Subgroups — C₂²×C₉×#208 #13153526

Order 1728.

1130 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₉×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃×#208	C₃	1	1	yes	maximal, characteristic
5	C₂²×C₉×M₁₆	—	3	1	no	maximal
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	C₄	1	6	yes	—
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	6	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₉×#208	C₂²	1	16	yes	direct product factor
12	C₂×C₉×C₃⋊C₈	C₂²	1	12	yes	complemented
13	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×#208	C₂×C₃	1	12	yes	—
15	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	2	yes	—
16	C₂³×C₄×C₉	—	2	1	no	—
17	C₂³×C₄×C₉	D₃	1	1	yes	characteristic
18	C₂×C₉×M₁₆	—	3	12	no	—
19	C₂²×C₈×C₉	—	3	2	no	—
20	C₂³×C₃×C₉	—	2	4	no	—
21	C₂³×C₃×C₉	C₂×C₄	1	6	yes	—
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	C₂×C₄	1	12	yes	—
24	C₂×C₃×C₄×C₉	C₂×C₄	1	6	yes	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
26	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
28	C₉×C₃⋊C₈	C₂³	1	8	yes	complemented
29	C₂²×C₃×M₁₆	—	3	1	no	—
30	C₂²×#208	C₉	1	1	yes	characteristic, direct product factor
31	C₂⁴×C₃²	C₃×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₃×C₄	1	6	yes	—
33	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
35	C₂²×C₃²×C₄	C₂²×C₃	1	6	yes	—
36	C₂⁴×C₉	—	2	1	no	—
37	C₂⁴×C₉	Dic₃	1	1	yes	characteristic
38	C₃×#208	C₂²×C₃	1	16	yes	—
39	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	12	yes	—
40	C₂²×C₄×C₉	—	2	6	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	—	2	6	no	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	Dic₃	1	1	yes	characteristic
45	C₂²×C₄×C₉	Dic₃	1	6	yes	—
46	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
47	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
48	C₉×M₁₆	—	3	16	no	—
49	C₂×C₈×C₉	—	3	12	no	—
50	C₂²×C₃×C₉	—	2	12	no	—
51	C₂²×C₃×C₉	M₁₆	1	4	yes	complemented
52	C₂²×C₃×C₉	C₂²×C₄	1	4	yes	—
53	C₂²×C₃×C₉	C₂²×C₄	1	3	yes	—
54	C₃×C₄×C₉	C₂²×C₄	1	3	yes	—
55	C₃×C₄×C₉	C₂²×C₄	1	4	yes	—
56	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
57	C₂³×C₃×C₄	—	2	1	no	—
58	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic
59	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
60	C₂×C₃×M₁₆	—	3	12	no	—
61	C₂²×C₃×C₈	—	3	2	no	—
62	C₂×#208	C₂×C₉	1	12	yes	direct product factor
63	C₂²×C₃⋊C₈	C₂×C₉	1	2	yes	complemented
64	C₂³×C₃²	—	2	4	no	—
65	C₂³×C₃²	C₂×C₃×C₄	1	6	yes	—
66	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
68	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
69	C₂×C₃²×C₄	C₂×C₃×C₄	1	6	yes	—
70	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
71	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
72	C₂³×C₉	—	2	8	no	—
73	C₂³×C₉	—	2	4	no	—
74	C₂³×C₉	—	2	6	no	—
75	C₂³×C₉	—	2	1	no	—
76	C₂³×C₉	C₂×Dic₃	1	6	yes	—
77	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
78	C₃×C₃⋊C₈	C₂³×C₃	1	8	yes	—
79	C₂×C₄×C₉	—	2	12	no	—
80	C₂×C₄×C₉	—	2	6	no	—
81	C₂×C₄×C₉	—	2	3	no	—
82	C₂×C₄×C₉	—	2	3	no	—
83	C₂×C₄×C₉	—	2	4	no	—
84	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
85	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
87	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
88	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
89	C₈×C₉	—	3	8	no	—
90	C₂²×M₁₆	—	3	1	no	Sylow
91	C₂×C₃×C₉	—	2	4	no	—
92	C₂×C₃×C₉	C₂×M₁₆	1	6	yes	complemented
93	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
94	C₂⁴×C₃	—	2	1	no	—
95	C₂⁴×C₃	C₄×C₉	1	1	yes	characteristic
96	C₂⁴×C₃	C₃×Dic₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	—	2	6	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	—	2	6	no	—
101	C₂²×C₃×C₄	C₄×C₉	1	6	yes	—
102	C₂²×C₃×C₄	C₄×C₉	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂²×C₉	1	6	yes	—
105	C₂²×C₃×C₄	C₃×Dic₃	1	6	yes	—
106	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×C₃×D₃	1	6	yes	—
108	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
109	C₃×M₁₆	—	3	16	no	—
110	C₂×C₃×C₈	—	3	12	no	—
111	#208	C₂²×C₉	1	16	yes	direct product factor
112	C₂×C₃⋊C₈	C₂²×C₉	1	12	yes	complemented
113	C₂²×C₃²	—	2	12	no	—
114	C₂²×C₃²	C₃×M₁₆	1	4	yes	—
115	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
116	C₂²×C₃²	C₂²×C₃×C₄	1	3	yes	—
117	C₃²×C₄	C₂²×C₃×C₄	1	3	yes	—
118	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
119	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
120	C₂²×C₉	—	2	24	no	—
121	C₂²×C₉	—	2	12	no	—
122	C₂²×C₉	—	2	4	no	—
123	C₂²×C₉	—	2	4	no	—
124	C₂²×C₉	—	2	3	no	—
125	C₂²×C₉	#208	1	4	yes	direct product factor
126	C₂²×C₉	C₂²×Dic₃	1	4	yes	—
127	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
128	C₄×C₉	—	2	1	no	—
129	C₄×C₉	—	2	3	no	—
130	C₄×C₉	—	2	4	no	—
131	C₄×C₉	C₂²×Dic₃	1	4	yes	—
132	C₄×C₉	C₂²×Dic₃	1	3	yes	—
133	C₄×C₉	C₂³×D₃	1	1	yes	characteristic
134	C₂³×C₄	D₃×C₉	1	1	yes	characteristic
135	C₂×M₁₆	—	3	12	no	—
136	C₂²×C₈	—	3	2	no	—
137	C₃×C₉	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₃	—	2	8	no	—
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	6	no	—
142	C₂³×C₃	—	2	1	no	—
143	C₂³×C₃	C₂×C₄×C₉	1	6	yes	—
144	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
145	C₂³×C₃	C₂×C₃×Dic₃	1	6	yes	—
146	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	12	no	—
148	C₂×C₃×C₄	—	2	3	no	—
149	C₂×C₃×C₄	—	2	6	no	—
150	C₂×C₃×C₄	—	2	3	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	C₂×C₄×C₉	1	12	yes	—
153	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
154	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
155	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
156	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
157	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
158	C₂×C₃×C₄	C₂×C₃×Dic₃	1	6	yes	—
159	C₂×C₃×C₄	C₂×C₃×Dic₃	1	3	yes	—
160	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
161	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
162	C₃×C₈	—	3	8	no	—
163	C₃⋊C₈	C₂³×C₉	1	8	yes	complemented
164	C₂×C₃²	—	2	4	no	—
165	C₂×C₃²	C₂×C₃×M₁₆	1	6	yes	—
166	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
167	C₂×C₉	—	2	8	no	—
168	C₂×C₉	—	2	4	no	—
169	C₂×C₉	—	2	6	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	C₂×#208	1	6	yes	direct product factor
172	C₂×C₉	C₂³×Dic₃	1	1	yes	characteristic
173	C₂⁴	C₉×Dic₃	1	1	yes	characteristic
174	C₂²×C₄	C₉×Dic₃	1	6	yes	—
175	C₂²×C₄	C₉×Dic₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₃×C₉	1	6	yes	—
178	M₁₆	—	3	16	no	—
179	C₂×C₈	—	3	12	no	—
180	C₂²×C₃	—	2	24	no	—
181	C₂²×C₃	—	2	12	no	—
182	C₂²×C₃	—	2	12	no	—
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	—	2	3	no	—
185	C₂²×C₃	—	2	4	no	—
186	C₂²×C₃	C₉×M₁₆	1	4	yes	complemented
187	C₂²×C₃	C₂²×C₄×C₉	1	4	yes	—
188	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
189	C₂²×C₃	C₃×#208	1	4	yes	—
190	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
191	C₂²×C₃	C₂²×C₃×Dic₃	1	3	yes	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	3	no	—
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	C₂²×C₄×C₉	1	3	yes	—
196	C₃×C₄	C₂²×C₄×C₉	1	4	yes	—
197	C₃×C₄	C₂⁴×C₉	1	1	yes	characteristic
198	C₃×C₄	C₂²×C₃×Dic₃	1	3	yes	—
199	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
200	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
201	C₃²	C₂²×C₃×M₁₆	1	1	yes	characteristic
202	C₉	—	2	1	no	—
203	C₉	C₂²×#208	1	1	yes	characteristic, direct product factor
204	C₂³	—	2	4	no	—
205	C₂³	C₂×C₉×Dic₃	1	6	yes	—
206	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₉×Dic₃	1	12	yes	—
208	C₂×C₄	C₂×C₉×Dic₃	1	3	yes	—
209	C₂×C₄	C₂×C₉×Dic₃	1	6	yes	—
210	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
211	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
212	C₈	—	3	8	no	—
213	C₂×C₃	—	2	8	no	—
214	C₂×C₃	—	2	4	no	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	6	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	C₂×C₉×M₁₆	1	6	yes	complemented
219	C₂×C₃	C₂³×C₄×C₉	1	1	yes	characteristic
220	C₂×C₃	C₂×C₃×#208	1	6	yes	—
221	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
222	C₂²	—	2	12	no	—
223	C₂²	C₉×#208	1	4	yes	direct product factor
224	C₂²	C₂²×C₉×Dic₃	1	4	yes	—
225	C₂²	C₂²×C₉×Dic₃	1	3	yes	—
226	C₄	C₂²×C₉×Dic₃	1	3	yes	—
227	C₄	C₂²×C₉×Dic₃	1	4	yes	—
228	C₄	C₂³×D₃×C₉	1	1	yes	characteristic
229	C₃	—	2	1	no	—
230	C₃	C₂²×C₉×M₁₆	1	1	yes	characteristic, complemented
231	C₃	C₂²×C₃×#208	1	1	yes	characteristic
232	C₂	—	2	4	no	—
233	C₂	C₂×C₉×#208	1	6	yes	direct product factor
234	C₂	C₂³×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂³×C₄×C₉
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂×C₃×C₄
prime radical	C₂⁴×C₃²	C₃×C₄
cosocle	C₃²×C₄	C₂⁴×C₃
hypercenter	C₂³×C₄×C₉	D₃
hypocenter	C₃	C₂²×C₉×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#208	C₂	direct (N × H)
3. C₂²×C₉×C₃⋊C₈	C₂	semidirect
11. C₉×#208	C₂²	direct (N × H)
12. C₂×C₉×C₃⋊C₈	C₂²	semidirect
28. C₉×C₃⋊C₈	C₂³	semidirect
30. C₂²×#208	C₉	direct (N × H)
51. C₂²×C₃×C₉	M₁₆	semidirect
62. C₂×#208	C₂×C₉	direct (N × H)
63. C₂²×C₃⋊C₈	C₂×C₉	semidirect
92. C₂×C₃×C₉	C₂×M₁₆	semidirect
111. #208	C₂²×C₉	direct (N × H)
112. C₂×C₃⋊C₈	C₂²×C₉	semidirect
125. C₂²×C₉	#208	direct (N × H)
137. C₃×C₉	C₂²×M₁₆	semidirect
163. C₃⋊C₈	C₂³×C₉	semidirect
171. C₂×C₉	C₂×#208	direct (N × H)
186. C₂²×C₃	C₉×M₁₆	semidirect
203. C₉	C₂²×#208	direct (N × H)
218. C₂×C₃	C₂×C₉×M₁₆	semidirect
223. C₂²	C₉×#208	direct (N × H)
230. C₃	C₂²×C₉×M₁₆	semidirect
233. C₂	C₂×C₉×#208	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×C₉×#208	2
C₂²×C₉×C₃⋊C₈	2
C₂²×C₃×#208	3
C₂²×C₉×M₁₆	3

Subgroups — C22×C9×#208 #13153526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×#208 #13153526