Subgroups — D₄×C₉×Q₁₆ #11768007

Order 1152.

708 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Q₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₉	C₂	1	2	yes	maximal
3	C₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#467	C₂	1	2	yes	maximal, complemented
5	C₉×#462	C₂	1	2	yes	maximal, complemented
6	C₉×#451	C₂	1	4	yes	maximal, complemented
7	C₄×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₉	C₂	1	1	yes	maximal, characteristic
9	C₃×D₄×Q₁₆	C₃	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₉	C₂²	1	4	yes	—
11	C₂×C₉×Q₁₆	—	2	8	no	—
12	C₂×C₉×Q₁₆	C₂²	1	4	yes	—
13	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₉	C₂²	1	2	yes	—
17	C₉×C₄⋊Q₈	—	2	2	no	—
18	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₉×C₂²⋊Q₈	—	2	4	no	—
20	C₉×C₂²⋊Q₈	C₂²	1	4	yes	—
21	C₄×Q₈×C₉	C₂²	1	2	yes	—
22	C₄×D₄×C₉	—	2	2	no	—
23	C₄×D₄×C₉	C₂²	1	1	yes	characteristic
24	C₉×#107	C₂²	1	1	yes	characteristic
25	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₉×#103	C₂²	1	2	yes	—
27	C₉×#103	C₂²	1	4	yes	—
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
30	C₂²×C₃×Q₁₆	C₂×C₃	1	2	yes	—
31	C₃×D₄×Q₈	C₂×C₃	1	2	yes	—
32	C₃×#494	C₂×C₃	1	1	yes	characteristic
33	C₃×#467	C₂×C₃	1	2	yes	—
34	C₃×#462	C₂×C₃	1	2	yes	—
35	C₃×#451	C₂×C₃	1	4	yes	—
36	C₃×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
37	C₃×C₈×D₄	C₂×C₃	1	1	yes	characteristic
38	C₂×Q₈×C₉	—	2	4	no	—
39	C₂×Q₈×C₉	—	2	8	no	—
40	C₂×Q₈×C₉	—	4	4	no	—
41	C₂×Q₈×C₉	C₂³	1	4	yes	—
42	C₂×Q₈×C₉	C₂³	1	2	yes	—
43	C₂×D₄×C₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	—	2	4	no	—
45	C₂²×C₄×C₉	C₂³	1	2	yes	—
46	C₉×Q₁₆	—	2	8	no	—
47	C₉×Q₁₆	—	2	4	no	—
48	C₉×Q₁₆	—	4	2	no	—
49	C₉×Q₁₆	D₄	1	4	yes	direct product factor
50	C₂×C₈×C₉	—	2	4	no	—
51	C₂×C₈×C₉	C₂³	1	2	yes	—
52	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
54	C₉×C₄⋊C₄	—	2	2	no	—
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	4	no	—
58	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
59	C₉×C₄⋊C₄	C₂³	1	2	yes	—
60	C₉×C₂²⋊C₄	—	2	4	no	—
61	C₉×C₂²⋊C₄	D₄	1	2	yes	—
62	C₄²×C₉	—	2	2	no	—
63	C₄²×C₉	C₂³	1	1	yes	characteristic
64	D₄×Q₁₆	C₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
66	C₂×C₃×Q₁₆	—	2	8	no	—
67	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
68	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
69	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
70	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
71	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
72	C₃×C₄⋊Q₈	—	2	2	no	—
73	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
74	C₃×C₂²⋊Q₈	—	2	4	no	—
75	C₃×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
76	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
77	C₃×C₄×D₄	—	2	2	no	—
78	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
79	C₃×#107	C₂²×C₃	1	1	yes	characteristic
80	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₃×#103	C₂²×C₃	1	2	yes	—
82	C₃×#103	C₂²×C₃	1	4	yes	—
83	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
85	C₂³×C₉	C₂×D₄	1	2	yes	—
86	Q₈×C₉	—	2	2	no	—
87	Q₈×C₉	—	2	4	no	—
88	Q₈×C₉	—	2	4	no	—
89	Q₈×C₉	—	4	8	no	—
90	Q₈×C₉	C₂×D₄	1	4	yes	—
91	D₄×C₉	Q₁₆	1	4	yes	direct product factor
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	4	no	—
97	C₂×C₄×C₉	—	4	8	no	—
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
102	C₈×C₉	—	2	2	no	—
103	C₈×C₉	—	2	1	no	—
104	C₈×C₉	C₂×D₄	1	2	yes	—
105	C₂²×Q₁₆	C₂×C₉	1	2	yes	complemented
106	D₄×Q₈	C₂×C₉	1	2	yes	—
107	#494	C₂×C₉	1	1	yes	characteristic, complemented
108	#467	C₂×C₉	1	2	yes	complemented
109	#462	C₂×C₉	1	2	yes	complemented
110	#451	C₂×C₉	1	4	yes	complemented
111	C₄×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
112	C₈×D₄	C₂×C₉	1	1	yes	characteristic
113	C₂×C₃×Q₈	—	2	4	no	—
114	C₂×C₃×Q₈	—	2	8	no	—
115	C₂×C₃×Q₈	—	4	4	no	—
116	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
117	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
118	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	—	2	4	no	—
120	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
121	C₃×Q₁₆	—	2	8	no	—
122	C₃×Q₁₆	—	2	4	no	—
123	C₃×Q₁₆	—	4	2	no	—
124	C₃×Q₁₆	C₃×D₄	1	4	yes	—
125	C₂×C₃×C₈	—	2	4	no	—
126	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
128	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	—	2	4	no	—
133	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
134	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
135	C₃×C₂²⋊C₄	—	2	4	no	—
136	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
137	C₃×C₄²	—	2	2	no	—
138	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
139	C₂²×C₉	—	2	2	no	—
140	C₂²×C₉	—	2	2	no	—
141	C₂²×C₉	C₂×Q₁₆	1	4	yes	complemented
142	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	4	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	4	4	no	—
147	C₄×C₉	—	4	2	no	—
148	C₄×C₉	C₂×Q₁₆	1	2	yes	complemented
149	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂²×Q₈	C₂²×C₉	1	4	yes	—
152	C₂×Q₁₆	—	2	8	no	—
153	C₂×Q₁₆	C₂²×C₉	1	4	yes	—
154	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
155	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
156	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
157	C₂²×C₈	C₂²×C₉	1	2	yes	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
160	C₂²⋊Q₈	—	2	4	no	—
161	C₂²⋊Q₈	C₂²×C₉	1	4	yes	—
162	C₄×Q₈	C₂²×C₉	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	C₂²×C₉	1	1	yes	characteristic
165	#107	C₂²×C₉	1	1	yes	characteristic
166	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
167	#103	C₂²×C₉	1	2	yes	—
168	#103	C₂²×C₉	1	4	yes	—
169	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
170	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
171	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
172	C₃×Q₈	—	2	2	no	—
173	C₃×Q₈	—	2	4	no	—
174	C₃×Q₈	—	2	4	no	—
175	C₃×Q₈	—	4	8	no	—
176	C₃×Q₈	C₂×C₃×D₄	1	4	yes	—
177	C₃×D₄	C₃×Q₁₆	1	4	yes	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	4	8	no	—
184	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	1	no	—
190	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
191	C₂×C₉	—	2	4	no	—
192	C₂×C₉	D₄²	1	1	yes	characteristic
193	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
194	C₂×C₉	#578	1	1	yes	characteristic
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	C₂³×C₉	1	4	yes	—
199	C₂×Q₈	C₂³×C₉	1	2	yes	—
200	C₂×D₄	D₄×C₉	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂³×C₉	1	2	yes	—
203	Q₁₆	—	2	8	no	—
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	4	2	no	—
206	Q₁₆	D₄×C₉	1	4	yes	direct product factor
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
209	C₂×C₈	C₂³×C₉	1	2	yes	—
210	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
216	C₄⋊C₄	C₂³×C₉	1	2	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	D₄×C₉	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	C₂³×C₉	1	1	yes	characteristic
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	C₂×C₃×Q₁₆	1	4	yes	—
224	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	—	4	4	no	—
230	C₃×C₄	C₂×C₃×Q₁₆	1	2	yes	—
231	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
232	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
233	C₉	D₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	C₂×D₄×C₉	1	2	yes	—
235	Q₈	—	2	2	no	—
236	Q₈	—	2	4	no	—
237	Q₈	—	2	4	no	—
238	Q₈	—	4	8	no	—
239	Q₈	C₂×D₄×C₉	1	4	yes	—
240	D₄	C₉×Q₁₆	1	4	yes	direct product factor
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
249	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
250	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	C₂×D₄×C₉	1	2	yes	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	C₃×D₄²	1	1	yes	characteristic
256	C₂×C₃	C₂²×C₃×Q₁₆	1	1	yes	characteristic
257	C₂×C₃	C₃×#578	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	C₂×C₉×Q₁₆	1	4	yes	complemented
261	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	1	no	—
265	C₄	—	4	4	no	—
266	C₄	—	4	2	no	—
267	C₄	C₂×C₉×Q₁₆	1	2	yes	complemented
268	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
269	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
270	C₃	C₃×D₄×Q₁₆	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	D₄²×C₉	1	1	yes	characteristic
273	C₂	C₂²×C₉×Q₁₆	1	1	yes	characteristic
274	C₂	C₉×#578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂×C₃×D₄	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄×D₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×Q₁₆	C₂	semidirect
3. C₉×#494	C₂	semidirect
4. C₉×#467	C₂	semidirect
5. C₉×#462	C₂	semidirect
6. C₉×#451	C₂	semidirect
7. C₄×C₉×Q₁₆	C₂	semidirect
49. C₉×Q₁₆	D₄	direct (N × H)
64. D₄×Q₁₆	C₉	direct (N × H)
91. D₄×C₉	Q₁₆	direct (N × H)
105. C₂²×Q₁₆	C₂×C₉	semidirect
107. #494	C₂×C₉	semidirect
108. #467	C₂×C₉	semidirect
109. #462	C₂×C₉	semidirect
110. #451	C₂×C₉	semidirect
111. C₄×Q₁₆	C₂×C₉	semidirect
141. C₂²×C₉	C₂×Q₁₆	semidirect
148. C₄×C₉	C₂×Q₁₆	semidirect
206. Q₁₆	D₄×C₉	direct (N × H)
233. C₉	D₄×Q₁₆	direct (N × H)
240. D₄	C₉×Q₁₆	direct (N × H)
260. C₂²	C₂×C₉×Q₁₆	semidirect
267. C₄	C₂×C₉×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Q₁₆	2
D₄×Q₈×C₉	2
C₉×#494	2
C₉×#467	2
C₉×#462	2
C₉×#451	2
C₄×C₉×Q₁₆	2
C₈×D₄×C₉	2
C₃×D₄×Q₁₆	3

Subgroups — D4×C9×Q16 #11768007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₉×Q₁₆ #11768007