Subgroups — C₃×#10588783 #13143077

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₃×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₉⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₃×#5102	—	3	1	no	maximal
6	#10588783	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₈×C₉	—	2	2	no	—
9	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
10	C₃×#3676	C₄	1	2	yes	—
11	C₃×C₄⋊Dic₉	C₄	1	2	yes	—
12	C₃×C₄×Dic₉	C₂²	1	2	yes	—
13	C₃×#3667	—	2	2	no	—
14	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
16	C₃×#926	—	3	1	no	—
17	C₂×C₃×#208	—	3	1	no	—
18	C₃×Dic₃⋊C₈	—	3	4	no	—
19	C₂²×C₈×C₉	—	2	1	no	—
20	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
21	#63488	C₂×C₃	1	1	yes	characteristic
22	C₂×#3667	C₂×C₃	1	1	yes	characteristic
23	Dic₉⋊C₈	C₂×C₃	1	4	yes	complemented
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
30	C₃×C₈×C₉	—	2	2	no	—
31	C₃×C₉⋊C₈	—	2	2	no	—
32	C₃×#424	—	9	1	no	—
33	#5102	—	3	1	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
37	C₃×#217	—	3	2	no	—
38	C₃×C₄⋊Dic₃	—	3	2	no	—
39	C₃×C₄×Dic₃	—	3	2	no	—
40	C₃×#208	—	6	2	no	—
41	C₂×C₃×C₃⋊C₈	—	3	2	no	—
42	C₂²×C₄×C₉	—	2	1	no	—
43	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₈×C₉	—	2	4	no	—
45	C₂×C₈×C₉	—	2	2	no	—
46	C₂×C₈×C₉	—	2	2	no	—
47	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
48	#3676	C₃×C₄	1	2	yes	—
49	C₄⋊Dic₉	C₃×C₄	1	2	yes	—
50	C₄×Dic₉	C₂²×C₃	1	2	yes	—
51	#3667	—	2	2	no	—
52	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
53	C₂²×C₃×C₉	—	2	2	no	—
54	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
57	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
58	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
59	C₃×Dic₉	—	2	4	no	—
60	C₂×C₃×M₁₆	—	9	1	no	—
61	C₂²×C₃×C₈	—	2	1	no	—
62	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
64	C₃×#117	—	9	1	no	—
65	#926	—	3	1	no	—
66	C₂×#208	—	3	1	no	—
67	C₃×C₄⋊C₈	—	9	4	no	—
68	Dic₃⋊C₈	—	3	4	no	—
69	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	Dic₆	1	2	yes	—
71	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	#67	1	2	yes	—
73	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₃×Dic₃	—	3	4	no	—
75	C₂³×C₉	—	2	1	no	—
76	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
77	C₃²×C₈	—	2	2	no	—
78	C₃×C₃⋊C₈	—	6	2	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	1	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
85	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
86	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
87	C₂×Dic₉	C₂×C₃×C₄	1	4	yes	—
88	C₈×C₉	—	2	4	no	—
89	C₈×C₉	—	2	2	no	—
90	C₉⋊C₈	—	2	2	no	—
91	#424	—	9	1	no	Sylow
92	C₂×C₃×C₉	—	2	2	no	—
93	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₉	#131	1	2	yes	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
98	C₃×M₁₆	—	18	2	no	—
99	C₂×C₃×C₈	—	2	4	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	9	2	no	—
104	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
105	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
106	C₃×C₄⋊C₄	—	9	2	no	—
107	C₃×C₂²⋊C₄	—	9	2	no	—
108	C₃×C₄²	—	9	2	no	—
109	#217	—	3	2	no	—
110	C₄⋊Dic₃	—	3	2	no	—
111	C₄×Dic₃	—	3	2	no	—
112	#208	—	6	2	no	—
113	C₂×C₃⋊C₈	—	3	2	no	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
116	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
117	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
118	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
119	C₃²×C₄	C₂×#67	1	1	yes	characteristic
120	C₃×Dic₃	—	6	4	no	—
121	C₂²×C₉	—	2	4	no	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	1	no	—
125	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
126	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₄×C₉	—	2	2	no	—
128	C₄×C₉	—	2	1	no	—
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
131	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
132	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
133	Dic₉	—	2	4	no	—
134	C₂×M₁₆	—	9	1	no	—
135	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
136	#117	—	9	1	no	—
137	C₄⋊C₈	—	9	4	no	—
138	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
141	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	9	4	no	—
147	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
148	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
149	C₂×C₃×C₄	#611	1	2	yes	—
150	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
151	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
152	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₃×#67	1	2	yes	—
154	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
155	C₂×Dic₃	—	3	4	no	—
156	C₃×C₈	—	2	4	no	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	18	2	no	—
160	C₃⋊C₈	—	6	2	no	—
161	C₂×C₃²	—	2	2	no	—
162	C₂×C₃²	#901	1	2	yes	—
163	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
164	C₂×C₉	—	2	4	no	—
165	C₂×C₉	—	2	2	no	—
166	C₂×C₉	—	2	2	no	—
167	C₂×C₉	—	2	1	no	—
168	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₉	C₃×#131	1	2	yes	—
170	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
171	M₁₆	—	18	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	9	2	no	—
174	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
175	C₄⋊C₄	—	9	2	no	—
176	C₂²⋊C₄	—	9	2	no	—
177	C₄²	—	9	2	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	Dic₉⋊C₄	1	2	yes	—
184	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
185	C₂²×C₃	C₃×Dic₃⋊C₄	1	2	yes	—
186	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	18	4	no	—
191	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
192	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
193	C₃×C₄	C₂×#611	1	1	yes	characteristic
194	C₃×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
195	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
196	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
197	Dic₃	—	6	4	no	—
198	C₃²	#5102	1	1	yes	characteristic
199	C₉	—	2	1	no	—
200	C₉	C₃×#424	1	1	yes	characteristic, complemented
201	C₂³	C₃×C₄×D₉	1	1	yes	characteristic
202	C₂×C₄	—	9	4	no	—
203	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
204	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
205	C₂×C₄	C₃×#611	1	2	yes	—
206	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	18	2	no	—
209	C₂×C₃	—	2	4	no	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	2	2	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	#63464	1	2	yes	—
215	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
216	C₂×C₃	C₃×#901	1	2	yes	—
217	C₂×C₃	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
218	C₂²	—	2	2	no	—
219	C₂²	C₃×Dic₉⋊C₄	1	2	yes	—
220	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
221	C₄	—	18	4	no	—
222	C₄	C₃×Dic₉⋊C₄	1	2	yes	—
223	C₄	C₂×C₃×Dic₁₈	1	1	yes	characteristic
224	C₄	C₂×C₃×#611	1	1	yes	characteristic
225	C₃	—	2	1	no	—
226	C₃	#10588783	1	1	yes	characteristic, direct product factor
227	C₃	C₃×#5102	1	1	yes	characteristic
228	C₂	—	2	2	no	—
229	C₂	C₃×#63464	1	2	yes	—
230	C₂	C₂×C₃×Dic₉⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₉
hypocenter	C₉	C₃×#424
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×Dic₉⋊C₈	C₂	semidirect
6. #10588783	C₃	direct (N × H)
23. Dic₉⋊C₈	C₂×C₃	semidirect
138. C₃×C₉	#424	semidirect
200. C₉	C₃×#424	semidirect
226. C₃	#10588783	direct (N × H)

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
C₃×#63488	2
C₂×C₃×#3667	2
C₃×Dic₉⋊C₈	2
C₃×#5102	3
#10588783	3

Subgroups — C3×#10588783 #13143077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588783 #13143077