Subgroups — C₂²×M₁₆×D₉ #11918430

Order 1152.

2580 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#3667	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×M₁₆×D₉	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#3663	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₉	C₂	1	2	yes	maximal, complemented
7	C₂²×D₃×M₁₆	—	3	1	no	maximal
8	C₂⁴×D₉	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂³×Dic₉	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₉	C₄	1	12	yes	—
12	C₂²×C₄×D₉	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₉	C₄	1	1	yes	characteristic
14	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₉	C₂²	1	12	yes	—
17	C₂×C₉×M₁₆	C₂²	1	12	yes	complemented
18	C₂²×C₈×C₉	C₂²	1	2	yes	complemented
19	C₂×#3667	C₂²	1	12	yes	complemented
20	C₂²×C₉⋊C₈	C₂²	1	2	yes	complemented
21	M₁₆×D₉	C₂²	1	64	yes	direct product factor
22	C₂×#3663	C₂²	1	24	yes	complemented
23	C₂×C₈×D₉	C₂²	1	24	yes	complemented
24	C₂³×C₄×D₃	—	3	1	no	—
25	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
26	C₂²×#208	—	3	1	no	—
27	C₂×D₃×M₁₆	—	3	24	no	—
28	C₂²×ΓL(1,𝔽₂₅)	—	3	2	no	—
29	C₂²×D₃×C₈	—	3	2	no	—
30	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₉	—	2	8	no	—
32	C₂³×D₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×D₉	C₂×C₄	1	12	yes	—
34	C₂³×D₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	6	yes	—
36	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₉	C₂³	1	6	yes	—
39	C₂²×Dic₉	C₂×C₄	1	12	yes	—
40	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₉	C₂×C₄	1	6	yes	—
43	C₂×C₄×D₉	C₂×C₄	1	48	yes	—
44	C₂×C₄×D₉	C₂×C₄	1	12	yes	—
45	C₂×C₄×D₉	C₂×C₄	1	6	yes	—
46	C₂×C₄×D₉	C₂×C₄	1	12	yes	—
47	C₂×C₄×D₉	C₂³	1	6	yes	—
48	C₂×C₄×D₉	C₂³	1	6	yes	—
49	C₂×C₄×D₉	C₂³	1	16	yes	—
50	C₉×M₁₆	C₂³	1	16	yes	complemented
51	C₂×C₈×C₉	C₂³	1	12	yes	complemented
52	#3667	C₂³	1	16	yes	complemented
53	C₂×C₉⋊C₈	C₂³	1	12	yes	complemented
54	#3663	C₂³	1	32	yes	complemented
55	C₈×D₉	C₂³	1	32	yes	complemented
56	C₂³×M₁₆	—	9	1	no	Sylow
57	C₂⁴×D₃	—	3	1	no	—
58	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
59	C₂³×Dic₃	—	3	1	no	—
60	C₂²×C₄×D₃	—	3	1	no	—
61	C₂²×C₄×D₃	—	3	12	no	—
62	C₂²×C₄×D₃	—	3	1	no	—
63	C₂²×C₄×D₃	—	3	1	no	—
64	C₂²×C₄×D₃	—	3	1	no	—
65	C₂²×C₄×D₃	—	3	12	no	—
66	C₂×C₃×M₁₆	C₂×D₃	1	12	yes	—
67	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
68	C₂×#208	—	3	12	no	—
69	C₂²×C₃⋊C₈	—	3	2	no	—
70	D₃×M₁₆	—	3	64	no	—
71	C₂×ΓL(1,𝔽₂₅)	—	3	24	no	—
72	C₂×D₃×C₈	—	3	24	no	—
73	C₂³×C₉	—	2	4	no	—
74	C₂³×C₉	C₂²×C₄	1	6	yes	—
75	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×D₉	—	2	4	no	—
77	C₂²×D₉	—	2	48	no	—
78	C₂²×D₉	M₁₆	1	8	yes	direct product factor
79	C₂²×D₉	C₂²×C₄	1	6	yes	—
80	C₂²×D₉	C₂²×C₄	1	16	yes	—
81	C₂²×D₉	C₂²×C₄	1	6	yes	—
82	C₂×C₄×C₉	C₂²×C₄	1	12	yes	—
83	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
84	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
85	C₂×C₄×C₉	C₂⁴	1	3	yes	—
86	C₂×C₄×C₉	C₂⁴	1	4	yes	—
87	C₂×Dic₉	C₂²×C₄	1	16	yes	—
88	C₂×Dic₉	C₂²×C₄	1	6	yes	—
89	C₂×Dic₉	C₂²×C₄	1	6	yes	—
90	C₄×D₉	C₂²×C₄	1	12	yes	—
91	C₄×D₉	C₂²×C₄	1	16	yes	—
92	C₄×D₉	C₂²×C₄	1	12	yes	—
93	C₄×D₉	C₂²×C₄	1	16	yes	—
94	C₄×D₉	C₂⁴	1	4	yes	—
95	C₄×D₉	C₂⁴	1	4	yes	—
96	C₈×C₉	C₂⁴	1	8	yes	complemented
97	C₉⋊C₈	C₂⁴	1	8	yes	complemented
98	C₂⁴×C₄	—	9	1	no	—
99	C₂²×M₁₆	—	9	1	no	—
100	C₂²×M₁₆	—	9	24	no	—
101	C₂²×M₁₆	—	9	2	no	—
102	C₂²×M₁₆	D₉	1	1	yes	characteristic, direct product factor
103	C₂³×C₈	—	9	2	no	—
104	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
105	C₂³×D₃	—	3	1	no	—
106	C₂³×D₃	—	3	12	no	—
107	C₂³×D₃	—	3	1	no	—
108	C₂³×D₃	—	6	8	no	—
109	C₂²×C₃×C₄	C₄×D₃	1	6	yes	—
110	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
112	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂²×Dic₃	—	3	12	no	—
114	C₂²×Dic₃	—	3	1	no	—
115	C₂²×Dic₃	—	3	1	no	—
116	C₂×C₄×D₃	—	3	6	no	—
117	C₂×C₄×D₃	—	3	12	no	—
118	C₂×C₄×D₃	—	3	6	no	—
119	C₂×C₄×D₃	—	3	48	no	—
120	C₂×C₄×D₃	—	3	12	no	—
121	C₂×C₄×D₃	—	3	6	no	—
122	C₂×C₄×D₃	—	3	6	no	—
123	C₂×C₄×D₃	—	3	16	no	—
124	C₃×M₁₆	C₂²×D₃	1	16	yes	—
125	C₂×C₃×C₈	C₂²×D₃	1	12	yes	—
126	#208	—	3	16	no	—
127	C₂×C₃⋊C₈	—	3	12	no	—
128	ΓL(1,𝔽₂₅)	—	3	32	no	—
129	D₃×C₈	—	3	32	no	—
130	C₂²×C₉	—	2	12	no	—
131	C₂²×C₉	C₂×M₁₆	1	4	yes	complemented
132	C₂²×C₉	C₂³×C₄	1	4	yes	—
133	C₂²×C₉	C₂³×C₄	1	3	yes	—
134	C₂×D₉	—	2	12	no	—
135	C₂×D₉	—	2	32	no	—
136	C₂×D₉	C₂×M₁₆	1	24	yes	direct product factor
137	C₂×D₉	C₂³×C₄	1	4	yes	—
138	C₂×D₉	C₂³×C₄	1	4	yes	—
139	C₄×C₉	C₂³×C₄	1	3	yes	—
140	C₄×C₉	C₂³×C₄	1	4	yes	—
141	C₄×C₉	C₂⁵	1	1	yes	characteristic
142	Dic₉	C₂³×C₄	1	4	yes	—
143	Dic₉	C₂³×C₄	1	4	yes	—
144	C₂⁵	—	9	1	no	—
145	C₂³×C₄	—	9	1	no	—
146	C₂³×C₄	—	9	1	no	—
147	C₂³×C₄	—	9	12	no	—
148	C₂³×C₄	—	9	1	no	—
149	C₂³×C₄	—	9	12	no	—
150	C₂³×C₄	—	9	1	no	—
151	C₂³×C₄	—	9	1	no	—
152	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
153	C₂×M₁₆	—	9	12	no	—
154	C₂×M₁₆	—	9	64	no	—
155	C₂×M₁₆	—	9	24	no	—
156	C₂×M₁₆	C₂×D₉	1	12	yes	direct product factor
157	C₂²×C₈	—	9	2	no	—
158	C₂²×C₈	—	9	24	no	—
159	C₂²×C₈	C₂×D₉	1	2	yes	complemented
160	C₂³×C₃	—	2	4	no	—
161	C₂³×C₃	C₂×C₄×D₃	1	6	yes	—
162	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂²×D₃	—	3	8	no	—
164	C₂²×D₃	—	3	6	no	—
165	C₂²×D₃	—	3	16	no	—
166	C₂²×D₃	—	3	6	no	—
167	C₂²×D₃	—	6	48	no	—
168	C₂²×D₃	—	6	4	no	—
169	C₂×C₃×C₄	C₂×C₄×D₃	1	12	yes	—
170	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
171	C₂×C₃×C₄	C₂×C₄×D₃	1	3	yes	—
172	C₂×C₃×C₄	C₂³×D₃	1	3	yes	—
173	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
174	C₂×Dic₃	—	3	16	no	—
175	C₂×Dic₃	—	3	6	no	—
176	C₂×Dic₃	—	3	6	no	—
177	C₄×D₃	—	3	12	no	—
178	C₄×D₃	—	3	16	no	—
179	C₄×D₃	—	3	12	no	—
180	C₄×D₃	—	3	16	no	—
181	C₄×D₃	—	3	4	no	—
182	C₄×D₃	—	3	4	no	—
183	C₃×C₈	C₂³×D₃	1	8	yes	—
184	C₃⋊C₈	—	3	8	no	—
185	C₂×C₉	—	2	4	no	—
186	C₂×C₉	C₂²×M₁₆	1	6	yes	complemented
187	C₂×C₉	C₂⁴×C₄	1	1	yes	characteristic
188	D₉	—	2	4	no	—
189	D₉	C₂²×M₁₆	1	8	yes	direct product factor
190	C₂⁴	—	9	1	no	—
191	C₂⁴	—	9	12	no	—
192	C₂⁴	—	9	1	no	—
193	C₂⁴	—	18	8	no	—
194	C₂⁴	C₄×D₉	1	1	yes	characteristic
195	C₂²×C₄	—	9	12	no	—
196	C₂²×C₄	—	9	1	no	—
197	C₂²×C₄	—	9	1	no	—
198	C₂²×C₄	—	9	12	no	—
199	C₂²×C₄	—	9	48	no	—
200	C₂²×C₄	—	9	12	no	—
201	C₂²×C₄	—	9	6	no	—
202	C₂²×C₄	—	9	6	no	—
203	C₂²×C₄	—	9	6	no	—
204	C₂²×C₄	—	9	16	no	—
205	C₂²×C₄	—	9	6	no	—
206	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
207	C₂²×C₄	C₄×D₉	1	6	yes	—
208	C₂²×C₄	C₂²×D₉	1	6	yes	—
209	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
210	M₁₆	—	9	16	no	—
211	M₁₆	—	9	32	no	—
212	M₁₆	C₂²×D₉	1	16	yes	direct product factor
213	C₂×C₈	—	9	12	no	—
214	C₂×C₈	—	9	32	no	—
215	C₂×C₈	C₂²×D₉	1	12	yes	complemented
216	C₂²×C₃	—	2	12	no	—
217	C₂²×C₃	D₃×M₁₆	1	4	yes	—
218	C₂²×C₃	C₂²×C₄×D₃	1	4	yes	—
219	C₂²×C₃	C₂²×C₄×D₃	1	3	yes	—
220	C₂×D₃	—	3	4	no	—
221	C₂×D₃	—	3	24	no	—
222	C₂×D₃	—	3	4	no	—
223	C₂×D₃	—	6	32	no	—
224	C₂×D₃	—	6	12	no	—
225	C₃×C₄	C₂²×C₄×D₃	1	3	yes	—
226	C₃×C₄	C₂²×C₄×D₃	1	4	yes	—
227	C₃×C₄	C₂⁴×D₃	1	1	yes	characteristic
228	Dic₃	—	3	4	no	—
229	Dic₃	—	3	4	no	—
230	C₉	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	2	4	no	—
232	C₂³	—	9	6	no	—
233	C₂³	—	9	8	no	—
234	C₂³	—	9	16	no	—
235	C₂³	—	9	6	no	—
236	C₂³	—	18	48	no	—
237	C₂³	—	18	4	no	—
238	C₂³	C₂×C₄×D₉	1	6	yes	—
239	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
240	C₂×C₄	—	9	16	no	—
241	C₂×C₄	—	9	6	no	—
242	C₂×C₄	—	9	6	no	—
243	C₂×C₄	—	9	4	no	—
244	C₂×C₄	—	9	4	no	—
245	C₂×C₄	—	9	12	no	—
246	C₂×C₄	—	9	16	no	—
247	C₂×C₄	—	9	16	no	—
248	C₂×C₄	—	9	12	no	—
249	C₂×C₄	C₂×C₄×D₉	1	12	yes	—
250	C₂×C₄	C₂×C₄×D₉	1	6	yes	—
251	C₂×C₄	C₂×C₄×D₉	1	3	yes	—
252	C₂×C₄	C₂³×D₉	1	3	yes	—
253	C₂×C₄	C₂³×D₉	1	4	yes	—
254	C₈	—	9	8	no	—
255	C₈	C₂³×D₉	1	8	yes	complemented
256	C₂×C₃	—	2	4	no	—
257	C₂×C₃	C₂×D₃×M₁₆	1	6	yes	—
258	C₂×C₃	C₂³×C₄×D₃	1	1	yes	characteristic
259	D₃	—	3	8	no	—
260	D₃	—	6	4	no	—
261	C₂²	—	2	12	no	—
262	C₂²	—	9	4	no	—
263	C₂²	—	9	24	no	—
264	C₂²	—	9	4	no	—
265	C₂²	—	18	32	no	—
266	C₂²	—	18	12	no	—
267	C₂²	M₁₆×D₉	1	4	yes	direct product factor
268	C₂²	C₂²×C₄×D₉	1	4	yes	—
269	C₂²	C₂²×C₄×D₉	1	3	yes	—
270	C₄	—	9	4	no	—
271	C₄	—	9	4	no	—
272	C₄	C₂²×C₄×D₉	1	4	yes	—
273	C₄	C₂²×C₄×D₉	1	3	yes	—
274	C₄	C₂⁴×D₉	1	1	yes	characteristic
275	C₃	C₂²×D₃×M₁₆	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	—	9	8	no	—
278	C₂	—	18	4	no	—
279	C₂	C₂×M₁₆×D₉	1	6	yes	direct product factor
280	C₂	C₂³×C₄×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂×C₉	C₂⁴×C₄
Frattini	C₃×C₄	C₂⁴×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂³×C₃	C₂×C₄×D₃
prime radical	C₂⁴×D₉	C₄
cosocle	C₄×C₉	C₂⁵
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	C₂³×M₁₆
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉×M₁₆	C₂	semidirect
3. C₂²×#3667	C₂	semidirect
4. C₂×M₁₆×D₉	C₂	direct (N × H)
5. C₂²×#3663	C₂	semidirect
6. C₂²×C₈×D₉	C₂	semidirect
17. C₂×C₉×M₁₆	C₂²	semidirect
18. C₂²×C₈×C₉	C₂²	semidirect
19. C₂×#3667	C₂²	semidirect
20. C₂²×C₉⋊C₈	C₂²	semidirect
21. M₁₆×D₉	C₂²	direct (N × H)
22. C₂×#3663	C₂²	semidirect
23. C₂×C₈×D₉	C₂²	semidirect
50. C₉×M₁₆	C₂³	semidirect
51. C₂×C₈×C₉	C₂³	semidirect
52. #3667	C₂³	semidirect
53. C₂×C₉⋊C₈	C₂³	semidirect
54. #3663	C₂³	semidirect
55. C₈×D₉	C₂³	semidirect
78. C₂²×D₉	M₁₆	direct (N × H)
96. C₈×C₉	C₂⁴	semidirect
97. C₉⋊C₈	C₂⁴	semidirect
102. C₂²×M₁₆	D₉	direct (N × H)
131. C₂²×C₉	C₂×M₁₆	semidirect
136. C₂×D₉	C₂×M₁₆	direct (N × H)
156. C₂×M₁₆	C₂×D₉	direct (N × H)
159. C₂²×C₈	C₂×D₉	semidirect
186. C₂×C₉	C₂²×M₁₆	semidirect
189. D₉	C₂²×M₁₆	direct (N × H)
212. M₁₆	C₂²×D₉	direct (N × H)
215. C₂×C₈	C₂²×D₉	semidirect
230. C₉	C₂³×M₁₆	semidirect
255. C₈	C₂³×D₉	semidirect
267. C₂²	M₁₆×D₉	direct (N × H)
279. C₂	C₂×M₁₆×D₉	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×C₄×D₉	2
C₂²×C₉×M₁₆	2
C₂²×#3667	2
C₂×M₁₆×D₉	2
C₂²×#3663	2
C₂²×C₈×D₉	2
C₂²×D₃×M₁₆	3

Subgroups — C22×M16×D9 #11918430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×D₉ #11918430