Subgroups — C₉×#4716 #13141520

Order 1728.

364 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×Dic₆	C₂	1	2	yes	maximal
3	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₉×#817	C₂	1	1	yes	maximal, characteristic
5	C₉×#816	C₂	1	2	yes	maximal
6	C₃×#4716	C₃	1	1	yes	maximal, characteristic
7	C₉×#441	—	3	1	no	maximal
8	C₂×C₉×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₉×Dic₃⋊C₄	—	2	2	no	—
14	C₄×C₉×Dic₃	—	2	2	no	—
15	C₉×Dic₁₂	C₄	1	4	yes	complemented
16	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
17	C₃×C₄×Dic₆	C₂×C₃	1	2	yes	—
18	C₃²×#97	C₂×C₃	1	1	yes	characteristic
19	C₃×#817	C₂×C₃	1	1	yes	characteristic
20	C₃×#816	C₂×C₃	1	2	yes	—
21	C₂×C₉×Q₁₆	—	3	1	no	—
22	C₄×Q₈×C₉	—	3	2	no	—
23	C₉×#106	—	3	1	no	—
24	C₉×#103	—	3	2	no	—
25	C₉×#97	—	2	1	no	—
26	C₉×#97	D₃	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₉×Dic₃	—	2	2	no	—
30	C₂×C₉×Dic₃	—	2	2	no	—
31	C₉×Dic₆	—	2	2	no	—
32	C₉×Dic₆	C₂×C₄	1	4	yes	—
33	C₃×C₈×C₉	—	2	1	no	—
34	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
35	C₃×#441	—	3	1	no	—
36	#4716	C₉	1	1	yes	characteristic, direct product factor
37	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
38	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
40	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
41	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
42	C₃×Dic₃⋊C₄	—	2	2	no	—
43	C₃×C₄×Dic₃	—	2	2	no	—
44	C₃×Dic₁₂	C₃×C₄	1	4	yes	—
45	C₂×Q₈×C₉	—	3	2	no	—
46	C₉×Q₁₆	—	3	4	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
50	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
51	C₉×C₄⋊C₄	—	3	2	no	—
52	C₉×C₄⋊C₄	—	6	2	no	—
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	—	6	2	no	—
55	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
57	C₃×C₄×C₉	—	2	2	no	—
58	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	#41	1	1	yes	characteristic
60	C₉×Dic₃	—	2	4	no	—
61	C₉×Dic₃	—	4	2	no	—
62	C₂×C₃×Q₁₆	—	3	1	no	—
63	C₃×C₄×Q₈	—	3	2	no	—
64	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic
65	C₄×Dic₆	C₂×C₉	1	2	yes	—
66	C₃×#106	—	3	1	no	—
67	C₃×#103	—	3	2	no	—
68	C₃×#97	—	2	1	no	—
69	C₃×#97	C₂×C₉	1	1	yes	characteristic
70	C₃×#97	C₃×D₃	1	1	yes	characteristic
71	#817	C₂×C₉	1	1	yes	characteristic
72	#816	C₂×C₉	1	2	yes	—
73	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
74	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₃×Dic₃	—	2	2	no	—
76	C₂×C₃×Dic₃	—	2	2	no	—
77	C₃×Dic₆	—	2	2	no	—
78	C₃×Dic₆	C₂×C₃×C₄	1	4	yes	—
79	C₃²×C₈	—	2	1	no	—
80	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
81	Q₈×C₉	—	3	4	no	—
82	Q₈×C₉	—	6	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	—	6	2	no	—
86	C₂×C₄×C₉	—	6	2	no	—
87	C₂×C₄×C₉	D₁₂	1	2	yes	—
88	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
89	C₈×C₉	—	2	2	no	—
90	C₈×C₉	—	2	1	no	—
91	C₈×C₉	—	2	2	no	—
92	C₈×C₉	C₄×D₃	1	2	yes	—
93	#441	—	3	1	no	Sylow
94	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₃×C₉	#137	1	2	yes	—
96	C₂×C₃×Q₈	—	3	2	no	—
97	C₂×Dic₆	C₂²×C₉	1	2	yes	—
98	C₃×Q₁₆	—	3	4	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	3	2	no	—
106	C₃×C₄⋊C₄	—	6	2	no	—
107	C₃×C₄²	—	2	1	no	—
108	C₃×C₄²	—	6	2	no	—
109	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
110	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
111	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
112	Dic₃⋊C₄	—	2	2	no	—
113	C₄×Dic₃	—	2	2	no	—
114	Dic₁₂	C₄×C₉	1	4	yes	complemented
115	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
116	C₃²×C₄	—	2	2	no	—
117	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₃²×C₄	C₃×#41	1	1	yes	characteristic
119	C₃×Dic₃	—	2	4	no	—
120	C₃×Dic₃	—	4	2	no	—
121	C₂²×C₉	—	2	1	no	—
122	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
123	C₄×C₉	—	2	2	no	—
124	C₄×C₉	—	2	1	no	—
125	C₄×C₉	—	2	1	no	—
126	C₄×C₉	—	4	2	no	—
127	C₄×C₉	—	6	4	no	—
128	C₄×C₉	—	12	2	no	—
129	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
130	C₄×C₉	#235	1	1	yes	characteristic
131	C₂×Q₁₆	—	3	1	no	—
132	C₄×Q₈	—	3	2	no	—
133	#106	—	3	1	no	—
134	#103	—	3	2	no	—
135	#97	D₃×C₉	1	1	yes	characteristic
136	C₃×C₉	#441	1	1	yes	characteristic, complemented, Sylow
137	C₃×Q₈	—	3	4	no	—
138	C₃×Q₈	—	6	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	6	2	no	—
142	C₂×C₃×C₄	—	6	2	no	—
143	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
144	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
145	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
146	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
147	C₂×Dic₃	—	2	2	no	—
148	C₂×Dic₃	—	2	2	no	—
149	Dic₆	—	2	2	no	—
150	Dic₆	C₂×C₄×C₉	1	4	yes	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	—	2	2	no	—
155	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
156	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
157	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
158	C₂×C₃²	C₃×#137	1	2	yes	—
159	C₂×C₉	—	2	2	no	—
160	C₂×C₉	—	2	1	no	—
161	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
162	C₂×C₉	#909	1	2	yes	—
163	C₂×Q₈	—	3	2	no	—
164	Q₁₆	—	3	4	no	—
165	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
166	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
167	C₄⋊C₄	—	3	2	no	—
168	C₄⋊C₄	—	6	2	no	—
169	C₄²	—	6	2	no	—
170	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
173	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	6	4	no	—
180	C₃×C₄	—	12	2	no	—
181	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
182	C₃×C₄	C₉×#41	1	1	yes	characteristic
183	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
184	C₃×C₄	C₃×#235	1	1	yes	characteristic
185	Dic₃	—	2	4	no	—
186	Dic₃	—	4	2	no	—
187	C₃²	C₃×#441	1	1	yes	characteristic
188	C₉	—	2	1	no	—
189	C₉	#4716	1	1	yes	characteristic, direct product factor
190	Q₈	—	3	4	no	—
191	Q₈	—	6	2	no	—
192	C₂×C₄	—	6	2	no	—
193	C₂×C₄	—	6	2	no	—
194	C₂×C₄	C₉×D₁₂	1	2	yes	—
195	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	C₄×D₃×C₉	1	2	yes	—
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
201	C₂×C₃	C₉×#137	1	2	yes	—
202	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
203	C₂×C₃	C₃×#909	1	2	yes	—
204	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	6	4	no	—
207	C₄	—	12	2	no	—
208	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
209	C₄	C₉×#235	1	1	yes	characteristic
210	C₃	—	2	1	no	—
211	C₃	C₉×#441	1	1	yes	characteristic, complemented
212	C₃	C₃×#4716	1	1	yes	characteristic
213	C₂	C₄×C₉×D₁₂	1	1	yes	characteristic
214	C₂	C₉×#909	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, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#97	D₃
hypocenter	C₃	C₉×#441
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
15. C₉×Dic₁₂	C₄	semidirect
36. #4716	C₉	direct (N × H)
114. Dic₁₂	C₄×C₉	semidirect
136. C₃×C₉	#441	semidirect
189. C₉	#4716	direct (N × H)
211. C₃	C₉×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×Dic₁₂	2
C₄×C₉×Dic₆	2
C₃×C₉×#97	2
C₉×#817	2
C₉×#816	2
C₃×#4716	3
C₉×#441	3

Subgroups — C9×#4716 #13141520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4716 #13141520