Subgroups — C₉×#5142 #13143252

Order 1728.

386 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#949	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#823	C₂	1	2	yes	maximal, complemented
5	C₉×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5142	C₃	1	1	yes	maximal, characteristic
7	C₉×#472	—	3	1	no	maximal
8	C₃×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂×C₉×D₁₂	C₂²	1	2	yes	—
10	C₃×C₉×M₁₆	—	2	1	no	—
11	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	—	2	1	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₉×#215	—	2	2	no	—
15	C₉×#213	—	2	2	no	—
16	C₉×#208	C₂²	1	2	yes	—
17	C₉×D₂₄	—	2	1	no	—
18	C₉×D₂₄	—	2	1	no	—
19	C₃²×#131	C₂×C₃	1	1	yes	characteristic
20	C₃×#949	C₂×C₃	1	2	yes	—
21	C₂×C₃×D₂₄	C₂×C₃	1	1	yes	characteristic
22	C₃×#823	C₂×C₃	1	2	yes	—
23	C₃×#819	C₂×C₃	1	1	yes	characteristic
24	C₉×AGL(1,ℤ/8)	—	3	2	no	—
25	C₂×C₉×D₈	—	3	1	no	—
26	C₉×#131	—	2	1	no	—
27	C₉×#131	D₃	1	1	yes	characteristic, complemented
28	C₉×#108	—	3	1	no	—
29	C₉×#100	—	3	2	no	—
30	C₂²×D₃×C₉	—	2	2	no	—
31	C₃×Q₈×C₉	D₄	1	1	yes	characteristic
32	C₃×D₄×C₉	—	2	1	no	—
33	C₃×D₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	—	2	1	no	—
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₉×D₁₂	—	2	2	no	—
37	C₉×D₁₂	—	2	2	no	—
38	C₃×C₈×C₉	—	2	1	no	—
39	C₃×C₈×C₉	D₄	1	1	yes	characteristic
40	C₃×C₈×C₉	D₄	1	1	yes	characteristic
41	C₉×C₃⋊C₈	—	2	2	no	—
42	C₃×#472	—	3	1	no	—
43	#5142	C₉	1	1	yes	characteristic, direct product factor
44	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
46	C₃²×M₁₆	—	2	1	no	—
47	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃²×C₈	—	2	1	no	—
49	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₃×#215	—	2	2	no	—
51	C₃×#213	—	2	2	no	—
52	C₃×#208	C₂²×C₃	1	2	yes	—
53	C₃×D₂₄	—	2	1	no	—
54	C₃×D₂₄	—	2	1	no	—
55	C₉×#41	—	2	1	no	—
56	C₉×#41	C₂×D₃	1	1	yes	characteristic
57	C₂×D₄×C₉	—	3	2	no	—
58	C₉×SD₁₆	—	6	2	no	—
59	C₉×D₈	—	6	2	no	—
60	C₉×D₈	—	6	1	no	—
61	C₉×D₈	—	6	1	no	—
62	C₉×M₁₆	—	2	2	no	—
63	C₉×M₁₆	—	2	1	no	—
64	C₉×M₁₆	—	2	1	no	—
65	C₉×M₁₆	—	3	2	no	—
66	C₉×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂×C₈×C₉	—	2	2	no	—
68	C₂×C₈×C₉	—	2	1	no	—
69	C₂×C₈×C₉	—	2	1	no	—
70	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
71	C₂²×C₃×C₉	—	2	1	no	—
72	C₂²×C₃×C₉	#41	1	1	yes	characteristic
73	C₂×D₃×C₉	—	4	2	no	—
74	C₂×D₃×C₉	—	8	2	no	—
75	C₃×C₄×C₉	—	2	1	no	—
76	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₃×AGL(1,ℤ/8)	—	3	2	no	—
79	C₂×C₃×D₈	—	3	1	no	—
80	C₃×#131	—	2	1	no	—
81	C₃×#131	C₂×C₉	1	1	yes	characteristic, complemented
82	C₃×#131	C₃×D₃	1	1	yes	characteristic
83	#949	C₂×C₉	1	2	yes	complemented
84	C₂×D₂₄	C₂×C₉	1	1	yes	characteristic, complemented
85	C₃×#108	—	3	1	no	—
86	C₃×#100	—	3	2	no	—
87	#823	C₂×C₉	1	2	yes	complemented
88	#819	C₂×C₉	1	1	yes	characteristic, complemented
89	C₂²×C₃×D₃	—	2	2	no	—
90	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
91	C₃²×D₄	—	2	1	no	—
92	C₃²×D₄	C₃×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	—	2	1	no	—
94	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₃×D₁₂	—	2	2	no	—
96	C₃×D₁₂	—	2	2	no	—
97	C₂³×C₉	—	6	2	no	—
98	C₃²×C₈	—	2	1	no	—
99	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
100	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
101	C₃×C₃⋊C₈	—	2	2	no	—
102	Q₈×C₉	—	2	1	no	—
103	Q₈×C₉	D₁₂	1	1	yes	characteristic
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	1	no	—
106	D₄×C₉	—	2	1	no	—
107	D₄×C₉	—	6	2	no	—
108	D₄×C₉	—	6	2	no	—
109	D₄×C₉	D₁₂	1	1	yes	characteristic
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	2	1	no	—
113	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
114	C₈×C₉	—	2	2	no	—
115	C₈×C₉	—	2	1	no	—
116	C₈×C₉	—	2	1	no	—
117	C₈×C₉	—	2	1	no	—
118	C₈×C₉	—	6	2	no	—
119	C₈×C₉	#67	1	1	yes	characteristic
120	C₈×C₉	#67	1	1	yes	characteristic
121	#472	—	3	1	no	Sylow
122	C₂×C₃×C₉	—	2	1	no	—
123	C₂×C₃×C₉	—	4	1	no	—
124	C₂×C₃×C₉	#121	1	1	yes	characteristic
125	D₃×C₉	—	8	2	no	—
126	C₃×#41	—	2	1	no	—
127	C₃×#41	C₂²×C₉	1	1	yes	characteristic
128	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
129	C₂×C₃×D₄	—	3	2	no	—
130	C₂×D₁₂	C₂²×C₉	1	2	yes	—
131	C₃×SD₁₆	—	6	2	no	—
132	C₃×D₈	—	6	2	no	—
133	C₃×D₈	—	6	1	no	—
134	C₃×D₈	—	6	1	no	—
135	C₃×M₁₆	—	2	2	no	—
136	C₃×M₁₆	—	2	1	no	—
137	C₃×M₁₆	—	2	1	no	—
138	C₃×M₁₆	—	2	1	no	—
139	C₃×M₁₆	—	3	2	no	—
140	C₃×M₁₆	C₂²×C₉	1	1	yes	characteristic
141	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
142	C₂×C₃×C₈	—	2	2	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
148	#215	—	2	2	no	—
149	#213	—	2	2	no	—
150	#208	C₂²×C₉	1	2	yes	—
151	D₂₄	—	2	1	no	—
152	D₂₄	—	2	1	no	—
153	C₂²×C₃²	—	2	1	no	—
154	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
155	C₂×C₃×D₃	—	4	2	no	—
156	C₂×C₃×D₃	—	8	2	no	—
157	C₃²×C₄	—	2	1	no	—
158	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
159	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	—	12	2	no	—
164	C₂²×C₉	—	24	2	no	—
165	C₂²×C₉	#235	1	1	yes	characteristic
166	C₄×C₉	—	2	2	no	—
167	C₄×C₉	—	2	1	no	—
168	C₄×C₉	—	2	1	no	—
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
171	C₄×C₉	C₂×#67	1	1	yes	characteristic
172	AGL(1,ℤ/8)	—	3	2	no	—
173	C₂×D₈	—	3	1	no	—
174	#131	D₃×C₉	1	1	yes	characteristic, complemented
175	#108	—	3	1	no	—
176	#100	—	3	2	no	—
177	C₃×C₉	#472	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₃	—	6	2	no	—
179	C₂²×D₃	—	2	2	no	—
180	C₃×Q₈	—	2	1	no	—
181	C₃×Q₈	D₄×C₉	1	1	yes	characteristic
182	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
183	C₃×D₄	—	2	2	no	—
184	C₃×D₄	—	2	1	no	—
185	C₃×D₄	—	2	1	no	—
186	C₃×D₄	—	2	1	no	—
187	C₃×D₄	—	6	2	no	—
188	C₃×D₄	—	6	2	no	—
189	C₃×D₄	D₄×C₉	1	1	yes	characteristic
190	C₃×D₄	C₃×D₁₂	1	1	yes	characteristic
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
197	D₁₂	—	2	2	no	—
198	D₁₂	—	2	2	no	—
199	C₃×C₈	—	2	2	no	—
200	C₃×C₈	—	2	1	no	—
201	C₃×C₈	—	2	1	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	6	2	no	—
205	C₃×C₈	D₄×C₉	1	1	yes	characteristic
206	C₃×C₈	D₄×C₉	1	1	yes	characteristic
207	C₃×C₈	C₃×#67	1	1	yes	characteristic
208	C₃×C₈	C₃×#67	1	1	yes	characteristic
209	C₃⋊C₈	—	2	2	no	—
210	C₂×C₃²	—	2	1	no	—
211	C₂×C₃²	—	4	1	no	—
212	C₂×C₃²	C₃×#121	1	1	yes	characteristic
213	C₃×D₃	—	8	2	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	4	1	no	—
217	C₂×C₉	—	4	2	no	—
218	C₂×C₉	—	4	1	no	—
219	C₂×C₉	—	24	2	no	—
220	C₂×C₉	#930	1	1	yes	characteristic
221	#41	C₂×D₃×C₉	1	1	yes	characteristic
222	C₂×D₄	—	3	2	no	—
223	SD₁₆	—	6	2	no	—
224	D₈	—	6	2	no	—
225	D₈	—	6	1	no	—
226	D₈	—	6	1	no	—
227	M₁₆	—	2	1	no	—
228	M₁₆	—	3	2	no	—
229	M₁₆	C₂×D₃×C₉	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	12	2	no	—
237	C₂²×C₃	—	24	2	no	—
238	C₂²×C₃	C₉×#41	1	1	yes	characteristic
239	C₂²×C₃	C₃×#235	1	1	yes	characteristic
240	C₂×D₃	—	4	2	no	—
241	C₂×D₃	—	8	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
248	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
249	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
250	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
251	C₃²	C₃×#472	1	1	yes	characteristic
252	C₉	—	2	1	no	—
253	C₉	#5142	1	1	yes	characteristic, direct product factor
254	C₂³	—	6	2	no	—
255	Q₈	C₉×D₁₂	1	1	yes	characteristic
256	D₄	—	2	1	no	—
257	D₄	—	6	2	no	—
258	D₄	—	6	2	no	—
259	D₄	C₉×D₁₂	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	6	2	no	—
264	C₈	C₉×#67	1	1	yes	characteristic
265	C₈	C₉×#67	1	1	yes	characteristic
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	—	4	2	no	—
271	C₂×C₃	—	4	1	no	—
272	C₂×C₃	—	4	1	no	—
273	C₂×C₃	—	24	2	no	—
274	C₂×C₃	C₉×#121	1	1	yes	characteristic
275	C₂×C₃	C₃×#930	1	1	yes	characteristic
276	D₃	—	8	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	12	2	no	—
279	C₂²	—	24	2	no	—
280	C₂²	C₉×#235	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
283	C₄	C₂×C₉×#67	1	1	yes	characteristic
284	C₃	—	2	1	no	—
285	C₃	C₉×#472	1	1	yes	characteristic, complemented
286	C₃	C₃×#5142	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	1	no	—
289	C₂	—	24	2	no	—
290	C₂	C₉×#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#930
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#131	C₂
socle	C₂×C₃²	C₃×#121
prime radical	C₃×#5142	C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#131	D₃
hypocenter	C₃	C₉×#472
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#131

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#131	C₂	semidirect
2. C₉×#949	C₂	semidirect
3. C₂×C₉×D₂₄	C₂	semidirect
4. C₉×#823	C₂	semidirect
5. C₉×#819	C₂	semidirect
27. C₉×#131	D₃	semidirect
43. #5142	C₉	direct (N × H)
81. C₃×#131	C₂×C₉	semidirect
83. #949	C₂×C₉	semidirect
84. C₂×D₂₄	C₂×C₉	semidirect
87. #823	C₂×C₉	semidirect
88. #819	C₂×C₉	semidirect
174. #131	D₃×C₉	semidirect
177. C₃×C₉	#472	semidirect
253. C₉	#5142	direct (N × H)
285. C₃	C₉×#472	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#131	2
C₉×#949	2
C₂×C₉×D₂₄	2
C₉×#823	2
C₉×#819	2
C₃×#5142	3
C₉×#472	3

Subgroups — C9×#5142 #13143252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5142 #13143252