Subgroups — #13135126

Order 1728.

350 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88532	C₂	1	1	yes	maximal, characteristic
2	C₉×#926	C₂	1	1	yes	maximal, characteristic
3	C₃×#63488	C₂	1	1	yes	maximal, characteristic
4	#10589157	—	3	1	no	maximal
5	#10588210	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	#88532	—	2	1	no	—
8	#88532	—	2	1	no	—
9	C₂×C₃⋊(C₉⋊C₈)	C₄	1	2	yes	complemented
10	C₉×#217	—	2	1	no	—
11	C₃×#3676	—	2	1	no	—
12	C₉×C₄⋊Dic₃	—	2	1	no	—
13	C₄×C₉×Dic₃	C₄	1	2	yes	complemented
14	C₃×C₄⋊Dic₉	—	2	1	no	—
15	C₃×C₄×Dic₉	C₄	1	2	yes	complemented
16	C₂×#3748	—	3	1	no	—
17	C₃×#926	—	3	1	no	—
18	C₃×#926	D₃	1	1	yes	characteristic
19	C₉×#117	—	3	1	no	—
20	#63488	D₃	1	1	yes	characteristic
21	C₂×#3667	—	3	1	no	—
22	C₂×#3667	—	6	1	no	—
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×C₉×Dic₃	—	2	2	no	—
31	C₂×C₃×Dic₉	—	2	2	no	—
32	C₃⋊(C₉⋊C₈)	—	2	2	no	—
33	#4531	—	9	1	no	—
34	#4531	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	#3748	—	6	1	no	—
37	#3748	—	6	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
39	C₃×#217	—	2	1	no	—
40	C₃×#217	—	6	1	no	—
41	C₃×C₄⋊Dic₃	—	2	1	no	—
42	C₃×C₄⋊Dic₃	—	6	1	no	—
43	C₃×C₄×Dic₃	—	3	2	no	—
44	C₃×C₄×Dic₃	Dic₃	1	2	yes	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
47	C₉×C₄⋊C₄	—	6	1	no	—
48	C₉×C₂²⋊C₄	—	6	1	no	—
49	C₄²×C₉	—	3	2	no	—
50	#3676	—	2	1	no	—
51	C₄⋊Dic₉	—	2	1	no	—
52	C₄×Dic₉	Dic₃	1	2	yes	complemented
53	#3667	—	6	2	no	—
54	#3667	—	6	2	no	—
55	#3667	—	6	1	no	—
56	#3667	—	6	1	no	—
57	C₂×C₉⋊C₈	—	3	2	no	—
58	C₂×C₉⋊C₈	—	6	2	no	—
59	C₂²×C₃×C₉	—	4	1	no	—
60	C₂²×C₃×C₉	C₂²⋊C₄	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	C₄²	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	C₉×Dic₃	—	4	2	no	—
68	C₃×Dic₉	—	4	2	no	—
69	C₃×#117	—	3	1	no	—
70	C₃×#117	—	9	1	no	—
71	#926	—	3	1	no	—
72	#926	D₉	1	1	yes	characteristic
73	C₂×#208	—	9	1	no	—
74	C₂×#208	—	9	1	no	—
75	C₂×#208	—	18	1	no	—
76	C₂³×C₃²	#67	1	1	yes	characteristic
77	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	#67	1	1	yes	characteristic
83	C₂×C₃×Dic₃	—	2	2	no	—
84	C₂×C₃×Dic₃	—	6	2	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	#67	1	1	yes	characteristic
87	C₃⋊(C₃⋊C₈)	—	6	2	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	—	2	1	no	—
94	C₂×C₄×C₉	—	6	2	no	—
95	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
96	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
97	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
98	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
100	C₂×C₄×C₉	#67	1	1	yes	characteristic
101	C₂×Dic₉	—	2	2	no	—
102	C₉⋊C₈	—	6	4	no	—
103	C₉⋊C₈	—	6	2	no	—
104	H(3,ℤ/4)	—	27	1	no	Sylow
105	C₂×C₃×C₉	—	2	1	no	—
106	C₂×C₃×C₉	—	2	1	no	—
107	C₂×C₃×C₉	—	2	1	no	—
108	C₂×C₃×C₉	#95	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
111	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	6	1	no	—
113	C₃×C₄⋊C₄	—	18	1	no	—
114	C₃×C₂²⋊C₄	—	6	1	no	—
115	C₃×C₂²⋊C₄	—	18	1	no	—
116	C₃×C₄²	—	3	2	no	—
117	C₃×C₄²	—	9	2	no	—
118	#217	—	2	1	no	—
119	#217	—	6	1	no	—
120	C₄⋊Dic₃	—	2	1	no	—
121	C₄⋊Dic₃	—	6	1	no	—
122	C₄×Dic₃	—	3	2	no	—
123	C₄×Dic₃	Dic₉	1	2	yes	complemented
124	#208	—	18	2	no	—
125	#208	—	18	2	no	—
126	#208	—	18	1	no	—
127	#208	—	18	1	no	—
128	#208	—	18	1	no	—
129	#208	—	18	1	no	—
130	C₂×C₃⋊C₈	—	9	2	no	—
131	C₂×C₃⋊C₈	—	9	2	no	—
132	C₂×C₃⋊C₈	—	18	2	no	—
133	C₂²×C₃²	—	4	1	no	—
134	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
135	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
136	C₂²×C₃²	#217	1	1	yes	characteristic
137	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
138	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
139	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
140	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
141	C₃×Dic₃	—	4	2	no	—
142	C₃×Dic₃	—	12	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	—	2	1	no	—
146	C₂²×C₉	—	4	1	no	—
147	C₂²×C₉	—	4	2	no	—
148	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
149	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
150	C₂²×C₉	#217	1	1	yes	characteristic
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	2	1	no	—
154	C₄×C₉	—	2	1	no	—
155	C₄×C₉	—	12	2	no	—
156	C₄×C₉	C₄×Dic₃	1	1	yes	characteristic
157	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
158	C₄×C₉	C₄⋊Dic₃	1	1	yes	characteristic
159	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	Dic₉	—	4	2	no	—
161	C₂×M₁₆	—	27	1	no	—
162	#117	—	3	1	no	—
163	#117	—	9	1	no	—
164	C₃×C₉	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	#611	1	1	yes	characteristic
167	C₂³×C₃	#625	1	1	yes	characteristic
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	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	6	2	no	—
175	C₂×C₃×C₄	—	18	2	no	—
176	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
177	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
178	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
179	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
181	C₂×C₃×C₄	#611	1	1	yes	characteristic
182	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
183	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
184	C₂×C₃×C₄	#624	1	1	yes	characteristic
185	C₂×C₃×C₄	#626	1	1	yes	characteristic
186	C₂×C₃×C₄	#626	1	1	yes	characteristic
187	C₂×C₃×C₄	#627	1	1	yes	characteristic
188	C₂×Dic₃	—	2	2	no	—
189	C₂×Dic₃	—	6	2	no	—
190	C₃⋊C₈	—	18	4	no	—
191	C₃⋊C₈	—	18	2	no	—
192	C₃⋊C₈	—	18	2	no	—
193	C₂×C₃²	—	2	1	no	—
194	C₂×C₃²	—	2	1	no	—
195	C₂×C₃²	—	2	1	no	—
196	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
197	C₂×C₉	—	2	2	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	2	1	no	—
201	C₂×C₉	—	2	1	no	—
202	C₂×C₉	—	4	1	no	—
203	C₂×C₉	—	4	1	no	—
204	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
205	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
206	M₁₆	—	54	1	no	—
207	M₁₆	—	54	1	no	—
208	C₂×C₈	—	27	2	no	—
209	C₄⋊C₄	—	6	1	no	—
210	C₄⋊C₄	—	18	1	no	—
211	C₂²⋊C₄	—	6	1	no	—
212	C₂²⋊C₄	—	18	1	no	—
213	C₄²	—	3	2	no	—
214	C₄²	—	9	2	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	4	1	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
222	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
223	C₂²×C₃	#3676	1	1	yes	characteristic
224	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
225	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
226	C₂²×C₃	#3724	1	1	yes	characteristic
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	12	2	no	—
232	C₃×C₄	—	36	2	no	—
233	C₃×C₄	C₄×Dic₉	1	1	yes	characteristic
234	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
235	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
236	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
237	C₃×C₄	Dic₃²	1	1	yes	characteristic
238	C₃×C₄	#3722	1	1	yes	characteristic
239	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
240	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
241	Dic₃	—	4	2	no	—
242	Dic₃	—	12	2	no	—
243	C₃²	#4531	1	1	yes	characteristic
244	C₉	—	2	1	no	—
245	C₉	#4531	1	1	yes	characteristic, complemented
246	C₂³	#6194	1	1	yes	characteristic
247	C₂×C₄	—	6	2	no	—
248	C₂×C₄	—	18	2	no	—
249	C₂×C₄	#6189	1	1	yes	characteristic
250	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
251	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
252	C₂×C₄	#6192	1	1	yes	characteristic
253	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
254	C₂×C₄	#6195	1	1	yes	characteristic
255	C₈	—	54	2	no	—
256	C₂×C₃	—	2	2	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	—	2	1	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	4	1	no	—
265	C₂×C₃	—	4	1	no	—
266	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
267	C₂×C₃	#63579	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	#88442	1	1	yes	characteristic
270	C₂²	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
271	C₂²	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
272	C₄	—	12	2	no	—
273	C₄	—	36	2	no	—
274	C₄	Dic₃×Dic₉	1	1	yes	characteristic
275	C₄	Dic₉⋊Dic₃	1	1	yes	characteristic
276	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
277	C₄	#88445	1	1	yes	characteristic
278	C₃	—	2	1	no	—
279	C₃	#10588210	1	1	yes	characteristic, complemented
280	C₃	#10589157	1	1	yes	characteristic
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center	C₄	Dic₃×Dic₉
commutator	C₃×C₄×C₉	C₄²
Frattini	C₂²×C₃×C₄	D₃²
Fitting, cosocle	C₂²×C₃×C₄×C₉	C₂²
socle	C₂×C₃²	C₃⋊#95
prime radical	C₂³×C₃²	#67
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	H(3,ℤ/4)
LCS	G ▷ C₃×C₄×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
13. C₄×C₉×Dic₃	C₄	semidirect
15. C₃×C₄×Dic₉	C₄	semidirect
52. C₄×Dic₉	Dic₃	semidirect
123. C₄×Dic₃	Dic₉	semidirect
164. C₃×C₉	H(3,ℤ/4)	semidirect
245. C₉	#4531	semidirect
279. C₃	#10588210	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88532	2
C₉×#926	2
C₃×#63488	2
#10589157	3
#10588210	3