Subgroups — C₉×#4711 #13141516

Order 1728.

436 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×D₁₂	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	2	yes	maximal, complemented
5	C₉×#817	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4711	C₃	1	1	yes	maximal, characteristic
7	C₉×#442	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₂²	1	2	yes	—
9	C₂×C₄×D₃×C₉	—	2	2	no	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₉×(C₂×D₃)⋊C₄	—	2	2	no	—
14	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₉×D₂₄	C₄	1	4	yes	complemented
16	C₂×C₃×D₂₄	C₂×C₃	1	1	yes	characteristic
17	C₃×C₄×D₁₂	C₂×C₃	1	2	yes	—
18	C₃²×#97	C₂×C₃	1	1	yes	characteristic
19	C₃×#821	C₂×C₃	1	2	yes	—
20	C₃×#817	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×D₈	—	3	1	no	—
22	C₄×D₄×C₉	—	3	2	no	—
23	C₉×#106	—	3	1	no	—
24	C₉×#102	—	3	2	no	—
25	C₉×#97	—	2	1	no	—
26	C₉×#97	D₃	1	1	yes	characteristic, complemented
27	C₂²×D₃×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₉×Dic₃	—	2	2	no	—
31	C₉×D₁₂	—	2	2	no	—
32	C₉×D₁₂	C₂×C₄	1	4	yes	complemented
33	C₄×D₃×C₉	—	4	4	no	—
34	C₃×C₈×C₉	—	2	1	no	—
35	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
36	C₃×#442	—	3	1	no	—
37	#4711	C₉	1	1	yes	characteristic, direct product factor
38	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
39	C₂×C₃×C₄×D₃	—	2	2	no	—
40	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
43	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
45	C₃×D₂₄	C₃×C₄	1	4	yes	—
46	C₂×D₄×C₉	—	3	2	no	—
47	C₂²×C₄×C₉	—	6	2	no	—
48	C₉×D₈	—	3	4	no	—
49	C₂×C₈×C₉	—	2	1	no	—
50	C₂×C₈×C₉	—	2	1	no	—
51	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
52	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
53	C₉×C₄⋊C₄	—	3	2	no	—
54	C₉×C₂²⋊C₄	—	6	2	no	—
55	C₄²×C₉	—	2	1	no	—
56	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
58	C₂×D₃×C₉	—	2	4	no	—
59	C₂×D₃×C₉	—	4	2	no	—
60	C₂×D₃×C₉	—	4	2	no	—
61	C₃×C₄×C₉	—	2	2	no	—
62	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	#41	1	1	yes	characteristic
64	C₉×Dic₃	—	4	2	no	—
65	C₂×C₃×D₈	—	3	1	no	—
66	C₃×C₄×D₄	—	3	2	no	—
67	C₂×D₂₄	C₂×C₉	1	1	yes	characteristic
68	C₄×D₁₂	C₂×C₉	1	2	yes	complemented
69	C₃×#106	—	3	1	no	—
70	C₃×#102	—	3	2	no	—
71	C₃×#97	—	2	1	no	—
72	C₃×#97	C₂×C₉	1	1	yes	characteristic, complemented
73	C₃×#97	C₃×D₃	1	1	yes	characteristic
74	#821	C₂×C₉	1	2	yes	complemented
75	#817	C₂×C₉	1	1	yes	characteristic, complemented
76	C₂²×C₃×D₃	—	2	2	no	—
77	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
78	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₂×C₃×Dic₃	—	2	2	no	—
80	C₃×D₁₂	—	2	2	no	—
81	C₃×D₁₂	C₂×C₃×C₄	1	4	yes	—
82	C₃×C₄×D₃	—	4	4	no	—
83	C₂³×C₉	—	6	2	no	—
84	C₃²×C₈	—	2	1	no	—
85	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
86	D₄×C₉	—	3	4	no	—
87	D₄×C₉	—	6	2	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	6	2	no	—
91	C₂×C₄×C₉	—	12	4	no	—
92	C₂×C₄×C₉	D₁₂	1	2	yes	—
93	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
94	C₈×C₉	—	2	2	no	—
95	C₈×C₉	—	2	1	no	—
96	C₈×C₉	—	2	2	no	—
97	C₈×C₉	C₄×D₃	1	2	yes	complemented
98	#442	—	3	1	no	Sylow
99	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₃×C₉	AGL(1,ℤ/8)	1	2	yes	—
101	D₃×C₉	—	4	4	no	—
102	C₂×C₃×D₄	—	3	2	no	—
103	C₂²×C₃×C₄	—	6	2	no	—
104	C₂×D₁₂	C₂²×C₉	1	2	yes	—
105	C₂×C₄×D₃	—	2	2	no	—
106	C₃×D₈	—	3	4	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	3	2	no	—
114	C₃×C₂²⋊C₄	—	6	2	no	—
115	C₃×C₄²	—	2	1	no	—
116	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
117	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
118	(C₂×D₃)⋊C₄	—	2	2	no	—
119	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
120	D₂₄	C₄×C₉	1	4	yes	complemented
121	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
122	C₂×C₃×D₃	—	2	4	no	—
123	C₂×C₃×D₃	—	4	2	no	—
124	C₂×C₃×D₃	—	4	2	no	—
125	C₃²×C₄	—	2	2	no	—
126	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₃²×C₄	C₃×#41	1	1	yes	characteristic
128	C₃×Dic₃	—	4	2	no	—
129	C₂²×C₉	—	2	1	no	—
130	C₂²×C₉	—	6	4	no	—
131	C₂²×C₉	—	12	2	no	—
132	C₂²×C₉	—	12	2	no	—
133	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
134	C₄×C₉	—	2	2	no	—
135	C₄×C₉	—	2	1	no	—
136	C₄×C₉	—	2	1	no	—
137	C₄×C₉	—	4	2	no	—
138	C₄×C₉	—	12	2	no	—
139	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
140	C₄×C₉	#235	1	1	yes	characteristic
141	C₂×D₈	—	3	1	no	—
142	C₄×D₄	—	3	2	no	—
143	#106	—	3	1	no	—
144	#102	—	3	2	no	—
145	#97	D₃×C₉	1	1	yes	characteristic, complemented
146	C₃×C₉	#442	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₃	—	6	2	no	—
148	C₂²×D₃	—	2	2	no	—
149	C₃×D₄	—	3	4	no	—
150	C₃×D₄	—	6	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	6	2	no	—
154	C₂×C₃×C₄	—	12	4	no	—
155	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
156	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
157	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
159	C₂×Dic₃	—	2	2	no	—
160	D₁₂	—	2	2	no	—
161	D₁₂	C₂×C₄×C₉	1	4	yes	complemented
162	C₄×D₃	—	4	4	no	—
163	C₃×C₈	—	2	2	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	2	2	no	—
167	C₃×C₈	C₂×C₄×C₉	1	2	yes	complemented
168	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
169	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
170	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	2	yes	—
171	C₃×D₃	—	4	4	no	—
172	C₂×C₉	—	2	2	no	—
173	C₂×C₉	—	2	1	no	—
174	C₂×C₉	—	12	4	no	—
175	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
176	C₂×C₉	#908	1	2	yes	—
177	C₂×D₄	—	3	2	no	—
178	C₂²×C₄	—	6	2	no	—
179	D₈	—	3	4	no	—
180	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
181	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
182	C₄⋊C₄	—	3	2	no	—
183	C₂²⋊C₄	—	6	2	no	—
184	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
185	C₂²×C₃	—	2	1	no	—
186	C₂²×C₃	—	6	4	no	—
187	C₂²×C₃	—	12	2	no	—
188	C₂²×C₃	—	12	2	no	—
189	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
190	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
191	C₂×D₃	—	2	4	no	—
192	C₂×D₃	—	4	2	no	—
193	C₂×D₃	—	4	2	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	—	12	2	no	—
200	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
201	C₃×C₄	C₉×#41	1	1	yes	characteristic
202	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
203	C₃×C₄	C₃×#235	1	1	yes	characteristic
204	Dic₃	—	4	2	no	—
205	C₃²	C₃×#442	1	1	yes	characteristic
206	C₉	—	2	1	no	—
207	C₉	#4711	1	1	yes	characteristic, direct product factor
208	C₂³	—	6	2	no	—
209	D₄	—	3	4	no	—
210	D₄	—	6	2	no	—
211	C₂×C₄	—	6	2	no	—
212	C₂×C₄	—	12	4	no	—
213	C₂×C₄	C₉×D₁₂	1	2	yes	—
214	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	C₄×D₃×C₉	1	2	yes	complemented
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	12	4	no	—
220	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
221	C₂×C₃	C₉×AGL(1,ℤ/8)	1	2	yes	—
222	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
223	C₂×C₃	C₃×#908	1	2	yes	—
224	D₃	—	4	4	no	—
225	C₂²	—	6	4	no	—
226	C₂²	—	12	2	no	—
227	C₂²	—	12	2	no	—
228	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	12	2	no	—
231	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
232	C₄	C₉×#235	1	1	yes	characteristic
233	C₃	—	2	1	no	—
234	C₃	C₉×#442	1	1	yes	characteristic, complemented
235	C₃	C₃×#4711	1	1	yes	characteristic
236	C₂	—	12	4	no	—
237	C₂	C₄×C₉×D₁₂	1	1	yes	characteristic
238	C₂	C₉×#908	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#97	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₂₄	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#97	D₃
hypocenter	C₃	C₉×#442
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#97

Complements

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

N	H ≅ G/N	type
2. C₄×C₉×D₁₂	C₂	semidirect
3. C₃×C₉×#97	C₂	semidirect
4. C₉×#821	C₂	semidirect
5. C₉×#817	C₂	semidirect
15. C₉×D₂₄	C₄	semidirect
26. C₉×#97	D₃	semidirect
32. C₉×D₁₂	C₂×C₄	semidirect
35. C₃×C₈×C₉	C₂×C₄	semidirect
37. #4711	C₉	direct (N × H)
68. C₄×D₁₂	C₂×C₉	semidirect
72. C₃×#97	C₂×C₉	semidirect
74. #821	C₂×C₉	semidirect
75. #817	C₂×C₉	semidirect
97. C₈×C₉	C₄×D₃	semidirect
120. D₂₄	C₄×C₉	semidirect
145. #97	D₃×C₉	semidirect
146. C₃×C₉	#442	semidirect
161. D₁₂	C₂×C₄×C₉	semidirect
167. C₃×C₈	C₂×C₄×C₉	semidirect
207. C₉	#4711	direct (N × H)
216. C₈	C₄×D₃×C₉	semidirect
234. C₃	C₉×#442	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×D₂₄	2
C₄×C₉×D₁₂	2
C₃×C₉×#97	2
C₉×#821	2
C₉×#817	2
C₃×#4711	3
C₉×#442	3

Subgroups — C9×#4711 #13141516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4711 #13141516