Subgroups — C₃×#10592631 #13162968

Order 1728.

1500 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×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₃×#63658	C₂	1	4	yes	maximal
4	#10592631	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×#4932	—	3	4	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	2	yes	—
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₃⋊Dic₃	C₂²	1	4	yes	—
9	C₃³×C₄²	C₄	1	4	yes	complemented
10	C₃×C₄×C₃⋊Dic₃	C₄	1	8	yes	complemented
11	C₂×C₃²×C₄²	—	2	4	no	—
12	C₂×C₃²×C₄²	D₃	1	4	yes	—
13	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₂×C₄×C₃⋊Dic₃	C₂×C₃	1	2	yes	—
15	C₂×C₃×C₄×Dic₃	—	3	8	no	—
16	#63658	C₂×C₃	1	4	yes	—
17	C₃×C₃⋊#95	—	3	16	no	—
18	C₂³×C₃³	C₂³	1	1	yes	characteristic
19	C₂×C₃³×C₄	—	2	4	no	—
20	C₂×C₃³×C₄	C₂×C₄	1	4	yes	—
21	C₂×C₃³×C₄	C₂×C₄	1	4	yes	—
22	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₃⋊Dic₃	—	2	8	no	—
24	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	8	yes	—
25	C₃×#376	—	9	1	no	—
26	#4932	—	3	4	no	—
27	C₂²×C₃²×C₄	—	2	8	no	—
28	C₂²×C₃²×C₄	—	2	4	no	—
29	C₂²×C₃²×C₄	C₂×D₃	1	8	yes	—
30	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
31	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
32	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
33	C₂²×C₃⋊Dic₃	C₂²×C₃	1	4	yes	—
34	C₂²×C₃×Dic₃	—	3	16	no	—
35	C₃²×C₄²	—	2	16	no	—
36	C₃²×C₄²	Dic₃	1	16	yes	complemented
37	C₃²×C₄²	C₃×C₄	1	4	yes	complemented
38	C₄×C₃⋊Dic₃	C₃×C₄	1	8	yes	complemented
39	C₃×C₄×Dic₃	—	3	32	no	—
40	C₂²×C₃³	C₂²×C₄	1	2	yes	—
41	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃³	#41	1	4	yes	—
43	C₃³×C₄	—	2	4	no	—
44	C₃³×C₄	C₄²	1	4	yes	—
45	C₃×C₃⋊Dic₃	—	2	8	no	—
46	C₂×C₃×C₄²	—	2	4	no	—
47	C₂×C₃×C₄²	—	9	2	no	—
48	C₂×C₃×C₄²	C₃×D₃	1	4	yes	—
49	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
50	C₂×C₄×Dic₃	—	3	8	no	—
51	C₃×#95	—	9	4	no	—
52	C₃⋊#95	—	3	16	no	—
53	C₂³×C₃²	—	2	4	no	—
54	C₂³×C₃²	C₂²×D₃	1	4	yes	—
55	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
56	C₂×C₃²×C₄	—	2	32	no	—
57	C₂×C₃²×C₄	—	2	16	no	—
58	C₂×C₃²×C₄	—	2	4	no	—
59	C₂×C₃²×C₄	—	2	16	no	—
60	C₂×C₃²×C₄	—	2	16	no	—
61	C₂×C₃²×C₄	—	2	8	no	—
62	C₂×C₃²×C₄	C₄×D₃	1	16	yes	—
63	C₂×C₃²×C₄	C₂×Dic₃	1	16	yes	—
64	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
65	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
66	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
67	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
68	C₂×C₃⋊Dic₃	—	2	8	no	—
69	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	8	yes	—
70	C₂×C₃×Dic₃	—	3	32	no	—
71	C₂×C₃×Dic₃	—	6	32	no	—
72	#376	—	9	1	no	Sylow
73	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₃³	#117	1	4	yes	—
75	C₂×C₃³	#117	1	2	yes	—
76	C₂²×C₃×C₄	—	2	8	no	—
77	C₂²×C₃×C₄	—	2	4	no	—
78	C₂²×C₃×C₄	—	9	4	no	—
79	C₂²×C₃×C₄	C₂×C₃×D₃	1	8	yes	—
80	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
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	16	no	—
84	C₃×C₄²	—	2	16	no	—
85	C₃×C₄²	—	9	8	no	—
86	C₃×C₄²	C₃×Dic₃	1	16	yes	complemented
87	C₃×C₄²	C₃⋊Dic₃	1	4	yes	complemented
88	C₄×Dic₃	—	3	32	no	—
89	C₂²×C₃²	—	2	8	no	—
90	C₂²×C₃²	—	2	16	no	—
91	C₂²×C₃²	—	2	4	no	—
92	C₂²×C₃²	C₂×C₄×D₃	1	8	yes	—
93	C₂²×C₃²	#235	1	16	yes	—
94	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
95	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
96	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₃²	C₃×#41	1	4	yes	—
98	C₃²×C₄	—	2	32	no	—
99	C₃²×C₄	—	2	16	no	—
100	C₃²×C₄	—	2	4	no	—
101	C₃²×C₄	—	2	16	no	—
102	C₃²×C₄	C₄×Dic₃	1	16	yes	—
103	C₃²×C₄	C₃×C₄²	1	4	yes	—
104	C₃⋊Dic₃	—	2	8	no	—
105	C₃×Dic₃	—	6	32	no	—
106	C₂×C₄²	—	9	2	no	—
107	C₂×C₄²	C₃×C₃⋊D₃	1	1	yes	characteristic
108	#95	—	9	4	no	—
109	C₃³	#376	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₃	—	2	4	no	—
111	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
112	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	32	no	—
114	C₂×C₃×C₄	—	2	16	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	16	no	—
117	C₂×C₃×C₄	—	2	16	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	9	8	no	—
120	C₂×C₃×C₄	—	18	8	no	—
121	C₂×C₃×C₄	C₃×C₄×D₃	1	16	yes	—
122	C₂×C₃×C₄	C₂×C₃×Dic₃	1	16	yes	—
123	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
124	C₂×C₃×C₄	C₄×C₃⋊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	32	no	—
128	C₂×Dic₃	—	6	32	no	—
129	C₂×C₃²	—	2	4	no	—
130	C₂×C₃²	—	2	8	no	—
131	C₂×C₃²	—	2	16	no	—
132	C₂×C₃²	#872	1	16	yes	—
133	C₂×C₃²	C₂×C₄×Dic₃	1	4	yes	—
134	C₂×C₃²	#926	1	8	yes	—
135	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
136	C₂×C₃²	C₃×#117	1	2	yes	—
137	C₂×C₃²	C₃×#117	1	4	yes	—
138	C₂²×C₄	—	9	4	no	—
139	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
140	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
141	C₄²	—	9	8	no	—
142	C₄²	C₃×C₃⋊Dic₃	1	4	yes	complemented
143	C₂²×C₃	—	2	8	no	—
144	C₂²×C₃	—	2	16	no	—
145	C₂²×C₃	—	2	4	no	—
146	C₂²×C₃	C₂×C₃×C₄×D₃	1	8	yes	—
147	C₂²×C₃	C₃×#235	1	16	yes	—
148	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
149	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	2	yes	—
150	C₂²×C₃	#3828	1	4	yes	—
151	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
152	C₃×C₄	—	2	32	no	—
153	C₃×C₄	—	2	16	no	—
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	2	16	no	—
156	C₃×C₄	—	18	8	no	—
157	C₃×C₄	C₃×C₄×Dic₃	1	16	yes	—
158	C₃×C₄	C₄×C₃⋊Dic₃	1	4	yes	—
159	Dic₃	—	6	32	no	—
160	C₃²	—	2	4	no	—
161	C₃²	#4932	1	4	yes	complemented
162	C₃²	C₃×#376	1	1	yes	characteristic, complemented
163	C₂³	C₂²×C₃×C₃⋊D₃	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×C₃⋊D₃	1	4	yes	—
168	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	4	yes	—
169	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
170	C₂×C₃	—	2	16	no	—
171	C₂×C₃	—	2	4	no	—
172	C₂×C₃	—	2	8	no	—
173	C₂×C₃	C₃×#872	1	16	yes	—
174	C₂×C₃	C₂×C₃×C₄×Dic₃	1	4	yes	—
175	C₂×C₃	C₃×#926	1	8	yes	—
176	C₂×C₃	#64081	1	4	yes	—
177	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
178	C₂×C₃	#64135	1	2	yes	—
179	C₂²	C₂×C₃×C₄×C₃⋊D₃	1	2	yes	—
180	C₂²	C₃×#3828	1	4	yes	—
181	C₂²	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	18	8	no	—
184	C₄	C₃×C₄×C₃⋊Dic₃	1	4	yes	—
185	C₃	—	2	4	no	—
186	C₃	C₃×#4932	1	4	yes	complemented
187	C₃	#10592631	1	1	yes	characteristic, direct product factor
188	C₂	C₃×#64081	1	4	yes	—
189	C₂	C₂×C₃×C₄×C₃⋊Dic₃	1	1	yes	characteristic
190	C₂	C₃×#64135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₃×C₄²
Frattini	C₂³	C₂²×C₃×C₃⋊D₃
Fitting	C₂×C₃³×C₄²	C₂
socle, prime radical	C₂³×C₃³	C₂³
cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₃×C₄²	C₃⋊D₃
hypocenter	C₃²	C₃×#376
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
4. #10592631	C₃	direct (N × H)
9. C₃³×C₄²	C₄	semidirect
10. C₃×C₄×C₃⋊Dic₃	C₄	semidirect
36. C₃²×C₄²	Dic₃	semidirect
37. C₃²×C₄²	C₃×C₄	semidirect
38. C₄×C₃⋊Dic₃	C₃×C₄	semidirect
86. C₃×C₄²	C₃×Dic₃	semidirect
87. C₃×C₄²	C₃⋊Dic₃	semidirect
109. C₃³	#376	semidirect
142. C₄²	C₃×C₃⋊Dic₃	semidirect
161. C₃²	#4932	semidirect
162. C₃²	C₃×#376	semidirect
186. C₃	C₃×#4932	semidirect
187. C₃	#10592631	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×C₃³×C₄²	2
C₂×C₃×C₄×C₃⋊Dic₃	2
C₃×#63658	2
#10592631	3
C₃×#4932	3

Subgroups — C3×#10592631 #13162968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592631 #13162968