Subgroups — C₉×#5819 #13153678

Order 1728.

938 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₉×#948	C₂	1	8	yes	maximal, complemented
3	C₂×C₉×#208	C₂	1	6	yes	maximal, complemented
4	C₃×#5819	C₃	1	1	yes	maximal, characteristic
5	C₉×#568	—	3	1	no	maximal
6	C₃×C₉×#41	C₄	1	4	yes	—
7	C₃×C₉×#41	C₂²	1	4	yes	—
8	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
9	C₂×C₃×D₄×C₉	C₄	1	3	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
11	C₉×#208	C₂²	1	4	yes	—
12	C₉×#208	C₂²	1	12	yes	complemented
13	C₂×C₉×C₃⋊C₈	C₂²	1	12	yes	complemented
14	C₂×C₃²×#41	C₂×C₃	1	1	yes	characteristic
15	C₃×#948	C₂×C₃	1	8	yes	—
16	C₂×C₃×#208	C₂×C₃	1	6	yes	—
17	C₂×C₉×#41	—	2	1	no	—
18	C₂×C₉×#41	D₃	1	1	yes	characteristic
19	C₉×#131	—	3	8	no	—
20	C₂×C₉×M₁₆	—	3	6	no	—
21	C₂³×C₃×C₉	C₂×C₄	1	3	yes	—
22	C₃×Q₈×C₉	C₂×C₄	1	4	yes	—
23	C₃×D₄×C₉	C₂×C₄	1	12	yes	—
24	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	6	yes	—
26	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₉×C₃⋊C₈	C₂³	1	8	yes	—
29	C₃×#568	—	3	1	no	—
30	#5819	C₉	1	1	yes	characteristic, direct product factor
31	C₃²×#41	C₃×C₄	1	4	yes	—
32	C₃²×#41	C₂²×C₃	1	4	yes	—
33	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
34	C₂×C₃²×D₄	C₃×C₄	1	3	yes	—
35	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
36	C₃×#208	C₂²×C₃	1	4	yes	—
37	C₃×#208	C₂²×C₃	1	12	yes	—
38	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	12	yes	—
39	C₉×#41	—	2	4	no	—
40	C₉×#41	—	2	4	no	—
41	C₉×#41	Dic₃	1	4	yes	—
42	C₉×#41	C₂×D₃	1	4	yes	—
43	C₂×Q₈×C₉	—	2	1	no	—
44	C₂×Q₈×C₉	Dic₃	1	1	yes	characteristic
45	C₂×D₄×C₉	—	2	3	no	—
46	C₂×D₄×C₉	Dic₃	1	3	yes	—
47	C₂²×C₄×C₉	—	2	3	no	—
48	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
49	C₉×M₁₆	—	3	4	no	—
50	C₉×M₁₆	—	3	12	no	—
51	C₂×C₈×C₉	—	3	12	no	—
52	C₂²×C₃×C₉	—	4	3	no	—
53	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
55	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×C₉	C₂²×C₄	1	6	yes	—
57	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
58	C₂×C₃×#41	—	2	1	no	—
59	C₂×C₃×#41	C₂×C₉	1	1	yes	characteristic
60	C₂×C₃×#41	C₃×D₃	1	1	yes	characteristic
61	C₃×#131	—	3	8	no	—
62	C₂×C₃×M₁₆	—	3	6	no	—
63	#948	C₂×C₉	1	8	yes	complemented
64	C₂×#208	C₂×C₉	1	6	yes	complemented
65	C₂³×C₃²	C₂×C₃×C₄	1	3	yes	—
66	C₃²×Q₈	C₂×C₃×C₄	1	4	yes	—
67	C₃²×D₄	C₂×C₃×C₄	1	12	yes	—
68	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
69	C₂×C₃²×C₄	C₂×C₃×C₄	1	6	yes	—
70	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
71	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂³×C₉	—	2	3	no	—
73	C₂³×C₉	C₂×Dic₃	1	3	yes	—
74	C₃×C₃⋊C₈	C₂³×C₃	1	8	yes	—
75	Q₈×C₉	—	2	4	no	—
76	Q₈×C₉	C₂×Dic₃	1	4	yes	—
77	D₄×C₉	—	2	12	no	—
78	D₄×C₉	C₂×Dic₃	1	12	yes	—
79	C₂×C₄×C₉	—	2	3	no	—
80	C₂×C₄×C₉	—	2	6	no	—
81	C₂×C₄×C₉	—	2	1	no	—
82	C₂×C₄×C₉	—	2	6	no	—
83	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
84	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
85	C₂×C₄×C₉	C₂²×D₃	1	6	yes	—
86	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
87	C₈×C₉	—	3	8	no	—
88	#568	—	3	1	no	Sylow
89	C₂×C₃×C₉	—	2	1	no	—
90	C₂×C₃×C₉	—	2	6	no	—
91	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
92	C₃×#41	—	2	4	no	—
93	C₃×#41	—	2	4	no	—
94	C₃×#41	C₄×C₉	1	4	yes	—
95	C₃×#41	C₂²×C₉	1	4	yes	—
96	C₃×#41	C₃×Dic₃	1	4	yes	—
97	C₃×#41	C₂×C₃×D₃	1	4	yes	—
98	C₂×C₃×Q₈	—	2	1	no	—
99	C₂×C₃×Q₈	C₄×C₉	1	1	yes	characteristic
100	C₂×C₃×Q₈	C₃×Dic₃	1	1	yes	characteristic
101	C₂×C₃×D₄	—	2	3	no	—
102	C₂×C₃×D₄	C₄×C₉	1	3	yes	—
103	C₂×C₃×D₄	C₃×Dic₃	1	3	yes	—
104	C₂²×C₃×C₄	—	2	3	no	—
105	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
106	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
107	C₃×M₁₆	—	3	4	no	—
108	C₃×M₁₆	—	3	12	no	—
109	C₂×C₃×C₈	—	3	12	no	—
110	#208	C₂²×C₉	1	4	yes	—
111	#208	C₂²×C₉	1	12	yes	complemented
112	C₂×C₃⋊C₈	C₂²×C₉	1	12	yes	complemented
113	C₂²×C₃²	—	4	3	no	—
114	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
115	C₂²×C₃²	C₂²×C₃×C₄	1	6	yes	—
116	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₃²×C₄	C₂²×C₃×C₄	1	6	yes	—
118	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
119	C₂²×C₉	—	2	1	no	—
120	C₂²×C₉	—	2	6	no	—
121	C₂²×C₉	—	4	3	no	—
122	C₂²×C₉	—	4	6	no	—
123	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
124	C₂²×C₉	C₂²×Dic₃	1	6	yes	—
125	C₄×C₉	—	2	1	no	—
126	C₄×C₉	—	2	1	no	—
127	C₄×C₉	—	2	6	no	—
128	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
129	C₄×C₉	C₂²×Dic₃	1	6	yes	—
130	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
131	C₂×#41	D₃×C₉	1	1	yes	characteristic
132	#131	—	3	8	no	—
133	C₂×M₁₆	—	3	6	no	—
134	C₃×C₉	#568	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₃	—	2	3	no	—
136	C₂³×C₃	C₂×C₄×C₉	1	3	yes	—
137	C₂³×C₃	C₂×C₃×Dic₃	1	3	yes	—
138	C₃×Q₈	—	2	4	no	—
139	C₃×Q₈	C₂×C₄×C₉	1	4	yes	—
140	C₃×Q₈	C₂×C₃×Dic₃	1	4	yes	—
141	C₃×D₄	—	2	12	no	—
142	C₃×D₄	C₂×C₄×C₉	1	12	yes	—
143	C₃×D₄	C₂×C₃×Dic₃	1	12	yes	—
144	C₂×C₃×C₄	—	2	3	no	—
145	C₂×C₃×C₄	—	2	6	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	6	no	—
148	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
149	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
150	C₂×C₃×C₄	C₂³×C₉	1	6	yes	—
151	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₃×Dic₃	1	3	yes	—
153	C₂×C₃×C₄	C₂×C₃×Dic₃	1	6	yes	—
154	C₂×C₃×C₄	C₂²×C₃×D₃	1	6	yes	—
155	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
156	C₃×C₈	—	3	8	no	—
157	C₃⋊C₈	C₂³×C₉	1	8	yes	—
158	C₂×C₃²	—	2	1	no	—
159	C₂×C₃²	—	2	6	no	—
160	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
161	C₂×C₉	—	2	2	no	—
162	C₂×C₉	—	2	1	no	—
163	C₂×C₉	—	2	6	no	—
164	C₂×C₉	—	2	1	no	—
165	C₂×C₉	—	4	6	no	—
166	C₂×C₉	C₂³×Dic₃	1	1	yes	characteristic
167	#41	C₉×Dic₃	1	4	yes	—
168	#41	C₂×D₃×C₉	1	4	yes	—
169	C₂×Q₈	C₉×Dic₃	1	1	yes	characteristic
170	C₂×D₄	C₉×Dic₃	1	3	yes	—
171	C₂²×C₄	C₂×D₃×C₉	1	3	yes	—
172	M₁₆	—	3	4	no	—
173	M₁₆	—	3	12	no	—
174	C₂×C₈	—	3	12	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	6	no	—
177	C₂²×C₃	—	4	3	no	—
178	C₂²×C₃	—	4	3	no	—
179	C₂²×C₃	—	4	6	no	—
180	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
181	C₂²×C₃	C₂²×C₄×C₉	1	6	yes	—
182	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
183	C₂²×C₃	C₂²×C₃×Dic₃	1	6	yes	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	6	no	—
187	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
188	C₃×C₄	C₂²×C₄×C₉	1	6	yes	—
189	C₃×C₄	C₂⁴×C₉	1	1	yes	characteristic
190	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
191	C₃×C₄	C₂²×C₃×Dic₃	1	6	yes	—
192	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
193	C₃²	C₃×#568	1	1	yes	characteristic
194	C₉	—	2	1	no	—
195	C₉	#5819	1	1	yes	characteristic, direct product factor
196	C₂³	C₂×C₉×Dic₃	1	3	yes	—
197	Q₈	C₂×C₉×Dic₃	1	4	yes	—
198	D₄	C₂×C₉×Dic₃	1	12	yes	—
199	C₂×C₄	C₂×C₉×Dic₃	1	3	yes	—
200	C₂×C₄	C₂×C₉×Dic₃	1	6	yes	—
201	C₂×C₄	C₂²×D₃×C₉	1	6	yes	—
202	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
203	C₈	—	3	8	no	—
204	C₂×C₃	—	2	2	no	—
205	C₂×C₃	—	2	1	no	—
206	C₂×C₃	—	2	6	no	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	6	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	4	6	no	—
211	C₂×C₃	C₂³×C₄×C₉	1	1	yes	characteristic
212	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
213	C₂²	—	4	3	no	—
214	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
215	C₂²	C₂²×C₉×Dic₃	1	6	yes	—
216	C₄	C₂²×C₉×Dic₃	1	1	yes	characteristic
217	C₄	C₂²×C₉×Dic₃	1	6	yes	—
218	C₄	C₂³×D₃×C₉	1	1	yes	characteristic
219	C₃	—	2	1	no	—
220	C₃	C₉×#568	1	1	yes	characteristic, complemented
221	C₃	C₃×#5819	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	6	no	—
224	C₂	C₂³×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄×C₉
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂×C₃²	C₂³×C₃×C₄
prime radical	C₂×C₃²×#41	C₂×C₃
cosocle	C₃²×C₄	C₂⁴×C₃
hypercenter	C₂×C₉×#41	D₃
hypocenter	C₃	C₉×#568
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₉×#41

Complements

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

N	H ≅ G/N	type
2. C₉×#948	C₂	semidirect
3. C₂×C₉×#208	C₂	semidirect
12. C₉×#208	C₂²	semidirect
13. C₂×C₉×C₃⋊C₈	C₂²	semidirect
30. #5819	C₉	direct (N × H)
63. #948	C₂×C₉	semidirect
64. C₂×#208	C₂×C₉	semidirect
111. #208	C₂²×C₉	semidirect
112. C₂×C₃⋊C₈	C₂²×C₉	semidirect
134. C₃×C₉	#568	semidirect
195. C₉	#5819	direct (N × H)
220. C₃	C₉×#568	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×#41	2
C₉×#948	2
C₂×C₉×#208	2
C₃×#5819	3
C₉×#568	3

Subgroups — C9×#5819 #13153678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5819 #13153678