Subgroups — C₉×#4745 #13141667

Order 1728.

496 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#237	C₂	1	1	yes	maximal, characteristic
2	D₃×C₉×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×#100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#833	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#824	C₂	1	2	yes	maximal, complemented
6	C₃×#4745	C₃	1	1	yes	maximal, characteristic
7	C₉×#413	—	3	1	no	maximal
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×#67	C₄	1	2	yes	—
10	C₂²×C₉×Dic₃	C₄	1	2	yes	—
11	C₉×#237	—	2	2	no	—
12	C₉×#237	—	2	2	no	—
13	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
15	C₃×C₉×M₁₆	C₂²	1	2	yes	—
16	C₉×#208	C₂²	1	2	yes	—
17	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
18	D₃×C₈×C₉	—	2	2	no	—
19	C₂×C₃×#237	C₂×C₃	1	1	yes	characteristic
20	C₃×D₃×M₁₆	C₂×C₃	1	2	yes	—
21	C₂×C₉×#41	—	3	1	no	—
22	C₃²×#100	C₂×C₃	1	1	yes	characteristic
23	C₃×#833	C₂×C₃	1	1	yes	characteristic
24	C₃×#824	C₂×C₃	1	2	yes	—
25	C₂×C₉×M₁₆	—	3	2	no	—
26	C₉×#101	—	3	2	no	—
27	C₉×#100	—	2	1	no	—
28	C₉×#100	—	3	1	no	—
29	C₉×#100	D₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
31	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
32	C₃×D₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₉×#67	—	2	4	no	—
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₂×C₉×Dic₃	—	2	2	no	—
37	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
38	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₄×D₃×C₉	D₄	1	2	yes	—
40	C₄×D₃×C₉	D₄	1	2	yes	—
41	C₉×Dic₆	—	2	2	no	—
42	C₃×C₈×C₉	—	2	2	no	—
43	C₉×C₃⋊C₈	—	2	2	no	—
44	C₃×#413	—	3	1	no	—
45	#4745	C₉	1	1	yes	characteristic, direct product factor
46	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃×#67	C₃×C₄	1	2	yes	—
48	C₂²×C₃×Dic₃	C₃×C₄	1	2	yes	—
49	C₃×#237	—	2	2	no	—
50	C₃×#237	—	2	2	no	—
51	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
53	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
54	C₃×#208	C₂²×C₃	1	2	yes	—
55	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
56	C₃×D₃×C₈	—	2	2	no	—
57	C₉×#41	—	6	2	no	—
58	C₉×#41	—	6	2	no	—
59	C₂×Q₈×C₉	—	3	1	no	—
60	C₂×D₄×C₉	—	2	1	no	—
61	C₂×D₄×C₉	—	3	2	no	—
62	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₉	—	3	1	no	—
64	C₂²×C₄×C₉	—	3	2	no	—
65	C₉×M₁₆	—	2	2	no	—
66	C₉×M₁₆	—	3	2	no	—
67	C₉×M₁₆	—	6	2	no	—
68	C₉×M₁₆	C₂×D₃	1	2	yes	—
69	C₂×C₈×C₉	—	6	2	no	—
70	C₂²×C₃×C₉	—	2	2	no	—
71	C₂²×C₃×C₉	—	4	2	no	—
72	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×D₃×C₉	—	4	1	no	—
74	C₂×D₃×C₉	C₂²⋊C₄	1	2	yes	—
75	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
76	C₉×Dic₃	—	2	2	no	—
77	C₉×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₉×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×#41	—	3	1	no	—
80	C₂×#237	C₂×C₉	1	1	yes	characteristic
81	C₂×C₃×M₁₆	—	3	2	no	—
82	D₃×M₁₆	C₂×C₉	1	2	yes	complemented
83	C₃×#101	—	3	2	no	—
84	C₃×#100	—	2	1	no	—
85	C₃×#100	—	3	1	no	—
86	C₃×#100	C₂×C₉	1	1	yes	characteristic, complemented
87	C₃×#100	C₃×D₃	1	1	yes	characteristic
88	#833	C₂×C₉	1	1	yes	characteristic, complemented
89	#824	C₂×C₉	1	2	yes	complemented
90	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
91	C₂²×C₃×D₃	C₂×C₃×C₄	1	1	yes	characteristic
92	C₃²×D₄	—	2	2	no	—
93	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₃×#67	—	2	4	no	—
95	C₂×C₃×Dic₃	—	2	2	no	—
96	C₂×C₃×Dic₃	—	2	2	no	—
97	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
98	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
99	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
100	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
101	C₃×Dic₆	—	2	2	no	—
102	C₂³×C₉	—	2	2	no	—
103	C₂³×C₉	—	3	1	no	—
104	C₂³×C₉	C₄×D₃	1	2	yes	—
105	C₃²×C₈	—	2	2	no	—
106	C₃×C₃⋊C₈	—	2	2	no	—
107	Q₈×C₉	—	6	2	no	—
108	D₄×C₉	—	2	2	no	—
109	D₄×C₉	—	4	2	no	—
110	D₄×C₉	—	6	4	no	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	3	1	no	—
113	C₂×C₄×C₉	—	3	2	no	—
114	C₂×C₄×C₉	—	3	2	no	—
115	C₂×C₄×C₉	—	3	2	no	—
116	C₂×C₄×C₉	—	6	2	no	—
117	C₂×C₄×C₉	—	6	2	no	—
118	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
119	C₈×C₉	—	2	2	no	—
120	C₈×C₉	—	4	2	no	—
121	C₈×C₉	—	6	2	no	—
122	#413	—	3	1	no	Sylow
123	C₂×C₃×C₉	—	2	1	no	—
124	C₂×C₃×C₉	—	4	2	no	—
125	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	D₃×C₉	—	2	2	no	—
127	C₃×#41	—	6	2	no	—
128	C₃×#41	—	6	2	no	—
129	C₂×C₃×Q₈	—	3	1	no	—
130	C₂×C₃×D₄	—	2	1	no	—
131	C₂×C₃×D₄	—	3	2	no	—
132	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
133	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
134	C₂²×C₃×C₄	—	3	1	no	—
135	C₂²×C₃×C₄	—	3	2	no	—
136	C₂×#67	C₄×C₉	1	2	yes	—
137	C₂²×Dic₃	C₄×C₉	1	2	yes	—
138	#237	—	2	2	no	—
139	#237	—	2	2	no	—
140	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
141	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
142	C₃×M₁₆	—	2	2	no	—
143	C₃×M₁₆	—	3	2	no	—
144	C₃×M₁₆	—	6	2	no	—
145	C₃×M₁₆	C₂²×C₉	1	2	yes	—
146	C₃×M₁₆	C₂×C₃×D₃	1	2	yes	—
147	C₂×C₃×C₈	—	6	2	no	—
148	#208	C₂²×C₉	1	2	yes	—
149	ΓL(1,𝔽₂₅)	—	2	2	no	—
150	D₃×C₈	—	2	2	no	—
151	C₂²×C₃²	—	2	2	no	—
152	C₂²×C₃²	—	4	2	no	—
153	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂×C₃×D₃	—	4	1	no	—
155	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	2	yes	—
156	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
157	C₃×Dic₃	—	2	2	no	—
158	C₃×Dic₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₃×Dic₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	3	2	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	—	8	2	no	—
166	C₂²×C₉	—	12	1	no	—
167	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	—	3	1	no	—
170	C₄×C₉	—	3	1	no	—
171	C₄×C₉	—	6	2	no	—
172	C₄×C₉	D₃×D₄	1	2	yes	—
173	C₂×#41	—	3	1	no	—
174	C₂×M₁₆	—	3	2	no	—
175	#101	—	3	2	no	—
176	#100	—	3	1	no	—
177	#100	D₃×C₉	1	1	yes	characteristic, complemented
178	C₃×C₉	#413	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	—	3	1	no	—
181	C₂³×C₃	C₂×C₄×C₉	1	2	yes	—
182	C₂³×C₃	C₃×C₄×D₃	1	2	yes	—
183	C₂²×D₃	C₂×C₄×C₉	1	1	yes	characteristic
184	C₃×Q₈	—	6	2	no	—
185	C₃×D₄	—	2	2	no	—
186	C₃×D₄	—	2	2	no	—
187	C₃×D₄	—	4	2	no	—
188	C₃×D₄	—	6	4	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	3	1	no	—
191	C₂×C₃×C₄	—	3	2	no	—
192	C₂×C₃×C₄	—	3	2	no	—
193	C₂×C₃×C₄	—	3	2	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	—	6	2	no	—
196	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
198	#67	—	2	4	no	—
199	C₂×Dic₃	—	2	2	no	—
200	C₂×Dic₃	—	2	2	no	—
201	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
202	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
203	C₄×D₃	D₄×C₉	1	2	yes	—
204	C₄×D₃	D₄×C₉	1	2	yes	—
205	Dic₆	—	2	2	no	—
206	C₃×C₈	—	2	2	no	—
207	C₃×C₈	—	2	2	no	—
208	C₃×C₈	—	4	2	no	—
209	C₃×C₈	—	6	2	no	—
210	C₃⋊C₈	—	2	2	no	—
211	C₂×C₃²	—	2	1	no	—
212	C₂×C₃²	—	4	2	no	—
213	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₃×D₃	—	2	2	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	—	4	1	no	—
218	C₂×C₉	—	4	2	no	—
219	C₂×C₉	—	6	2	no	—
220	C₂×C₉	—	8	2	no	—
221	C₂×C₉	D₃×C₂²⋊C₄	1	1	yes	characteristic
222	#41	—	6	2	no	—
223	#41	—	6	2	no	—
224	C₂×Q₈	—	3	1	no	—
225	C₂×D₄	—	3	2	no	—
226	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
227	C₂²×C₄	—	3	1	no	—
228	C₂²×C₄	—	3	2	no	—
229	M₁₆	—	3	2	no	—
230	M₁₆	—	6	2	no	—
231	M₁₆	C₂×D₃×C₉	1	2	yes	—
232	C₂×C₈	—	6	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	3	2	no	—
237	C₂²×C₃	—	4	2	no	—
238	C₂²×C₃	—	4	2	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	—	8	2	no	—
241	C₂²×C₃	—	12	1	no	—
242	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
243	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
244	C₂×D₃	—	4	1	no	—
245	C₂×D₃	C₉×C₂²⋊C₄	1	2	yes	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	3	1	no	—
248	C₃×C₄	—	3	1	no	—
249	C₃×C₄	—	6	2	no	—
250	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
251	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
252	Dic₃	—	2	2	no	—
253	Dic₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
254	Dic₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
255	C₃²	C₃×#413	1	1	yes	characteristic
256	C₉	—	2	1	no	—
257	C₉	#4745	1	1	yes	characteristic, direct product factor
258	C₂³	—	3	1	no	—
259	C₂³	C₄×D₃×C₉	1	2	yes	—
260	Q₈	—	6	2	no	—
261	D₄	—	2	2	no	—
262	D₄	—	6	4	no	—
263	C₂×C₄	—	3	1	no	—
264	C₂×C₄	—	3	2	no	—
265	C₂×C₄	—	3	2	no	—
266	C₂×C₄	—	3	2	no	—
267	C₂×C₄	—	6	2	no	—
268	C₂×C₄	—	6	2	no	—
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	—	4	1	no	—
276	C₂×C₃	—	4	2	no	—
277	C₂×C₃	—	4	2	no	—
278	C₂×C₃	—	6	2	no	—
279	C₂×C₃	—	8	2	no	—
280	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
282	D₃	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	3	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	12	1	no	—
287	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
288	C₄	—	3	1	no	—
289	C₄	—	3	1	no	—
290	C₄	—	6	2	no	—
291	C₄	D₃×D₄×C₉	1	2	yes	—
292	C₃	—	2	1	no	—
293	C₃	C₉×#413	1	1	yes	characteristic, complemented
294	C₃	C₃×#4745	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	6	2	no	—
298	C₂	D₃×C₉×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#100	C₂
socle	C₂×C₃²	C₂×C₃×C₂²⋊C₄
prime radical	C₂×C₃×#237	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#100	D₃
hypocenter	C₃	C₉×#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#100

Complements

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

N	H ≅ G/N	type
2. D₃×C₉×M₁₆	C₂	semidirect
3. C₃×C₉×#100	C₂	semidirect
4. C₉×#833	C₂	semidirect
5. C₉×#824	C₂	semidirect
29. C₉×#100	D₃	semidirect
45. #4745	C₉	direct (N × H)
82. D₃×M₁₆	C₂×C₉	semidirect
86. C₃×#100	C₂×C₉	semidirect
88. #833	C₂×C₉	semidirect
89. #824	C₂×C₉	semidirect
177. #100	D₃×C₉	semidirect
178. C₃×C₉	#413	semidirect
257. C₉	#4745	direct (N × H)
293. C₃	C₉×#413	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#237	2
D₃×C₉×M₁₆	2
C₃×C₉×#100	2
C₉×#833	2
C₉×#824	2
C₃×#4745	3
C₉×#413	3

Subgroups — C9×#4745 #13141667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4745 #13141667