Subgroups — C₉×#4530 #13135565

Order 1728.

386 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	2	yes	maximal
3	C₃×#4530	C₃	1	1	yes	maximal, characteristic
4	C₉×#343	—	3	1	no	maximal
5	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₉×C₄⋊C₄	—	2	2	no	—
8	C₉×#208	—	2	2	no	—
9	C₉×#208	C₄	1	4	yes	complemented
10	C₂×C₉×C₃⋊C₈	—	2	2	no	—
11	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×#208	C₂×C₃	1	2	yes	—
13	C₂×C₉×M₁₆	—	3	2	no	—
14	C₂×C₉×C₄⋊C₄	—	2	1	no	—
15	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic
16	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₉	—	2	2	no	—
18	C₂×C₃×C₄×C₉	—	4	2	no	—
19	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
22	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
23	C₉×C₃⋊C₈	—	2	4	no	—
24	C₃×#343	—	3	1	no	—
25	#4530	C₉	1	1	yes	characteristic, direct product factor
26	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₃²×C₄⋊C₄	—	2	2	no	—
29	C₃×#208	—	2	2	no	—
30	C₃×#208	C₃×C₄	1	4	yes	—
31	C₂×C₃×C₃⋊C₈	—	2	2	no	—
32	C₂²×C₄×C₉	—	2	1	no	—
33	C₂²×C₄×C₉	—	2	2	no	—
34	C₂²×C₄×C₉	Dic₃	1	2	yes	—
35	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
36	C₉×M₁₆	—	3	4	no	—
37	C₉×M₁₆	—	6	2	no	—
38	C₂×C₈×C₉	—	6	2	no	—
39	C₉×C₄⋊C₄	—	2	4	no	—
40	C₉×C₄⋊C₄	—	2	2	no	—
41	C₂²×C₃×C₉	—	2	1	no	—
42	C₂²×C₃×C₉	—	2	1	no	—
43	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
44	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
46	C₃×C₄×C₉	—	4	2	no	—
47	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
48	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₃×M₁₆	—	3	2	no	—
50	C₂×C₃×C₄⋊C₄	—	2	1	no	—
51	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
52	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
53	C₂×#208	C₂×C₉	1	2	yes	—
54	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	—	2	2	no	—
56	C₂×C₃²×C₄	—	4	2	no	—
57	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
58	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
60	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
61	C₂³×C₉	—	2	1	no	—
62	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
63	C₃×C₃⋊C₈	—	2	4	no	—
64	C₂×C₄×C₉	—	2	4	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	1	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	4	4	no	—
71	C₂×C₄×C₉	—	4	2	no	—
72	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
73	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
74	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
75	C₂×C₄×C₉	#67	1	2	yes	—
76	C₈×C₉	—	6	4	no	—
77	#343	—	3	1	no	Sylow
78	C₂×C₃×C₉	—	2	1	no	—
79	C₂×C₃×C₉	—	2	1	no	—
80	C₂×C₃×C₉	#95	1	1	yes	characteristic
81	C₂×C₃×C₉	#100	1	1	yes	characteristic
82	C₂×C₃×C₉	#101	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	2	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	C₄×C₉	1	2	yes	—
86	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
88	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
89	C₃×M₁₆	—	3	4	no	—
90	C₃×M₁₆	—	6	2	no	—
91	C₂×C₃×C₈	—	6	2	no	—
92	C₃×C₄⋊C₄	—	2	4	no	—
93	C₃×C₄⋊C₄	—	2	2	no	—
94	C₃×C₄⋊C₄	—	2	2	no	—
95	#208	—	2	2	no	—
96	#208	C₄×C₉	1	4	yes	complemented
97	C₂×C₃⋊C₈	—	2	2	no	—
98	C₂²×C₃²	—	2	1	no	—
99	C₂²×C₃²	—	2	1	no	—
100	C₂²×C₃²	C₃×C₄²	1	1	yes	characteristic
101	C₂²×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
103	C₃²×C₄	—	4	2	no	—
104	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
105	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	—	2	2	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	—	2	1	no	—
110	C₂²×C₉	—	2	1	no	—
111	C₂²×C₉	—	2	1	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	C₄×Dic₃	1	1	yes	characteristic
114	C₂²×C₉	C₄⋊Dic₃	1	1	yes	characteristic
115	C₂²×C₉	#217	1	1	yes	characteristic
116	C₄×C₉	—	2	2	no	—
117	C₄×C₉	—	2	2	no	—
118	C₄×C₉	—	4	4	no	—
119	C₄×C₉	—	4	2	no	—
120	C₄×C₉	Dic₃⋊C₄	1	2	yes	—
121	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
122	C₂×M₁₆	—	3	2	no	—
123	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic
124	C₃×C₉	#343	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
127	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	4	4	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
139	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
140	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
141	C₂×C₃×C₄	Q₈×C₉	1	1	yes	characteristic
142	C₂×C₃×C₄	C₃×Dic₆	1	1	yes	characteristic
143	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
144	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
145	C₂×C₃×C₄	C₃×#67	1	2	yes	—
146	C₃×C₈	—	6	4	no	—
147	C₃⋊C₈	—	2	4	no	—
148	C₂×C₃²	—	2	1	no	—
149	C₂×C₃²	—	2	1	no	—
150	C₂×C₃²	C₃×#95	1	1	yes	characteristic
151	C₂×C₃²	C₃×#100	1	1	yes	characteristic
152	C₂×C₃²	C₃×#101	1	1	yes	characteristic
153	C₂×C₉	—	2	2	no	—
154	C₂×C₉	—	2	2	no	—
155	C₂×C₉	—	2	1	no	—
156	C₂×C₉	—	2	1	no	—
157	C₂×C₉	—	2	1	no	—
158	C₂×C₉	—	2	1	no	—
159	C₂×C₉	—	2	1	no	—
160	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
161	C₂×C₉	#833	1	1	yes	characteristic
162	C₂×C₉	#836	1	1	yes	characteristic
163	C₂²×C₄	C₉×Dic₃	1	2	yes	—
164	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
165	M₁₆	—	3	4	no	—
166	M₁₆	—	6	2	no	—
167	C₂×C₈	—	6	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	C₄²×C₉	1	1	yes	characteristic
179	C₂²×C₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₃	C₉×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₃	C₃×C₄×Dic₃	1	1	yes	characteristic
182	C₂²×C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
183	C₂²×C₃	C₃×#217	1	1	yes	characteristic
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	—	4	2	no	—
189	C₃×C₄	C₉×C₂²⋊C₄	1	2	yes	—
190	C₃×C₄	C₉×C₄⋊C₄	1	2	yes	—
191	C₃×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
192	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
193	C₃²	C₃×#343	1	1	yes	characteristic
194	C₉	—	2	1	no	—
195	C₉	#4530	1	1	yes	characteristic, direct product factor
196	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₉×Dic₆	1	1	yes	characteristic
200	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
201	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
202	C₂×C₄	C₉×#67	1	2	yes	—
203	C₈	—	6	4	no	—
204	C₂×C₃	—	2	2	no	—
205	C₂×C₃	—	2	2	no	—
206	C₂×C₃	—	2	1	no	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	C₉×#95	1	1	yes	characteristic
214	C₂×C₃	C₉×#100	1	1	yes	characteristic
215	C₂×C₃	C₉×#101	1	1	yes	characteristic
216	C₂×C₃	C₃×C₃⋊#95	1	1	yes	characteristic
217	C₂×C₃	C₃×#833	1	1	yes	characteristic
218	C₂×C₃	C₃×#836	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₄×C₉×Dic₃	1	1	yes	characteristic
222	C₂²	C₉×C₄⋊Dic₃	1	1	yes	characteristic
223	C₂²	C₉×#217	1	1	yes	characteristic
224	C₄	—	4	2	no	—
225	C₄	C₉×Dic₃⋊C₄	1	2	yes	—
226	C₄	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
227	C₃	—	2	1	no	—
228	C₃	C₉×#343	1	1	yes	characteristic, complemented
229	C₃	C₃×#4530	1	1	yes	characteristic
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	C₉×C₃⋊#95	1	1	yes	characteristic
233	C₂	C₉×#833	1	1	yes	characteristic
234	C₂	C₉×#836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	#217
commutator	C₂²×C₃	C₄²×C₉
Frattini	C₂²×C₃×C₄	C₂×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle	C₂²×C₃²	C₃×C₂²⋊C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂²×C₃²×C₄	C₂²×C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#343
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₉×#208	C₄	semidirect
25. #4530	C₉	direct (N × H)
96. #208	C₄×C₉	semidirect
124. C₃×C₉	#343	semidirect
195. C₉	#4530	direct (N × H)
228. C₃	C₉×#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₉×#208	2
C₃×#4530	3
C₉×#343	3

Subgroups — C9×#4530 #13135565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4530 #13135565