Subgroups — Dic₉×#208 #13139345

Order 1728.

494 conjugacy classes of subgroups.

The subgroup lattice has 262 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	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
4	#11720274	C₂	1	2	yes	maximal, complemented
5	C₃⋊C₈×Dic₉	C₂	1	2	yes	maximal, complemented
6	Dic₃×#208	—	3	1	no	maximal
7	M₁₆×Dic₉	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₉	C₄	1	2	yes	—
10	#88532	C₄	1	4	yes	complemented
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
12	C₃×C₄×Dic₉	C₄	1	2	yes	—
13	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₉×#208	C₄	1	4	yes	complemented
16	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×#3748	—	3	1	no	—
18	C₂×C₃×C₄×Dic₃	—	3	1	no	—
19	C₂×C₃×#208	D₃	1	1	yes	characteristic
20	#63554	—	3	2	no	—
21	Dic₃×C₃⋊C₈	—	3	2	no	—
22	C₂×C₉×M₁₆	—	3	1	no	—
23	C₂×C₄×Dic₉	D₃	1	1	yes	characteristic
24	C₂×#3667	—	3	1	no	—
25	C₂×#3667	—	6	1	no	—
26	#63375	—	3	2	no	—
27	C₈×Dic₉	—	3	2	no	—
28	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃×Dic₉	—	2	4	no	—
34	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
35	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
36	C₃⋊(C₉⋊C₈)	C₂×C₄	1	4	yes	complemented
37	C₉×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
38	Dic₃×M₁₆	—	9	1	no	—
39	C₄×#208	—	9	1	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃×Dic₃	—	3	2	no	—
42	#3748	—	3	4	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
44	C₃×C₄×Dic₃	—	3	2	no	—
45	C₃×C₄×Dic₃	—	3	1	no	—
46	C₃×C₄×Dic₃	—	3	1	no	—
47	C₃×#208	Dic₃	1	4	yes	—
48	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
51	C₂²×Dic₉	Dic₃	1	2	yes	—
52	C₉×M₁₆	—	3	4	no	—
53	C₂×C₈×C₉	—	3	2	no	—
54	C₄×Dic₉	Dic₃	1	2	yes	—
55	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
56	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
57	#3667	—	3	4	no	—
58	#3667	—	6	4	no	—
59	C₂×C₉⋊C₈	—	3	2	no	—
60	C₂×C₉⋊C₈	—	6	2	no	—
61	C₂²×C₃×C₉	—	2	2	no	—
62	C₂²×C₃×C₉	C₄²	1	2	yes	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₄²	1	2	yes	—
65	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₃×Dic₉	—	2	2	no	—
68	C₃×Dic₉	M₁₆	1	4	yes	complemented
69	C₂×C₃×M₁₆	—	3	1	no	—
70	C₂×C₃×C₄²	—	9	1	no	—
71	C₂×C₄×Dic₃	—	3	1	no	—
72	C₂×#208	—	9	1	no	—
73	C₂×#208	—	9	1	no	—
74	C₂×#208	—	18	1	no	—
75	C₂×#208	D₉	1	1	yes	characteristic
76	#815	—	9	2	no	—
77	C₈×Dic₃	—	9	2	no	—
78	#803	—	9	2	no	—
79	C₄×C₃⋊C₈	—	9	2	no	—
80	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
83	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₃×Dic₃	—	3	2	no	—
86	C₂×C₃×Dic₃	—	3	2	no	—
87	C₂×C₃×Dic₃	—	6	4	no	—
88	C₂³×C₉	—	2	1	no	—
89	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
90	C₃⋊(C₃⋊C₈)	—	3	4	no	—
91	C₃×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
97	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
98	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
100	C₂×Dic₉	—	2	4	no	—
101	C₂×Dic₉	C₂×Dic₃	1	2	yes	—
102	C₂×Dic₉	C₂×Dic₃	1	2	yes	—
103	C₈×C₉	—	3	4	no	—
104	C₉⋊C₈	—	3	4	no	—
105	C₉⋊C₈	—	6	4	no	—
106	C₄×M₁₆	—	27	1	no	Sylow
107	C₂×C₃×C₉	—	2	2	no	—
108	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	9	2	no	—
112	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
113	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
114	C₂²×Dic₃	—	3	2	no	—
115	C₃×M₁₆	—	3	4	no	—
116	C₂×C₃×C₈	—	3	2	no	—
117	C₃×C₄²	—	9	2	no	—
118	C₃×C₄²	—	9	1	no	—
119	C₃×C₄²	—	9	1	no	—
120	C₄×Dic₃	—	3	2	no	—
121	C₄×Dic₃	—	3	1	no	—
122	C₄×Dic₃	—	3	1	no	—
123	#208	—	9	4	no	—
124	#208	—	9	4	no	—
125	#208	—	18	4	no	—
126	#208	Dic₉	1	4	yes	direct product factor
127	C₂×C₃⋊C₈	—	9	2	no	—
128	C₂×C₃⋊C₈	—	9	2	no	—
129	C₂×C₃⋊C₈	—	18	2	no	—
130	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
133	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
134	C₃²×C₄	C₄×Dic₃	1	2	yes	—
135	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
137	C₃×Dic₃	—	3	4	no	—
138	C₃×Dic₃	—	6	2	no	—
139	C₂²×C₉	—	2	2	no	—
140	C₂²×C₉	—	2	2	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	—	4	2	no	—
143	C₂²×C₉	C₄×Dic₃	1	2	yes	—
144	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
145	C₄×C₉	—	2	2	no	—
146	C₄×C₉	—	2	1	no	—
147	C₄×C₉	—	2	1	no	—
148	C₄×C₉	C₄×Dic₃	1	2	yes	—
149	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
150	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
151	Dic₉	—	2	2	no	—
152	Dic₉	#208	1	4	yes	direct product factor
153	C₂×M₁₆	—	3	1	no	—
154	C₂×M₁₆	—	27	1	no	—
155	C₂×C₄²	—	9	1	no	—
156	#97	—	27	2	no	—
157	C₄×C₈	—	27	2	no	—
158	C₃×C₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
161	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	9	2	no	—
167	C₂×C₃×C₄	—	9	2	no	—
168	C₂×C₃×C₄	—	18	4	no	—
169	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
170	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
171	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
173	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
174	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
175	C₂×C₃×C₄	#624	1	2	yes	—
176	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
177	C₂×Dic₃	—	3	2	no	—
178	C₂×Dic₃	—	3	2	no	—
179	C₂×Dic₃	—	6	4	no	—
180	C₃×C₈	—	3	4	no	—
181	C₃⋊C₈	—	9	4	no	—
182	C₃⋊C₈	—	9	4	no	—
183	C₃⋊C₈	—	18	4	no	—
184	C₃⋊C₈	C₂×Dic₉	1	4	yes	complemented
185	C₂×C₃²	—	2	2	no	—
186	C₂×C₃²	D₃×M₁₆	1	2	yes	—
187	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
188	C₂×C₉	—	2	2	no	—
189	C₂×C₉	—	2	2	no	—
190	C₂×C₉	—	2	1	no	—
191	C₂×C₉	—	4	2	no	—
192	C₂×C₉	C₂×#208	1	2	yes	—
193	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
194	C₂²×C₄	—	9	2	no	—
195	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
196	M₁₆	—	3	4	no	—
197	M₁₆	—	27	4	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	27	2	no	—
200	C₄²	—	9	2	no	—
201	C₄²	—	9	1	no	—
202	C₄²	—	9	1	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	C₄×Dic₉	1	2	yes	—
209	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
210	C₂²×C₃	Dic₃²	1	2	yes	—
211	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	9	4	no	—
216	C₃×C₄	—	18	2	no	—
217	C₃×C₄	C₄×Dic₉	1	2	yes	—
218	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
219	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
220	C₃×C₄	Dic₃²	1	2	yes	—
221	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
222	C₃×C₄	C₂×#624	1	1	yes	characteristic
223	Dic₃	—	3	4	no	—
224	Dic₃	—	6	2	no	—
225	C₃²	Dic₃×M₁₆	1	1	yes	characteristic
226	C₉	—	2	1	no	—
227	C₉	C₄×#208	1	1	yes	characteristic, complemented
228	C₂³	Dic₃×D₉	1	1	yes	characteristic
229	C₂×C₄	—	9	2	no	—
230	C₂×C₄	—	9	2	no	—
231	C₂×C₄	—	18	4	no	—
232	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
233	C₂×C₄	(C₃⋊D₉)⋊C₄	1	2	yes	—
234	C₂×C₄	D₃×Dic₉	1	2	yes	—
235	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
236	C₈	—	3	4	no	—
237	C₈	—	27	4	no	—
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	—	2	2	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	M₁₆×D₉	1	2	yes	—
244	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
245	C₂×C₃	D₃×#208	1	2	yes	—
246	C₂×C₃	C₂×Dic₃²	1	1	yes	characteristic
247	C₂²	—	2	2	no	—
248	C₂²	Dic₃×Dic₉	1	2	yes	—
249	C₂²	C₂×Dic₃×D₉	1	1	yes	characteristic
250	C₄	—	9	4	no	—
251	C₄	—	18	2	no	—
252	C₄	Dic₃×Dic₉	1	2	yes	—
253	C₄	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
254	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
255	C₃	—	2	1	no	—
256	C₃	M₁₆×Dic₉	1	1	yes	characteristic, complemented
257	C₃	Dic₃×#208	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	D₉×#208	1	2	yes	—
260	C₂	C₂×Dic₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₄×M₁₆
LCS	G ▷ 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
4. #11720274	C₂	semidirect
5. C₃⋊C₈×Dic₉	C₂	semidirect
10. #88532	C₄	semidirect
15. C₉×#208	C₄	semidirect
36. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
37. C₉×C₃⋊C₈	C₂×C₄	semidirect
68. C₃×Dic₉	M₁₆	semidirect
126. #208	Dic₉	direct (N × H)
152. Dic₉	#208	direct (N × H)
158. C₃×C₉	C₄×M₁₆	semidirect
184. C₃⋊C₈	C₂×Dic₉	semidirect
227. C₉	C₄×#208	semidirect
256. C₃	M₁₆×Dic₉	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
C₂×C₃×C₄×Dic₉	2
C₂×C₉×#208	2
#11720274	2
C₃⋊C₈×Dic₉	2
Dic₃×#208	3
M₁₆×Dic₉	3

Subgroups — Dic9×#208 #13139345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₉×#208 #13139345