Subgroups — C₃×#10592635 #13162973

Order 1728.

1356 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×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	#10592635	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×#4936	—	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)	—	2	4	no	—
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	2	yes	—
22	C₂×C₃³×C₄	D₄	1	2	yes	—
23	C₂×C₃³×C₄	Q₈	1	2	yes	—
24	C₂×C₃×C₃⋊Dic₃	—	2	4	no	—
25	C₂×C₃×C₃⋊Dic₃	—	2	8	no	—
26	C₃×#382	—	9	1	no	—
27	#4936	—	3	4	no	—
28	C₂²×C₃²×C₄	—	2	8	no	—
29	C₂²×C₃²×C₄	—	2	4	no	—
30	C₂²×C₃²×C₄	C₂×D₃	1	8	yes	—
31	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
32	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
33	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₂²×C₃⋊Dic₃	C₂²×C₃	1	4	yes	—
35	C₂²×C₃×Dic₃	—	3	16	no	—
36	C₃²×C₄²	—	2	16	no	—
37	C₃²×C₄²	Dic₃	1	16	yes	complemented
38	C₃²×C₄²	C₃×C₄	1	4	yes	complemented
39	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
40	C₃×C₄⋊Dic₃	—	6	16	no	—
41	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₃³	#41	1	4	yes	—
45	C₃³×C₄	—	2	4	no	—
46	C₃³×C₄	C₄⋊C₄	1	4	yes	—
47	C₃×C₃⋊Dic₃	—	4	4	no	—
48	C₂×C₃×C₄⋊C₄	—	9	2	no	—
49	C₂×C₃×C₄²	—	2	4	no	—
50	C₂×C₃×C₄²	C₃×D₃	1	4	yes	—
51	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₃	—	3	8	no	—
53	C₃×#95	—	9	4	no	—
54	C₃⋊#95	—	3	16	no	—
55	C₂³×C₃²	—	2	4	no	—
56	C₂³×C₃²	C₂²×D₃	1	4	yes	—
57	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
58	C₂×C₃²×C₄	—	2	32	no	—
59	C₂×C₃²×C₄	—	2	16	no	—
60	C₂×C₃²×C₄	—	2	4	no	—
61	C₂×C₃²×C₄	—	2	8	no	—
62	C₂×C₃²×C₄	—	2	8	no	—
63	C₂×C₃²×C₄	—	2	16	no	—
64	C₂×C₃²×C₄	—	2	8	no	—
65	C₂×C₃²×C₄	Dic₆	1	8	yes	—
66	C₂×C₃²×C₄	D₁₂	1	8	yes	—
67	C₂×C₃²×C₄	C₂×Dic₃	1	16	yes	—
68	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
69	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
70	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
71	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
72	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
73	C₂×C₃⋊Dic₃	—	2	4	no	—
74	C₂×C₃⋊Dic₃	—	2	8	no	—
75	C₂×C₃×Dic₃	—	6	16	no	—
76	C₂×C₃×Dic₃	—	6	32	no	—
77	#382	—	9	1	no	Sylow
78	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃³	#117	1	2	yes	—
80	C₂×C₃³	#124	1	2	yes	—
81	C₂×C₃³	#125	1	2	yes	—
82	C₂²×C₃×C₄	—	2	8	no	—
83	C₂²×C₃×C₄	—	2	4	no	—
84	C₂²×C₃×C₄	—	9	4	no	—
85	C₂²×C₃×C₄	C₂×C₃×D₃	1	8	yes	—
86	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
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	16	no	—
90	C₃×C₄⋊C₄	—	18	4	no	—
91	C₃×C₄²	—	2	16	no	—
92	C₃×C₄²	C₃×Dic₃	1	16	yes	complemented
93	C₃×C₄²	C₃⋊Dic₃	1	4	yes	complemented
94	C₄⋊Dic₃	—	6	16	no	—
95	C₂²×C₃²	—	2	4	no	—
96	C₂²×C₃²	—	2	4	no	—
97	C₂²×C₃²	—	2	16	no	—
98	C₂²×C₃²	—	2	4	no	—
99	C₂²×C₃²	C₂×Dic₆	1	4	yes	—
100	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
101	C₂²×C₃²	#235	1	16	yes	—
102	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
103	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
104	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
105	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
106	C₂²×C₃²	C₃×#41	1	4	yes	—
107	C₃²×C₄	—	2	32	no	—
108	C₃²×C₄	—	2	16	no	—
109	C₃²×C₄	—	2	4	no	—
110	C₃²×C₄	—	2	16	no	—
111	C₃²×C₄	C₄⋊Dic₃	1	16	yes	—
112	C₃²×C₄	C₃×C₄⋊C₄	1	4	yes	—
113	C₃⋊Dic₃	—	4	4	no	—
114	C₃×Dic₃	—	12	16	no	—
115	C₂×C₄⋊C₄	—	9	2	no	—
116	C₂×C₄²	C₃×C₃⋊D₃	1	1	yes	characteristic
117	#95	—	9	4	no	—
118	C₃³	#382	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₃	—	2	4	no	—
120	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
121	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	32	no	—
123	C₂×C₃×C₄	—	2	16	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	8	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	16	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	—	18	4	no	—
130	C₂×C₃×C₄	—	18	8	no	—
131	C₂×C₃×C₄	C₃×Dic₆	1	8	yes	—
132	C₂×C₃×C₄	C₃×D₁₂	1	8	yes	—
133	C₂×C₃×C₄	C₂×C₃×Dic₃	1	16	yes	—
134	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
135	C₂×C₃×C₄	#634	1	2	yes	—
136	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	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	16	no	—
140	C₂×Dic₃	—	6	32	no	—
141	C₂×C₃²	—	2	4	no	—
142	C₂×C₃²	—	2	8	no	—
143	C₂×C₃²	—	2	8	no	—
144	C₂×C₃²	—	2	8	no	—
145	C₂×C₃²	#870	1	8	yes	—
146	C₂×C₃²	#875	1	8	yes	—
147	C₂×C₃²	C₂×C₄⋊Dic₃	1	4	yes	—
148	C₂×C₃²	#926	1	8	yes	—
149	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₃²	C₃×#117	1	2	yes	—
151	C₂×C₃²	C₃×#124	1	2	yes	—
152	C₂×C₃²	C₃×#125	1	2	yes	—
153	C₂²×C₄	—	9	4	no	—
154	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
155	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
156	C₄⋊C₄	—	18	4	no	—
157	C₄²	C₃×C₃⋊Dic₃	1	4	yes	complemented
158	C₂²×C₃	—	2	4	no	—
159	C₂²×C₃	—	2	4	no	—
160	C₂²×C₃	—	2	16	no	—
161	C₂²×C₃	—	2	4	no	—
162	C₂²×C₃	C₂×C₃×Dic₆	1	4	yes	—
163	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
164	C₂²×C₃	C₃×#235	1	16	yes	—
165	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
166	C₂²×C₃	C₂×#634	1	1	yes	characteristic
167	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
168	C₂²×C₃	#3828	1	4	yes	—
169	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
170	C₃×C₄	—	2	32	no	—
171	C₃×C₄	—	2	16	no	—
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	2	16	no	—
174	C₃×C₄	—	36	4	no	—
175	C₃×C₄	C₃×C₄⋊Dic₃	1	16	yes	—
176	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
177	Dic₃	—	12	16	no	—
178	C₃²	—	2	4	no	—
179	C₃²	#4936	1	4	yes	complemented
180	C₃²	C₃×#382	1	1	yes	characteristic, complemented
181	C₂³	C₂²×C₃×C₃⋊D₃	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×#634	1	2	yes	—
186	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	2	yes	—
187	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	4	yes	—
188	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
189	C₂×C₃	—	2	8	no	—
190	C₂×C₃	—	2	8	no	—
191	C₂×C₃	—	2	4	no	—
192	C₂×C₃	—	2	8	no	—
193	C₂×C₃	C₃×#870	1	8	yes	—
194	C₂×C₃	C₃×#875	1	8	yes	—
195	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
196	C₂×C₃	C₃×#926	1	8	yes	—
197	C₂×C₃	#64079	1	2	yes	—
198	C₂×C₃	#64084	1	2	yes	—
199	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
200	C₂×C₃	#64135	1	2	yes	—
201	C₂²	C₂×C₃×#634	1	1	yes	characteristic
202	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
203	C₂²	C₃×#3828	1	4	yes	—
204	C₂²	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
205	C₄	—	2	4	no	—
206	C₄	—	36	4	no	—
207	C₄	C₃×C₄⋊(C₃⋊Dic₃)	1	4	yes	—
208	C₃	—	2	4	no	—
209	C₃	C₃×#4936	1	4	yes	complemented
210	C₃	#10592635	1	1	yes	characteristic, direct product factor
211	C₂	C₃×#64079	1	2	yes	—
212	C₂	C₃×#64084	1	2	yes	—
213	C₂	C₂×C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
214	C₂	C₃×#64135	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₃×C₄
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×#382
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
4. #10592635	C₃	direct (N × H)
9. C₃³×C₄²	C₄	semidirect
37. C₃²×C₄²	Dic₃	semidirect
38. C₃²×C₄²	C₃×C₄	semidirect
92. C₃×C₄²	C₃×Dic₃	semidirect
93. C₃×C₄²	C₃⋊Dic₃	semidirect
118. C₃³	#382	semidirect
157. C₄²	C₃×C₃⋊Dic₃	semidirect
179. C₃²	#4936	semidirect
180. C₃²	C₃×#382	semidirect
209. C₃	C₃×#4936	semidirect
210. C₃	#10592635	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
#10592635	3
C₃×#4936	3

Subgroups — C3×#10592635 #13162973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592635 #13162973