Subgroups — #10588604⋊C₃ #13142431

Order 1728.

432 conjugacy classes of subgroups.

The subgroup lattice has 218 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	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₃×#4924	—	3	1	no	maximal
5	#10588604	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₄	1	2	yes	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₄²×3₋¹⁺²	C₄	1	1	yes	characteristic
9	C₄²×3₋¹⁺²	C₄	1	1	yes	characteristic
10	C₄²×3₋¹⁺²	C₂²	1	2	yes	—
11	#3667⋊C₃	—	2	4	no	—
12	C₂×3₋¹⁺²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
14	C₂×C₃×#208	—	3	2	no	—
15	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
16	C₂×C₄²×C₉	—	2	1	no	—
17	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
18	C₂×#3667	C₂×C₃	1	2	yes	—
19	C₄⋊(C₉⋊C₈)	C₂×C₃	1	4	yes	complemented
20	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×3₋¹⁺²	—	2	4	no	—
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
23	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
25	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
26	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
27	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
28	3₋¹⁺²⋊C₈	—	2	4	no	—
29	C₃×#423	—	9	1	no	—
30	#4924	—	3	1	no	—
31	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
32	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
33	C₃²×C₄²	Dic₃	1	1	yes	characteristic
34	C₃²×C₄²	Dic₃	1	1	yes	characteristic
35	C₃²×C₄²	C₂×D₃	1	2	yes	—
36	C₃×#208	—	6	4	no	—
37	C₂×C₃×C₃⋊C₈	—	3	4	no	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	C₃×C₄	1	2	yes	—
41	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
42	C₄²×C₉	—	2	1	no	—
43	C₄²×C₉	—	2	1	no	—
44	C₄²×C₉	—	2	2	no	—
45	C₄²×C₉	C₃×C₄	1	1	yes	characteristic
46	C₄²×C₉	C₃×C₄	1	1	yes	characteristic
47	C₄²×C₉	C₂²×C₃	1	2	yes	—
48	#3667	—	2	4	no	—
49	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	—
50	C₂²×3₋¹⁺²	—	2	2	no	—
51	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
52	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
53	C₄×3₋¹⁺²	—	2	2	no	—
54	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
55	C₄×3₋¹⁺²	M₁₆	1	4	yes	complemented
56	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
57	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₃×M₁₆	—	9	2	no	—
59	C₂×C₃×C₄²	—	3	1	no	—
60	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
61	C₂×#208	—	3	2	no	—
62	C₃×C₄⋊C₈	—	9	4	no	—
63	C₄⋊(C₃⋊C₈)	—	3	4	no	—
64	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
65	C₂×C₃²×C₄	—	2	4	no	—
66	C₂×C₃²×C₄	Dic₆	1	2	yes	—
67	C₂×C₃²×C₄	D₁₂	1	2	yes	—
68	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
69	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
70	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂³×C₉	—	2	1	no	—
73	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
74	C₃×C₃⋊C₈	—	6	4	no	—
75	C₂×C₄×C₉	—	2	8	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	1	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
84	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
86	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
87	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
88	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
89	C₉⋊C₈	—	2	4	no	—
90	#423	—	9	1	no	Sylow
91	C₂×3₋¹⁺²	—	2	2	no	—
92	C₂×3₋¹⁺²	C₂×C₄⋊C₄	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	—	18	4	no	—
99	C₂×C₃×C₈	—	9	4	no	—
100	C₃×C₄²	—	3	1	no	—
101	C₃×C₄²	—	3	1	no	—
102	C₃×C₄²	—	3	2	no	—
103	C₃×C₄²	C₃×Dic₃	1	1	yes	characteristic
104	C₃×C₄²	C₃×Dic₃	1	1	yes	characteristic
105	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
106	#208	—	6	4	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×Dic₆	1	1	yes	characteristic
115	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
116	C₂²×C₉	—	2	4	no	—
117	C₂²×C₉	—	2	2	no	—
118	C₂²×C₉	—	2	2	no	—
119	C₂²×C₉	—	2	1	no	—
120	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
121	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₄×C₉	—	2	4	no	—
123	C₄×C₉	—	2	2	no	—
124	C₄×C₉	—	2	4	no	—
125	C₄×C₉	—	2	2	no	—
126	C₄×C₉	—	2	1	no	—
127	C₄×C₉	—	2	1	no	—
128	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
129	C₄×C₉	C₃×M₁₆	1	4	yes	complemented
130	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
131	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
132	C₂×M₁₆	—	9	2	no	—
133	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
134	C₄⋊C₈	—	9	4	no	—
135	3₋¹⁺²	#423	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₃	—	3	1	no	—
137	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	—	3	2	no	—
140	C₂×C₃×C₄	—	3	2	no	—
141	C₂×C₃×C₄	—	3	2	no	—
142	C₂×C₃×C₄	—	3	1	no	—
143	C₂×C₃×C₄	—	3	2	no	—
144	C₂×C₃×C₄	—	3	1	no	—
145	C₂×C₃×C₄	—	6	4	no	—
146	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
147	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
148	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
150	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
152	C₃×C₈	—	18	4	no	—
153	C₃⋊C₈	—	6	4	no	—
154	C₂×C₃²	—	2	2	no	—
155	C₂×C₃²	C₂×#208	1	2	yes	—
156	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
157	C₂×C₉	—	2	4	no	—
158	C₂×C₉	—	2	2	no	—
159	C₂×C₉	—	2	2	no	—
160	C₂×C₉	—	2	1	no	—
161	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₉	C₂×C₃×M₁₆	1	2	yes	—
163	C₂²×C₄	3₋¹⁺²⋊C₄	1	2	yes	—
164	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
165	M₁₆	—	18	4	no	—
166	C₂×C₈	—	9	4	no	—
167	C₄²	3₋¹⁺²⋊C₄	1	1	yes	characteristic
168	C₄²	3₋¹⁺²⋊C₄	1	1	yes	characteristic
169	C₄²	C₂×AGL(1,ℤ/9)	1	2	yes	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	3	1	no	—
172	C₂²×C₃	—	3	2	no	—
173	C₂²×C₃	—	6	2	no	—
174	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
175	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	3	1	no	—
178	C₃×C₄	—	3	1	no	—
179	C₃×C₄	—	3	4	no	—
180	C₃×C₄	—	3	2	no	—
181	C₃×C₄	—	6	2	no	—
182	C₃×C₄	C₃×#208	1	4	yes	—
183	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
184	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
185	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
186	C₃²	#4924	1	1	yes	characteristic
187	C₉	—	2	1	no	—
188	C₉	C₃×#423	1	1	yes	characteristic, complemented
189	C₂³	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	3₋¹⁺²⋊Q₈	1	2	yes	—
192	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
193	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
194	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
195	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
197	C₈	—	18	4	no	—
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	—	3	2	no	—
200	C₂×C₃	—	3	1	no	—
201	C₂×C₃	—	6	2	no	—
202	C₂×C₃	C₂×C₃×#208	1	2	yes	—
203	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
206	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	#3667⋊C₃	1	4	yes	complemented
209	C₄	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
210	C₄	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
211	C₄	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
212	C₃	—	3	1	no	—
213	C₃	C₃×#4924	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₂×#3667⋊C₃	1	2	yes	—
216	C₂	C₂×C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂²×C₃×C₄
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×#423
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(3₋¹⁺²⋊C₈)	C₂	semidirect
5. #10588604	C₃	semidirect
19. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
55. C₄×3₋¹⁺²	M₁₆	semidirect
129. C₄×C₉	C₃×M₁₆	semidirect
135. 3₋¹⁺²	#423	semidirect
188. C₉	C₃×#423	semidirect
208. C₄	#3667⋊C₃	semidirect

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
C₄⋊(3₋¹⁺²⋊C₈)	2
C₃×#4924	3
#10588604	3

Subgroups — #10588604⋊C3 #13142431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588604⋊C₃ #13142431