Subgroups — #13135144

Order 1728.

338 conjugacy classes of subgroups.

The subgroup lattice has 280 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	C₃×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
4	#10589169	—	3	1	no	maximal
5	#10588236	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	#88532	—	2	1	no	—
8	#88532	C₄	1	2	yes	complemented
9	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
10	C₃×#3676	C₄	1	2	yes	—
11	C₃×C₄⋊Dic₉	—	2	1	no	—
12	C₃×C₄×Dic₉	—	2	1	no	—
13	C₉×#208	—	2	1	no	—
14	C₉×#208	—	2	1	no	—
15	C₂×C₉×C₃⋊C₈	C₄	1	2	yes	complemented
16	C₂×#3748	—	3	1	no	—
17	C₃×#926	—	3	1	no	—
18	C₂×C₃×#208	D₃	1	1	yes	characteristic
19	C₂×C₉×M₁₆	—	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	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	C₃⋊(C₉⋊C₈)	—	2	2	no	—
32	C₉×C₃⋊C₈	—	2	2	no	—
33	#4556	—	9	1	no	—
34	#4533	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	#3748	—	3	2	no	—
37	#3748	—	6	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
39	C₃×#217	—	3	2	no	—
40	C₃×C₄⋊Dic₃	—	6	1	no	—
41	C₃×C₄×Dic₃	—	6	1	no	—
42	C₃×#208	—	2	1	no	—
43	C₃×#208	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	Dic₃	1	2	yes	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
47	C₉×M₁₆	—	6	1	no	—
48	C₉×M₁₆	—	6	1	no	—
49	C₂×C₈×C₉	—	3	2	no	—
50	#3676	Dic₃	1	2	yes	—
51	C₄⋊Dic₉	—	2	1	no	—
52	C₄×Dic₉	—	2	1	no	—
53	#3667	—	3	2	no	—
54	#3667	—	6	2	no	—
55	#3667	—	6	2	no	—
56	#3667	—	6	1	no	—
57	C₂×C₉⋊C₈	—	6	2	no	—
58	C₂×C₉⋊C₈	—	6	1	no	—
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	C₃×#117	—	9	1	no	—
70	#926	—	3	1	no	—
71	C₂×#208	—	9	1	no	—
72	C₂×#208	—	9	1	no	—
73	C₂×#208	—	18	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	C₂×Dic₃	1	1	yes	characteristic
85	C₃⋊(C₃⋊C₈)	—	6	2	no	—
86	C₃×C₃⋊C₈	—	2	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	C₄×D₃	1	1	yes	characteristic
96	C₂×C₄×C₉	D₁₂	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	—	2	2	no	—
100	C₈×C₉	—	6	2	no	—
101	C₉⋊C₈	—	6	4	no	—
102	C₉⋊C₈	—	6	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	—	6	1	no	—
112	C₃×M₁₆	—	6	1	no	—
113	C₂×C₃×C₈	—	3	2	no	—
114	C₃×C₄⋊C₄	—	18	1	no	—
115	C₃×C₂²⋊C₄	—	9	2	no	—
116	C₃×C₄²	—	18	1	no	—
117	#217	—	3	2	no	—
118	C₄⋊Dic₃	—	6	1	no	—
119	C₄×Dic₃	—	6	1	no	—
120	#208	—	2	1	no	—
121	#208	—	2	1	no	—
122	#208	—	9	2	no	—
123	#208	—	9	2	no	—
124	#208	—	18	2	no	—
125	#208	—	18	2	no	—
126	#208	—	18	1	no	—
127	#208	—	18	1	no	—
128	C₂×C₃⋊C₈	—	18	2	no	—
129	C₂×C₃⋊C₈	—	18	1	no	—
130	C₂×C₃⋊C₈	—	18	1	no	—
131	C₂×C₃⋊C₈	Dic₉	1	2	yes	complemented
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	C₄⋊Dic₃	1	1	yes	characteristic
148	C₂²×C₉	#217	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	Dic₃⋊C₄	1	1	yes	characteristic
155	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
156	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
157	Dic₉	—	4	2	no	—
158	C₂×M₁₆	—	3	1	no	—
159	C₂×M₁₆	—	27	1	no	—
160	#117	—	9	1	no	—
161	C₃×C₉	#344	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₃	—	2	1	no	—
163	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
164	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	1	no	—
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	—	18	2	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	#624	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	—	6	2	no	—
185	C₃×C₈	—	6	2	no	—
186	C₃⋊C₈	—	2	2	no	—
187	C₃⋊C₈	—	18	4	no	—
188	C₃⋊C₈	—	18	2	no	—
189	C₃⋊C₈	—	18	2	no	—
190	C₂×C₃²	—	2	1	no	—
191	C₂×C₃²	—	2	1	no	—
192	C₂×C₃²	—	2	1	no	—
193	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
194	C₂×C₉	—	2	2	no	—
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	4	1	no	—
200	C₂×C₉	—	4	1	no	—
201	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
202	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
203	M₁₆	—	6	1	no	—
204	M₁₆	—	6	1	no	—
205	M₁₆	—	27	2	no	—
206	M₁₆	—	54	1	no	—
207	C₂×C₈	—	3	2	no	—
208	C₂×C₈	—	54	1	no	—
209	C₄⋊C₄	—	18	1	no	—
210	C₂²⋊C₄	—	9	2	no	—
211	C₄²	—	18	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	C₄×Dic₉	1	1	yes	characteristic
219	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
220	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
221	C₂²×C₃	Dic₃²	1	1	yes	characteristic
222	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
223	C₂²×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	36	2	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	#3722	1	1	yes	characteristic
235	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
236	C₃×C₄	#3724	1	1	yes	characteristic
237	Dic₃	—	12	2	no	—
238	C₃²	#4556	1	1	yes	characteristic
239	C₉	—	2	1	no	—
240	C₉	#4533	1	1	yes	characteristic, complemented
241	C₂³	Dic₃×D₉	1	1	yes	characteristic
242	C₂×C₄	—	18	2	no	—
243	C₂×C₄	#6189	1	1	yes	characteristic
244	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	#6192	1	1	yes	characteristic
246	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
247	C₂×C₄	#6194	1	1	yes	characteristic
248	C₂×C₄	#6195	1	1	yes	characteristic
249	C₈	—	6	2	no	—
250	C₈	—	54	2	no	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	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	—	4	1	no	—
260	C₂×C₃	—	4	1	no	—
261	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
262	C₂×C₃	#63579	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	Dic₃×Dic₉	1	1	yes	characteristic
265	C₂²	Dic₉⋊Dic₃	1	1	yes	characteristic
266	C₂²	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
267	C₄	—	36	2	no	—
268	C₄	#88442	1	1	yes	characteristic
269	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
270	C₄	#88445	1	1	yes	characteristic
271	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
272	C₃	—	2	1	no	—
273	C₃	#10588236	1	1	yes	characteristic, complemented
274	C₃	#10589169	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₃)⋊Dic₉
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
8. #88532	C₄	semidirect
15. C₂×C₉×C₃⋊C₈	C₄	semidirect
131. C₂×C₃⋊C₈	Dic₉	semidirect
161. C₃×C₉	#344	semidirect
240. C₉	#4533	semidirect
273. C₃	#10588236	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
C₃×#63488	2
C₂×C₉×#208	2
#10589169	3
#10588236	3