Subgroups — C₉×#5137 #13143195

Order 1728.

398 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₉×#924	C₂	1	2	yes	maximal
3	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#817	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#816	C₂	1	2	yes	maximal, complemented
6	C₃×#5137	C₃	1	1	yes	maximal, characteristic
7	C₉×#470	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Dic₆	C₂²	1	2	yes	—
10	C₃×C₉×M₁₆	—	2	2	no	—
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₉×#217	—	2	2	no	—
14	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₉×Dic₃⋊C₄	—	2	2	no	—
16	C₉×Dic₁₂	—	2	2	no	—
17	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
18	C₃×#924	C₂×C₃	1	2	yes	—
19	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
20	C₃×#817	C₂×C₃	1	1	yes	characteristic
21	C₃×#816	C₂×C₃	1	2	yes	—
22	C₂×C₉×Q₁₆	—	3	1	no	—
23	C₂×C₉×M₁₆	—	2	1	no	—
24	C₂×C₉×M₁₆	D₃	1	1	yes	characteristic
25	C₉×C₂²⋊Q₈	—	3	2	no	—
26	C₉×#106	—	3	1	no	—
27	C₉×#103	—	3	2	no	—
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₉×Dic₃	—	2	2	no	—
35	C₉×Dic₆	—	2	2	no	—
36	C₉×Dic₆	—	2	2	no	—
37	C₃×C₈×C₉	—	2	1	no	—
38	C₃×C₈×C₉	D₄	1	2	yes	—
39	C₃×#470	—	3	1	no	—
40	#5137	C₉	1	1	yes	characteristic, direct product factor
41	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
43	C₃²×M₁₆	—	2	2	no	—
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃×#217	—	2	2	no	—
47	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
48	C₃×Dic₃⋊C₄	—	2	2	no	—
49	C₃×Dic₁₂	—	2	2	no	—
50	C₂×Q₈×C₉	—	3	2	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
53	C₉×Q₁₆	—	6	2	no	—
54	C₉×M₁₆	—	2	4	no	—
55	C₉×M₁₆	—	2	2	no	—
56	C₂×C₈×C₉	—	2	1	no	—
57	C₂×C₈×C₉	—	2	1	no	—
58	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
59	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
60	C₉×C₄⋊C₄	—	3	2	no	—
61	C₉×C₄⋊C₄	—	6	2	no	—
62	C₉×C₂²⋊C₄	—	6	2	no	—
63	C₂²×C₃×C₉	—	2	1	no	—
64	C₂²×C₃×C₉	—	2	2	no	—
65	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×C₉	—	2	1	no	—
67	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×C₉	#41	1	1	yes	characteristic
69	C₉×Dic₃	—	4	2	no	—
70	C₉×Dic₃	—	4	2	no	—
71	C₂×C₃×Q₁₆	—	3	1	no	—
72	C₂×C₃×M₁₆	—	2	1	no	—
73	C₂×C₃×M₁₆	C₂×C₉	1	1	yes	characteristic
74	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
75	C₃×C₂²⋊Q₈	—	3	2	no	—
76	#924	C₂×C₉	1	2	yes	—
77	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
78	C₃×#106	—	3	1	no	—
79	C₃×#103	—	3	2	no	—
80	#817	C₂×C₉	1	1	yes	characteristic, complemented
81	#816	C₂×C₉	1	2	yes	complemented
82	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	—	2	1	no	—
84	C₂×C₃²×C₄	—	2	1	no	—
85	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₃×Dic₃	—	2	2	no	—
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₃×Dic₆	—	2	2	no	—
90	C₃×Dic₆	—	2	2	no	—
91	C₂³×C₉	—	2	1	no	—
92	C₂³×C₉	D₁₂	1	1	yes	characteristic
93	C₃²×C₈	—	2	1	no	—
94	C₃²×C₈	C₃×D₄	1	2	yes	—
95	Q₈×C₉	—	6	2	no	—
96	Q₈×C₉	—	6	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	1	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	6	2	no	—
104	C₂×C₄×C₉	—	6	2	no	—
105	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
107	C₈×C₉	—	2	2	no	—
108	C₈×C₉	—	2	1	no	—
109	C₈×C₉	—	2	2	no	—
110	C₈×C₉	#67	1	2	yes	—
111	#470	—	3	1	no	Sylow
112	C₂×C₃×C₉	—	4	1	no	—
113	C₂×C₃×C₉	#121	1	1	yes	characteristic
114	C₂×C₃×C₉	#137	1	2	yes	—
115	C₂×C₃×Q₈	—	3	2	no	—
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
119	C₂×Dic₆	C₂²×C₉	1	2	yes	—
120	C₃×Q₁₆	—	6	2	no	—
121	C₃×M₁₆	—	2	4	no	—
122	C₃×M₁₆	—	2	2	no	—
123	C₃×M₁₆	—	2	2	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
129	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	3	2	no	—
131	C₃×C₄⋊C₄	—	6	2	no	—
132	C₃×C₂²⋊C₄	—	6	2	no	—
133	#217	—	2	2	no	—
134	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
135	Dic₃⋊C₄	—	2	2	no	—
136	Dic₁₂	—	2	2	no	—
137	C₂²×C₃²	—	2	1	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
140	C₃²×C₄	—	2	1	no	—
141	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
142	C₃²×C₄	C₃×#41	1	1	yes	characteristic
143	C₃×Dic₃	—	4	2	no	—
144	C₃×Dic₃	—	4	2	no	—
145	C₂²×C₉	—	2	2	no	—
146	C₂²×C₉	—	2	2	no	—
147	C₂²×C₉	—	2	1	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	—	4	2	no	—
150	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
151	C₄×C₉	—	2	2	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	—	12	2	no	—
157	C₄×C₉	#235	1	1	yes	characteristic
158	C₄×C₉	C₂×#67	1	1	yes	characteristic
159	C₂×Q₁₆	—	3	1	no	—
160	C₂×M₁₆	D₃×C₉	1	1	yes	characteristic
161	C₂²⋊Q₈	—	3	2	no	—
162	#106	—	3	1	no	—
163	#103	—	3	2	no	—
164	C₃×C₉	#470	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
167	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
168	C₃×Q₈	—	6	2	no	—
169	C₃×Q₈	—	6	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	1	no	—
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	—	6	2	no	—
179	C₂×C₃×C₄	—	6	2	no	—
180	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
182	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	2	2	no	—
186	Dic₆	—	2	2	no	—
187	Dic₆	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	1	no	—
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	D₄×C₉	1	2	yes	—
193	C₃×C₈	C₃×#67	1	2	yes	—
194	C₂×C₃²	—	4	1	no	—
195	C₂×C₃²	C₃×#121	1	1	yes	characteristic
196	C₂×C₃²	C₃×#137	1	2	yes	—
197	C₂×C₉	—	2	2	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	4	2	no	—
200	C₂×C₉	—	4	1	no	—
201	C₂×C₉	#909	1	2	yes	—
202	C₂×C₉	#930	1	1	yes	characteristic
203	C₂×Q₈	—	3	2	no	—
204	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
205	Q₁₆	—	6	2	no	—
206	M₁₆	—	2	2	no	—
207	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
208	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
209	C₄⋊C₄	—	3	2	no	—
210	C₄⋊C₄	—	6	2	no	—
211	C₂²⋊C₄	—	6	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
220	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	12	2	no	—
227	C₃×C₄	—	12	2	no	—
228	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
229	C₃×C₄	C₉×#41	1	1	yes	characteristic
230	C₃×C₄	C₃×#235	1	1	yes	characteristic
231	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
232	Dic₃	—	4	2	no	—
233	Dic₃	—	4	2	no	—
234	C₃²	C₃×#470	1	1	yes	characteristic
235	C₉	—	2	1	no	—
236	C₉	#5137	1	1	yes	characteristic, direct product factor
237	C₂³	C₉×D₁₂	1	1	yes	characteristic
238	Q₈	—	6	2	no	—
239	Q₈	—	6	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	6	2	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	C₉×#67	1	2	yes	—
248	C₂×C₃	—	2	2	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	4	2	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	—	4	1	no	—
253	C₂×C₃	C₉×#121	1	1	yes	characteristic
254	C₂×C₃	C₉×#137	1	2	yes	—
255	C₂×C₃	C₃×#909	1	2	yes	—
256	C₂×C₃	C₃×#930	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	12	2	no	—
262	C₄	—	12	2	no	—
263	C₄	C₉×#235	1	1	yes	characteristic
264	C₄	C₂×C₉×#67	1	1	yes	characteristic
265	C₃	—	2	1	no	—
266	C₃	C₉×#470	1	1	yes	characteristic, complemented
267	C₃	C₃×#5137	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	C₉×#909	1	2	yes	—
270	C₂	C₉×#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×M₁₆	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂³×C₃²	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₉×M₁₆	D₃
hypocenter	C₃	C₉×#470
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂×C₉×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₉×Dic₁₂	C₂	semidirect
4. C₉×#817	C₂	semidirect
5. C₉×#816	C₂	semidirect
40. #5137	C₉	direct (N × H)
77. C₂×Dic₁₂	C₂×C₉	semidirect
80. #817	C₂×C₉	semidirect
81. #816	C₂×C₉	semidirect
164. C₃×C₉	#470	semidirect
236. C₉	#5137	direct (N × H)
266. C₃	C₉×#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×M₁₆	2
C₉×#924	2
C₂×C₉×Dic₁₂	2
C₉×#817	2
C₉×#816	2
C₃×#5137	3
C₉×#470	3

Subgroups — C9×#5137 #13143195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5137 #13143195