Subgroups — #13144801

Order 1728.

1090 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	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88551	C₂	1	2	yes	maximal
3	#11720572	C₂	1	4	yes	maximal, complemented
4	#10592910	—	3	1	no	maximal
5	#10588891	—	3	3	no	maximal
6	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃⋊Dic₉	C₂²	1	4	yes	—
11	#88551	—	2	4	no	—
12	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
13	C₂×#3757	—	3	2	no	—
14	C₂³×C₄×C₉	D₃	1	3	yes	—
15	#63658	—	3	4	no	—
16	C₂×#3676	—	3	6	no	—
17	C₉⋊#95	—	3	12	no	—
18	C₂³×C₃×C₉	—	2	4	no	—
19	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
20	C₂³×C₃×C₉	D₄	1	4	yes	—
21	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	—	2	8	no	—
23	C₂×C₃×C₄×C₉	—	2	4	no	—
24	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
25	C₂×C₃⋊Dic₉	—	2	4	no	—
26	C₂×C₃⋊Dic₉	—	2	8	no	—
27	#5211	—	9	3	no	—
28	#5211	—	9	1	no	—
29	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
30	C₂²×C₃²×C₄	—	2	4	no	—
31	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
32	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
33	C₂²×C₃⋊Dic₃	—	3	4	no	—
34	C₂⁴×C₉	C₂×D₃	1	3	yes	—
35	#3757	—	6	4	no	—
36	C₂²×C₄×C₉	—	2	12	no	—
37	C₂²×C₄×C₉	Dic₃	1	12	yes	complemented
38	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
39	C₂²×Dic₉	—	3	12	no	—
40	#3676	—	6	12	no	—
41	C₂²×C₃×C₉	—	2	4	no	—
42	C₂²×C₃×C₉	—	2	8	no	—
43	C₂²×C₃×C₉	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
46	C₂²×C₃×C₉	#41	1	4	yes	—
47	C₃×C₄×C₉	—	2	4	no	—
48	C₃⋊Dic₉	—	4	4	no	—
49	C₂³×C₃×C₄	D₉	1	3	yes	—
50	C₂³×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
51	C₂×#217	—	9	6	no	—
52	C₂×#217	—	9	2	no	—
53	C₃⋊#95	—	9	12	no	—
54	C₃⋊#95	—	9	4	no	—
55	C₂³×C₃²	—	2	4	no	—
56	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
57	C₂³×C₃²	#67	1	4	yes	—
58	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	8	no	—
60	C₂×C₃²×C₄	—	2	4	no	—
61	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
62	C₂×C₃⋊Dic₃	—	6	8	no	—
63	C₂×C₃⋊Dic₃	—	6	4	no	—
64	C₂³×C₉	—	2	12	no	—
65	C₂³×C₉	C₂×Dic₃	1	6	yes	—
66	C₂³×C₉	#67	1	12	yes	—
67	C₂³×C₉	C₂²×D₃	1	3	yes	—
68	C₂×C₄×C₉	—	2	24	no	—
69	C₂×C₄×C₉	—	2	12	no	—
70	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
71	C₂×Dic₉	—	6	24	no	—
72	C₂×Dic₉	—	6	12	no	—
73	#381	—	27	1	no	Sylow
74	C₂×C₃×C₉	—	2	4	no	—
75	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₉	#117	1	2	yes	—
77	C₂×C₃×C₉	#123	1	4	yes	—
78	C₂⁴×C₃	C₂×D₉	1	3	yes	—
79	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	2	12	no	—
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	Dic₉	1	12	yes	complemented
83	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
84	C₂²×C₃×C₄	C₃⋊Dic₃	1	4	yes	—
85	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
86	C₂²×Dic₃	—	9	12	no	—
87	C₂²×Dic₃	—	9	4	no	—
88	#217	—	18	12	no	—
89	#217	—	18	4	no	—
90	C₂²×C₃²	—	2	4	no	—
91	C₂²×C₃²	—	2	8	no	—
92	C₂²×C₃²	#217	1	4	yes	—
93	C₂²×C₃²	#235	1	4	yes	—
94	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
95	C₂²×C₃²	C₂×#67	1	2	yes	—
96	C₃²×C₄	—	2	4	no	—
97	C₃⋊Dic₃	—	12	4	no	—
98	C₂²×C₉	—	2	12	no	—
99	C₂²×C₉	—	2	24	no	—
100	C₂²×C₉	#217	1	12	yes	—
101	C₂²×C₉	#235	1	12	yes	—
102	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
103	C₂²×C₉	C₂×#67	1	6	yes	—
104	C₄×C₉	—	2	12	no	—
105	Dic₉	—	12	12	no	—
106	C₂³×C₄	C₃⋊D₉	1	1	yes	characteristic
107	C₂×C₂²⋊C₄	—	27	2	no	—
108	#95	—	27	4	no	—
109	C₃×C₉	#381	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₃	—	2	12	no	—
111	C₂³×C₃	—	2	4	no	—
112	C₂³×C₃	C₂×Dic₉	1	6	yes	—
113	C₂³×C₃	#611	1	12	yes	—
114	C₂³×C₃	C₂²×D₉	1	3	yes	—
115	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
116	C₂³×C₃	#638	1	4	yes	—
117	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	24	no	—
119	C₂×C₃×C₄	—	2	12	no	—
120	C₂×C₃×C₄	—	2	8	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
123	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
124	C₂×Dic₃	—	18	24	no	—
125	C₂×Dic₃	—	18	12	no	—
126	C₂×Dic₃	—	18	8	no	—
127	C₂×Dic₃	—	18	4	no	—
128	C₂×C₃²	—	2	4	no	—
129	C₂×C₃²	#926	1	2	yes	—
130	C₂×C₃²	#929	1	4	yes	—
131	C₂×C₃²	C₂×#217	1	1	yes	characteristic
132	C₂×C₉	—	2	12	no	—
133	C₂×C₉	#926	1	6	yes	—
134	C₂×C₉	#929	1	12	yes	—
135	C₂×C₉	C₂×#217	1	3	yes	—
136	C₂⁴	C₂×C₃⋊D₉	1	1	yes	characteristic
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	27	4	no	—
139	C₂²×C₄	C₃⋊Dic₉	1	4	yes	complemented
140	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
141	C₂²⋊C₄	—	54	4	no	—
142	C₂²×C₃	—	2	12	no	—
143	C₂²×C₃	—	2	24	no	—
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	2	8	no	—
146	C₂²×C₃	#3676	1	12	yes	—
147	C₂²×C₃	#3697	1	12	yes	—
148	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
149	C₂²×C₃	C₂×#611	1	6	yes	—
150	C₂²×C₃	#3757	1	4	yes	—
151	C₂²×C₃	#3828	1	4	yes	—
152	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
153	C₂²×C₃	C₂×#638	1	2	yes	—
154	C₃×C₄	—	2	12	no	—
155	C₃×C₄	—	2	4	no	—
156	Dic₃	—	36	12	no	—
157	Dic₃	—	36	4	no	—
158	C₃²	#5211	1	1	yes	characteristic
159	C₉	#5211	1	3	yes	complemented
160	C₂³	—	2	4	no	—
161	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
162	C₂³	#6233	1	4	yes	—
163	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	54	8	no	—
167	C₂×C₄	—	54	4	no	—
168	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
169	C₂×C₃	—	2	4	no	—
170	C₂×C₃	—	2	12	no	—
171	C₂×C₃	#63488	1	6	yes	—
172	C₂×C₃	#63491	1	12	yes	—
173	C₂×C₃	C₂×#3676	1	3	yes	—
174	C₂×C₃	#64135	1	2	yes	—
175	C₂×C₃	#64138	1	4	yes	—
176	C₂×C₃	C₂×#3757	1	1	yes	characteristic
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	#88551	1	4	yes	—
180	C₂²	#88736	1	4	yes	—
181	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
182	C₂²	C₂×#6233	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	108	4	no	—
185	C₃	#10588891	1	3	yes	complemented
186	C₃	#10592910	1	1	yes	characteristic
187	C₂	—	2	4	no	—
188	C₂	#11721815	1	2	yes	—
189	C₂	#11721818	1	4	yes	—
190	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂³×C₄	C₃⋊D₉
hypocenter	C₃×C₉	#381
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. #11720572	C₂	semidirect
8. C₂²×C₃×C₄×C₉	C₄	semidirect
37. C₂²×C₄×C₉	Dic₃	semidirect
82. C₂²×C₃×C₄	Dic₉	semidirect
109. C₃×C₉	#381	semidirect
139. C₂²×C₄	C₃⋊Dic₉	semidirect
159. C₉	#5211	semidirect
185. C₃	#10588891	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×#88551	2
#11720572	2
#10592910	3
#10588891	3