Subgroups — #13144228

Order 1728.

630 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	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	4	yes	maximal, complemented
4	#10592623	—	3	1	no	maximal
5	#10588604	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
9	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
10	C₃×C₄²×C₉	C₂²	1	2	yes	—
11	#88532	—	2	4	no	—
12	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
14	C₂×#3748	—	3	2	no	—
15	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	4	no	—
16	C₂×C₄²×C₉	D₃	1	3	yes	—
17	C₂×#3667	—	3	6	no	—
18	C₄⋊(C₉⋊C₈)	—	3	12	no	—
19	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	—	2	4	no	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
26	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₃⋊(C₉⋊C₈)	—	2	4	no	—
28	#4924	—	9	3	no	—
29	#4924	—	9	1	no	—
30	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
31	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
32	C₃²×C₄²	Dic₃	1	1	yes	characteristic
33	C₃²×C₄²	Dic₃	1	1	yes	characteristic
34	C₃²×C₄²	C₂×D₃	1	2	yes	—
35	#3748	—	6	4	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
37	C₂²×C₄×C₉	Dic₃	1	6	yes	—
38	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
39	C₄²×C₉	Dic₃	1	3	yes	—
40	C₄²×C₉	Dic₃	1	3	yes	—
41	C₄²×C₉	C₂×D₃	1	6	yes	—
42	#3667	—	6	12	no	—
43	C₂×C₉⋊C₈	—	3	12	no	—
44	C₂²×C₃×C₉	—	2	2	no	—
45	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
46	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₄×C₉	—	2	2	no	—
48	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
49	C₃×C₄×C₉	M₁₆	1	4	yes	complemented
50	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
51	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₃×C₄²	D₉	1	3	yes	—
53	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
54	C₂×#208	—	9	6	no	—
55	C₂×#208	—	9	2	no	—
56	C₄⋊(C₃⋊C₈)	—	9	12	no	—
57	C₄⋊(C₃⋊C₈)	—	9	4	no	—
58	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	Dic₆	1	2	yes	—
61	C₂×C₃²×C₄	D₁₂	1	2	yes	—
62	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
63	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
64	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
66	C₂³×C₉	C₂×Dic₃	1	3	yes	—
67	C₃⋊(C₃⋊C₈)	—	6	4	no	—
68	C₂×C₄×C₉	—	2	12	no	—
69	C₂×C₄×C₉	Dic₆	1	6	yes	—
70	C₂×C₄×C₉	D₁₂	1	6	yes	—
71	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
72	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
73	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
74	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
75	C₉⋊C₈	—	6	12	no	—
76	#423	—	27	1	no	Sylow
77	C₂×C₃×C₉	—	2	2	no	—
78	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
80	C₂²×C₃×C₄	Dic₉	1	6	yes	—
81	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
82	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
83	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
84	C₃×C₄²	Dic₉	1	3	yes	—
85	C₃×C₄²	Dic₉	1	3	yes	—
86	C₃×C₄²	C₂×D₉	1	6	yes	—
87	C₃×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
88	C₃×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
89	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
90	#208	—	18	12	no	—
91	#208	—	18	4	no	—
92	C₂×C₃⋊C₈	—	9	12	no	—
93	C₂×C₃⋊C₈	—	9	4	no	—
94	C₂²×C₃²	—	2	2	no	—
95	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
96	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	#208	1	4	yes	—
99	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
100	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
101	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
102	C₂²×C₉	—	2	6	no	—
103	C₂²×C₉	C₄⋊Dic₃	1	6	yes	—
104	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
105	C₄×C₉	—	2	6	no	—
106	C₄×C₉	#208	1	12	yes	complemented
107	C₄×C₉	C₄⋊Dic₃	1	6	yes	—
108	C₄×C₉	C₂×Dic₆	1	3	yes	—
109	C₄×C₉	C₂×D₁₂	1	3	yes	—
110	C₂×M₁₆	—	27	2	no	—
111	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
112	C₄⋊C₈	—	27	4	no	—
113	C₃×C₉	#423	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₃	C₂×Dic₉	1	3	yes	—
115	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	12	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
119	C₂×C₃×C₄	D₃₆	1	6	yes	—
120	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
121	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
122	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
123	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
124	C₂×C₃×C₄	#634	1	2	yes	—
125	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
126	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
127	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
128	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
130	C₃⋊C₈	—	18	12	no	—
131	C₃⋊C₈	—	18	4	no	—
132	C₂×C₃²	—	2	2	no	—
133	C₂×C₃²	C₂×#208	1	2	yes	—
134	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
135	C₂×C₉	—	2	6	no	—
136	C₂×C₉	C₂×#208	1	6	yes	—
137	C₂×C₉	C₂×C₄⋊Dic₃	1	3	yes	—
138	C₂²×C₄	C₃⋊Dic₉	1	2	yes	—
139	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
140	M₁₆	—	54	4	no	—
141	C₂×C₈	—	27	4	no	—
142	C₄²	C₃⋊Dic₉	1	1	yes	characteristic
143	C₄²	C₃⋊Dic₉	1	1	yes	characteristic
144	C₄²	C₂×C₃⋊D₉	1	2	yes	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	—	2	6	no	—
147	C₂²×C₃	C₄⋊Dic₉	1	6	yes	—
148	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
149	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
150	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	—	2	6	no	—
153	C₃×C₄	#3667	1	12	yes	complemented
154	C₃×C₄	C₄⋊Dic₉	1	6	yes	—
155	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
156	C₃×C₄	C₂×D₃₆	1	3	yes	—
157	C₃×C₄	#3748	1	4	yes	—
158	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
159	C₃×C₄	C₂×#634	1	1	yes	characteristic
160	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
161	C₃²	#4924	1	1	yes	characteristic
162	C₉	#4924	1	3	yes	complemented
163	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	#6226	1	2	yes	—
166	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
167	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
168	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
169	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
171	C₈	—	54	4	no	—
172	C₂×C₃	—	2	2	no	—
173	C₂×C₃	—	2	6	no	—
174	C₂×C₃	C₂×#3667	1	6	yes	—
175	C₂×C₃	C₂×C₄⋊Dic₉	1	3	yes	—
176	C₂×C₃	C₂×#3748	1	2	yes	—
177	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	C₄⋊(C₃⋊Dic₉)	1	2	yes	—
180	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	#88532	1	4	yes	complemented
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	#10588604	1	3	yes	complemented
187	C₃	#10592623	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₂×#88532	1	2	yes	—
190	C₂	C₂×C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₄²	C₃⋊D₉
hypocenter	C₃×C₉	#423
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×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⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
49. C₃×C₄×C₉	M₁₆	semidirect
106. C₄×C₉	#208	semidirect
113. C₃×C₉	#423	semidirect
153. C₃×C₄	#3667	semidirect
162. C₉	#4924	semidirect
182. C₄	#88532	semidirect
186. C₃	#10588604	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×#88532	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592623	3
#10588604	3