Subgroups — #13135128

Order 1728.

438 conjugacy classes of subgroups.

The subgroup lattice has 296 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	C₉×#926	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#10589160	—	3	1	no	maximal
5	#10588208	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₉	—	2	1	no	—
8	C₄×C₃⋊Dic₉	—	2	1	no	—
9	C₄×C₃⋊Dic₉	C₄	1	2	yes	complemented
10	C₉×#217	—	2	1	no	—
11	C₉×C₄⋊Dic₃	C₄	1	2	yes	complemented
12	C₄×C₉×Dic₃	—	2	1	no	—
13	C₃×#3667	—	2	1	no	—
14	C₃×#3667	C₄	1	2	yes	complemented
15	C₂×C₃×C₉⋊C₈	—	2	1	no	—
16	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
17	C₃×#926	D₃	1	1	yes	characteristic
18	C₂×C₃×#208	—	3	1	no	—
19	C₉×#117	—	3	1	no	—
20	C₂×C₄×Dic₉	—	3	1	no	—
21	C₂×C₄×Dic₉	—	6	1	no	—
22	C₂×#3667	D₃	1	1	yes	characteristic
23	C₂³×C₃×C₉	D₄	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	C₂×C₄	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	4	no	—
31	C₂×C₃⋊Dic₉	—	2	2	no	—
32	C₂×C₉×Dic₃	—	2	2	no	—
33	C₃×C₉⋊C₈	—	2	2	no	—
34	#4555	—	9	1	no	—
35	#4532	—	9	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	—	6	1	no	—
38	C₄×C₃⋊Dic₃	—	3	2	no	—
39	C₄×C₃⋊Dic₃	—	6	1	no	—
40	C₃×#217	—	2	1	no	—
41	C₃×C₄⋊Dic₃	Dic₃	1	2	yes	—
42	C₃×C₄×Dic₃	—	2	1	no	—
43	C₃×#208	—	3	2	no	—
44	C₃×#208	—	6	1	no	—
45	C₂×C₃×C₃⋊C₈	—	6	1	no	—
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
48	C₂²×Dic₉	—	6	2	no	—
49	C₂²×Dic₉	—	6	1	no	—
50	C₉×C₄⋊C₄	—	3	2	no	—
51	C₉×C₂²⋊C₄	—	6	1	no	—
52	C₄²×C₉	—	6	1	no	—
53	C₄×Dic₉	—	3	2	no	—
54	C₄×Dic₉	—	6	2	no	—
55	C₄×Dic₉	—	6	2	no	—
56	C₄×Dic₉	—	6	1	no	—
57	#3667	—	2	1	no	—
58	#3667	Dic₃	1	2	yes	complemented
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	C₂²⋊C₄	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	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
68	C₃⋊Dic₉	—	2	4	no	—
69	C₉×Dic₃	—	4	2	no	—
70	C₂×C₃×M₁₆	—	9	1	no	—
71	C₃×#117	—	3	1	no	—
72	#926	D₉	1	1	yes	characteristic
73	C₂×C₄×Dic₃	—	9	1	no	—
74	C₂×C₄×Dic₃	—	9	1	no	—
75	C₂×C₄×Dic₃	—	18	1	no	—
76	C₂×#208	—	3	1	no	—
77	C₂³×C₃²	#67	1	1	yes	characteristic
78	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₄×D₃	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×C₃⋊Dic₃	—	6	4	no	—
85	C₂×C₃⋊Dic₃	—	6	2	no	—
86	C₂×C₃×Dic₃	—	2	2	no	—
87	C₂³×C₉	—	2	1	no	—
88	C₂³×C₉	D₁₂	1	1	yes	characteristic
89	C₃×C₃⋊C₈	—	6	2	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	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	6	2	no	—
97	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
98	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
99	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
101	C₂×C₄×C₉	#67	1	1	yes	characteristic
102	C₂×C₄×C₉	#67	1	1	yes	characteristic
103	C₂×Dic₉	—	6	4	no	—
104	C₂×Dic₉	—	6	8	no	—
105	C₂×Dic₉	—	6	4	no	—
106	C₂×Dic₉	—	6	2	no	—
107	C₉⋊C₈	—	2	2	no	—
108	#339	—	27	1	no	Sylow
109	C₂×C₃×C₉	—	2	1	no	—
110	C₂×C₃×C₉	—	2	1	no	—
111	C₂×C₃×C₉	#95	1	1	yes	characteristic
112	C₂×C₃×C₉	#104	1	2	yes	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
115	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
116	C₂²×Dic₃	—	18	2	no	—
117	C₂²×Dic₃	—	18	1	no	—
118	C₂²×Dic₃	—	18	1	no	—
119	C₃×M₁₆	—	9	2	no	—
120	C₃×M₁₆	—	18	1	no	—
121	C₂×C₃×C₈	—	18	1	no	—
122	C₃×C₄⋊C₄	—	3	2	no	—
123	C₃×C₂²⋊C₄	—	6	1	no	—
124	C₃×C₄²	—	6	1	no	—
125	#217	—	2	1	no	—
126	C₄⋊Dic₃	Dic₉	1	2	yes	complemented
127	C₄×Dic₃	—	2	1	no	—
128	C₄×Dic₃	—	9	2	no	—
129	C₄×Dic₃	—	9	2	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	#208	—	3	2	no	—
135	#208	—	6	1	no	—
136	C₂×C₃⋊C₈	—	6	1	no	—
137	C₂²×C₃²	—	2	2	no	—
138	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
139	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
140	C₂²×C₃²	#217	1	1	yes	characteristic
141	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
142	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
143	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
144	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
145	C₃⋊Dic₃	—	6	4	no	—
146	C₃×Dic₃	—	4	2	no	—
147	C₂²×C₉	—	2	4	no	—
148	C₂²×C₉	—	2	2	no	—
149	C₂²×C₉	—	2	1	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	2	1	no	—
152	C₂²×C₉	C₄⋊Dic₃	1	1	yes	characteristic
153	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
154	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
155	C₄×C₉	—	2	1	no	—
156	C₄×C₉	—	2	1	no	—
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	12	2	no	—
160	C₄×C₉	C₄×Dic₃	1	1	yes	characteristic
161	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
162	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
163	C₄×C₉	#217	1	1	yes	characteristic
164	Dic₉	—	6	8	no	—
165	Dic₉	—	6	4	no	—
166	C₂×M₁₆	—	9	1	no	—
167	#117	—	3	1	no	—
168	C₂×C₄²	—	27	1	no	—
169	C₃×C₉	#339	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	#611	1	1	yes	characteristic
172	C₂³×C₃	#626	1	1	yes	characteristic
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	6	2	no	—
180	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
181	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
182	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
183	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
185	C₂×C₃×C₄	#611	1	1	yes	characteristic
186	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
187	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
188	C₂×C₃×C₄	#624	1	1	yes	characteristic
189	C₂×C₃×C₄	#625	1	1	yes	characteristic
190	C₂×C₃×C₄	#626	1	1	yes	characteristic
191	C₂×C₃×C₄	#627	1	1	yes	characteristic
192	C₂×Dic₃	—	2	2	no	—
193	C₂×Dic₃	—	18	4	no	—
194	C₂×Dic₃	—	18	4	no	—
195	C₂×Dic₃	—	18	8	no	—
196	C₂×Dic₃	—	18	4	no	—
197	C₂×Dic₃	—	18	2	no	—
198	C₂×Dic₃	—	18	2	no	—
199	C₃×C₈	—	18	2	no	—
200	C₃⋊C₈	—	6	2	no	—
201	C₂×C₃²	—	2	1	no	—
202	C₂×C₃²	—	2	1	no	—
203	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
204	C₂×C₃²	#837	1	2	yes	—
205	C₂×C₉	—	2	2	no	—
206	C₂×C₉	—	2	2	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	—	2	1	no	—
209	C₂×C₉	—	2	2	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	#825	1	2	yes	—
212	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
213	C₂²×C₄	—	54	1	no	—
214	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
215	M₁₆	—	9	2	no	—
216	M₁₆	—	18	1	no	—
217	C₂×C₈	—	18	1	no	—
218	C₄⋊C₄	—	3	2	no	—
219	C₂²⋊C₄	—	6	1	no	—
220	C₄²	—	6	1	no	—
221	C₄²	—	27	2	no	—
222	C₄²	—	54	1	no	—
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
230	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
231	C₂²×C₃	#3676	1	1	yes	characteristic
232	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
233	C₂²×C₃	#3722	1	1	yes	characteristic
234	C₂²×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	12	2	no	—
240	C₃×C₄	C₄×Dic₉	1	1	yes	characteristic
241	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
242	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
243	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
244	C₃×C₄	Dic₃²	1	1	yes	characteristic
245	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
246	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
247	C₃×C₄	#3724	1	1	yes	characteristic
248	Dic₃	—	4	2	no	—
249	Dic₃	—	18	4	no	—
250	Dic₃	—	18	8	no	—
251	Dic₃	—	18	4	no	—
252	C₃²	#4532	1	1	yes	characteristic
253	C₉	—	2	1	no	—
254	C₉	#4555	1	1	yes	characteristic, complemented
255	C₂³	#6195	1	1	yes	characteristic
256	C₂×C₄	—	6	2	no	—
257	C₂×C₄	—	54	4	no	—
258	C₂×C₄	—	54	2	no	—
259	C₂×C₄	#6189	1	1	yes	characteristic
260	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
261	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
262	C₂×C₄	#6192	1	1	yes	characteristic
263	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
264	C₂×C₄	#6194	1	1	yes	characteristic
265	C₈	—	18	2	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
275	C₂×C₃	#63396	1	2	yes	—
276	C₂×C₃	#63568	1	2	yes	—
277	C₂×C₃	#63579	1	1	yes	characteristic
278	C₂²	—	2	2	no	—
279	C₂²	Dic₉⋊Dic₃	1	1	yes	characteristic
280	C₂²	#88445	1	1	yes	characteristic
281	C₂²	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
282	C₄	—	12	2	no	—
283	C₄	—	54	4	no	—
284	C₄	Dic₃×Dic₉	1	1	yes	characteristic
285	C₄	#88442	1	1	yes	characteristic
286	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
287	C₄	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	#10588208	1	1	yes	characteristic, complemented
290	C₃	#10589160	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	#11720300	1	2	yes	—
294	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6194
commutator	C₃×C₄×C₉	C₄²
Frattini	C₂²×C₃×C₄	D₃²
Fitting, cosocle	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	#217
prime radical	C₂³×C₃²	#67
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#339
LCS	G ▷ C₃×C₄×C₉ ▷ 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
9. C₄×C₃⋊Dic₉	C₄	semidirect
11. C₉×C₄⋊Dic₃	C₄	semidirect
14. C₃×#3667	C₄	semidirect
58. #3667	Dic₃	semidirect
126. C₄⋊Dic₃	Dic₉	semidirect
169. C₃×C₉	#339	semidirect
254. C₉	#4555	semidirect
289. C₃	#10588208	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
C₉×#926	2
C₂×C₃×#3667	2
#10589160	3
#10588208	3