Subgroups — #13135928

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589416	—	3	1	no	maximal
5	#10588228	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
8	C₂×C₃×C₈×C₉	—	2	1	no	—
9	C₂×C₃×C₈×C₉	—	2	1	no	—
10	C₂×C₃×C₈×C₉	C₄	1	2	yes	complemented
11	C₄×C₃⋊Dic₉	—	2	1	no	—
12	C₄×C₃⋊Dic₉	—	2	1	no	—
13	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
14	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₄	1	2	yes	complemented
16	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
17	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
19	C₂²×C₈×C₉	D₃	1	3	yes	—
20	C₂×C₄×Dic₉	—	3	3	no	—
21	C₂²×C₉⋊C₈	—	3	3	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
29	C₂×C₃⋊Dic₉	—	2	2	no	—
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₂×C₃⋊Dic₉	C₈	1	4	yes	complemented
32	C₃×C₈×C₉	—	2	2	no	—
33	C₃⋊(C₉⋊C₈)	—	2	2	no	—
34	#4550	—	9	3	no	—
35	#4550	—	9	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	—	3	2	no	—
38	C₂×C₃²×C₈	—	2	1	no	—
39	C₂×C₃²×C₈	—	2	1	no	—
40	C₂×C₃²×C₈	Dic₃	1	2	yes	—
41	C₄×C₃⋊Dic₃	—	6	1	no	—
42	C₄×C₃⋊Dic₃	—	6	1	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
45	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
46	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
47	C₂²×Dic₉	—	3	6	no	—
48	C₂×C₈×C₉	—	2	3	no	—
49	C₂×C₈×C₉	—	2	3	no	—
50	C₂×C₈×C₉	Dic₃	1	6	yes	complemented
51	C₄×Dic₉	—	6	3	no	—
52	C₄×Dic₉	—	6	3	no	—
53	C₂×C₉⋊C₈	—	3	6	no	—
54	C₂×C₉⋊C₈	—	6	3	no	—
55	C₂×C₉⋊C₈	—	6	3	no	—
56	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
60	C₂²×C₃×C₉	M₁₆	1	2	yes	—
61	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
65	C₃⋊Dic₉	—	2	4	no	—
66	C₂²×C₃×C₈	D₉	1	3	yes	—
67	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
68	C₂×C₄×Dic₃	—	9	3	no	—
69	C₂×C₄×Dic₃	—	9	1	no	—
70	C₂²×C₃⋊C₈	—	9	3	no	—
71	C₂²×C₃⋊C₈	—	9	1	no	—
72	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
73	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
77	C₂×C₃²×C₄	#67	1	1	yes	characteristic
78	C₂×C₃²×C₄	#67	1	1	yes	characteristic
79	C₂×C₃⋊Dic₃	—	3	4	no	—
80	C₂×C₃⋊Dic₃	—	6	2	no	—
81	C₂×C₃⋊Dic₃	—	6	2	no	—
82	C₂³×C₉	C₄×D₃	1	3	yes	—
83	C₃²×C₈	—	2	2	no	—
84	C₃⋊(C₃⋊C₈)	—	6	2	no	—
85	C₂×C₄×C₉	Dic₆	1	3	yes	—
86	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
87	C₂×C₄×C₉	D₁₂	1	3	yes	—
88	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
89	C₂×C₄×C₉	#67	1	3	yes	—
90	C₂×C₄×C₉	#67	1	3	yes	—
91	C₂×Dic₉	—	3	12	no	—
92	C₂×Dic₉	—	6	6	no	—
93	C₂×Dic₉	—	6	6	no	—
94	C₈×C₉	—	2	6	no	—
95	C₉⋊C₈	—	6	6	no	—
96	#336	—	27	1	no	Sylow
97	C₂×C₃×C₉	#95	1	1	yes	characteristic
98	C₂×C₃×C₉	C₄×C₈	1	1	yes	characteristic
99	C₂×C₃×C₉	#97	1	1	yes	characteristic
100	C₂×C₃×C₉	C₂²⋊C₈	1	2	yes	—
101	C₂×C₃×C₉	C₄⋊C₈	1	2	yes	—
102	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
103	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₂²×Dic₃	—	9	6	no	—
105	C₂²×Dic₃	—	9	2	no	—
106	C₂×C₃×C₈	—	2	3	no	—
107	C₂×C₃×C₈	—	2	3	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	Dic₉	1	6	yes	complemented
111	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	—
112	C₄×Dic₃	—	18	3	no	—
113	C₄×Dic₃	—	18	3	no	—
114	C₄×Dic₃	—	18	1	no	—
115	C₄×Dic₃	—	18	1	no	—
116	C₂×C₃⋊C₈	—	9	6	no	—
117	C₂×C₃⋊C₈	—	9	2	no	—
118	C₂×C₃⋊C₈	—	18	3	no	—
119	C₂×C₃⋊C₈	—	18	3	no	—
120	C₂×C₃⋊C₈	—	18	1	no	—
121	C₂×C₃⋊C₈	—	18	1	no	—
122	C₂²×C₃²	D₃×C₈	1	2	yes	—
123	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	2	yes	—
124	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
125	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
126	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
127	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
128	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
129	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
130	C₃²×C₄	#217	1	1	yes	characteristic
131	C₃⋊Dic₃	—	6	4	no	—
132	C₂²×C₉	D₃×C₈	1	6	yes	—
133	C₂²×C₉	ΓL(1,𝔽₂₅)	1	6	yes	—
134	C₂²×C₉	C₄×Dic₃	1	3	yes	—
135	C₂²×C₉	Dic₃⋊C₄	1	3	yes	—
136	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
137	C₄×C₉	Dic₃⋊C₄	1	3	yes	—
138	C₄×C₉	C₄⋊Dic₃	1	3	yes	—
139	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
140	C₄×C₉	#217	1	3	yes	—
141	Dic₉	—	6	12	no	—
142	C₂²×C₈	—	27	1	no	—
143	C₂²×C₈	C₃⋊D₉	1	1	yes	characteristic
144	C₂×C₄²	—	27	1	no	—
145	C₃×C₉	#336	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₃	C₄×D₉	1	3	yes	—
147	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
148	C₂×C₃×C₄	Dic₁₈	1	3	yes	—
149	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
150	C₂×C₃×C₄	D₃₆	1	3	yes	—
151	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
152	C₂×C₃×C₄	#611	1	3	yes	—
153	C₂×C₃×C₄	#611	1	3	yes	—
154	C₂×C₃×C₄	#634	1	1	yes	characteristic
155	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
158	C₂×C₃×C₄	#638	1	1	yes	characteristic
159	C₂×C₃×C₄	#638	1	1	yes	characteristic
160	C₂×Dic₃	—	9	12	no	—
161	C₂×Dic₃	—	9	4	no	—
162	C₂×Dic₃	—	18	6	no	—
163	C₂×Dic₃	—	18	6	no	—
164	C₂×Dic₃	—	18	2	no	—
165	C₂×Dic₃	—	18	2	no	—
166	C₃×C₈	—	2	6	no	—
167	C₃×C₈	—	2	2	no	—
168	C₃⋊C₈	—	18	6	no	—
169	C₃⋊C₈	—	18	2	no	—
170	C₂×C₃²	C₈×Dic₃	1	1	yes	characteristic
171	C₂×C₃²	Dic₃⋊C₈	1	2	yes	—
172	C₂×C₃²	#815	1	1	yes	characteristic
173	C₂×C₃²	(C₂×D₃)⋊C₈	1	2	yes	—
174	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
175	C₂×C₉	C₈×Dic₃	1	3	yes	—
176	C₂×C₉	Dic₃⋊C₈	1	6	yes	—
177	C₂×C₉	#815	1	3	yes	—
178	C₂×C₉	(C₂×D₃)⋊C₈	1	6	yes	—
179	C₂×C₉	C₃⋊#95	1	3	yes	—
180	C₂²×C₄	—	27	2	no	—
181	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	27	2	no	—
185	C₂×C₈	—	54	1	no	—
186	C₂×C₈	—	54	1	no	—
187	C₂×C₈	C₃⋊Dic₉	1	2	yes	complemented
188	C₄²	—	54	1	no	—
189	C₄²	—	54	1	no	—
190	C₂²×C₃	C₈×D₉	1	6	yes	—
191	C₂²×C₃	#3663	1	6	yes	—
192	C₂²×C₃	C₄×Dic₉	1	3	yes	—
193	C₂²×C₃	Dic₉⋊C₄	1	3	yes	—
194	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
195	C₂²×C₃	C₈×C₃⋊D₃	1	2	yes	—
196	C₂²×C₃	#3743	1	2	yes	—
197	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
198	C₂²×C₃	#3750	1	1	yes	characteristic
199	C₂²×C₃	#3752	1	1	yes	characteristic
200	C₃×C₄	Dic₉⋊C₄	1	3	yes	—
201	C₃×C₄	C₄⋊Dic₉	1	3	yes	—
202	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
203	C₃×C₄	#3676	1	3	yes	—
204	C₃×C₄	#3750	1	1	yes	characteristic
205	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
206	C₃×C₄	#3752	1	1	yes	characteristic
207	C₃×C₄	#3757	1	1	yes	characteristic
208	Dic₃	—	18	12	no	—
209	Dic₃	—	18	4	no	—
210	C₃²	#4550	1	1	yes	characteristic
211	C₉	#4550	1	3	yes	complemented
212	C₂³	C₄×C₃⋊D₉	1	1	yes	characteristic
213	C₂×C₄	—	27	4	no	—
214	C₂×C₄	—	54	2	no	—
215	C₂×C₄	—	54	2	no	—
216	C₂×C₄	#6226	1	1	yes	characteristic
217	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
218	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
219	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
220	C₂×C₄	#6233	1	1	yes	characteristic
221	C₂×C₄	#6233	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	54	2	no	—
224	C₂×C₃	C₈×Dic₉	1	3	yes	—
225	C₂×C₃	Dic₉⋊C₈	1	6	yes	—
226	C₂×C₃	#63375	1	3	yes	—
227	C₂×C₃	(C₂×D₉)⋊C₈	1	6	yes	—
228	C₂×C₃	C₉⋊#95	1	3	yes	—
229	C₂×C₃	C₈×C₃⋊Dic₃	1	1	yes	characteristic
230	C₂×C₃	#63641	1	2	yes	—
231	C₂×C₃	#63642	1	1	yes	characteristic
232	C₂×C₃	#63647	1	2	yes	—
233	C₂×C₃	#63658	1	1	yes	characteristic
234	C₂²	C₈×C₃⋊D₉	1	2	yes	—
235	C₂²	#88523	1	2	yes	—
236	C₂²	C₄×C₃⋊Dic₉	1	1	yes	characteristic
237	C₂²	#88534	1	1	yes	characteristic
238	C₂²	#88536	1	1	yes	characteristic
239	C₄	—	54	4	no	—
240	C₄	#88534	1	1	yes	characteristic
241	C₄	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
242	C₄	#88536	1	1	yes	characteristic
243	C₄	#88551	1	1	yes	characteristic
244	C₃	#10588228	1	3	yes	complemented
245	C₃	#10589416	1	1	yes	characteristic
246	C₂	C₈×C₃⋊Dic₉	1	1	yes	characteristic
247	C₂	#11720537	1	2	yes	—
248	C₂	#11720538	1	1	yes	characteristic
249	C₂	#11720542	1	2	yes	—
250	C₂	#11720572	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₄×C₈
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂²×C₃×C₄×C₉	C₂²
hypercenter	C₂²×C₈	C₃⋊D₉
hypocenter	C₃×C₉	#336
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
10. C₂×C₃×C₈×C₉	C₄	semidirect
15. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
31. C₂×C₃⋊Dic₉	C₈	semidirect
50. C₂×C₈×C₉	Dic₃	semidirect
110. C₂×C₃×C₈	Dic₉	semidirect
145. C₃×C₉	#336	semidirect
187. C₂×C₈	C₃⋊Dic₉	semidirect
211. C₉	#4550	semidirect
244. C₃	#10588228	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₂×C₄×C₃⋊Dic₉	2
C₂²×C₃⋊(C₉⋊C₈)	2
#10589416	3
#10588228	3