Subgroups — C₄×#3667⋊C₃ #13142429

Order 1728.

480 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	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×#3667⋊C₃	C₂	1	2	yes	maximal
3	#63362⋊C₃	C₂	1	2	yes	maximal, complemented
4	C₄×3₋¹⁺²⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#208	—	3	1	no	maximal
6	C₄×#3667	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×3₋¹⁺²	C₄	1	2	yes	—
8	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₄²×3₋¹⁺²	C₄	1	2	yes	—
10	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	#3667⋊C₃	C₄	1	8	yes	direct product factor
13	C₂×3₋¹⁺²⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
15	C₂×C₃×#208	—	3	2	no	—
16	C₃×#803	—	3	2	no	—
17	C₃×C₄×C₃⋊C₈	—	3	2	no	—
18	C₂×C₄²×C₉	—	2	1	no	—
19	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
20	C₂×#3667	C₂×C₃	1	2	yes	—
21	#63362	C₂×C₃	1	2	yes	complemented
22	C₄×C₉⋊C₈	C₂×C₃	1	2	yes	complemented
23	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×3₋¹⁺²	—	2	4	no	—
25	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
26	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
28	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
29	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
30	3₋¹⁺²⋊C₈	C₂×C₄	1	8	yes	complemented
31	C₃×C₄×M₁₆	—	9	1	no	—
32	C₄×#208	—	3	1	no	—
33	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₃²×C₄²	Dic₃	1	2	yes	—
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
38	C₃×#208	—	3	8	no	—
39	C₂×C₃×C₃⋊C₈	—	3	4	no	—
40	C₂²×C₄×C₉	—	2	2	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	C₃×C₄	1	2	yes	—
43	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
44	C₄²×C₉	—	2	2	no	—
45	C₄²×C₉	—	2	1	no	—
46	C₄²×C₉	—	2	1	no	—
47	C₄²×C₉	C₃×C₄	1	2	yes	—
48	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
49	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
50	#3667	C₃×C₄	1	8	yes	complemented
51	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	—
52	C₂²×3₋¹⁺²	—	2	2	no	—
53	C₂²×3₋¹⁺²	C₄²	1	2	yes	—
54	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
55	C₄×3₋¹⁺²	—	2	2	no	—
56	C₄×3₋¹⁺²	C₄²	1	2	yes	—
57	C₄×3₋¹⁺²	M₁₆	1	4	yes	complemented
58	C₄×3₋¹⁺²	C₂²×C₄	1	2	yes	—
59	C₂×C₃×M₁₆	—	9	2	no	—
60	C₂×C₃×C₄²	—	3	1	no	—
61	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
62	C₂×#208	—	3	2	no	—
63	C₃×#97	—	9	2	no	—
64	C₃×C₄×C₈	—	9	2	no	—
65	#803	—	3	2	no	—
66	C₄×C₃⋊C₈	—	3	2	no	—
67	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
68	C₂×C₃²×C₄	—	2	4	no	—
69	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
70	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
71	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
72	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂³×C₉	—	2	1	no	—
75	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
76	C₃×C₃⋊C₈	—	3	8	no	—
77	C₂×C₄×C₉	—	2	8	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
85	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
86	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
89	C₉⋊C₈	C₂×C₃×C₄	1	8	yes	complemented
90	C₄×M₁₆	—	9	1	no	Sylow
91	C₂×3₋¹⁺²	—	2	2	no	—
92	C₂×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
93	C₂×3₋¹⁺²	C₂×M₁₆	1	2	yes	—
94	C₂²×C₃×C₄	—	3	2	no	—
95	C₂²×C₃×C₄	—	3	1	no	—
96	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
97	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
98	C₃×M₁₆	—	9	8	no	—
99	C₂×C₃×C₈	—	9	4	no	—
100	C₃×C₄²	—	3	2	no	—
101	C₃×C₄²	—	3	1	no	—
102	C₃×C₄²	—	3	1	no	—
103	C₃×C₄²	C₃×Dic₃	1	2	yes	—
104	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
105	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
106	#208	—	3	8	no	—
107	C₂×C₃⋊C₈	—	3	4	no	—
108	C₂²×C₃²	—	2	2	no	—
109	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
110	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
111	C₃²×C₄	—	2	2	no	—
112	C₃²×C₄	#208	1	4	yes	—
113	C₃²×C₄	C₄×Dic₃	1	2	yes	—
114	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
115	C₂²×C₉	—	2	4	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	2	2	no	—
118	C₂²×C₉	—	2	1	no	—
119	C₂²×C₉	C₃×C₄²	1	2	yes	—
120	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
121	C₄×C₉	—	2	4	no	—
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	—	2	4	no	—
124	C₄×C₉	—	2	2	no	—
125	C₄×C₉	—	2	2	no	—
126	C₄×C₉	C₃×C₄²	1	2	yes	—
127	C₄×C₉	C₃×M₁₆	1	4	yes	complemented
128	C₄×C₉	C₂²×C₃×C₄	1	2	yes	—
129	C₂×M₁₆	—	9	2	no	—
130	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
131	#97	—	9	2	no	—
132	C₄×C₈	—	9	2	no	—
133	3₋¹⁺²	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₃	—	3	1	no	—
135	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	3	4	no	—
138	C₂×C₃×C₄	—	3	2	no	—
139	C₂×C₃×C₄	—	3	1	no	—
140	C₂×C₃×C₄	—	3	2	no	—
141	C₂×C₃×C₄	—	3	1	no	—
142	C₂×C₃×C₄	—	6	4	no	—
143	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
144	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
145	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
148	C₃×C₈	—	9	8	no	—
149	C₃⋊C₈	—	3	8	no	—
150	C₂×C₃²	—	2	2	no	—
151	C₂×C₃²	C₂×#208	1	2	yes	—
152	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
153	C₂×C₉	—	2	4	no	—
154	C₂×C₉	—	2	2	no	—
155	C₂×C₉	—	2	2	no	—
156	C₂×C₉	—	2	1	no	—
157	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
158	C₂×C₉	C₂×C₃×M₁₆	1	2	yes	—
159	C₂²×C₄	3₋¹⁺²⋊C₄	1	2	yes	—
160	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
161	M₁₆	—	9	8	no	—
162	C₂×C₈	—	9	4	no	—
163	C₄²	3₋¹⁺²⋊C₄	1	2	yes	—
164	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
165	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	3	1	no	—
168	C₂²×C₃	—	3	2	no	—
169	C₂²×C₃	—	6	2	no	—
170	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
171	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	3	2	no	—
174	C₃×C₄	—	3	4	no	—
175	C₃×C₄	—	3	2	no	—
176	C₃×C₄	—	6	2	no	—
177	C₃×C₄	C₃×#208	1	4	yes	—
178	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
179	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
180	C₃²	C₄×#208	1	1	yes	characteristic
181	C₉	—	2	1	no	—
182	C₉	C₃×C₄×M₁₆	1	1	yes	characteristic, complemented
183	C₂³	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₄×AGL(1,ℤ/9)	1	4	yes	—
186	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
187	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
188	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
189	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
190	C₈	—	9	8	no	—
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	3	2	no	—
193	C₂×C₃	—	3	1	no	—
194	C₂×C₃	—	6	2	no	—
195	C₂×C₃	C₂×C₃×#208	1	2	yes	—
196	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
197	C₂²	—	2	2	no	—
198	C₂²	C₄×3₋¹⁺²⋊C₄	1	2	yes	—
199	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	#3667⋊C₃	1	4	yes	direct product factor
202	C₄	C₄×3₋¹⁺²⋊C₄	1	2	yes	—
203	C₄	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
204	C₃	—	3	1	no	—
205	C₃	C₃×C₄×#208	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#3667⋊C₃	1	2	yes	—
208	C₂	C₂×C₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂×C₃×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂²×C₃×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×C₄×M₁₆
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. #63362⋊C₃	C₂	semidirect
4. C₄×3₋¹⁺²⋊C₈	C₂	semidirect
6. C₄×#3667	C₃	semidirect
12. #3667⋊C₃	C₄	direct (N × H)
21. #63362	C₂×C₃	semidirect
22. C₄×C₉⋊C₈	C₂×C₃	semidirect
30. 3₋¹⁺²⋊C₈	C₂×C₄	semidirect
50. #3667	C₃×C₄	semidirect
57. C₄×3₋¹⁺²	M₁₆	semidirect
89. C₉⋊C₈	C₂×C₃×C₄	semidirect
127. C₄×C₉	C₃×M₁₆	semidirect
133. 3₋¹⁺²	C₄×M₁₆	semidirect
182. C₉	C₃×C₄×M₁₆	semidirect
201. C₄	#3667⋊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
C₂×C₄²×3₋¹⁺²	2
C₂×#3667⋊C₃	2
#63362⋊C₃	2
C₄×3₋¹⁺²⋊C₈	2
C₃×C₄×#208	3
C₄×#3667	3

Subgroups — C4×#3667⋊C3 #13142429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#3667⋊C₃ #13142429