Subgroups — C₉×#4837 #13142004

Order 1728.

416 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal
2	C₄²×D₃×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4837	C₃	1	1	yes	maximal, characteristic
7	C₉×#423	—	3	1	no	maximal
8	C₂×C₄×D₃×C₉	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃×C₉	C₄	1	1	yes	characteristic
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₄×C₉×Dic₃	C₄	1	1	yes	characteristic
14	C₄×C₉×Dic₃	C₄	1	1	yes	characteristic
15	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
17	C₉×ΓL(1,𝔽₂₅)	—	2	4	no	—
18	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	2	yes	—
19	C₃×C₄²×D₃	C₂×C₃	1	1	yes	characteristic
20	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₃×Dic₃⋊C₈	C₂×C₃	1	2	yes	—
22	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
23	C₂×C₉×M₁₆	—	3	2	no	—
24	C₂×C₄²×C₉	—	3	1	no	—
25	C₉×C₄⋊C₈	—	2	1	no	—
26	C₉×C₄⋊C₈	—	3	2	no	—
27	C₉×C₄⋊C₈	—	3	1	no	—
28	C₉×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
29	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₄×D₃×C₉	—	2	2	no	—
37	C₄×D₃×C₉	—	2	2	no	—
38	C₄×D₃×C₉	D₄	1	2	yes	—
39	C₄×D₃×C₉	Q₈	1	2	yes	—
40	C₃×C₈×C₉	—	2	2	no	—
41	C₉×C₃⋊C₈	—	2	2	no	—
42	C₃×#423	—	3	1	no	—
43	#4837	C₉	1	1	yes	characteristic, direct product factor
44	C₂×C₃×C₄×D₃	C₃×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×D₃	C₃×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
48	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₄×Dic₃	C₃×C₄	1	1	yes	characteristic
50	C₃×C₄×Dic₃	C₃×C₄	1	1	yes	characteristic
51	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
53	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
54	C₂²×C₄×C₉	—	3	1	no	—
55	C₂²×C₄×C₉	—	3	1	no	—
56	C₂²×C₄×C₉	—	3	1	no	—
57	C₉×M₁₆	—	6	4	no	—
58	C₂×C₈×C₉	—	2	2	no	—
59	C₂×C₈×C₉	—	3	2	no	—
60	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
61	C₄²×C₉	—	2	1	no	—
62	C₄²×C₉	—	3	1	no	—
63	C₄²×C₉	—	3	1	no	—
64	C₄²×C₉	—	3	1	no	—
65	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₃×C₉	—	2	1	no	—
68	C₂×D₃×C₉	—	2	1	no	—
69	C₂×D₃×C₉	C₄⋊C₄	1	2	yes	—
70	C₃×C₄×C₉	—	2	1	no	—
71	C₃×C₄×C₉	M₁₆	1	2	yes	complemented
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
74	C₉×Dic₃	—	2	1	no	—
75	C₉×Dic₃	C₄⋊C₄	1	2	yes	—
76	C₉×Dic₃	M₁₆	1	2	yes	complemented
77	C₂×C₃×M₁₆	—	3	2	no	—
78	C₂×C₃×C₄²	—	3	1	no	—
79	C₂×ΓL(1,𝔽₂₅)	C₂×C₉	1	2	yes	—
80	C₄²×D₃	C₂×C₉	1	1	yes	characteristic
81	C₃×C₄⋊C₈	—	2	1	no	—
82	C₃×C₄⋊C₈	—	3	2	no	—
83	C₃×C₄⋊C₈	—	3	1	no	—
84	C₃×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
85	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
86	Dic₃⋊C₈	C₂×C₉	1	2	yes	complemented
87	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic, complemented
88	C₂²×C₃×D₃	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
95	C₃×C₄×D₃	—	2	2	no	—
96	C₃×C₄×D₃	—	2	2	no	—
97	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
98	C₃×C₄×D₃	C₃×Q₈	1	2	yes	—
99	C₂³×C₉	—	3	1	no	—
100	C₃²×C₈	—	2	2	no	—
101	C₃×C₃⋊C₈	—	2	2	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	3	1	no	—
106	C₂×C₄×C₉	—	3	1	no	—
107	C₂×C₄×C₉	—	3	2	no	—
108	C₂×C₄×C₉	—	3	2	no	—
109	C₂×C₄×C₉	—	3	1	no	—
110	C₂×C₄×C₉	—	6	2	no	—
111	C₂×C₄×C₉	—	6	2	no	—
112	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
113	C₂×C₄×C₉	C₄×D₃	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	C₈×C₉	—	6	2	no	—
118	#423	—	3	1	no	Sylow
119	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
122	D₃×C₉	—	2	2	no	—
123	C₂²×C₃×C₄	—	3	1	no	—
124	C₂²×C₃×C₄	—	3	1	no	—
125	C₂²×C₃×C₄	—	3	1	no	—
126	C₂×C₄×D₃	C₄×C₉	1	1	yes	characteristic
127	C₂×C₄×D₃	C₄×C₉	1	1	yes	characteristic
128	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
129	C₃×M₁₆	—	6	4	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	3	2	no	—
132	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
133	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
134	C₃×C₄²	—	2	1	no	—
135	C₃×C₄²	—	3	1	no	—
136	C₃×C₄²	—	3	1	no	—
137	C₃×C₄²	—	3	1	no	—
138	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
139	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
140	C₄×Dic₃	C₄×C₉	1	1	yes	characteristic
141	C₄×Dic₃	C₄×C₉	1	1	yes	characteristic
142	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
143	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
144	ΓL(1,𝔽₂₅)	—	2	4	no	—
145	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₂×C₃×D₃	—	2	1	no	—
147	C₂×C₃×D₃	—	2	1	no	—
148	C₂×C₃×D₃	C₃×C₄⋊C₄	1	2	yes	—
149	C₃²×C₄	—	2	1	no	—
150	C₃²×C₄	C₃×M₁₆	1	2	yes	—
151	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
152	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
153	C₃×Dic₃	—	2	1	no	—
154	C₃×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
155	C₃×Dic₃	C₃×M₁₆	1	2	yes	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	3	2	no	—
158	C₂²×C₉	—	6	1	no	—
159	C₂²×C₉	—	6	1	no	—
160	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	3	2	no	—
166	C₄×C₉	—	3	2	no	—
167	C₄×C₉	—	4	1	no	—
168	C₄×C₉	—	6	1	no	—
169	C₄×C₉	ΓL(1,𝔽₂₅)	1	2	yes	complemented
170	C₄×C₉	D₃×D₄	1	1	yes	characteristic
171	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
172	C₂×M₁₆	—	3	2	no	—
173	C₂×C₄²	—	3	1	no	—
174	C₄⋊C₈	—	3	2	no	—
175	C₄⋊C₈	—	3	1	no	—
176	C₄⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
177	C₃×C₉	#423	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₃	—	3	1	no	—
179	C₂²×D₃	C₂×C₄×C₉	1	1	yes	characteristic
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	3	1	no	—
184	C₂×C₃×C₄	—	3	1	no	—
185	C₂×C₃×C₄	—	3	2	no	—
186	C₂×C₃×C₄	—	3	2	no	—
187	C₂×C₃×C₄	—	3	1	no	—
188	C₂×C₃×C₄	—	6	2	no	—
189	C₂×C₃×C₄	—	6	2	no	—
190	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
193	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
196	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
197	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
198	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
199	C₄×D₃	—	2	2	no	—
200	C₄×D₃	—	2	2	no	—
201	C₄×D₃	D₄×C₉	1	2	yes	—
202	C₄×D₃	Q₈×C₉	1	2	yes	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	4	2	no	—
206	C₃×C₈	—	6	2	no	—
207	C₃⋊C₈	—	2	2	no	—
208	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
210	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
211	C₃×D₃	—	2	2	no	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	6	2	no	—
216	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₉	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
218	C₂×C₉	D₃×M₁₆	1	1	yes	characteristic
219	C₂²×C₄	—	3	1	no	—
220	C₂²×C₄	—	3	1	no	—
221	C₂²×C₄	—	3	1	no	—
222	M₁₆	—	6	4	no	—
223	C₂×C₈	—	3	2	no	—
224	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
225	C₄²	—	3	1	no	—
226	C₄²	—	3	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	3	2	no	—
231	C₂²×C₃	—	6	1	no	—
232	C₂²×C₃	—	6	1	no	—
233	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
234	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
235	C₂×D₃	—	2	1	no	—
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	C₉×C₄⋊C₄	1	2	yes	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	3	2	no	—
244	C₃×C₄	—	3	2	no	—
245	C₃×C₄	—	4	1	no	—
246	C₃×C₄	—	6	1	no	—
247	C₃×C₄	C₉×M₁₆	1	2	yes	complemented
248	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
249	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
250	C₃×C₄	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
251	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
252	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
253	Dic₃	—	2	1	no	—
254	Dic₃	C₉×C₄⋊C₄	1	2	yes	—
255	Dic₃	C₉×M₁₆	1	2	yes	complemented
256	C₃²	C₃×#423	1	1	yes	characteristic
257	C₉	—	2	1	no	—
258	C₉	#4837	1	1	yes	characteristic, direct product factor
259	C₂³	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	2	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	3	1	no	—
265	C₂×C₄	—	6	2	no	—
266	C₂×C₄	—	6	2	no	—
267	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
268	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	6	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	6	2	no	—
276	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₃	C₂×C₉×M₁₆	1	1	yes	characteristic
278	C₂×C₃	C₂×C₉×M₁₆	1	1	yes	characteristic
279	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₃	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
281	C₂×C₃	C₃×D₃×M₁₆	1	1	yes	characteristic
282	D₃	—	2	2	no	—
283	C₂²	—	3	2	no	—
284	C₂²	—	6	1	no	—
285	C₂²	—	6	1	no	—
286	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	3	2	no	—
289	C₄	—	3	2	no	—
290	C₄	—	6	1	no	—
291	C₄	C₉×ΓL(1,𝔽₂₅)	1	2	yes	complemented
292	C₄	D₃×D₄×C₉	1	1	yes	characteristic
293	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
294	C₃	—	2	1	no	—
295	C₃	C₉×#423	1	1	yes	characteristic, complemented
296	C₃	C₃×#4837	1	1	yes	characteristic
297	C₂	—	6	2	no	—
298	C₂	D₃×C₉×C₄⋊C₄	1	1	yes	characteristic
299	C₂	C₂×C₉×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
300	C₂	D₃×C₉×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂²×C₃×D₃	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₄⋊C₈	D₃
hypocenter	C₃	C₉×#423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₉×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
4. C₉×Dic₃⋊C₈	C₂	semidirect
5. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
28. C₉×C₄⋊C₈	D₃	semidirect
43. #4837	C₉	direct (N × H)
71. C₃×C₄×C₉	M₁₆	semidirect
76. C₉×Dic₃	M₁₆	semidirect
84. C₃×C₄⋊C₈	C₂×C₉	semidirect
86. Dic₃⋊C₈	C₂×C₉	semidirect
87. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
169. C₄×C₉	ΓL(1,𝔽₂₅)	semidirect
176. C₄⋊C₈	D₃×C₉	semidirect
177. C₃×C₉	#423	semidirect
247. C₃×C₄	C₉×M₁₆	semidirect
255. Dic₃	C₉×M₁₆	semidirect
258. C₉	#4837	direct (N × H)
291. C₄	C₉×ΓL(1,𝔽₂₅)	semidirect
295. C₃	C₉×#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×ΓL(1,𝔽₂₅)	2
C₄²×D₃×C₉	2
C₃×C₉×C₄⋊C₈	2
C₉×Dic₃⋊C₈	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#4837	3
C₉×#423	3

Subgroups — C9×#4837 #13142004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4837 #13142004