Subgroups — #10588616⋊C₃ #13142443

Order 1728.

516 conjugacy classes of subgroups.

The subgroup lattice has 195 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×C₄⋊(3₋¹⁺²⋊C₄)	C₂	1	2	yes	maximal
3	3₋¹⁺²⋊#95	C₂	1	4	yes	maximal
4	C₃×#4936	—	3	1	no	maximal
5	#10588616	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	2	yes	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂²×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
9	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
10	C₄⋊(3₋¹⁺²⋊C₄)	—	2	4	no	—
11	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₃	—	3	2	no	—
13	C₃×C₃⋊#95	—	3	4	no	—
14	C₂×C₄²×C₉	—	2	1	no	—
15	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
16	C₂×C₄⋊Dic₉	C₂×C₃	1	2	yes	—
17	C₉⋊#95	C₂×C₃	1	4	yes	—
18	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	—	2	4	no	—
20	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
22	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
23	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
24	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
25	C₂×3₋¹⁺²⋊C₄	—	2	8	no	—
26	C₃×#382	—	9	1	no	—
27	#4936	—	3	1	no	—
28	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
29	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	—	3	4	no	—
31	C₃²×C₄²	Dic₃	1	4	yes	—
32	C₃×C₄⋊Dic₃	—	6	4	no	—
33	C₂²×C₄×C₉	—	2	2	no	—
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
36	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
38	C₄²×C₉	—	2	4	no	—
39	C₄²×C₉	C₃×C₄	1	4	yes	complemented
40	C₄⋊Dic₉	—	2	4	no	—
41	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
42	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
43	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
44	C₂²×3₋¹⁺²	#41	1	4	yes	—
45	C₄×3₋¹⁺²	—	2	4	no	—
46	C₄×3₋¹⁺²	C₄⋊C₄	1	4	yes	—
47	3₋¹⁺²⋊C₄	—	4	4	no	—
48	C₂×C₃×C₄⋊C₄	—	9	2	no	—
49	C₂×C₃×C₄²	—	3	1	no	—
50	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₃	—	3	2	no	—
52	C₃×#95	—	9	4	no	—
53	C₃⋊#95	—	3	4	no	—
54	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
55	C₂×C₃²×C₄	—	2	4	no	—
56	C₂×C₃²×C₄	Dic₆	1	2	yes	—
57	C₂×C₃²×C₄	D₁₂	1	2	yes	—
58	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
59	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
60	C₂×C₃×Dic₃	—	6	4	no	—
61	C₂×C₃×Dic₃	—	6	8	no	—
62	C₂³×C₉	—	2	1	no	—
63	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
64	C₂×C₄×C₉	—	2	8	no	—
65	C₂×C₄×C₉	—	2	4	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
71	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
72	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
73	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
74	C₂×Dic₉	—	2	4	no	—
75	C₂×Dic₉	—	2	8	no	—
76	#382	—	9	1	no	Sylow
77	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×3₋¹⁺²	#117	1	2	yes	—
79	C₂×3₋¹⁺²	#124	1	2	yes	—
80	C₂×3₋¹⁺²	#125	1	2	yes	—
81	C₂²×C₃×C₄	—	3	1	no	—
82	C₂²×C₃×C₄	—	3	2	no	—
83	C₂²×C₃×C₄	—	9	4	no	—
84	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
85	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
86	C₂²×Dic₃	—	3	4	no	—
87	C₃×C₄⋊C₄	—	18	4	no	—
88	C₃×C₄²	—	3	4	no	—
89	C₃×C₄²	C₃×Dic₃	1	4	yes	—
90	C₄⋊Dic₃	—	6	4	no	—
91	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
92	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
93	C₂²×C₃²	#235	1	4	yes	—
94	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
95	C₃²×C₄	—	2	4	no	—
96	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
97	C₃×Dic₃	—	12	4	no	—
98	C₂²×C₉	—	2	1	no	—
99	C₂²×C₉	—	2	1	no	—
100	C₂²×C₉	—	2	4	no	—
101	C₂²×C₉	—	2	1	no	—
102	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
103	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
104	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
105	C₂²×C₉	C₃×#41	1	4	yes	—
106	C₄×C₉	—	2	8	no	—
107	C₄×C₉	—	2	4	no	—
108	C₄×C₉	—	2	4	no	—
109	C₄×C₉	C₃×C₄⋊C₄	1	4	yes	—
110	Dic₉	—	4	4	no	—
111	C₂×C₄⋊C₄	—	9	2	no	—
112	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
113	#95	—	9	4	no	—
114	3₋¹⁺²	#382	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₃	—	3	1	no	—
116	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	—	3	2	no	—
119	C₂×C₃×C₄	—	3	2	no	—
120	C₂×C₃×C₄	—	3	2	no	—
121	C₂×C₃×C₄	—	3	4	no	—
122	C₂×C₃×C₄	—	6	4	no	—
123	C₂×C₃×C₄	—	18	4	no	—
124	C₂×C₃×C₄	—	18	8	no	—
125	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
126	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
127	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
128	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
129	C₂×Dic₃	—	6	4	no	—
130	C₂×Dic₃	—	6	8	no	—
131	C₂×C₃²	#870	1	2	yes	—
132	C₂×C₃²	#875	1	2	yes	—
133	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
134	C₂×C₃²	#926	1	2	yes	—
135	C₂×C₉	—	2	1	no	—
136	C₂×C₉	—	2	2	no	—
137	C₂×C₉	—	2	2	no	—
138	C₂×C₉	—	2	2	no	—
139	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₉	C₃×#117	1	2	yes	—
141	C₂×C₉	C₃×#124	1	2	yes	—
142	C₂×C₉	C₃×#125	1	2	yes	—
143	C₂²×C₄	—	9	4	no	—
144	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
145	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
146	C₄⋊C₄	—	18	4	no	—
147	C₄²	3₋¹⁺²⋊C₄	1	4	yes	complemented
148	C₂²×C₃	—	3	1	no	—
149	C₂²×C₃	—	3	1	no	—
150	C₂²×C₃	—	3	4	no	—
151	C₂²×C₃	—	3	1	no	—
152	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
153	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
154	C₂²×C₃	C₃×#235	1	4	yes	—
155	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	3	4	no	—
158	C₃×C₄	—	6	4	no	—
159	C₃×C₄	—	36	4	no	—
160	C₃×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
161	Dic₃	—	12	4	no	—
162	C₃²	#4936	1	1	yes	characteristic
163	C₉	—	2	1	no	—
164	C₉	C₃×#382	1	1	yes	characteristic, complemented
165	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	18	4	no	—
168	C₂×C₄	—	18	8	no	—
169	C₂×C₄	3₋¹⁺²⋊Q₈	1	2	yes	—
170	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
171	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	4	yes	—
172	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
173	C₂×C₃	—	3	1	no	—
174	C₂×C₃	—	3	2	no	—
175	C₂×C₃	—	3	2	no	—
176	C₂×C₃	—	3	2	no	—
177	C₂×C₃	C₃×#870	1	2	yes	—
178	C₂×C₃	C₃×#875	1	2	yes	—
179	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
180	C₂×C₃	C₃×#926	1	2	yes	—
181	C₂²	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
182	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
183	C₂²	#3697⋊C₃	1	4	yes	—
184	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	36	4	no	—
187	C₄	C₄⋊(3₋¹⁺²⋊C₄)	1	4	yes	—
188	C₃	—	3	1	no	—
189	C₃	C₃×#4936	1	1	yes	characteristic
190	C₂	#63432⋊C₃	1	2	yes	—
191	C₂	#63437⋊C₃	1	2	yes	—
192	C₂	C₂×C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic
193	C₂	#63488⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#382
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. #10588616	C₃	semidirect
9. C₄²×3₋¹⁺²	C₄	semidirect
39. C₄²×C₉	C₃×C₄	semidirect
114. 3₋¹⁺²	#382	semidirect
147. C₄²	3₋¹⁺²⋊C₄	semidirect
164. C₉	C₃×#382	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×C₄⋊(3₋¹⁺²⋊C₄)	2
3₋¹⁺²⋊#95	2
C₃×#4936	3
#10588616	3

Subgroups — #10588616⋊C3 #13142443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588616⋊C₃ #13142443