Subgroups — C₉×#5106 #13143088

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#926	C₂	1	2	yes	maximal
3	C₉×#818	C₂	1	2	yes	maximal, complemented
4	C₉×#817	C₂	1	2	yes	maximal, complemented
5	C₃×#5106	C₃	1	1	yes	maximal, characteristic
6	C₉×#427	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#217	—	2	2	no	—
12	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
13	C₄×C₉×Dic₃	—	2	2	no	—
14	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
15	C₃×#926	C₂×C₃	1	2	yes	—
16	C₃×#818	C₂×C₃	1	2	yes	—
17	C₃×#817	C₂×C₃	1	2	yes	—
18	C₂²×C₈×C₉	—	2	1	no	—
19	C₂²×C₈×C₉	D₃	1	1	yes	characteristic
20	C₉×#117	—	3	2	no	—
21	C₉×#107	—	3	2	no	—
22	C₉×#106	—	3	2	no	—
23	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₉×Dic₃	—	2	4	no	—
29	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
30	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
31	C₃×#427	—	3	1	no	—
32	#5106	C₉	1	1	yes	characteristic, direct product factor
33	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
35	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
37	C₃×#217	—	2	2	no	—
38	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
39	C₃×C₄×Dic₃	—	2	2	no	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
42	C₂×C₈×C₉	—	2	4	no	—
43	C₂×C₈×C₉	—	2	1	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	Dic₃	1	4	yes	complemented
46	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
47	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
48	C₉×C₄⋊C₄	—	3	4	no	—
49	C₉×C₂²⋊C₄	—	6	2	no	—
50	C₄²×C₉	—	6	2	no	—
51	C₂²×C₃×C₉	—	2	2	no	—
52	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
54	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
55	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
57	C₉×Dic₃	—	4	4	no	—
58	C₂²×C₃×C₈	—	2	1	no	—
59	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
61	C₃×#117	—	3	2	no	—
62	#926	C₂×C₉	1	2	yes	—
63	C₃×#107	—	3	2	no	—
64	C₃×#106	—	3	2	no	—
65	#818	C₂×C₉	1	2	yes	complemented
66	#817	C₂×C₉	1	2	yes	complemented
67	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
69	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
71	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂×C₃×Dic₃	—	2	4	no	—
73	C₂³×C₉	—	2	1	no	—
74	C₂³×C₉	D₁₂	1	1	yes	characteristic
75	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
76	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	1	no	—
81	C₂×C₄×C₉	—	6	4	no	—
82	C₂×C₄×C₉	Dic₆	1	2	yes	—
83	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
85	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
86	C₈×C₉	—	2	2	no	—
87	C₈×C₉	—	2	2	no	—
88	C₈×C₉	C₂×Dic₃	1	2	yes	—
89	C₈×C₉	C₂×Dic₃	1	2	yes	—
90	#427	—	3	1	no	Sylow
91	C₂×C₃×C₉	—	2	2	no	—
92	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₃×C₉	#135	1	2	yes	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	—	2	4	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	C₄×C₉	1	4	yes	complemented
101	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
103	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
104	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	3	4	no	—
107	C₃×C₂²⋊C₄	—	6	2	no	—
108	C₃×C₄²	—	6	2	no	—
109	#217	—	2	2	no	—
110	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
111	C₄×Dic₃	—	2	2	no	—
112	C₂²×C₃²	—	2	2	no	—
113	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
114	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
115	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
116	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
118	C₃×Dic₃	—	4	4	no	—
119	C₂²×C₉	—	2	4	no	—
120	C₂²×C₉	—	2	2	no	—
121	C₂²×C₉	—	2	2	no	—
122	C₂²×C₉	—	2	1	no	—
123	C₂²×C₉	C₄⋊Dic₃	1	2	yes	—
124	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
125	C₄×C₉	—	2	2	no	—
126	C₄×C₉	—	2	1	no	—
127	C₄×C₉	—	2	1	no	—
128	C₄×C₉	—	12	4	no	—
129	C₄×C₉	C₄⋊Dic₃	1	2	yes	—
130	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
131	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
132	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
133	#117	—	3	2	no	—
134	#107	—	3	2	no	—
135	#106	—	3	2	no	—
136	C₃×C₉	#427	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
139	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	6	4	no	—
145	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
146	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
147	C₂×C₃×C₄	Q₈×C₉	1	2	yes	—
148	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
149	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
150	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
152	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
153	C₂×Dic₃	—	2	4	no	—
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	—	2	2	no	—
156	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
157	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
158	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
159	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
160	C₂×C₃²	—	2	2	no	—
161	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₃²	C₃×#135	1	2	yes	—
163	C₂×C₉	—	2	4	no	—
164	C₂×C₉	—	2	2	no	—
165	C₂×C₉	—	2	2	no	—
166	C₂×C₉	—	2	1	no	—
167	C₂×C₉	#904	1	2	yes	—
168	C₂×C₉	C₂×C₄⋊Dic₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
170	C₂×C₈	C₉×Dic₃	1	4	yes	complemented
171	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
172	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
173	C₄⋊C₄	—	3	4	no	—
174	C₂²⋊C₄	—	6	2	no	—
175	C₄²	—	6	2	no	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	C₉×C₄⋊C₄	1	2	yes	—
182	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
183	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
184	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	12	4	no	—
189	C₃×C₄	C₉×C₄⋊C₄	1	2	yes	—
190	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
191	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
192	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
193	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
194	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
195	Dic₃	—	4	4	no	—
196	C₃²	C₃×#427	1	1	yes	characteristic
197	C₉	—	2	1	no	—
198	C₉	#5106	1	1	yes	characteristic, direct product factor
199	C₂³	C₉×D₁₂	1	1	yes	characteristic
200	C₂×C₄	—	6	4	no	—
201	C₂×C₄	C₉×Dic₆	1	2	yes	—
202	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
203	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
204	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
205	C₈	C₂×C₉×Dic₃	1	2	yes	—
206	C₈	C₂×C₉×Dic₃	1	2	yes	—
207	C₂×C₃	—	2	4	no	—
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₃	C₉×#135	1	2	yes	—
214	C₂×C₃	C₃×#904	1	2	yes	—
215	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	C₉×C₄⋊Dic₃	1	2	yes	—
218	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
219	C₄	—	12	4	no	—
220	C₄	C₉×C₄⋊Dic₃	1	2	yes	—
221	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
222	C₄	C₂²×C₉×Dic₃	1	1	yes	characteristic
223	C₃	—	2	1	no	—
224	C₃	C₉×#427	1	1	yes	characteristic, complemented
225	C₃	C₃×#5106	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	C₉×#904	1	2	yes	—
228	C₂	C₂×C₉×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂³×C₃²	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×C₈×C₉	D₃
hypocenter	C₃	C₉×#427
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
3. C₉×#818	C₂	semidirect
4. C₉×#817	C₂	semidirect
8. C₂×C₃×C₈×C₉	C₄	semidirect
32. #5106	C₉	direct (N × H)
45. C₂×C₈×C₉	Dic₃	semidirect
65. #818	C₂×C₉	semidirect
66. #817	C₂×C₉	semidirect
100. C₂×C₃×C₈	C₄×C₉	semidirect
136. C₃×C₉	#427	semidirect
170. C₂×C₈	C₉×Dic₃	semidirect
198. C₉	#5106	direct (N × H)
224. C₃	C₉×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₉×#926	2
C₉×#818	2
C₉×#817	2
C₃×#5106	3
C₉×#427	3

Subgroups — C9×#5106 #13143088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5106 #13143088