Subgroups — C₃×#10588616 #13142387

Order 1728.

570 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×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊Dic₉	C₂	1	2	yes	maximal
3	C₃×C₉⋊#95	C₂	1	4	yes	maximal
4	C₃×#4936	—	3	1	no	maximal
5	#10588616	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×Dic₉	C₂²	1	4	yes	—
9	C₃×C₄²×C₉	C₄	1	4	yes	complemented
10	C₃×C₄⋊Dic₉	—	2	4	no	—
11	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₃	—	3	2	no	—
13	C₃×C₃⋊#95	—	3	4	no	—
14	C₂×C₄²×C₉	—	2	1	no	—
15	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
16	C₂×C₄⋊Dic₉	C₂×C₃	1	2	yes	—
17	C₉⋊#95	C₂×C₃	1	4	yes	—
18	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
19	C₂×C₃×C₄×C₉	—	2	4	no	—
20	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
23	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
24	C₂×C₃×Dic₉	—	2	4	no	—
25	C₂×C₃×Dic₉	—	2	8	no	—
26	C₃×#382	—	9	1	no	—
27	#4936	—	3	1	no	—
28	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
29	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	—	3	4	no	—
31	C₃²×C₄²	Dic₃	1	4	yes	—
32	C₃×C₄⋊Dic₃	—	6	4	no	—
33	C₂²×C₄×C₉	—	2	2	no	—
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
36	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
38	C₄²×C₉	—	2	4	no	—
39	C₄²×C₉	C₃×C₄	1	4	yes	complemented
40	C₄⋊Dic₉	—	2	4	no	—
41	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₃×C₉	#41	1	4	yes	—
45	C₃×C₄×C₉	—	2	4	no	—
46	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	—
47	C₃×Dic₉	—	4	4	no	—
48	C₂×C₃×C₄⋊C₄	—	9	2	no	—
49	C₂×C₃×C₄²	—	2	1	no	—
50	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
51	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₃	—	3	2	no	—
53	C₃×#95	—	9	4	no	—
54	C₃⋊#95	—	3	4	no	—
55	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₃²×C₄	—	2	4	no	—
57	C₂×C₃²×C₄	Dic₆	1	2	yes	—
58	C₂×C₃²×C₄	D₁₂	1	2	yes	—
59	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
60	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
61	C₂×C₃×Dic₃	—	6	4	no	—
62	C₂×C₃×Dic₃	—	6	8	no	—
63	C₂³×C₉	—	2	1	no	—
64	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
65	C₂×C₄×C₉	—	2	8	no	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
72	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
74	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
75	C₂×Dic₉	—	2	4	no	—
76	C₂×Dic₉	—	2	8	no	—
77	#382	—	9	1	no	Sylow
78	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₉	#117	1	2	yes	—
80	C₂×C₃×C₉	#124	1	2	yes	—
81	C₂×C₃×C₉	#125	1	2	yes	—
82	C₂²×C₃×C₄	—	2	2	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	—	9	4	no	—
85	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
86	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
88	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
89	C₂²×Dic₃	—	3	4	no	—
90	C₃×C₄⋊C₄	—	18	4	no	—
91	C₃×C₄²	—	2	4	no	—
92	C₃×C₄²	Dic₉	1	4	yes	complemented
93	C₃×C₄²	C₃×Dic₃	1	4	yes	—
94	C₄⋊Dic₃	—	6	4	no	—
95	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
96	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
97	C₂²×C₃²	#235	1	4	yes	—
98	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
99	C₃²×C₄	—	2	4	no	—
100	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
101	C₃×Dic₃	—	12	4	no	—
102	C₂²×C₉	—	2	1	no	—
103	C₂²×C₉	—	2	1	no	—
104	C₂²×C₉	—	2	4	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
107	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
108	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
109	C₂²×C₉	C₃×#41	1	4	yes	—
110	C₄×C₉	—	2	8	no	—
111	C₄×C₉	—	2	4	no	—
112	C₄×C₉	—	2	4	no	—
113	C₄×C₉	C₃×C₄⋊C₄	1	4	yes	—
114	Dic₉	—	4	4	no	—
115	C₂×C₄⋊C₄	—	9	2	no	—
116	C₂×C₄²	C₃×D₉	1	1	yes	characteristic
117	#95	—	9	4	no	—
118	C₃×C₉	#382	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
121	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	18	4	no	—
130	C₂×C₃×C₄	—	18	8	no	—
131	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
132	C₂×C₃×C₄	D₃₆	1	2	yes	—
133	C₂×C₃×C₄	C₂×Dic₉	1	4	yes	—
134	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
135	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
136	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
138	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
139	C₂×Dic₃	—	6	4	no	—
140	C₂×Dic₃	—	6	8	no	—
141	C₂×C₃²	#870	1	2	yes	—
142	C₂×C₃²	#875	1	2	yes	—
143	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
144	C₂×C₃²	#926	1	2	yes	—
145	C₂×C₉	—	2	1	no	—
146	C₂×C₉	—	2	2	no	—
147	C₂×C₉	—	2	2	no	—
148	C₂×C₉	—	2	2	no	—
149	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₉	C₃×#117	1	2	yes	—
151	C₂×C₉	C₃×#124	1	2	yes	—
152	C₂×C₉	C₃×#125	1	2	yes	—
153	C₂²×C₄	—	9	4	no	—
154	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
155	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
156	C₄⋊C₄	—	18	4	no	—
157	C₄²	C₃×Dic₉	1	4	yes	complemented
158	C₂²×C₃	—	2	1	no	—
159	C₂²×C₃	—	2	1	no	—
160	C₂²×C₃	—	2	4	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
163	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
164	C₂²×C₃	#3697	1	4	yes	—
165	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
166	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
167	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
168	C₂²×C₃	C₃×#235	1	4	yes	—
169	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
170	C₃×C₄	—	2	8	no	—
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	2	4	no	—
174	C₃×C₄	—	36	4	no	—
175	C₃×C₄	C₄⋊Dic₉	1	4	yes	—
176	C₃×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
177	Dic₃	—	12	4	no	—
178	C₃²	#4936	1	1	yes	characteristic
179	C₉	—	2	1	no	—
180	C₉	C₃×#382	1	1	yes	characteristic, complemented
181	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	18	4	no	—
184	C₂×C₄	—	18	8	no	—
185	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
186	C₂×C₄	C₃×D₃₆	1	2	yes	—
187	C₂×C₄	C₂×C₃×Dic₉	1	4	yes	—
188	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	2	no	—
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	2	2	no	—
193	C₂×C₃	#63432	1	2	yes	—
194	C₂×C₃	#63437	1	2	yes	—
195	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
196	C₂×C₃	#63488	1	2	yes	—
197	C₂×C₃	C₃×#870	1	2	yes	—
198	C₂×C₃	C₃×#875	1	2	yes	—
199	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
200	C₂×C₃	C₃×#926	1	2	yes	—
201	C₂²	C₂×C₃×Dic₁₈	1	1	yes	characteristic
202	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
203	C₂²	C₃×#3697	1	4	yes	—
204	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
205	C₄	—	2	4	no	—
206	C₄	—	36	4	no	—
207	C₄	C₃×C₄⋊Dic₉	1	4	yes	—
208	C₃	—	2	1	no	—
209	C₃	#10588616	1	1	yes	characteristic, direct product factor
210	C₃	C₃×#4936	1	1	yes	characteristic
211	C₂	C₃×#63432	1	2	yes	—
212	C₂	C₃×#63437	1	2	yes	—
213	C₂	C₂×C₃×C₄⋊Dic₉	1	1	yes	characteristic
214	C₂	C₃×#63488	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₃×#382
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
5. #10588616	C₃	direct (N × H)
9. C₃×C₄²×C₉	C₄	semidirect
39. C₄²×C₉	C₃×C₄	semidirect
92. C₃×C₄²	Dic₉	semidirect
118. C₃×C₉	#382	semidirect
157. C₄²	C₃×Dic₉	semidirect
180. C₉	C₃×#382	semidirect
209. C₃	#10588616	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₃×C₄⋊Dic₉	2
C₃×C₉⋊#95	2
C₃×#4936	3
#10588616	3

Subgroups — C3×#10588616 #13142387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588616 #13142387