃	—	3	1	no	—
123	C₂²×Dic₃	—	3	1	no	—
124	C₂×C₃×C₈	—	3	2	no	—
125	C₃×C₄²	—	9	1	no	—
126	C₃×C₄²	—	9	1	no	—
127	C₃×C₄²	—	9	1	no	—
128	C₃×C₄²	—	9	1	no	—
129	C₄×Dic₃	—	3	1	no	—
130	C₄×Dic₃	—	3	1	no	—
131	C₄×Dic₃	—	3	1	no	—
132	C₄×Dic₃	—	3	1	no	—
133	C₂×C₃⋊C₈	—	9	2	no	—
134	C₂×C₃⋊C₈	—	9	2	no	—
135	C₂×C₃⋊C₈	—	18	2	no	—
136	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
137	C₂²×C₃²	—	2	1	no	—
138	C₂²×C₃²	—	2	1	no	—
139	C₂²×C₃²	D₃×C₈	1	2	yes	—
140	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
141	C₃²×C₄	—	2	1	no	—
142	C₃²×C₄	#237	1	2	yes	—
143	C₃×Dic₃	—	3	2	no	—
144	C₃×Dic₃	—	3	2	no	—
145	C₃×Dic₃	—	3	2	no	—
146	C₃×Dic₃	—	6	1	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	4	1	no	—
152	C₂²×C₉	—	4	1	no	—
153	C₂²×C₉	C₂×C₃⋊C₈	1	2	yes	—
154	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
155	C₄×C₉	—	2	1	no	—
156	C₄×C₉	—	2	2	no	—
157	C₄×C₉	—	4	1	no	—
158	C₄×C₉	#235	1	2	yes	—
159	Dic₉	—	2	1	no	—
160	Dic₉	C₂×C₃⋊C₈	1	2	yes	—
161	Dic₉	C₂×C₃⋊C₈	1	2	yes	—
162	Dic₉	#208	1	2	yes	—
163	C₂×C₄²	—	9	1	no	—
164	C₄⋊C₈	—	27	2	no	—
165	C₂²⋊C₈	—	3	1	no	—
166	C₂²⋊C₈	—	27	1	no	—
167	C₄×C₈	—	27	2	no	—
168	C₃×C₉	#431	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
171	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	9	4	no	—
178	C₂×C₃×C₄	—	9	1	no	—
179	C₂×C₃×C₄	—	9	1	no	—
180	C₂×C₃×C₄	—	9	1	no	—
181	C₂×C₃×C₄	—	9	1	no	—
182	C₂×C₃×C₄	—	18	2	no	—
183	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
185	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
187	C₂×Dic₃	—	3	4	no	—
188	C₂×Dic₃	—	3	1	no	—
189	C₂×Dic₃	—	3	1	no	—
190	C₂×Dic₃	—	3	1	no	—
191	C₂×Dic₃	—	3	1	no	—
192	C₂×Dic₃	—	6	2	no	—
193	C₃×C₈	—	6	2	no	—
194	C₃⋊C₈	—	2	2	no	—
195	C₃⋊C₈	—	18	2	no	—
196	C₃⋊C₈	—	18	2	no	—
197	C₃⋊C₈	—	36	2	no	—
198	C₂×C₃²	—	2	2	no	—
199	C₂×C₃²	#877	1	1	yes	characteristic
200	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
201	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
202	C₂×C₉	—	2	2	no	—
203	C₂×C₉	—	2	1	no	—
204	C₂×C₉	—	2	1	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	4	2	no	—
207	C₂×C₉	C₂²×C₃⋊C₈	1	1	yes	characteristic
208	C₂×C₉	C₂×#208	1	1	yes	characteristic
209	C₂×C₉	#926	1	1	yes	characteristic
210	C₂²×C₄	—	9	1	no	—
211	C₂²×C₄	—	9	1	no	—
212	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
213	C₂×C₈	—	3	2	no	—
214	C₂×C₈	—	27	2	no	—
215	C₄²	—	9	1	no	—
216	C₄²	—	9	1	no	—
217	C₄²	—	9	1	no	—
218	C₄²	—	9	1	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	4	1	no	—
226	C₂²×C₃	—	4	1	no	—
227	C₂²×C₃	C₈×D₉	1	2	yes	—
228	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
229	C₂²×C₃	D₃×C₃⋊C₈	1	2	yes	—
230	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	—	9	2	no	—
236	C₃×C₄	—	9	2	no	—
237	C₃×C₄	—	9	2	no	—
238	C₃×C₄	—	18	1	no	—
239	C₃×C₄	#3699	1	2	yes	—
240	C₃×C₄	#3804	1	2	yes	—
241	Dic₃	—	3	2	no	—
242	Dic₃	—	3	2	no	—
243	Dic₃	—	3	2	no	—
244	Dic₃	—	6	1	no	—
245	C₃²	#4718	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	#4925	1	1	yes	characteristic, complemented
248	C₂³	Dic₃×D₉	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	9	1	no	—
251	C₂×C₄	—	9	1	no	—
252	C₂×C₄	—	9	1	no	—
253	C₂×C₄	—	9	4	no	—
254	C₂×C₄	—	9	1	no	—
255	C₂×C₄	—	18	2	no	—
256	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
257	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
258	C₈	—	6	2	no	—
259	C₈	—	54	2	no	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	4	2	no	—
266	C₂×C₃	#63439	1	1	yes	characteristic
267	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
268	C₂×C₃	M₁₆×D₉	1	1	yes	characteristic
269	C₂×C₃	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
270	C₂×C₃	D₃×#208	1	1	yes	characteristic
271	C₂×C₃	#63955	1	1	yes	characteristic
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	D₉×C₃⋊C₈	1	2	yes	—
275	C₂²	C₂×Dic₃×D₉	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	9	2	no	—
278	C₄	—	9	2	no	—
279	C₄	—	9	2	no	—
280	C₄	—	18	1	no	—
281	C₄	#88662	1	2	yes	—
282	C₃	—	2	1	no	—
283	C₃	#10588398	1	1	yes	characteristic, complemented
284	C₃	#10591053	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	C₂×D₉×C₃⋊C₈	1	1	yes	characteristic
287	C₂	D₉×#208	1	1	yes	characteristic
288	C₂	#11721245	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#431
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. Dic₉⋊(C₃⋊C₈)	C₂	semidirect
5. C₃⋊C₈×Dic₉	C₂	semidirect
33. C₂×C₃×Dic₉	C₈	semidirect
104. C₂×Dic₉	C₃⋊C₈	semidirect
168. C₃×C₉	#431	semidirect
247. C₉	#4925	semidirect
283. C₃	#10588398	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×Dic₉	2
#11720571	2
C₉×#830	2
Dic₉⋊(C₃⋊C₈)	2
C₃⋊C₈×Dic₉	2
#10591053	3
#10588398	3