Subgroups — #13135149

Order 1728.

346 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	#11721815	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#10589171	—	3	1	no	maximal
5	#10588232	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	#88551	C₄	1	2	yes	—
8	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
9	C₄×C₃⋊Dic₉	—	2	1	no	—
10	C₉×#208	—	2	1	no	—
11	C₉×#208	C₄	1	2	yes	complemented
12	C₂×C₉×C₃⋊C₈	—	2	1	no	—
13	C₃×#3667	—	2	1	no	—
14	C₃×#3667	—	2	1	no	—
15	C₂×C₃×C₉⋊C₈	C₄	1	2	yes	complemented
16	#64135	—	3	1	no	—
17	C₂×C₃×#208	—	3	1	no	—
18	C₂×C₃×#208	D₃	1	1	yes	characteristic
19	C₂×C₉×M₁₆	—	3	1	no	—
20	#63488	—	3	1	no	—
21	#63488	—	6	1	no	—
22	C₂×#3667	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	C₉×C₃⋊C₈	—	2	2	no	—
32	C₃×C₉⋊C₈	—	2	2	no	—
33	#4559	—	9	1	no	—
34	#4556	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	#3757	—	3	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
38	C₄×C₃⋊Dic₃	—	6	1	no	—
39	C₃×#208	—	2	1	no	—
40	C₃×#208	—	6	1	no	—
41	C₃×#208	—	6	1	no	—
42	C₃×#208	Dic₃	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	3	2	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
47	C₉×M₁₆	—	3	2	no	—
48	C₉×M₁₆	—	6	1	no	—
49	C₂×C₈×C₉	—	6	1	no	—
50	#3676	—	3	2	no	—
51	#3676	—	6	2	no	—
52	C₄⋊Dic₉	—	6	2	no	—
53	C₄⋊Dic₉	—	6	1	no	—
54	C₄×Dic₉	—	6	2	no	—
55	C₄×Dic₉	—	6	1	no	—
56	#3667	—	2	1	no	—
57	#3667	—	2	1	no	—
58	C₂×C₉⋊C₈	Dic₃	1	2	yes	complemented
59	C₂²×C₃×C₉	—	4	1	no	—
60	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
67	C₃⋊Dic₉	—	4	2	no	—
68	C₂×C₃×M₁₆	—	3	1	no	—
69	C₂×C₃×M₁₆	—	9	1	no	—
70	#926	—	9	1	no	—
71	#926	—	9	1	no	—
72	#926	—	18	1	no	—
73	C₂×#208	—	3	1	no	—
74	C₂×#208	D₉	1	1	yes	characteristic
75	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	#67	1	1	yes	characteristic
81	C₂×C₃²×C₄	#67	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	—	6	2	no	—
83	C₂³×C₉	—	2	1	no	—
84	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
85	C₃×C₃⋊C₈	—	2	2	no	—
86	C₃×C₃⋊C₈	—	6	2	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	1	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
94	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
95	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
97	C₂×C₄×C₉	#67	1	1	yes	characteristic
98	C₂×C₄×C₉	#67	1	1	yes	characteristic
99	C₂×Dic₉	—	6	4	no	—
100	C₂×Dic₉	—	6	2	no	—
101	C₈×C₉	—	6	2	no	—
102	C₉⋊C₈	—	2	2	no	—
103	#344	—	27	1	no	Sylow
104	C₂×C₃×C₉	—	2	1	no	—
105	C₂×C₃×C₉	—	2	1	no	—
106	C₂×C₃×C₉	—	2	1	no	—
107	C₂×C₃×C₉	#95	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
110	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
111	C₃×M₁₆	—	3	2	no	—
112	C₃×M₁₆	—	6	1	no	—
113	C₃×M₁₆	—	18	1	no	—
114	C₃×M₁₆	—	18	1	no	—
115	C₂×C₃×C₈	—	6	1	no	—
116	C₂×C₃×C₈	—	9	2	no	—
117	#217	—	9	2	no	—
118	#217	—	9	2	no	—
119	#217	—	18	2	no	—
120	C₄⋊Dic₃	—	18	2	no	—
121	C₄⋊Dic₃	—	18	1	no	—
122	C₄⋊Dic₃	—	18	1	no	—
123	C₄×Dic₃	—	18	2	no	—
124	C₄×Dic₃	—	18	1	no	—
125	C₄×Dic₃	—	18	1	no	—
126	#208	—	2	1	no	—
127	#208	—	6	1	no	—
128	#208	—	6	1	no	—
129	#208	Dic₉	1	2	yes	complemented
130	C₂×C₃⋊C₈	—	2	1	no	—
131	C₂×C₃⋊C₈	—	3	2	no	—
132	C₂²×C₃²	—	4	1	no	—
133	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
134	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
135	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
136	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
137	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
138	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	#217	1	1	yes	characteristic
140	C₃⋊Dic₃	—	12	2	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	—	2	1	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	4	1	no	—
145	C₂²×C₉	—	4	2	no	—
146	C₂²×C₉	C₄×Dic₃	1	1	yes	characteristic
147	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
148	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
154	C₄×C₉	C₄⋊Dic₃	1	1	yes	characteristic
155	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
156	C₄×C₉	#217	1	1	yes	characteristic
157	Dic₉	—	12	4	no	—
158	Dic₉	—	12	2	no	—
159	C₂×M₁₆	—	3	1	no	—
160	C₂×M₁₆	—	9	1	no	—
161	#117	—	27	1	no	—
162	C₃×C₉	#344	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
165	C₂³×C₃	#624	1	1	yes	characteristic
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
174	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
176	C₂×C₃×C₄	#611	1	1	yes	characteristic
177	C₂×C₃×C₄	#611	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
179	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
180	C₂×C₃×C₄	#625	1	1	yes	characteristic
181	C₂×C₃×C₄	#626	1	1	yes	characteristic
182	C₂×C₃×C₄	#626	1	1	yes	characteristic
183	C₂×C₃×C₄	#627	1	1	yes	characteristic
184	C₂×Dic₃	—	18	4	no	—
185	C₂×Dic₃	—	18	2	no	—
186	C₂×Dic₃	—	18	2	no	—
187	C₃×C₈	—	6	2	no	—
188	C₃×C₈	—	18	2	no	—
189	C₃⋊C₈	—	2	2	no	—
190	C₃⋊C₈	—	6	2	no	—
191	C₂×C₃²	—	2	1	no	—
192	C₂×C₃²	—	2	1	no	—
193	C₂×C₃²	—	2	1	no	—
194	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
195	C₂×C₉	—	2	2	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	4	1	no	—
201	C₂×C₉	—	4	1	no	—
202	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
203	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
204	M₁₆	—	3	2	no	—
205	M₁₆	—	6	1	no	—
206	M₁₆	—	18	1	no	—
207	M₁₆	—	18	1	no	—
208	C₂×C₈	—	6	1	no	—
209	C₂×C₈	—	9	2	no	—
210	C₄⋊C₄	—	54	1	no	—
211	C₂²⋊C₄	—	27	2	no	—
212	C₄²	—	54	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	1	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	C₄×Dic₉	1	1	yes	characteristic
220	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
222	C₂²×C₃	Dic₃²	1	1	yes	characteristic
223	C₂²×C₃	#3722	1	1	yes	characteristic
224	C₂²×C₃	#3724	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
230	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
231	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
232	C₃×C₄	#3676	1	1	yes	characteristic
233	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
234	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
235	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
236	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
237	Dic₃	—	36	2	no	—
238	Dic₃	—	36	4	no	—
239	Dic₃	—	36	2	no	—
240	C₃²	#4559	1	1	yes	characteristic
241	C₉	—	2	1	no	—
242	C₉	#4556	1	1	yes	characteristic, complemented
243	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
244	C₂×C₄	—	54	2	no	—
245	C₂×C₄	#6189	1	1	yes	characteristic
246	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
247	C₂×C₄	#6192	1	1	yes	characteristic
248	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
249	C₂×C₄	#6194	1	1	yes	characteristic
250	C₂×C₄	#6195	1	1	yes	characteristic
251	C₈	—	6	2	no	—
252	C₈	—	18	2	no	—
253	C₂×C₃	—	2	2	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	4	1	no	—
262	C₂×C₃	—	4	1	no	—
263	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
264	C₂×C₃	#63579	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	Dic₃×Dic₉	1	1	yes	characteristic
267	C₂²	#88442	1	1	yes	characteristic
268	C₂²	#88445	1	1	yes	characteristic
269	C₄	—	108	2	no	—
270	C₄	Dic₉⋊Dic₃	1	1	yes	characteristic
271	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
272	C₄	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
273	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
274	C₃	—	2	1	no	—
275	C₃	#10588232	1	1	yes	characteristic, complemented
276	C₃	#10589171	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₉)⋊Dic₃
commutator	C₂²×C₃×C₉	C₄²
Frattini	C₂²×C₃×C₄	D₃²
Fitting, cosocle	C₂²×C₃×C₄×C₉	C₂²
socle	C₂×C₃²	C₃⋊#95
prime radical	C₂³×C₃²	C₄×D₃
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#344
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
11. C₉×#208	C₄	semidirect
15. C₂×C₃×C₉⋊C₈	C₄	semidirect
58. C₂×C₉⋊C₈	Dic₃	semidirect
129. #208	Dic₉	semidirect
162. C₃×C₉	#344	semidirect
242. C₉	#4556	semidirect
275. C₃	#10588232	semidirect

Maximal subgroups

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

subgroup	index
#11721815	2
C₂×C₉×#208	2
C₂×C₃×#3667	2
#10589171	3
#10588232	3