Subgroups — C₂×C₃×#63371 #13143069

Order 1728.

346 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	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#63371	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#812	—	3	1	no	maximal
5	C₂×#63371	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈×C₉	C₄	1	2	yes	—
8	C₂×C₃×C₈×C₉	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₃×C₉⋊C₁₆	C₂²	1	4	yes	complemented
12	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
13	C₃×#812	—	3	4	no	—
14	C₂×C₃×C₃⋊C₁₆	—	3	2	no	—
15	C₂²×C₈×C₉	—	2	1	no	—
16	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
17	#63371	C₂×C₃	1	4	yes	direct product factor
18	C₂×C₉⋊C₁₆	C₂×C₃	1	2	yes	complemented
19	C₂³×C₃×C₉	C₈	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	C₈	1	2	yes	—
21	C₂×C₃×C₄×C₉	C₈	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
25	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
26	C₃×C₈×C₉	C₂³	1	1	yes	characteristic
27	C₂×C₃×M₃₂	—	9	1	no	—
28	C₂×#812	—	3	1	no	—
29	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
30	C₂×C₃²×C₈	Dic₃	1	1	yes	characteristic
31	C₂×C₃²×C₈	Dic₃	1	2	yes	—
32	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
33	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
36	C₃×C₃⋊C₁₆	—	3	4	no	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	1	no	—
39	C₂×C₈×C₉	—	2	1	no	—
40	C₂×C₈×C₉	—	2	2	no	—
41	C₂×C₈×C₉	C₃×C₄	1	2	yes	—
42	C₂×C₈×C₉	C₃×C₄	1	1	yes	characteristic
43	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
45	C₉⋊C₁₆	C₂²×C₃	1	4	yes	complemented
46	C₂²×C₃×C₉	—	2	2	no	—
47	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
48	C₂²×C₃×C₉	C₂×C₈	1	1	yes	characteristic
49	C₃×C₄×C₉	C₂×C₈	1	1	yes	characteristic
50	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
51	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
55	C₃×M₃₂	—	9	4	no	—
56	C₂×C₃×C₁₆	—	9	2	no	—
57	#812	—	3	4	no	—
58	C₂×C₃⋊C₁₆	—	3	2	no	—
59	C₂³×C₃²	C₃⋊C₈	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₃⋊C₈	1	2	yes	—
61	C₂×C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
63	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	C₃×C₈	1	1	yes	characteristic
66	C₃²×C₈	C₂×Dic₃	1	2	yes	—
67	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
68	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	C₃×C₈	1	2	yes	—
74	C₂×C₄×C₉	C₃×C₈	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
77	C₈×C₉	—	2	1	no	—
78	C₈×C₉	—	2	2	no	—
79	C₈×C₉	—	2	1	no	—
80	C₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
81	C₈×C₉	C₂×C₃×C₄	1	2	yes	—
82	C₈×C₉	C₂³×C₃	1	1	yes	characteristic
83	C₂×M₃₂	—	9	1	no	Sylow
84	C₂×C₃×C₉	—	2	2	no	—
85	C₂×C₃×C₉	M₃₂	1	2	yes	complemented
86	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	Dic₉	1	1	yes	characteristic
95	C₂×C₃×C₈	Dic₉	1	2	yes	—
96	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
97	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
98	C₂×C₃×C₈	C₃×Dic₃	1	2	yes	—
99	C₂×C₃×C₈	C₃×Dic₃	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
102	C₃×C₁₆	—	9	4	no	—
103	C₃⋊C₁₆	—	3	4	no	—
104	C₂²×C₃²	—	2	2	no	—
105	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
106	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
107	C₃²×C₄	C₂×C₃⋊C₈	1	2	yes	—
108	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
109	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
110	C₂²×C₉	—	2	4	no	—
111	C₂²×C₉	—	2	2	no	—
112	C₂²×C₉	—	2	2	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	C₂×C₃×C₈	1	2	yes	—
115	C₂²×C₉	C₂×C₃×C₈	1	1	yes	characteristic
116	C₄×C₉	—	2	1	no	—
117	C₄×C₉	—	2	2	no	—
118	C₄×C₉	—	2	1	no	—
119	C₄×C₉	C₂×C₃×C₈	1	1	yes	characteristic
120	C₄×C₉	C₂×C₃×C₈	1	2	yes	—
121	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
123	M₃₂	—	9	4	no	—
124	C₂×C₁₆	—	9	2	no	—
125	C₃×C₉	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₃	—	2	1	no	—
127	C₂³×C₃	C₉⋊C₈	1	1	yes	characteristic
128	C₂³×C₃	C₃×C₃⋊C₈	1	1	yes	characteristic
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	C₉⋊C₈	1	2	yes	—
134	C₂×C₃×C₄	C₉⋊C₈	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
136	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
137	C₂×C₃×C₄	C₃×C₃⋊C₈	1	2	yes	—
138	C₂×C₃×C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	C₂×Dic₉	1	2	yes	—
145	C₃×C₈	C₂×Dic₉	1	1	yes	characteristic
146	C₃×C₈	C₂²×D₉	1	1	yes	characteristic
147	C₃×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
148	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
149	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
150	C₂×C₃²	—	2	2	no	—
151	C₂×C₃²	#812	1	2	yes	—
152	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
153	C₂×C₉	—	2	4	no	—
154	C₂×C₉	—	2	2	no	—
155	C₂×C₉	—	2	1	no	—
156	C₂×C₉	—	2	2	no	—
157	C₂×C₉	C₃×M₃₂	1	2	yes	complemented
158	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
159	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
160	C₂×C₈	C₃×Dic₉	1	2	yes	—
161	C₂×C₈	C₃×Dic₉	1	1	yes	characteristic
162	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
163	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
164	C₁₆	—	9	4	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	C₂×C₉⋊C₈	1	2	yes	—
171	C₂²×C₃	C₂×C₉⋊C₈	1	1	yes	characteristic
172	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	2	yes	—
173	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	C₂×C₉⋊C₈	1	2	yes	—
178	C₃×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
179	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
180	C₃×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
181	C₃×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
182	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
183	C₃²	C₂×#812	1	1	yes	characteristic
184	C₉	—	2	1	no	—
185	C₉	C₂×C₃×M₃₂	1	1	yes	characteristic, complemented
186	C₂³	C₃×C₉⋊C₈	1	1	yes	characteristic
187	C₂×C₄	C₃×C₉⋊C₈	1	2	yes	—
188	C₂×C₄	C₃×C₉⋊C₈	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
190	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
191	C₈	C₂×C₃×Dic₉	1	1	yes	characteristic
192	C₈	C₂×C₃×Dic₉	1	2	yes	—
193	C₈	C₂²×C₃×D₉	1	1	yes	characteristic
194	C₂×C₃	—	2	4	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	—	2	2	no	—
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	#63371	1	2	yes	direct product factor
200	C₂×C₃	C₂²×C₉⋊C₈	1	1	yes	characteristic
201	C₂×C₃	C₃×#812	1	2	yes	—
202	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	C₂×C₃×C₉⋊C₈	1	2	yes	—
205	C₂²	C₂×C₃×C₉⋊C₈	1	1	yes	characteristic
206	C₄	C₂×C₃×C₉⋊C₈	1	1	yes	characteristic
207	C₄	C₂×C₃×C₉⋊C₈	1	2	yes	—
208	C₄	C₂²×C₃×Dic₉	1	1	yes	characteristic
209	C₃	—	2	1	no	—
210	C₃	C₂×#63371	1	1	yes	characteristic, direct product factor
211	C₃	C₂×C₃×#812	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	C₃×#63371	1	2	yes	direct product factor
214	C₂	C₂²×C₃×C₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₉
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₃×C₈	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃⋊C₈
prime radical	C₂³×C₃²	C₃⋊C₈
cosocle	C₈×C₉	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₉
hypocenter	C₉	C₂×C₃×M₃₂
LCS	G ▷ C₂×C₉ ▷ C₉
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. C₃×#63371	C₂	direct (N × H)
3. C₂×C₃×C₉⋊C₁₆	C₂	semidirect
5. C₂×#63371	C₃	direct (N × H)
11. C₃×C₉⋊C₁₆	C₂²	semidirect
17. #63371	C₂×C₃	direct (N × H)
18. C₂×C₉⋊C₁₆	C₂×C₃	semidirect
45. C₉⋊C₁₆	C₂²×C₃	semidirect
85. C₂×C₃×C₉	M₃₂	semidirect
125. C₃×C₉	C₂×M₃₂	semidirect
157. C₂×C₉	C₃×M₃₂	semidirect
185. C₉	C₂×C₃×M₃₂	semidirect
199. C₂×C₃	#63371	direct (N × H)
210. C₃	C₂×#63371	direct (N × H)
213. C₂	C₃×#63371	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×C₃×C₈×C₉	2
C₃×#63371	2
C₂×C₃×C₉⋊C₁₆	2
C₂×C₃×#812	3
C₂×#63371	3

Subgroups — C2×C3×#63371 #13143069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63371 #13143069