Subgroups — #13144802

Order 1728.

1170 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88551	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	2	yes	maximal, complemented
4	#11720572	C₂	1	2	yes	maximal, complemented
5	#10592911	—	3	1	no	maximal
6	#10588892	—	3	3	no	maximal
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	—	2	4	no	—
9	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊Dic₉	C₂²	1	4	yes	—
13	#88551	—	2	4	no	—
14	C₄⋊(C₃⋊Dic₉)	—	2	4	no	—
15	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
16	C₂×#3757	—	3	2	no	—
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
18	C₂³×C₄×C₉	D₃	1	3	yes	—
19	#63658	—	3	2	no	—
20	C₂×#3676	—	3	6	no	—
21	C₂×C₄⋊Dic₉	—	3	6	no	—
22	C₉⋊#95	—	3	6	no	—
23	C₂³×C₃×C₉	—	2	2	no	—
24	C₂³×C₃×C₉	—	2	2	no	—
25	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
26	C₂³×C₃×C₉	D₄	1	2	yes	—
27	C₂³×C₃×C₉	Q₈	1	2	yes	—
28	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	8	no	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
34	C₂×C₃⋊Dic₉	—	2	4	no	—
35	C₂×C₃⋊Dic₉	—	2	4	no	—
36	C₂×C₃⋊Dic₉	—	2	4	no	—
37	#5212	—	9	3	no	—
38	#5212	—	9	1	no	—
39	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃²×C₄	—	2	4	no	—
41	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃⋊Dic₃	—	3	4	no	—
45	C₂⁴×C₉	C₂×D₃	1	3	yes	—
46	#3757	—	6	4	no	—
47	C₄⋊(C₃⋊Dic₃)	—	6	4	no	—
48	C₂²×C₄×C₉	—	2	12	no	—
49	C₂²×C₄×C₉	Dic₃	1	12	yes	complemented
50	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
51	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
52	C₂²×Dic₉	—	3	12	no	—
53	#3676	—	6	12	no	—
54	C₄⋊Dic₉	—	6	12	no	—
55	C₂²×C₃×C₉	—	2	2	no	—
56	C₂²×C₃×C₉	—	2	2	no	—
57	C₂²×C₃×C₉	—	2	4	no	—
58	C₂²×C₃×C₉	—	2	4	no	—
59	C₂²×C₃×C₉	C₄⋊C₄	1	4	yes	complemented
60	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
62	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₃×C₉	#41	1	2	yes	—
65	C₃×C₄×C₉	—	2	2	no	—
66	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	complemented
67	C₃⋊Dic₉	—	4	4	no	—
68	C₂³×C₃×C₄	D₉	1	3	yes	—
69	C₂³×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
70	C₂×#217	—	9	6	no	—
71	C₂×#217	—	9	2	no	—
72	C₂×C₄⋊Dic₃	—	9	6	no	—
73	C₂×C₄⋊Dic₃	—	9	2	no	—
74	C₃⋊#95	—	9	6	no	—
75	C₃⋊#95	—	9	2	no	—
76	C₂³×C₃²	—	2	2	no	—
77	C₂³×C₃²	—	2	2	no	—
78	C₂³×C₃²	Dic₆	1	2	yes	—
79	C₂³×C₃²	D₁₂	1	2	yes	—
80	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
81	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
85	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
86	C₂×C₃²×C₄	#67	1	4	yes	—
87	C₂×C₃⋊Dic₃	—	6	4	no	—
88	C₂×C₃⋊Dic₃	—	6	4	no	—
89	C₂×C₃⋊Dic₃	—	6	4	no	—
90	C₂³×C₉	—	2	6	no	—
91	C₂³×C₉	—	2	6	no	—
92	C₂³×C₉	Dic₆	1	6	yes	—
93	C₂³×C₉	D₁₂	1	6	yes	—
94	C₂³×C₉	C₂×Dic₃	1	6	yes	—
95	C₂³×C₉	C₂²×D₃	1	3	yes	—
96	C₂×C₄×C₉	—	2	6	no	—
97	C₂×C₄×C₉	—	2	24	no	—
98	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
99	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
100	C₂×C₄×C₉	#67	1	12	yes	—
101	C₂×Dic₉	—	6	12	no	—
102	C₂×Dic₉	—	6	12	no	—
103	C₂×Dic₉	—	6	12	no	—
104	#380	—	27	1	no	Sylow
105	C₂×C₃×C₉	—	2	4	no	—
106	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₉	#117	1	1	yes	characteristic
109	C₂×C₃×C₉	#121	1	2	yes	—
110	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
111	C₂⁴×C₃	C₂×D₉	1	3	yes	—
112	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	—	2	12	no	—
114	C₂²×C₃×C₄	—	2	4	no	—
115	C₂²×C₃×C₄	Dic₉	1	12	yes	complemented
116	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
117	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
118	C₂²×C₃×C₄	C₃⋊Dic₃	1	4	yes	—
119	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
121	C₂²×Dic₃	—	9	12	no	—
122	C₂²×Dic₃	—	9	4	no	—
123	#217	—	18	12	no	—
124	#217	—	18	4	no	—
125	C₄⋊Dic₃	—	18	12	no	—
126	C₄⋊Dic₃	—	18	4	no	—
127	C₂²×C₃²	—	2	4	no	—
128	C₂²×C₃²	—	2	2	no	—
129	C₂²×C₃²	—	2	4	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
132	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
133	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
134	C₂²×C₃²	#235	1	2	yes	—
135	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
136	C₂²×C₃²	C₂×#67	1	2	yes	—
137	C₃²×C₄	—	2	2	no	—
138	C₃²×C₄	#217	1	4	yes	—
139	C₃⋊Dic₃	—	12	4	no	—
140	C₂²×C₉	—	2	12	no	—
141	C₂²×C₉	—	2	6	no	—
142	C₂²×C₉	—	2	12	no	—
143	C₂²×C₉	—	2	6	no	—
144	C₂²×C₉	C₄⋊Dic₃	1	12	yes	complemented
145	C₂²×C₉	C₂×Dic₆	1	3	yes	—
146	C₂²×C₉	C₂×D₁₂	1	3	yes	—
147	C₂²×C₉	#235	1	6	yes	—
148	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
149	C₂²×C₉	C₂×#67	1	6	yes	—
150	C₄×C₉	—	2	6	no	—
151	C₄×C₉	#217	1	12	yes	complemented
152	Dic₉	—	12	12	no	—
153	C₂³×C₄	C₃⋊D₉	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	27	2	no	—
155	C₂×C₂²⋊C₄	—	27	2	no	—
156	#95	—	27	2	no	—
157	C₃×C₉	#380	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₃	—	2	6	no	—
159	C₂³×C₃	—	2	6	no	—
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	Dic₁₈	1	6	yes	—
163	C₂³×C₃	D₃₆	1	6	yes	—
164	C₂³×C₃	C₂×Dic₉	1	6	yes	—
165	C₂³×C₃	C₂²×D₉	1	3	yes	—
166	C₂³×C₃	#634	1	2	yes	—
167	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	2	yes	—
168	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
169	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	6	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	24	no	—
173	C₂×C₃×C₄	—	2	8	no	—
174	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
175	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
176	C₂×C₃×C₄	#611	1	12	yes	—
177	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
179	C₂×C₃×C₄	#638	1	4	yes	—
180	C₂×Dic₃	—	18	12	no	—
181	C₂×Dic₃	—	18	12	no	—
182	C₂×Dic₃	—	18	12	no	—
183	C₂×Dic₃	—	18	4	no	—
184	C₂×Dic₃	—	18	4	no	—
185	C₂×Dic₃	—	18	4	no	—
186	C₂×C₃²	—	2	4	no	—
187	C₂×C₃²	#924	1	2	yes	—
188	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
189	C₂×C₃²	#926	1	1	yes	characteristic
190	C₂×C₃²	#930	1	2	yes	—
191	C₂×C₃²	C₂×#217	1	1	yes	characteristic
192	C₂×C₉	—	2	12	no	—
193	C₂×C₉	#924	1	6	yes	—
194	C₂×C₉	C₂×C₄⋊Dic₃	1	3	yes	—
195	C₂×C₉	#926	1	3	yes	—
196	C₂×C₉	#930	1	6	yes	—
197	C₂×C₉	C₂×#217	1	3	yes	—
198	C₂⁴	C₂×C₃⋊D₉	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	27	4	no	—
201	C₂²×C₄	C₃⋊Dic₉	1	4	yes	complemented
202	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
204	C₄⋊C₄	—	54	4	no	—
205	C₂²⋊C₄	—	54	4	no	—
206	C₂²×C₃	—	2	6	no	—
207	C₂²×C₃	—	2	12	no	—
208	C₂²×C₃	—	2	6	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	12	no	—
214	C₂²×C₃	C₄⋊Dic₉	1	12	yes	complemented
215	C₂²×C₃	C₂×Dic₁₈	1	3	yes	—
216	C₂²×C₃	C₂×D₃₆	1	3	yes	—
217	C₂²×C₃	#3697	1	6	yes	—
218	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
219	C₂²×C₃	C₂×#611	1	6	yes	—
220	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
221	C₂²×C₃	C₂×#634	1	1	yes	characteristic
222	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
223	C₂²×C₃	#3828	1	2	yes	—
224	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
225	C₂²×C₃	C₂×#638	1	2	yes	—
226	C₃×C₄	—	2	6	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	#3676	1	12	yes	complemented
229	C₃×C₄	#3757	1	4	yes	—
230	Dic₃	—	36	12	no	—
231	Dic₃	—	36	4	no	—
232	C₃²	#5212	1	1	yes	characteristic
233	C₉	#5212	1	3	yes	complemented
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	#6226	1	2	yes	—
237	C₂³	C₄⋊(C₃⋊D₉)	1	2	yes	—
238	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
239	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	54	4	no	—
243	C₂×C₄	—	54	4	no	—
244	C₂×C₄	—	54	4	no	—
245	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
246	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
247	C₂×C₄	#6233	1	4	yes	—
248	C₂×C₃	—	2	4	no	—
249	C₂×C₃	—	2	12	no	—
250	C₂×C₃	#63486	1	6	yes	—
251	C₂×C₃	C₂×C₄⋊Dic₉	1	3	yes	—
252	C₂×C₃	#63488	1	3	yes	—
253	C₂×C₃	#63492	1	6	yes	—
254	C₂×C₃	C₂×#3676	1	3	yes	—
255	C₂×C₃	#64133	1	2	yes	—
256	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
257	C₂×C₃	#64135	1	1	yes	characteristic
258	C₂×C₃	#64139	1	2	yes	—
259	C₂×C₃	C₂×#3757	1	1	yes	characteristic
260	C₂²	—	2	4	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₄⋊(C₃⋊Dic₉)	1	4	yes	complemented
265	C₂²	C₂×#6226	1	1	yes	characteristic
266	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
267	C₂²	#88736	1	2	yes	—
268	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
269	C₂²	C₂×#6233	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	108	4	no	—
272	C₄	#88551	1	4	yes	complemented
273	C₃	#10588892	1	3	yes	complemented
274	C₃	#10592911	1	1	yes	characteristic
275	C₂	—	2	4	no	—
276	C₂	#11721813	1	2	yes	—
277	C₂	C₂×C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
278	C₂	#11721815	1	1	yes	characteristic
279	C₂	#11721819	1	2	yes	—
280	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂³×C₄	C₃⋊D₉
hypocenter	C₃×C₉	#380
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊(C₃⋊Dic₉)	C₂	semidirect
4. #11720572	C₂	semidirect
9. C₂²×C₃×C₄×C₉	C₄	semidirect
49. C₂²×C₄×C₉	Dic₃	semidirect
59. C₂²×C₃×C₉	C₄⋊C₄	semidirect
66. C₃×C₄×C₉	C₂²⋊C₄	semidirect
115. C₂²×C₃×C₄	Dic₉	semidirect
144. C₂²×C₉	C₄⋊Dic₃	semidirect
151. C₄×C₉	#217	semidirect
157. C₃×C₉	#380	semidirect
201. C₂²×C₄	C₃⋊Dic₉	semidirect
214. C₂²×C₃	C₄⋊Dic₉	semidirect
228. C₃×C₄	#3676	semidirect
233. C₉	#5212	semidirect
264. C₂²	C₄⋊(C₃⋊Dic₉)	semidirect
272. C₄	#88551	semidirect
273. C₃	#10588892	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×#88551	2
C₂×C₄⋊(C₃⋊Dic₉)	2
#11720572	2
#10592911	3
#10588892	3