Subgroups — C₂×#63371⋊C₃ #13143132

Order 1728.

308 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	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	#63371⋊C₃	C₂	1	4	yes	maximal, direct product factor
3	C₂×3₋¹⁺²⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×#812	—	3	1	no	maximal
5	C₂×#63371	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₂×C₈×3₋¹⁺²	C₄	1	2	yes	—
8	C₂×C₈×3₋¹⁺²	C₄	1	1	yes	characteristic
9	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
11	3₋¹⁺²⋊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	complemented
18	C₂×C₉⋊C₁₆	C₂×C₃	1	2	yes	complemented
19	C₂³×3₋¹⁺²	C₈	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	C₈	1	2	yes	—
21	C₂×C₄×3₋¹⁺²	C₈	1	1	yes	characteristic
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
23	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
24	C₈×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
25	C₈×3₋¹⁺²	C₂×C₄	1	2	yes	—
26	C₈×3₋¹⁺²	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	2	yes	—
31	C₂×C₃²×C₈	Dic₃	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
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	1	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₂×C₈×C₉	—	2	1	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	C₂²×3₋¹⁺²	—	2	2	no	—
47	C₂²×3₋¹⁺²	C₂×C₈	1	2	yes	—
48	C₂²×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
49	C₄×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
50	C₄×3₋¹⁺²	C₂×C₈	1	2	yes	—
51	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₈	—	3	1	no	—
53	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
54	C₃×M₃₂	—	9	4	no	—
55	C₂×C₃×C₁₆	—	9	2	no	—
56	#812	—	3	4	no	—
57	C₂×C₃⋊C₁₆	—	3	2	no	—
58	C₂³×C₃²	C₃⋊C₈	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₃⋊C₈	1	2	yes	—
60	C₂×C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
62	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
63	C₂³×C₉	—	2	1	no	—
64	C₂³×C₉	C₃×C₈	1	1	yes	characteristic
65	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
66	C₃²×C₈	C₂×Dic₃	1	2	yes	—
67	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	C₃×C₈	1	2	yes	—
73	C₂×C₄×C₉	C₃×C₈	1	1	yes	characteristic
74	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
75	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
76	C₈×C₉	—	2	2	no	—
77	C₈×C₉	—	2	1	no	—
78	C₈×C₉	—	2	1	no	—
79	C₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
80	C₈×C₉	C₂×C₃×C₄	1	2	yes	—
81	C₈×C₉	C₂³×C₃	1	1	yes	characteristic
82	C₂×M₃₂	—	9	1	no	Sylow
83	C₂×3₋¹⁺²	—	2	2	no	—
84	C₂×3₋¹⁺²	M₃₂	1	2	yes	complemented
85	C₂×3₋¹⁺²	C₂²×C₈	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
88	C₂×C₃×C₈	—	3	1	no	—
89	C₂×C₃×C₈	—	3	2	no	—
90	C₂×C₃×C₈	—	3	2	no	—
91	C₂×C₃×C₈	—	3	1	no	—
92	C₂×C₃×C₈	C₃×Dic₃	1	1	yes	characteristic
93	C₂×C₃×C₈	C₃×Dic₃	1	2	yes	—
94	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
95	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
96	C₃×C₁₆	—	9	4	no	—
97	C₃⋊C₁₆	—	3	4	no	—
98	C₂²×C₃²	—	2	2	no	—
99	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
100	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
101	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
102	C₃²×C₄	C₂×C₃⋊C₈	1	2	yes	—
103	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
104	C₂²×C₉	—	2	4	no	—
105	C₂²×C₉	—	2	2	no	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	—	2	1	no	—
108	C₂²×C₉	C₂×C₃×C₈	1	2	yes	—
109	C₂²×C₉	C₂×C₃×C₈	1	1	yes	characteristic
110	C₄×C₉	—	2	2	no	—
111	C₄×C₉	—	2	1	no	—
112	C₄×C₉	—	2	1	no	—
113	C₄×C₉	C₂×C₃×C₈	1	1	yes	characteristic
114	C₄×C₉	C₂×C₃×C₈	1	2	yes	—
115	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
117	M₃₂	—	9	4	no	—
118	C₂×C₁₆	—	9	2	no	—
119	3₋¹⁺²	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₃	—	3	1	no	—
121	C₂³×C₃	C₃×C₃⋊C₈	1	1	yes	characteristic
122	C₂×C₃×C₄	—	3	1	no	—
123	C₂×C₃×C₄	—	3	2	no	—
124	C₂×C₃×C₄	—	3	2	no	—
125	C₂×C₃×C₄	—	3	1	no	—
126	C₂×C₃×C₄	C₃×C₃⋊C₈	1	2	yes	—
127	C₂×C₃×C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
129	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
130	C₃×C₈	—	3	2	no	—
131	C₃×C₈	—	3	1	no	—
132	C₃×C₈	—	3	1	no	—
133	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
134	C₃×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
135	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
136	C₂×C₃²	—	2	2	no	—
137	C₂×C₃²	#812	1	2	yes	—
138	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
139	C₂×C₉	—	2	4	no	—
140	C₂×C₉	—	2	2	no	—
141	C₂×C₉	—	2	1	no	—
142	C₂×C₉	—	2	2	no	—
143	C₂×C₉	C₃×M₃₂	1	2	yes	complemented
144	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
145	C₂²×C₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
146	C₂×C₈	3₋¹⁺²⋊C₄	1	1	yes	characteristic
147	C₂×C₈	3₋¹⁺²⋊C₄	1	2	yes	—
148	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
149	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
150	C₁₆	—	9	4	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	3	1	no	—
153	C₂²×C₃	—	3	2	no	—
154	C₂²×C₃	—	6	2	no	—
155	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	2	yes	—
156	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
157	C₃×C₄	—	3	1	no	—
158	C₃×C₄	—	3	2	no	—
159	C₃×C₄	—	3	1	no	—
160	C₃×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
161	C₃×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
162	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
163	C₃²	C₂×#812	1	1	yes	characteristic
164	C₉	—	2	1	no	—
165	C₉	C₂×C₃×M₃₂	1	1	yes	characteristic, complemented
166	C₂³	3₋¹⁺²⋊C₈	1	1	yes	characteristic
167	C₂×C₄	3₋¹⁺²⋊C₈	1	2	yes	—
168	C₂×C₄	3₋¹⁺²⋊C₈	1	1	yes	characteristic
169	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
170	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
171	C₈	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
172	C₈	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
173	C₈	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
174	C₂×C₃	—	2	2	no	—
175	C₂×C₃	—	3	1	no	—
176	C₂×C₃	—	3	2	no	—
177	C₂×C₃	—	6	2	no	—
178	C₂×C₃	C₃×#812	1	2	yes	—
179	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	C₂×3₋¹⁺²⋊C₈	1	2	yes	—
182	C₂²	C₂×3₋¹⁺²⋊C₈	1	1	yes	characteristic
183	C₄	C₂×3₋¹⁺²⋊C₈	1	2	yes	—
184	C₄	C₂×3₋¹⁺²⋊C₈	1	1	yes	characteristic
185	C₄	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
186	C₃	—	3	1	no	—
187	C₃	C₂×C₃×#812	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	#63371⋊C₃	1	2	yes	direct product factor
190	C₂	C₂²×3₋¹⁺²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂²×C₃×C₈
Frattini	C₃×C₈	C₂²×C₃×D₃
Fitting	C₂²×C₈×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×C₃⋊C₈
prime radical	C₂³×C₃²	C₃⋊C₈
cosocle	C₈×C₉	C₂³×C₃
hypercenter	C₂²×C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×M₃₂
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #63371⋊C₃	C₂	direct (N × H)
3. C₂×3₋¹⁺²⋊C₁₆	C₂	semidirect
5. C₂×#63371	C₃	semidirect
11. 3₋¹⁺²⋊C₁₆	C₂²	semidirect
17. #63371	C₂×C₃	semidirect
18. C₂×C₉⋊C₁₆	C₂×C₃	semidirect
45. C₉⋊C₁₆	C₂²×C₃	semidirect
84. C₂×3₋¹⁺²	M₃₂	semidirect
119. 3₋¹⁺²	C₂×M₃₂	semidirect
143. C₂×C₉	C₃×M₃₂	semidirect
165. C₉	C₂×C₃×M₃₂	semidirect
189. C₂	#63371⋊C₃	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×C₈×3₋¹⁺²	2
#63371⋊C₃	2
C₂×3₋¹⁺²⋊C₁₆	2
C₂×C₃×#812	3
C₂×#63371	3

Subgroups — C2×#63371⋊C3 #13143132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63371⋊C₃ #13143132