Subgroups — #13144229

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#11720571	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	2	yes	maximal, complemented
4	C₄×C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#10592624	—	3	1	no	maximal
6	#10588605	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	4	yes	—
15	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
16	#63657	—	3	2	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	2	no	—
18	C₄×C₃⋊(C₃⋊C₈)	—	3	2	no	—
19	C₂×C₄²×C₉	D₃	1	3	yes	—
20	#63389	—	3	6	no	—
21	C₄⋊(C₉⋊C₈)	—	3	6	no	—
22	C₄×C₉⋊C₈	—	3	6	no	—
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	—	2	2	no	—
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	C₈	1	4	yes	complemented
27	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₃⋊(C₉⋊C₈)	—	2	4	no	—
34	#4925	—	9	3	no	—
35	#4925	—	9	1	no	—
36	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
37	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₃²×C₄²	Dic₃	1	1	yes	characteristic
40	C₃²×C₄²	Dic₃	1	1	yes	characteristic
41	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
44	C₂²×C₄×C₉	Dic₃	1	3	yes	—
45	C₂²×C₄×C₉	Dic₃	1	3	yes	—
46	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
47	C₄²×C₉	Dic₃	1	3	yes	—
48	C₄²×C₉	Dic₃	1	3	yes	—
49	C₄²×C₉	C₂×D₃	1	3	yes	—
50	C₄²×C₉	C₂×D₃	1	3	yes	—
51	C₂×C₉⋊C₈	—	3	12	no	—
52	C₂²×C₃×C₉	—	2	1	no	—
53	C₂²×C₃×C₉	—	2	1	no	—
54	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
55	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×C₉	—	2	1	no	—
57	C₃×C₄×C₉	—	2	1	no	—
58	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
59	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
60	C₃×C₄×C₉	M₁₆	1	2	yes	—
61	C₃×C₄×C₉	#41	1	2	yes	—
62	C₂×C₃×C₄²	D₉	1	3	yes	—
63	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
64	#830	—	9	6	no	—
65	#830	—	9	2	no	—
66	C₄⋊(C₃⋊C₈)	—	9	6	no	—
67	C₄⋊(C₃⋊C₈)	—	9	2	no	—
68	C₄×C₃⋊C₈	—	9	6	no	—
69	C₄×C₃⋊C₈	—	9	2	no	—
70	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₃⋊C₈	1	4	yes	—
74	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂³×C₉	C₂×Dic₃	1	3	yes	—
81	C₃⋊(C₃⋊C₈)	—	6	4	no	—
82	C₂×C₄×C₉	—	2	6	no	—
83	C₂×C₄×C₉	—	2	6	no	—
84	C₂×C₄×C₉	C₃⋊C₈	1	12	yes	complemented
85	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
87	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
88	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
89	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
90	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
91	C₉⋊C₈	—	6	12	no	—
92	#431	—	27	1	no	Sylow
93	C₂×C₃×C₉	—	2	2	no	—
94	C₂×C₃×C₉	#117	1	1	yes	characteristic
95	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
97	C₂²×C₃×C₄	Dic₉	1	3	yes	—
98	C₂²×C₃×C₄	Dic₉	1	3	yes	—
99	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
100	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
103	C₃×C₄²	Dic₉	1	3	yes	—
104	C₃×C₄²	Dic₉	1	3	yes	—
105	C₃×C₄²	C₂×D₉	1	3	yes	—
106	C₃×C₄²	C₂×D₉	1	3	yes	—
107	C₃×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
108	C₃×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
109	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
110	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₂×C₃⋊C₈	—	9	12	no	—
112	C₂×C₃⋊C₈	—	9	4	no	—
113	C₂²×C₃²	—	2	1	no	—
114	C₂²×C₃²	—	2	1	no	—
115	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
116	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
117	C₃²×C₄	—	2	1	no	—
118	C₃²×C₄	—	2	1	no	—
119	C₃²×C₄	C₂×C₃⋊C₈	1	2	yes	—
120	C₃²×C₄	C₂×C₃⋊C₈	1	2	yes	—
121	C₃²×C₄	#208	1	2	yes	—
122	C₃²×C₄	#235	1	2	yes	—
123	C₂²×C₉	—	2	3	no	—
124	C₂²×C₉	—	2	3	no	—
125	C₂²×C₉	C₂×C₃⋊C₈	1	6	yes	—
126	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
127	C₄×C₉	—	2	3	no	—
128	C₄×C₉	—	2	3	no	—
129	C₄×C₉	C₂×C₃⋊C₈	1	6	yes	—
130	C₄×C₉	C₂×C₃⋊C₈	1	6	yes	—
131	C₄×C₉	#208	1	6	yes	—
132	C₄×C₉	#235	1	6	yes	—
133	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
134	C₄⋊C₈	—	27	2	no	—
135	C₂²⋊C₈	—	27	2	no	—
136	C₄×C₈	—	27	2	no	—
137	C₃×C₉	#431	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₃	C₂×Dic₉	1	3	yes	—
139	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	6	no	—
141	C₂×C₃×C₄	—	2	6	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	C₉⋊C₈	1	12	yes	complemented
145	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
146	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
147	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
148	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
149	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
150	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	4	yes	—
151	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
152	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
158	C₃⋊C₈	—	18	12	no	—
159	C₃⋊C₈	—	18	4	no	—
160	C₂×C₃²	—	2	2	no	—
161	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
162	C₂×C₃²	C₂×#208	1	1	yes	characteristic
163	C₂×C₃²	#926	1	1	yes	characteristic
164	C₂×C₉	—	2	6	no	—
165	C₂×C₉	C₂²×C₃⋊C₈	1	3	yes	—
166	C₂×C₉	C₂×#208	1	3	yes	—
167	C₂×C₉	#926	1	3	yes	—
168	C₂²×C₄	C₃⋊Dic₉	1	1	yes	characteristic
169	C₂²×C₄	C₃⋊Dic₉	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
171	C₂×C₈	—	27	4	no	—
172	C₄²	C₃⋊Dic₉	1	1	yes	characteristic
173	C₄²	C₃⋊Dic₉	1	1	yes	characteristic
174	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
175	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
176	C₂²×C₃	—	2	3	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	3	no	—
180	C₂²×C₃	C₂×C₉⋊C₈	1	6	yes	—
181	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
182	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
183	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
184	C₃×C₄	—	2	3	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	3	no	—
188	C₃×C₄	C₂×C₉⋊C₈	1	6	yes	—
189	C₃×C₄	C₂×C₉⋊C₈	1	6	yes	—
190	C₃×C₄	#3667	1	6	yes	—
191	C₃×C₄	#3697	1	6	yes	—
192	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
193	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
194	C₃×C₄	#3748	1	2	yes	—
195	C₃×C₄	#3828	1	2	yes	—
196	C₃²	#4925	1	1	yes	characteristic
197	C₉	#4925	1	3	yes	complemented
198	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	C₃⋊(C₉⋊C₈)	1	4	yes	complemented
202	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
205	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
206	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
208	C₈	—	54	4	no	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	2	6	no	—
211	C₂×C₃	C₂²×C₉⋊C₈	1	3	yes	—
212	C₂×C₃	C₂×#3667	1	3	yes	—
213	C₂×C₃	#63488	1	3	yes	—
214	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
215	C₂×C₃	C₂×#3748	1	1	yes	characteristic
216	C₂×C₃	#64135	1	1	yes	characteristic
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	2	yes	—
220	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	C₂×C₃⋊(C₉⋊C₈)	1	2	yes	—
224	C₄	C₂×C₃⋊(C₉⋊C₈)	1	2	yes	—
225	C₄	#88532	1	2	yes	—
226	C₄	#88736	1	2	yes	—
227	C₃	#10588605	1	3	yes	complemented
228	C₃	#10592624	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂²×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
231	C₂	C₂×#88532	1	1	yes	characteristic
232	C₂	#11721815	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂²×C₈
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	#431
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄² ◁ 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
4. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
26. C₂×C₃×C₄×C₉	C₈	semidirect
84. C₂×C₄×C₉	C₃⋊C₈	semidirect
137. C₃×C₉	#431	semidirect
144. C₂×C₃×C₄	C₉⋊C₈	semidirect
197. C₉	#4925	semidirect
201. C₂×C₄	C₃⋊(C₉⋊C₈)	semidirect
227. C₃	#10588605	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
#11720571	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₄×C₃⋊(C₉⋊C₈)	2
#10592624	3
#10588605	3