Subgroups — C₃×#10588612 #13142382

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 192 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×#4932	—	3	1	no	maximal
5	#10588612	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	C₄	1	8	yes	complemented
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	4	yes	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
23	C₂×C₃×Dic₉	—	2	8	no	—
24	C₂×C₃×Dic₉	C₂×C₄	1	8	yes	—
25	C₃×#376	—	9	1	no	—
26	#4932	—	3	1	no	—
27	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
28	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	3	4	no	—
30	C₃²×C₄²	Dic₃	1	4	yes	—
31	C₃×C₄×Dic₃	—	3	8	no	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
35	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
36	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
37	C₄²×C₉	—	2	4	no	—
38	C₄²×C₉	C₃×C₄	1	4	yes	complemented
39	C₄×Dic₉	C₃×C₄	1	8	yes	complemented
40	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
41	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₉	#41	1	4	yes	—
43	C₃×C₄×C₉	—	2	4	no	—
44	C₃×C₄×C₉	C₄²	1	4	yes	—
45	C₃×Dic₉	—	2	8	no	—
46	C₂×C₃×C₄²	—	2	1	no	—
47	C₂×C₃×C₄²	—	9	2	no	—
48	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
50	C₂×C₄×Dic₃	—	3	2	no	—
51	C₃×#95	—	9	4	no	—
52	C₃⋊#95	—	3	4	no	—
53	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
54	C₂×C₃²×C₄	—	2	4	no	—
55	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
56	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
57	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
58	C₂×C₃×Dic₃	—	3	8	no	—
59	C₂×C₃×Dic₃	—	6	8	no	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
62	C₂×C₄×C₉	—	2	8	no	—
63	C₂×C₄×C₉	—	2	4	no	—
64	C₂×C₄×C₉	—	2	4	no	—
65	C₂×C₄×C₉	—	2	4	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
68	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
69	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
70	C₂×Dic₉	—	2	8	no	—
71	C₂×Dic₉	C₂×C₃×C₄	1	8	yes	—
72	#376	—	9	1	no	Sylow
73	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₃×C₉	#117	1	4	yes	—
75	C₂×C₃×C₉	#117	1	2	yes	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	9	4	no	—
79	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
80	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
82	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
83	C₂²×Dic₃	—	3	4	no	—
84	C₃×C₄²	—	2	4	no	—
85	C₃×C₄²	—	9	8	no	—
86	C₃×C₄²	Dic₉	1	4	yes	complemented
87	C₃×C₄²	C₃×Dic₃	1	4	yes	—
88	C₄×Dic₃	—	3	8	no	—
89	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
90	C₂²×C₃²	#235	1	4	yes	—
91	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
92	C₃²×C₄	—	2	4	no	—
93	C₃²×C₄	C₄×Dic₃	1	4	yes	—
94	C₃×Dic₃	—	6	8	no	—
95	C₂²×C₉	—	2	2	no	—
96	C₂²×C₉	—	2	4	no	—
97	C₂²×C₉	—	2	1	no	—
98	C₂²×C₉	C₂²×C₃×C₄	1	2	yes	—
99	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₂²×C₉	C₃×#41	1	4	yes	—
101	C₄×C₉	—	2	8	no	—
102	C₄×C₉	—	2	4	no	—
103	C₄×C₉	—	2	4	no	—
104	C₄×C₉	C₃×C₄²	1	4	yes	—
105	Dic₉	—	2	8	no	—
106	C₂×C₄²	—	9	2	no	—
107	C₂×C₄²	C₃×D₉	1	1	yes	characteristic
108	#95	—	9	4	no	—
109	C₃×C₉	#376	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₃	—	2	1	no	—
111	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
112	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	8	no	—
114	C₂×C₃×C₄	—	2	4	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	9	8	no	—
120	C₂×C₃×C₄	—	18	8	no	—
121	C₂×C₃×C₄	C₄×D₉	1	4	yes	—
122	C₂×C₃×C₄	C₂×Dic₉	1	4	yes	—
123	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
124	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
125	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
126	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
127	C₂×Dic₃	—	3	8	no	—
128	C₂×Dic₃	—	6	8	no	—
129	C₂×C₃²	#872	1	4	yes	—
130	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
131	C₂×C₃²	#926	1	2	yes	—
132	C₂×C₉	—	2	1	no	—
133	C₂×C₉	—	2	2	no	—
134	C₂×C₉	—	2	4	no	—
135	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
136	C₂×C₉	C₃×#117	1	2	yes	—
137	C₂×C₉	C₃×#117	1	4	yes	—
138	C₂²×C₄	—	9	4	no	—
139	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
140	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
141	C₄²	—	9	8	no	—
142	C₄²	C₃×Dic₉	1	4	yes	complemented
143	C₂²×C₃	—	2	2	no	—
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	C₂×C₄×D₉	1	2	yes	—
147	C₂²×C₃	#3697	1	4	yes	—
148	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
149	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
150	C₂²×C₃	C₃×#235	1	4	yes	—
151	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
152	C₃×C₄	—	2	8	no	—
153	C₃×C₄	—	2	4	no	—
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	18	8	no	—
157	C₃×C₄	C₄×Dic₉	1	4	yes	—
158	C₃×C₄	C₃×C₄×Dic₃	1	4	yes	—
159	Dic₃	—	6	8	no	—
160	C₃²	#4932	1	1	yes	characteristic
161	C₉	—	2	1	no	—
162	C₉	C₃×#376	1	1	yes	characteristic, complemented
163	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	9	8	no	—
166	C₂×C₄	—	18	8	no	—
167	C₂×C₄	C₃×C₄×D₉	1	4	yes	—
168	C₂×C₄	C₂×C₃×Dic₉	1	4	yes	—
169	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	—	2	2	no	—
172	C₂×C₃	—	2	4	no	—
173	C₂×C₃	#63434	1	4	yes	—
174	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
175	C₂×C₃	#63488	1	2	yes	—
176	C₂×C₃	C₃×#872	1	4	yes	—
177	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
178	C₂×C₃	C₃×#926	1	2	yes	—
179	C₂²	C₂×C₃×C₄×D₉	1	2	yes	—
180	C₂²	C₃×#3697	1	4	yes	—
181	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	18	8	no	—
184	C₄	C₃×C₄×Dic₉	1	4	yes	—
185	C₃	—	2	1	no	—
186	C₃	#10588612	1	1	yes	characteristic, direct product factor
187	C₃	C₃×#4932	1	1	yes	characteristic
188	C₂	C₃×#63434	1	4	yes	—
189	C₂	C₂×C₃×C₄×Dic₉	1	1	yes	characteristic
190	C₂	C₃×#63488	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₉
commutator	C₂×C₉	C₂×C₃×C₄²
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×#376
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
5. #10588612	C₃	direct (N × H)
9. C₃×C₄²×C₉	C₄	semidirect
10. C₃×C₄×Dic₉	C₄	semidirect
38. C₄²×C₉	C₃×C₄	semidirect
39. C₄×Dic₉	C₃×C₄	semidirect
86. C₃×C₄²	Dic₉	semidirect
109. C₃×C₉	#376	semidirect
142. C₄²	C₃×Dic₉	semidirect
162. C₉	C₃×#376	semidirect
186. C₃	#10588612	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×#4932	3
#10588612	3

Subgroups — C3×#10588612 #13142382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588612 #13142382