Subgroups — #13144240

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	2	yes	maximal
3	#11720572	C₂	1	4	yes	maximal
4	#10592635	—	3	1	no	maximal
5	#10588616	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₂²	1	4	yes	—
9	C₃×C₄²×C₉	C₄	1	4	yes	complemented
10	C₄⋊(C₃⋊Dic₉)	—	2	4	no	—
11	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
12	C₂×C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
13	#63658	—	3	4	no	—
14	C₂×C₄²×C₉	D₃	1	3	yes	—
15	C₂×C₄⋊Dic₉	—	3	6	no	—
16	C₉⋊#95	—	3	12	no	—
17	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄×C₉	—	2	4	no	—
19	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
20	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
22	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
23	C₂×C₃⋊Dic₉	—	2	4	no	—
24	C₂×C₃⋊Dic₉	—	2	8	no	—
25	#4936	—	9	3	no	—
26	#4936	—	9	1	no	—
27	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
28	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃⋊Dic₃	—	3	4	no	—
30	C₃²×C₄²	Dic₃	1	4	yes	—
31	C₄⋊(C₃⋊Dic₃)	—	6	4	no	—
32	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
33	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
34	C₂²×Dic₉	—	3	12	no	—
35	C₄²×C₉	Dic₃	1	12	yes	complemented
36	C₄⋊Dic₉	—	6	12	no	—
37	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₃×C₉	#41	1	4	yes	—
41	C₃×C₄×C₉	—	2	4	no	—
42	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	—
43	C₃⋊Dic₉	—	4	4	no	—
44	C₂×C₃×C₄²	D₉	1	3	yes	—
45	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
46	C₂×C₄⋊Dic₃	—	9	6	no	—
47	C₂×C₄⋊Dic₃	—	9	2	no	—
48	C₃⋊#95	—	9	12	no	—
49	C₃⋊#95	—	9	4	no	—
50	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
51	C₂×C₃²×C₄	—	2	4	no	—
52	C₂×C₃²×C₄	Dic₆	1	2	yes	—
53	C₂×C₃²×C₄	D₁₂	1	2	yes	—
54	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
55	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
56	C₂×C₃⋊Dic₃	—	6	4	no	—
57	C₂×C₃⋊Dic₃	—	6	8	no	—
58	C₂³×C₉	C₂²×D₃	1	3	yes	—
59	C₂×C₄×C₉	—	2	12	no	—
60	C₂×C₄×C₉	Dic₆	1	6	yes	—
61	C₂×C₄×C₉	D₁₂	1	6	yes	—
62	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
63	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
64	C₂×Dic₉	—	6	12	no	—
65	C₂×Dic₉	—	6	24	no	—
66	#382	—	27	1	no	Sylow
67	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃×C₉	#117	1	2	yes	—
69	C₂×C₃×C₉	#124	1	2	yes	—
70	C₂×C₃×C₉	#125	1	2	yes	—
71	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
72	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
73	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
74	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
75	C₂²×Dic₃	—	9	12	no	—
76	C₂²×Dic₃	—	9	4	no	—
77	C₃×C₄²	Dic₉	1	12	yes	complemented
78	C₃×C₄²	C₃⋊Dic₃	1	4	yes	—
79	C₄⋊Dic₃	—	18	12	no	—
80	C₄⋊Dic₃	—	18	4	no	—
81	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
82	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
83	C₂²×C₃²	#235	1	4	yes	—
84	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
85	C₃²×C₄	—	2	4	no	—
86	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
87	C₃⋊Dic₃	—	12	4	no	—
88	C₂²×C₉	C₂×Dic₆	1	3	yes	—
89	C₂²×C₉	C₂×D₁₂	1	3	yes	—
90	C₂²×C₉	#235	1	12	yes	—
91	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
92	C₄×C₉	—	2	12	no	—
93	C₄×C₉	C₄⋊Dic₃	1	12	yes	—
94	Dic₉	—	12	12	no	—
95	C₂×C₄⋊C₄	—	27	2	no	—
96	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
97	#95	—	27	4	no	—
98	C₃×C₉	#382	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₃	C₂²×D₉	1	3	yes	—
100	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
101	C₂×C₃×C₄	—	2	12	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
104	C₂×C₃×C₄	D₃₆	1	6	yes	—
105	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
106	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
107	C₂×C₃×C₄	#634	1	2	yes	—
108	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
109	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
110	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
111	C₂×Dic₃	—	18	12	no	—
112	C₂×Dic₃	—	18	24	no	—
113	C₂×Dic₃	—	18	4	no	—
114	C₂×Dic₃	—	18	8	no	—
115	C₂×C₃²	#870	1	2	yes	—
116	C₂×C₃²	#875	1	2	yes	—
117	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
118	C₂×C₃²	#926	1	2	yes	—
119	C₂×C₉	#870	1	6	yes	—
120	C₂×C₉	#875	1	6	yes	—
121	C₂×C₉	C₂×C₄⋊Dic₃	1	3	yes	—
122	C₂×C₉	#926	1	6	yes	—
123	C₂²×C₄	—	27	4	no	—
124	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
125	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
126	C₄⋊C₄	—	54	4	no	—
127	C₄²	C₃⋊Dic₉	1	4	yes	complemented
128	C₂²×C₃	C₂×Dic₁₈	1	3	yes	—
129	C₂²×C₃	C₂×D₃₆	1	3	yes	—
130	C₂²×C₃	#3697	1	12	yes	—
131	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
132	C₂²×C₃	C₂×#634	1	1	yes	characteristic
133	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
134	C₂²×C₃	#3828	1	4	yes	—
135	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
136	C₃×C₄	—	2	12	no	—
137	C₃×C₄	—	2	4	no	—
138	C₃×C₄	C₄⋊Dic₉	1	12	yes	—
139	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
140	Dic₃	—	36	4	no	—
141	Dic₃	—	36	12	no	—
142	C₃²	#4936	1	1	yes	characteristic
143	C₉	#4936	1	3	yes	complemented
144	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	54	4	no	—
147	C₂×C₄	—	54	8	no	—
148	C₂×C₄	#6226	1	2	yes	—
149	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
150	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
151	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
152	C₂×C₃	#63432	1	6	yes	—
153	C₂×C₃	#63437	1	6	yes	—
154	C₂×C₃	C₂×C₄⋊Dic₉	1	3	yes	—
155	C₂×C₃	#63488	1	6	yes	—
156	C₂×C₃	#64079	1	2	yes	—
157	C₂×C₃	#64084	1	2	yes	—
158	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
159	C₂×C₃	#64135	1	2	yes	—
160	C₂²	C₂×#6226	1	1	yes	characteristic
161	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
162	C₂²	#88736	1	4	yes	—
163	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
164	C₄	—	2	4	no	—
165	C₄	—	108	4	no	—
166	C₄	C₄⋊(C₃⋊Dic₉)	1	4	yes	—
167	C₃	#10588616	1	3	yes	complemented
168	C₃	#10592635	1	1	yes	characteristic
169	C₂	#11721707	1	2	yes	—
170	C₂	#11721712	1	2	yes	—
171	C₂	C₂×C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
172	C₂	#11721815	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂×C₄²	C₃⋊D₉
hypocenter	C₃×C₉	#382
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
9. C₃×C₄²×C₉	C₄	semidirect
35. C₄²×C₉	Dic₃	semidirect
77. C₃×C₄²	Dic₉	semidirect
98. C₃×C₉	#382	semidirect
127. C₄²	C₃⋊Dic₉	semidirect
143. C₉	#4936	semidirect
167. C₃	#10588616	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₄⋊(C₃⋊Dic₉)	2
#11720572	2
#10592635	3
#10588616	3