Subgroups — C₃×#10589415 #13145973

Order 1728.

700 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	#10589415	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×#4549	—	3	4	no	maximal
5	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
6	C₂×C₃³×C₈	—	2	2	no	—
7	C₂×C₃³×C₈	C₄	1	1	yes	characteristic
8	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
10	C₂²×C₃²×C₈	—	2	4	no	—
11	C₂²×C₃²×C₈	D₃	1	4	yes	—
12	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₃	1	2	yes	complemented
14	C₂×C₃×C₃⋊C₁₆	—	3	8	no	—
15	C₂³×C₃³	C₈	1	1	yes	characteristic
16	C₂×C₃³×C₄	—	2	1	no	—
17	C₂×C₃³×C₄	—	2	1	no	—
18	C₂×C₃³×C₄	C₈	1	1	yes	characteristic
19	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
20	C₃³×C₈	—	2	1	no	—
21	C₃³×C₈	D₄	1	2	yes	—
22	C₃×C₂²⋊C₁₆	—	9	1	no	—
23	#4549	—	3	4	no	—
24	C₂²×C₃²×C₄	—	2	4	no	—
25	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
26	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	—	2	16	no	—
28	C₂×C₃²×C₈	—	2	4	no	—
29	C₂×C₃²×C₈	—	2	4	no	—
30	C₂×C₃²×C₈	—	2	8	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	Dic₃	1	4	yes	—
33	C₂×C₃²×C₈	C₃×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
35	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
36	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
37	C₃×C₃⋊C₁₆	—	6	8	no	—
38	C₂²×C₃³	—	2	1	no	—
39	C₂²×C₃³	—	2	1	no	—
40	C₂²×C₃³	C₁₆	1	2	yes	complemented
41	C₂²×C₃³	C₂×C₈	1	1	yes	characteristic
42	C₃³×C₄	—	2	1	no	—
43	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
44	C₃³×C₄	M₁₆	1	1	yes	characteristic
45	C₂²×C₃×C₈	—	2	4	no	—
46	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
47	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
48	C₂×C₃×C₁₆	—	9	2	no	—
49	C₂×C₃⋊C₁₆	—	3	8	no	—
50	C₂³×C₃²	—	2	4	no	—
51	C₂³×C₃²	C₃⋊C₈	1	4	yes	—
52	C₂³×C₃²	C₃×C₈	1	1	yes	characteristic
53	C₂×C₃²×C₄	—	2	8	no	—
54	C₂×C₃²×C₄	—	2	8	no	—
55	C₂×C₃²×C₄	—	2	4	no	—
56	C₂×C₃²×C₄	—	2	4	no	—
57	C₂×C₃²×C₄	—	2	1	no	—
58	C₂×C₃²×C₄	—	2	1	no	—
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	—	2	4	no	—
61	C₂×C₃²×C₄	C₃⋊C₈	1	4	yes	—
62	C₂×C₃²×C₄	C₃×C₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
64	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
65	C₃²×C₈	—	2	8	no	—
66	C₃²×C₈	—	2	4	no	—
67	C₃²×C₈	—	2	1	no	—
68	C₃²×C₈	—	2	8	no	—
69	C₃²×C₈	#67	1	8	yes	—
70	C₃²×C₈	C₃×D₄	1	2	yes	—
71	C₂²⋊C₁₆	—	9	1	no	Sylow
72	C₂×C₃³	—	2	2	no	—
73	C₂×C₃³	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₃³	C₂×C₁₆	1	1	yes	characteristic
75	C₂×C₃³	M₃₂	1	1	yes	characteristic
76	C₂²×C₃×C₄	—	2	4	no	—
77	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
78	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
79	C₂×C₃×C₈	—	2	16	no	—
80	C₂×C₃×C₈	—	2	8	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	4	no	—
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
85	C₂×C₃×C₈	C₃⋊Dic₃	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
87	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
88	C₃×C₁₆	—	18	2	no	—
89	C₃⋊C₁₆	—	6	8	no	—
90	C₂²×C₃²	—	2	8	no	—
91	C₂²×C₃²	—	2	8	no	—
92	C₂²×C₃²	—	2	4	no	—
93	C₂²×C₃²	—	2	4	no	—
94	C₂²×C₃²	—	2	1	no	—
95	C₂²×C₃²	—	2	1	no	—
96	C₂²×C₃²	—	2	8	no	—
97	C₂²×C₃²	—	2	4	no	—
98	C₂²×C₃²	C₃⋊C₁₆	1	8	yes	complemented
99	C₂²×C₃²	C₃×C₁₆	1	2	yes	complemented
100	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
101	C₂²×C₃²	C₂×C₃×C₈	1	1	yes	characteristic
102	C₃²×C₄	—	2	8	no	—
103	C₃²×C₄	—	2	4	no	—
104	C₃²×C₄	—	2	1	no	—
105	C₃²×C₄	—	2	4	no	—
106	C₃²×C₄	—	2	4	no	—
107	C₃²×C₄	#208	1	4	yes	—
108	C₃²×C₄	#217	1	4	yes	—
109	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
110	C₃²×C₄	C₃×M₁₆	1	1	yes	characteristic
111	C₂²×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
112	C₂×C₁₆	—	9	2	no	—
113	C₃³	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₃	—	2	4	no	—
115	C₂³×C₃	C₃×C₃⋊C₈	1	4	yes	—
116	C₂³×C₃	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
117	C₂×C₃×C₄	—	2	8	no	—
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	C₃×C₃⋊C₈	1	4	yes	—
126	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
128	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
129	C₃×C₈	—	2	8	no	—
130	C₃×C₈	—	2	4	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	8	no	—
133	C₃×C₈	C₃×#67	1	8	yes	—
134	C₃×C₈	#638	1	2	yes	—
135	C₂×C₃²	—	2	16	no	—
136	C₂×C₃²	—	2	8	no	—
137	C₂×C₃²	—	2	2	no	—
138	C₂×C₃²	—	2	4	no	—
139	C₂×C₃²	—	2	4	no	—
140	C₂×C₃²	—	2	4	no	—
141	C₂×C₃²	C₂×C₃⋊C₁₆	1	4	yes	—
142	C₂×C₃²	#812	1	4	yes	—
143	C₂×C₃²	#830	1	4	yes	—
144	C₂×C₃²	C₃×C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₃²	C₂×C₃×C₁₆	1	1	yes	characteristic
146	C₂×C₃²	C₃×M₃₂	1	1	yes	characteristic
147	C₂²×C₄	C₃×C₃⋊Dic₃	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	C₃×C₃⋊Dic₃	1	1	yes	characteristic
150	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
151	C₁₆	—	18	2	no	—
152	C₂²×C₃	—	2	8	no	—
153	C₂²×C₃	—	2	8	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	2	4	no	—
159	C₂²×C₃	—	2	8	no	—
160	C₂²×C₃	C₃×C₃⋊C₁₆	1	8	yes	complemented
161	C₂²×C₃	C₃⋊(C₃⋊C₁₆)	1	2	yes	complemented
162	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	4	yes	—
163	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
164	C₃×C₄	—	2	8	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	2	4	no	—
169	C₃×C₄	C₃×#208	1	4	yes	—
170	C₃×C₄	C₃×#217	1	4	yes	—
171	C₃×C₄	#3748	1	1	yes	characteristic
172	C₃×C₄	#3757	1	1	yes	characteristic
173	C₃²	—	2	4	no	—
174	C₃²	#4549	1	4	yes	complemented
175	C₃²	C₃×C₂²⋊C₁₆	1	1	yes	characteristic, complemented
176	C₂³	C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
180	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
181	C₈	—	2	1	no	—
182	C₈	C₃×#638	1	2	yes	—
183	C₂×C₃	—	2	16	no	—
184	C₂×C₃	—	2	8	no	—
185	C₂×C₃	—	2	2	no	—
186	C₂×C₃	—	2	4	no	—
187	C₂×C₃	—	2	4	no	—
188	C₂×C₃	—	2	4	no	—
189	C₂×C₃	C₂×C₃×C₃⋊C₁₆	1	4	yes	—
190	C₂×C₃	C₃×#812	1	4	yes	—
191	C₂×C₃	C₃×#830	1	4	yes	—
192	C₂×C₃	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
193	C₂×C₃	#63639	1	1	yes	characteristic
194	C₂×C₃	#63657	1	1	yes	characteristic
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	C₃×C₃⋊(C₃⋊C₁₆)	1	2	yes	complemented
198	C₂²	C₂×C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	C₃×#3748	1	1	yes	characteristic
201	C₄	C₃×#3757	1	1	yes	characteristic
202	C₃	—	2	4	no	—
203	C₃	C₃×#4549	1	4	yes	complemented
204	C₃	#10589415	1	1	yes	characteristic, direct product factor
205	C₂	—	2	2	no	—
206	C₂	C₂×C₃×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
207	C₂	C₃×#63639	1	1	yes	characteristic
208	C₂	C₃×#63657	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₃×C₁₆
Frattini	C₂×C₈	C₂×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₂×C₈
prime radical	C₂³×C₃³	C₈
cosocle	C₂×C₃²×C₈	C₂²×C₃
hypercenter	C₂²×C₃×C₈	C₃⋊D₃
hypocenter	C₃²	C₃×C₂²⋊C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
3. #10589415	C₃	direct (N × H)
13. C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₃	semidirect
40. C₂²×C₃³	C₁₆	semidirect
98. C₂²×C₃²	C₃⋊C₁₆	semidirect
99. C₂²×C₃²	C₃×C₁₆	semidirect
113. C₃³	C₂²⋊C₁₆	semidirect
160. C₂²×C₃	C₃×C₃⋊C₁₆	semidirect
161. C₂²×C₃	C₃⋊(C₃⋊C₁₆)	semidirect
174. C₃²	#4549	semidirect
175. C₃²	C₃×C₂²⋊C₁₆	semidirect
197. C₂²	C₃×C₃⋊(C₃⋊C₁₆)	semidirect
203. C₃	C₃×#4549	semidirect
204. C₃	#10589415	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×C₃³×C₈	2
C₂×C₃×C₃⋊(C₃⋊C₁₆)	2
#10589415	3
C₃×#4549	3

Subgroups — C3×#10589415 #13145973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589415 #13145973