Subgroups — C₉×#4848 #13142041

Order 1728.

422 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#204	C₂	1	2	yes	maximal, complemented
2	C₉×#874	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#821	C₂	1	2	yes	maximal, complemented
6	C₃×#4848	C₃	1	1	yes	maximal, characteristic
7	C₉×#460	—	3	1	no	maximal
8	C₂×C₉×D₁₂	—	2	1	no	—
9	C₂×C₉×D₁₂	—	2	1	no	—
10	C₂×C₉×D₁₂	C₂²	1	2	yes	—
11	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₉×Dic₃⋊C₄	—	2	1	no	—
16	C₄×C₉×Dic₃	—	2	1	no	—
17	C₉×#204	—	2	4	no	—
18	C₂×C₃×#204	C₂×C₃	1	2	yes	—
19	C₃×#874	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×#821	C₂×C₃	1	2	yes	—
23	C₂×C₉×SD₁₆	—	3	2	no	—
24	C₉×#127	—	3	1	no	—
25	C₄×Q₈×C₉	—	3	1	no	—
26	C₉×C₄⋊C₈	—	2	1	no	—
27	C₉×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
28	C₉×#102	—	3	2	no	—
29	C₂²×D₃×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
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	1	no	—
34	C₂×C₉×Dic₃	—	2	1	no	—
35	C₉×D₁₂	—	2	2	no	—
36	C₉×D₁₂	—	2	2	no	—
37	C₉×D₁₂	—	4	2	no	—
38	C₉×D₁₂	—	4	2	no	—
39	C₉×Dic₆	—	2	1	no	—
40	C₉×Dic₆	D₄	1	2	yes	complemented
41	C₃×C₈×C₉	—	2	2	no	—
42	C₃×#460	—	3	1	no	—
43	#4848	C₉	1	1	yes	characteristic, direct product factor
44	C₂×C₃×D₁₂	—	2	1	no	—
45	C₂×C₃×D₁₂	—	2	1	no	—
46	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
47	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
49	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
51	C₃×Dic₃⋊C₄	—	2	1	no	—
52	C₃×C₄×Dic₃	—	2	1	no	—
53	C₃×#204	—	2	4	no	—
54	C₂×Q₈×C₉	—	3	1	no	—
55	C₂×D₄×C₉	—	3	2	no	—
56	C₂×D₄×C₉	—	6	1	no	—
57	C₂×D₄×C₉	—	6	1	no	—
58	C₉×SD₁₆	—	6	4	no	—
59	C₂×C₈×C₉	—	2	2	no	—
60	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
61	C₉×C₄⋊C₄	—	3	1	no	—
62	C₉×C₄⋊C₄	—	6	1	no	—
63	C₄²×C₉	—	2	1	no	—
64	C₄²×C₉	—	6	1	no	—
65	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×D₃×C₉	—	4	2	no	—
68	C₂×D₃×C₉	—	4	2	no	—
69	C₂×D₃×C₉	—	4	2	no	—
70	C₃×C₄×C₉	—	2	1	no	—
71	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×C₉	#41	1	1	yes	characteristic
74	C₉×Dic₃	—	2	2	no	—
75	C₉×Dic₃	—	4	1	no	—
76	C₂×C₃×SD₁₆	—	3	2	no	—
77	C₃×#127	—	3	1	no	—
78	C₃×C₄×Q₈	—	3	1	no	—
79	C₂×#204	C₂×C₉	1	2	yes	complemented
80	#874	C₂×C₉	1	1	yes	characteristic
81	C₄×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
82	C₃×C₄⋊C₈	—	2	1	no	—
83	C₃×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
84	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
85	C₃×#102	—	3	2	no	—
86	#821	C₂×C₉	1	2	yes	complemented
87	C₂²×C₃×D₃	—	2	2	no	—
88	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₃×Dic₃	—	2	1	no	—
92	C₂×C₃×Dic₃	—	2	1	no	—
93	C₃×D₁₂	—	2	2	no	—
94	C₃×D₁₂	—	2	2	no	—
95	C₃×D₁₂	—	4	2	no	—
96	C₃×D₁₂	—	4	2	no	—
97	C₃×Dic₆	—	2	1	no	—
98	C₃×Dic₆	C₃×D₄	1	2	yes	—
99	C₂³×C₉	—	6	2	no	—
100	C₃²×C₈	—	2	2	no	—
101	Q₈×C₉	—	3	2	no	—
102	Q₈×C₉	—	6	1	no	—
103	D₄×C₉	—	6	2	no	—
104	D₄×C₉	—	6	2	no	—
105	D₄×C₉	—	12	2	no	—
106	D₄×C₉	—	12	2	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	6	1	no	—
111	C₂×C₄×C₉	—	6	1	no	—
112	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
113	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	4	2	no	—
117	#460	—	3	1	no	Sylow
118	C₂×C₃×C₉	#121	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	D₃×C₉	—	8	2	no	—
122	C₂×C₃×Q₈	—	3	1	no	—
123	C₂×C₃×D₄	—	3	2	no	—
124	C₂×C₃×D₄	—	6	1	no	—
125	C₂×C₃×D₄	—	6	1	no	—
126	C₂×D₁₂	—	2	1	no	—
127	C₂×D₁₂	—	2	1	no	—
128	C₂×D₁₂	C₂²×C₉	1	2	yes	—
129	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
130	C₃×SD₁₆	—	6	4	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
133	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
134	C₃×C₄⋊C₄	—	3	1	no	—
135	C₃×C₄⋊C₄	—	6	1	no	—
136	C₃×C₄²	—	2	1	no	—
137	C₃×C₄²	—	6	1	no	—
138	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
139	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
140	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
141	Dic₃⋊C₄	—	2	1	no	—
142	C₄×Dic₃	—	2	1	no	—
143	#204	—	2	4	no	—
144	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₃×D₃	—	4	2	no	—
146	C₂×C₃×D₃	—	4	2	no	—
147	C₂×C₃×D₃	—	4	2	no	—
148	C₃²×C₄	—	2	1	no	—
149	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
150	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
151	C₃²×C₄	C₃×#41	1	1	yes	characteristic
152	C₃×Dic₃	—	2	2	no	—
153	C₃×Dic₃	—	4	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	12	2	no	—
156	C₂²×C₉	—	12	2	no	—
157	C₂²×C₉	—	12	2	no	—
158	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	2	2	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	4	1	no	—
164	C₄×C₉	—	6	2	no	—
165	C₄×C₉	—	12	1	no	—
166	C₄×C₉	#204	1	2	yes	complemented
167	C₄×C₉	D₃×D₄	1	1	yes	characteristic
168	C₄×C₉	#239	1	1	yes	characteristic
169	C₂×SD₁₆	—	3	2	no	—
170	#127	—	3	1	no	—
171	C₄×Q₈	—	3	1	no	—
172	C₄⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
173	#102	—	3	2	no	—
174	C₃×C₉	#460	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	6	2	no	—
176	C₂²×D₃	—	2	2	no	—
177	C₃×Q₈	—	3	2	no	—
178	C₃×Q₈	—	6	1	no	—
179	C₃×D₄	—	6	2	no	—
180	C₃×D₄	—	6	2	no	—
181	C₃×D₄	—	12	2	no	—
182	C₃×D₄	—	12	2	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	6	1	no	—
187	C₂×C₃×C₄	—	6	1	no	—
188	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
189	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
191	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
192	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
194	C₂×Dic₃	—	2	1	no	—
195	C₂×Dic₃	—	2	1	no	—
196	D₁₂	—	2	2	no	—
197	D₁₂	—	2	2	no	—
198	D₁₂	—	4	2	no	—
199	D₁₂	—	4	2	no	—
200	Dic₆	—	2	1	no	—
201	Dic₆	D₄×C₉	1	2	yes	complemented
202	C₃×C₈	—	2	2	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	4	2	no	—
205	C₂×C₃²	C₃×#121	1	1	yes	characteristic
206	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
207	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₃×D₃	—	8	2	no	—
209	C₂×C₉	—	2	1	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	24	2	no	—
213	C₂×C₉	#895	1	1	yes	characteristic
214	C₂×C₉	C₂×#204	1	1	yes	characteristic
215	C₂×C₉	#908	1	1	yes	characteristic
216	C₂×Q₈	—	3	1	no	—
217	C₂×D₄	—	3	2	no	—
218	C₂×D₄	—	6	1	no	—
219	C₂×D₄	—	6	1	no	—
220	SD₁₆	—	6	4	no	—
221	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄²	—	6	1	no	—
225	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	12	2	no	—
228	C₂²×C₃	—	12	2	no	—
229	C₂²×C₃	—	12	2	no	—
230	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
232	C₂×D₃	—	4	2	no	—
233	C₂×D₃	—	4	2	no	—
234	C₂×D₃	—	4	2	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	6	2	no	—
242	C₃×C₄	—	12	1	no	—
243	C₃×C₄	C₉×SD₁₆	1	2	yes	complemented
244	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
245	C₃×C₄	C₉×#41	1	1	yes	characteristic
246	C₃×C₄	C₃×#204	1	2	yes	—
247	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
248	C₃×C₄	C₃×#239	1	1	yes	characteristic
249	Dic₃	—	2	2	no	—
250	Dic₃	—	4	1	no	—
251	C₃²	C₃×#460	1	1	yes	characteristic
252	C₉	—	2	1	no	—
253	C₉	#4848	1	1	yes	characteristic, direct product factor
254	C₂³	—	6	2	no	—
255	Q₈	—	3	2	no	—
256	Q₈	—	6	1	no	—
257	D₄	—	6	2	no	—
258	D₄	—	6	2	no	—
259	D₄	—	12	2	no	—
260	D₄	—	12	2	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	6	1	no	—
263	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
264	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	24	2	no	—
271	C₂×C₃	C₉×#121	1	1	yes	characteristic
272	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
273	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂×C₃	C₃×#895	1	1	yes	characteristic
275	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
276	C₂×C₃	C₃×#908	1	1	yes	characteristic
277	D₃	—	8	2	no	—
278	C₂²	—	12	2	no	—
279	C₂²	—	12	2	no	—
280	C₂²	—	12	2	no	—
281	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	6	2	no	—
284	C₄	—	12	1	no	—
285	C₄	C₉×#204	1	2	yes	complemented
286	C₄	D₃×D₄×C₉	1	1	yes	characteristic
287	C₄	C₉×#239	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	C₉×#460	1	1	yes	characteristic, complemented
290	C₃	C₃×#4848	1	1	yes	characteristic
291	C₂	—	24	2	no	—
292	C₂	C₉×#895	1	1	yes	characteristic
293	C₂	C₂×C₉×#204	1	1	yes	characteristic
294	C₂	C₉×#908	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	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#874	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₄⋊C₈	D₃
hypocenter	C₃	C₉×#460
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#204	C₂	semidirect
3. C₄×C₉×Dic₆	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₉×#821	C₂	semidirect
27. C₉×C₄⋊C₈	D₃	semidirect
40. C₉×Dic₆	D₄	semidirect
43. #4848	C₉	direct (N × H)
71. C₃×C₄×C₉	SD₁₆	semidirect
79. C₂×#204	C₂×C₉	semidirect
81. C₄×Dic₆	C₂×C₉	semidirect
83. C₃×C₄⋊C₈	C₂×C₉	semidirect
86. #821	C₂×C₉	semidirect
166. C₄×C₉	#204	semidirect
172. C₄⋊C₈	D₃×C₉	semidirect
174. C₃×C₉	#460	semidirect
201. Dic₆	D₄×C₉	semidirect
243. C₃×C₄	C₉×SD₁₆	semidirect
253. C₉	#4848	direct (N × H)
285. C₄	C₉×#204	semidirect
289. C₃	C₉×#460	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#204	2
C₉×#874	2
C₄×C₉×Dic₆	2
C₃×C₉×C₄⋊C₈	2
C₉×#821	2
C₃×#4848	3
C₉×#460	3

Subgroups — C9×#4848 #13142041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4848 #13142041