Subgroups — #13144377

Order 1728.

570 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	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	4	yes	maximal, complemented
5	#10592698	—	3	1	no	maximal
6	#10588677	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
9	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	2	yes	—
11	#88532	—	2	2	no	—
12	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
15	C₃²×#117	D₃	1	1	yes	characteristic
16	C₂×#3748	—	3	1	no	—
17	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
18	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	4	no	—
19	C₉×#117	D₃	1	3	yes	—
20	C₂×#3667	—	3	3	no	—
21	C₂²×C₉⋊C₈	—	3	3	no	—
22	C₄⋊(C₉⋊C₈)	—	3	12	no	—
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₃⋊(C₉⋊C₈)	—	2	2	no	—
30	C₃⋊(C₉⋊C₈)	—	2	2	no	—
31	#4999	—	9	3	no	—
32	#4999	—	9	1	no	—
33	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
34	C₃²×C₄⋊C₄	Dic₃	1	2	yes	—
35	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
36	C₃²×C₄²	C₂×D₃	1	2	yes	—
37	#3748	—	6	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
41	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
42	C₉×C₄⋊C₄	Dic₃	1	6	yes	—
43	C₉×C₂²⋊C₄	Dic₃	1	6	yes	—
44	C₄²×C₉	C₂×D₃	1	6	yes	—
45	#3667	—	6	6	no	—
46	C₂×C₉⋊C₈	—	3	6	no	—
47	C₂×C₉⋊C₈	—	3	6	no	—
48	C₂×C₉⋊C₈	—	6	6	no	—
49	C₂²×C₃×C₉	—	2	2	no	—
50	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
52	C₃×C₄×C₉	—	2	4	no	—
53	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
54	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
55	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₃×#117	D₉	1	3	yes	—
57	C₃×#117	C₃⋊D₃	1	1	yes	characteristic
58	C₂×#208	—	9	3	no	—
59	C₂×#208	—	9	1	no	—
60	C₂²×C₃⋊C₈	—	9	3	no	—
61	C₂²×C₃⋊C₈	—	9	1	no	—
62	C₄⋊(C₃⋊C₈)	—	9	12	no	—
63	C₄⋊(C₃⋊C₈)	—	9	4	no	—
64	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
65	C₂×C₃²×C₄	Dic₆	1	2	yes	—
66	C₂×C₃²×C₄	D₁₂	1	2	yes	—
67	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
68	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂³×C₉	C₂×Dic₃	1	3	yes	—
71	C₃⋊(C₃⋊C₈)	—	6	2	no	—
72	C₃⋊(C₃⋊C₈)	—	6	2	no	—
73	C₂×C₄×C₉	Dic₆	1	6	yes	—
74	C₂×C₄×C₉	D₁₂	1	6	yes	—
75	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
76	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
77	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
78	C₉⋊C₈	—	6	6	no	—
79	C₉⋊C₈	—	6	6	no	—
80	#424	—	27	1	no	Sylow
81	C₂×C₃×C₉	—	2	2	no	—
82	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₉	#131	1	2	yes	—
84	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
85	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
86	C₃×C₄⋊C₄	Dic₉	1	6	yes	—
87	C₃×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
88	C₃×C₂²⋊C₄	Dic₉	1	6	yes	—
89	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	2	yes	—
90	C₃×C₄²	C₂×D₉	1	6	yes	—
91	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
92	#208	—	18	6	no	—
93	#208	—	18	2	no	—
94	C₂×C₃⋊C₈	—	9	6	no	—
95	C₂×C₃⋊C₈	—	9	6	no	—
96	C₂×C₃⋊C₈	—	9	2	no	—
97	C₂×C₃⋊C₈	—	9	2	no	—
98	C₂×C₃⋊C₈	—	18	6	no	—
99	C₂×C₃⋊C₈	—	18	2	no	—
100	C₂²×C₃²	—	2	2	no	—
101	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
102	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
103	C₃²×C₄	—	2	4	no	—
104	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
105	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
106	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
107	C₂²×C₉	—	2	6	no	—
108	C₂²×C₉	C₄⋊Dic₃	1	6	yes	—
109	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
110	C₄×C₉	—	2	12	no	—
111	C₄×C₉	C₄⋊Dic₃	1	6	yes	—
112	C₄×C₉	C₂×Dic₆	1	3	yes	—
113	C₄×C₉	C₂×D₁₂	1	3	yes	—
114	C₂×M₁₆	—	27	1	no	—
115	C₂²×C₈	—	27	1	no	—
116	#117	C₃⋊D₉	1	1	yes	characteristic
117	C₄⋊C₈	—	27	4	no	—
118	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₃	C₂×Dic₉	1	3	yes	—
120	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
121	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
122	C₂×C₃×C₄	D₃₆	1	6	yes	—
123	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
124	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
125	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
126	C₂×C₃×C₄	#634	1	2	yes	—
127	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
128	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
129	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
131	C₃⋊C₈	—	18	6	no	—
132	C₃⋊C₈	—	18	6	no	—
133	C₃⋊C₈	—	18	2	no	—
134	C₃⋊C₈	—	18	2	no	—
135	C₂×C₃²	—	2	2	no	—
136	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
137	C₂×C₃²	#948	1	2	yes	—
138	C₂×C₉	—	2	6	no	—
139	C₂×C₉	C₂×C₄⋊Dic₃	1	3	yes	—
140	C₂×C₉	#948	1	6	yes	—
141	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
142	M₁₆	—	54	2	no	—
143	C₂×C₈	—	27	2	no	—
144	C₂×C₈	—	27	2	no	—
145	C₂×C₈	—	54	2	no	—
146	C₄⋊C₄	C₃⋊Dic₉	1	2	yes	—
147	C₂²⋊C₄	C₃⋊Dic₉	1	2	yes	—
148	C₄²	C₂×C₃⋊D₉	1	2	yes	—
149	C₂²×C₃	—	2	6	no	—
150	C₂²×C₃	—	2	2	no	—
151	C₂²×C₃	C₄⋊Dic₉	1	6	yes	—
152	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
153	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
154	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
155	C₃×C₄	—	2	12	no	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	C₄⋊Dic₉	1	6	yes	—
158	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
159	C₃×C₄	C₂×D₃₆	1	3	yes	—
160	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
161	C₃×C₄	C₂×#634	1	1	yes	characteristic
162	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
163	C₃²	#4999	1	1	yes	characteristic
164	C₉	#4999	1	3	yes	complemented
165	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
166	C₂×C₄	#6226	1	2	yes	—
167	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
168	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
169	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
171	C₈	—	54	2	no	—
172	C₈	—	54	2	no	—
173	C₂×C₃	—	2	6	no	—
174	C₂×C₃	—	2	2	no	—
175	C₂×C₃	C₂×C₄⋊Dic₉	1	3	yes	—
176	C₂×C₃	#63510	1	6	yes	—
177	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
178	C₂×C₃	#64157	1	2	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₄⋊(C₃⋊Dic₉)	1	2	yes	—
181	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	C₄⋊(C₃⋊Dic₉)	1	2	yes	—
184	C₄	C₂×#6226	1	1	yes	characteristic
185	C₄	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
186	C₃	#10588677	1	3	yes	complemented
187	C₃	#10592698	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₂×C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
190	C₂	#11721865	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#117	C₃⋊D₉
hypocenter	C₃×C₉	#424
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
118. C₃×C₉	#424	semidirect
164. C₉	#4999	semidirect
186. C₃	#10588677	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
C₂×#88532	2
C₂²×C₃⋊(C₉⋊C₈)	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592698	3
#10588677	3