Subgroups — C₂²×C₉×#217 #13153760

Order 1728.

2770 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₉×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×#217	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×#217	C₃	1	1	yes	maximal, characteristic
5	C₂²×C₉×C₂²⋊C₄	—	3	1	no	maximal
6	C₂⁴×C₃×C₉	—	2	8	no	—
7	C₂⁴×C₃×C₉	C₄	1	8	yes	complemented
8	C₂⁴×C₃×C₉	C₂²	1	6	yes	—
9	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₉×Dic₃	—	2	8	no	—
11	C₂²×C₉×Dic₃	C₂²	1	12	yes	complemented
12	C₉×#217	C₂²	1	16	yes	direct product factor
13	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic
14	C₂³×C₃×Dic₃	C₂×C₃	1	2	yes	—
15	C₂⁵×C₉	—	2	1	no	—
16	C₂⁵×C₉	D₃	1	1	yes	characteristic
17	C₂×C₃×#217	C₂×C₃	1	12	yes	—
18	C₂³×C₄×C₉	—	3	2	no	—
19	C₂×C₉×C₂²⋊C₄	—	3	12	no	—
20	C₂³×C₃×C₉	—	2	32	no	—
21	C₂³×C₃×C₉	—	2	24	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
23	C₂³×C₃×C₉	C₂×C₄	1	24	yes	complemented
24	C₂³×C₃×C₉	D₄	1	8	yes	—
25	C₂³×C₃×C₉	C₂³	1	4	yes	—
26	C₂³×C₃×C₉	C₂³	1	3	yes	—
27	C₂×C₉×Dic₃	—	2	24	no	—
28	C₂×C₉×Dic₃	C₂³	1	8	yes	complemented
29	C₂²×C₃×C₂²⋊C₄	—	3	1	no	—
30	C₂²×#217	C₉	1	1	yes	characteristic, direct product factor
31	C₂⁴×C₃²	—	2	8	no	—
32	C₂⁴×C₃²	C₃×C₄	1	8	yes	—
33	C₂⁴×C₃²	C₂²×C₃	1	6	yes	—
34	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
35	C₂²×C₃×Dic₃	—	2	8	no	—
36	C₂²×C₃×Dic₃	C₂²×C₃	1	12	yes	—
37	C₂⁴×C₉	—	2	16	no	—
38	C₂⁴×C₉	—	2	8	no	—
39	C₂⁴×C₉	—	2	6	no	—
40	C₂⁴×C₉	—	2	1	no	—
41	C₂⁴×C₉	—	2	8	no	—
42	C₂⁴×C₉	Dic₃	1	8	yes	complemented
43	C₂⁴×C₉	C₂×D₃	1	6	yes	—
44	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
45	C₃×#217	C₂²×C₃	1	16	yes	—
46	C₂²×C₄×C₉	—	3	12	no	—
47	C₂²×C₄×C₉	—	6	8	no	—
48	C₉×C₂²⋊C₄	—	3	16	no	—
49	C₂²×C₃×C₉	—	2	48	no	—
50	C₂²×C₃×C₉	—	2	8	no	—
51	C₂²×C₃×C₉	C₂²⋊C₄	1	16	yes	complemented
52	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
53	C₂²×C₃×C₉	C₂²×C₄	1	8	yes	complemented
54	C₂²×C₃×C₉	C₂×D₄	1	12	yes	—
55	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
56	C₉×Dic₃	—	2	8	no	—
57	C₂⁵×C₃	—	2	1	no	—
58	C₂⁵×C₃	C₂×C₉	1	1	yes	characteristic
59	C₂⁵×C₃	C₃×D₃	1	1	yes	characteristic
60	C₂³×C₃×C₄	—	3	2	no	—
61	C₂³×Dic₃	C₂×C₉	1	2	yes	complemented
62	C₂×C₃×C₂²⋊C₄	—	3	12	no	—
63	C₂×#217	C₂×C₉	1	12	yes	direct product factor
64	C₂³×C₃²	—	2	32	no	—
65	C₂³×C₃²	—	2	24	no	—
66	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
67	C₂³×C₃²	C₂×C₃×C₄	1	24	yes	—
68	C₂³×C₃²	C₃×D₄	1	8	yes	—
69	C₂³×C₃²	C₂³×C₃	1	4	yes	—
70	C₂³×C₃²	C₂³×C₃	1	3	yes	—
71	C₂×C₃×Dic₃	—	2	24	no	—
72	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
73	C₂³×C₉	—	2	64	no	—
74	C₂³×C₉	—	2	48	no	—
75	C₂³×C₉	—	2	32	no	—
76	C₂³×C₉	—	2	24	no	—
77	C₂³×C₉	—	2	4	no	—
78	C₂³×C₉	—	2	24	no	—
79	C₂³×C₉	—	2	8	no	—
80	C₂³×C₉	—	2	4	no	—
81	C₂³×C₉	—	2	3	no	—
82	C₂³×C₉	C₂×Dic₃	1	4	yes	—
83	C₂³×C₉	C₂×Dic₃	1	24	yes	complemented
84	C₂³×C₉	#67	1	8	yes	—
85	C₂³×C₉	C₂²×D₃	1	4	yes	—
86	C₂³×C₉	C₂²×D₃	1	3	yes	—
87	C₂×C₄×C₉	—	3	8	no	—
88	C₂×C₄×C₉	—	6	24	no	—
89	C₂²×C₂²⋊C₄	—	3	1	no	Sylow
90	C₂×C₃×C₉	—	2	8	no	—
91	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	12	yes	complemented
92	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₃×C₉	C₂²×D₄	1	2	yes	—
94	C₂⁴×C₃	—	2	16	no	—
95	C₂⁴×C₃	—	2	8	no	—
96	C₂⁴×C₃	—	2	8	no	—
97	C₂⁴×C₃	—	2	8	no	—
98	C₂⁴×C₃	—	2	6	no	—
99	C₂⁴×C₃	—	2	1	no	—
100	C₂⁴×C₃	C₄×C₉	1	8	yes	complemented
101	C₂⁴×C₃	C₂²×C₉	1	6	yes	—
102	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
103	C₂⁴×C₃	C₃×Dic₃	1	8	yes	—
104	C₂⁴×C₃	C₂×C₃×D₃	1	6	yes	—
105	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	—	3	12	no	—
107	C₂²×C₃×C₄	—	6	8	no	—
108	C₂²×Dic₃	—	2	8	no	—
109	C₂²×Dic₃	C₂²×C₉	1	12	yes	complemented
110	C₃×C₂²⋊C₄	—	3	16	no	—
111	#217	C₂²×C₉	1	16	yes	direct product factor
112	C₂²×C₃²	—	2	48	no	—
113	C₂²×C₃²	—	2	8	no	—
114	C₂²×C₃²	C₃×C₂²⋊C₄	1	16	yes	—
115	C₂²×C₃²	C₂²×C₃×C₄	1	6	yes	—
116	C₂²×C₃²	C₂²×C₃×C₄	1	8	yes	—
117	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
118	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
119	C₃×Dic₃	—	2	8	no	—
120	C₂²×C₉	—	2	96	no	—
121	C₂²×C₉	—	2	16	no	—
122	C₂²×C₉	—	2	48	no	—
123	C₂²×C₉	—	2	8	no	—
124	C₂²×C₉	—	2	1	no	—
125	C₂²×C₉	—	2	16	no	—
126	C₂²×C₉	—	2	6	no	—
127	C₂²×C₉	—	2	8	no	—
128	C₂²×C₉	—	2	12	no	—
129	C₂²×C₉	#217	1	16	yes	direct product factor
130	C₂²×C₉	C₂²×Dic₃	1	6	yes	—
131	C₂²×C₉	C₂²×Dic₃	1	8	yes	complemented
132	C₂²×C₉	C₂×#67	1	12	yes	—
133	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
134	C₄×C₉	—	6	8	no	—
135	C₂⁵	D₃×C₉	1	1	yes	characteristic
136	C₂³×C₄	—	3	2	no	—
137	C₂×C₂²⋊C₄	—	3	12	no	—
138	C₃×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	48	no	—
140	C₂³×C₃	—	2	64	no	—
141	C₂³×C₃	—	2	32	no	—
142	C₂³×C₃	—	2	24	no	—
143	C₂³×C₃	—	2	32	no	—
144	C₂³×C₃	—	2	24	no	—
145	C₂³×C₃	—	2	4	no	—
146	C₂³×C₃	—	2	24	no	—
147	C₂³×C₃	—	2	8	no	—
148	C₂³×C₃	—	2	4	no	—
149	C₂³×C₃	—	2	3	no	—
150	C₂³×C₃	C₂×C₄×C₉	1	4	yes	—
151	C₂³×C₃	C₂×C₄×C₉	1	24	yes	complemented
152	C₂³×C₃	D₄×C₉	1	8	yes	—
153	C₂³×C₃	C₂³×C₉	1	4	yes	—
154	C₂³×C₃	C₂³×C₉	1	3	yes	—
155	C₂³×C₃	C₂×C₃×Dic₃	1	4	yes	—
156	C₂³×C₃	C₂×C₃×Dic₃	1	24	yes	—
157	C₂³×C₃	C₃×#67	1	8	yes	—
158	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
159	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
160	C₂×C₃×C₄	—	3	8	no	—
161	C₂×C₃×C₄	—	6	24	no	—
162	C₂×Dic₃	—	2	24	no	—
163	C₂×Dic₃	C₂³×C₉	1	8	yes	complemented
164	C₂×C₃²	—	2	8	no	—
165	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	12	yes	—
166	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
167	C₂×C₃²	C₂²×C₃×D₄	1	2	yes	—
168	C₂×C₉	—	2	16	no	—
169	C₂×C₉	—	2	8	no	—
170	C₂×C₉	—	2	12	no	—
171	C₂×C₉	—	2	1	no	—
172	C₂×C₉	—	2	2	no	—
173	C₂×C₉	C₂×#217	1	12	yes	direct product factor
174	C₂×C₉	C₂³×Dic₃	1	1	yes	characteristic
175	C₂×C₉	C₂²×#67	1	2	yes	—
176	C₂⁴	—	2	8	no	—
177	C₂⁴	C₉×Dic₃	1	8	yes	complemented
178	C₂⁴	C₂×D₃×C₉	1	6	yes	—
179	C₂⁴	C₂×D₃×C₉	1	1	yes	characteristic
180	C₂²×C₄	—	3	12	no	—
181	C₂²×C₄	—	6	8	no	—
182	C₂²⋊C₄	—	3	16	no	—
183	C₂²×C₃	—	2	16	no	—
184	C₂²×C₃	—	2	96	no	—
185	C₂²×C₃	—	2	48	no	—
186	C₂²×C₃	—	2	48	no	—
187	C₂²×C₃	—	2	8	no	—
188	C₂²×C₃	—	2	8	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	2	6	no	—
191	C₂²×C₃	—	2	8	no	—
192	C₂²×C₃	—	2	12	no	—
193	C₂²×C₃	—	2	16	no	—
194	C₂²×C₃	C₉×C₂²⋊C₄	1	16	yes	complemented
195	C₂²×C₃	C₂²×C₄×C₉	1	6	yes	—
196	C₂²×C₃	C₂²×C₄×C₉	1	8	yes	complemented
197	C₂²×C₃	C₂×D₄×C₉	1	12	yes	—
198	C₂²×C₃	C₃×#217	1	16	yes	—
199	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
200	C₂²×C₃	C₂²×C₃×Dic₃	1	6	yes	—
201	C₂²×C₃	C₂²×C₃×Dic₃	1	8	yes	—
202	C₂²×C₃	C₂×C₃×#67	1	12	yes	—
203	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
204	C₃×C₄	—	6	8	no	—
205	Dic₃	—	2	8	no	—
206	C₃²	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
207	C₉	—	2	1	no	—
208	C₉	C₂²×#217	1	1	yes	characteristic, direct product factor
209	C₂³	—	2	24	no	—
210	C₂³	—	2	32	no	—
211	C₂³	C₂×C₉×Dic₃	1	4	yes	—
212	C₂³	C₂×C₉×Dic₃	1	24	yes	complemented
213	C₂³	C₉×#67	1	8	yes	—
214	C₂³	C₂²×D₃×C₉	1	4	yes	—
215	C₂³	C₂²×D₃×C₉	1	3	yes	—
216	C₂×C₄	—	3	8	no	—
217	C₂×C₄	—	6	24	no	—
218	C₂×C₃	—	2	16	no	—
219	C₂×C₃	—	2	8	no	—
220	C₂×C₃	—	2	8	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	2	12	no	—
224	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	12	yes	complemented
225	C₂×C₃	C₂³×C₄×C₉	1	1	yes	characteristic
226	C₂×C₃	C₂²×D₄×C₉	1	2	yes	—
227	C₂×C₃	C₂×C₃×#217	1	12	yes	—
228	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
229	C₂×C₃	C₂²×C₃×#67	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	48	no	—
232	C₂²	C₉×#217	1	16	yes	direct product factor
233	C₂²	C₂²×C₉×Dic₃	1	6	yes	—
234	C₂²	C₂²×C₉×Dic₃	1	8	yes	complemented
235	C₂²	C₂×C₉×#67	1	12	yes	—
236	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
237	C₄	—	6	8	no	—
238	C₃	—	2	1	no	—
239	C₃	C₂²×C₉×C₂²⋊C₄	1	1	yes	characteristic, complemented
240	C₃	C₂²×C₃×#217	1	1	yes	characteristic
241	C₂	—	2	8	no	—
242	C₂	C₂×C₉×#217	1	12	yes	direct product factor
243	C₂	C₂³×C₉×Dic₃	1	1	yes	characteristic
244	C₂	C₂²×C₉×#67	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₉	C₂×D₃
commutator	C₂×C₃	C₂³×C₄×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂⁵×C₃×C₉	C₂
socle	C₂⁴×C₃²	C₂²×C₃
prime radical	C₂⁵×C₃²	C₂×C₃
cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₂⁵×C₉	D₃
hypocenter	C₃	C₂²×C₉×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₉ ◁ C₂⁵×C₉

Complements

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

N	H ≅ G/N	type
2. C₂³×C₉×Dic₃	C₂	semidirect
3. C₂×C₉×#217	C₂	direct (N × H)
7. C₂⁴×C₃×C₉	C₄	semidirect
11. C₂²×C₉×Dic₃	C₂²	semidirect
12. C₉×#217	C₂²	direct (N × H)
23. C₂³×C₃×C₉	C₂×C₄	semidirect
28. C₂×C₉×Dic₃	C₂³	semidirect
30. C₂²×#217	C₉	direct (N × H)
42. C₂⁴×C₉	Dic₃	semidirect
51. C₂²×C₃×C₉	C₂²⋊C₄	semidirect
53. C₂²×C₃×C₉	C₂²×C₄	semidirect
61. C₂³×Dic₃	C₂×C₉	semidirect
63. C₂×#217	C₂×C₉	direct (N × H)
83. C₂³×C₉	C₂×Dic₃	semidirect
91. C₂×C₃×C₉	C₂×C₂²⋊C₄	semidirect
100. C₂⁴×C₃	C₄×C₉	semidirect
109. C₂²×Dic₃	C₂²×C₉	semidirect
111. #217	C₂²×C₉	direct (N × H)
129. C₂²×C₉	#217	direct (N × H)
131. C₂²×C₉	C₂²×Dic₃	semidirect
138. C₃×C₉	C₂²×C₂²⋊C₄	semidirect
151. C₂³×C₃	C₂×C₄×C₉	semidirect
163. C₂×Dic₃	C₂³×C₉	semidirect
173. C₂×C₉	C₂×#217	direct (N × H)
177. C₂⁴	C₉×Dic₃	semidirect
194. C₂²×C₃	C₉×C₂²⋊C₄	semidirect
196. C₂²×C₃	C₂²×C₄×C₉	semidirect
208. C₉	C₂²×#217	direct (N × H)
212. C₂³	C₂×C₉×Dic₃	semidirect
224. C₂×C₃	C₂×C₉×C₂²⋊C₄	semidirect
232. C₂²	C₉×#217	direct (N × H)
234. C₂²	C₂²×C₉×Dic₃	semidirect
239. C₃	C₂²×C₉×C₂²⋊C₄	semidirect
242. C₂	C₂×C₉×#217	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃×C₉	2
C₂³×C₉×Dic₃	2
C₂×C₉×#217	2
C₂²×C₃×#217	3
C₂²×C₉×C₂²⋊C₄	3

Subgroups — C22×C9×#217 #13153760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×#217 #13153760