Subgroups — C₂×Q₁₆×AGL(1,ℤ/9) #13153467

Order 1728.

804 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×AGL(1,ℤ/9)	C₂	1	2	yes	maximal
2	C₂×Q₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#3674⋊C₃	C₂	1	2	yes	maximal, complemented
4	Q₁₆×AGL(1,ℤ/9)	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₆⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×D₃×Q₁₆	—	3	1	no	maximal
8	C₂×Q₁₆×D₉	C₃	1	1	yes	maximal, characteristic, complemented
9	C₂×Q₈×3₋¹⁺²	C₂²	1	2	yes	—
10	Q₈×AGL(1,ℤ/9)	—	2	4	no	—
11	Q₈×AGL(1,ℤ/9)	C₂²	1	8	yes	—
12	C₂×C₄×AGL(1,ℤ/9)	—	2	2	no	—
13	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
14	C₂×3₋¹⁺²⋊Q₈	—	2	2	no	—
15	C₂×3₋¹⁺²⋊Q₈	C₂²	1	2	yes	—
16	Q₁₆×3₋¹⁺²	C₂²	1	4	yes	complemented
17	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
18	#3674⋊C₃	C₂²	1	8	yes	complemented
19	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×AGL(1,ℤ/9)	C₂²	1	4	yes	—
21	Dic₃₆⋊C₃	C₂²	1	4	yes	complemented
22	C₂×C₃×D₃×Q₈	—	3	2	no	—
23	C₂×C₃²×Q₁₆	D₃	1	1	yes	characteristic
24	C₂×C₃×#216	—	3	2	no	—
25	C₃×D₃×Q₁₆	—	3	8	no	—
26	C₂×C₃×Dic₁₂	—	3	1	no	—
27	C₂×C₃×D₃×C₈	—	3	1	no	—
28	C₂×Q₈×D₉	C₂×C₃	1	2	yes	—
29	C₂×C₉×Q₁₆	—	2	1	no	—
30	C₂×C₉×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#3674	C₂×C₃	1	2	yes	complemented
32	Q₁₆×D₉	C₂×C₃	1	8	yes	complemented
33	C₂×C₈×D₉	C₂×C₃	1	1	yes	characteristic
34	C₂×Dic₃₆	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂²×AGL(1,ℤ/9)	D₄	1	1	yes	characteristic
36	Q₈×3₋¹⁺²	—	2	2	no	—
37	Q₈×3₋¹⁺²	C₂³	1	4	yes	—
38	C₂×C₄×3₋¹⁺²	—	2	2	no	—
39	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
40	C₂×3₋¹⁺²⋊C₄	—	2	2	no	—
41	C₂×3₋¹⁺²⋊C₄	D₄	1	1	yes	characteristic
42	C₄×AGL(1,ℤ/9)	—	2	8	no	—
43	C₄×AGL(1,ℤ/9)	D₄	1	2	yes	—
44	C₄×AGL(1,ℤ/9)	C₂³	1	2	yes	—
45	3₋¹⁺²⋊Q₈	—	2	2	no	—
46	3₋¹⁺²⋊Q₈	—	2	8	no	—
47	3₋¹⁺²⋊Q₈	C₂³	1	4	yes	—
48	C₈×3₋¹⁺²	C₂³	1	2	yes	—
49	3₋¹⁺²⋊C₈	C₂³	1	2	yes	—
50	C₂²×C₃×Q₁₆	—	9	1	no	—
51	C₂×D₃×Q₁₆	—	3	1	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
53	C₃×D₃×Q₈	—	3	8	no	—
54	C₃×D₃×Q₈	—	6	4	no	—
55	C₂×C₃×C₄×D₃	—	3	1	no	—
56	C₂×C₃×C₄×D₃	—	6	2	no	—
57	C₂×C₃×Dic₆	—	3	2	no	—
58	C₂×C₃×Dic₆	—	6	2	no	—
59	C₃²×Q₁₆	C₂×D₃	1	4	yes	—
60	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#216	—	3	8	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₃×Dic₁₂	—	3	4	no	—
64	C₃×D₃×C₈	—	3	4	no	—
65	C₂×Q₈×C₉	—	2	2	no	—
66	C₂×Q₈×C₉	C₂²×C₃	1	2	yes	—
67	Q₈×D₉	—	2	4	no	—
68	Q₈×D₉	C₂²×C₃	1	8	yes	—
69	C₂×C₄×D₉	—	2	2	no	—
70	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
71	C₂×Dic₁₈	—	2	2	no	—
72	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
73	C₉×Q₁₆	—	2	4	no	—
74	C₉×Q₁₆	C₂²×C₃	1	4	yes	complemented
75	C₂×C₈×C₉	—	2	1	no	—
76	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
77	#3674	C₂²×C₃	1	8	yes	complemented
78	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
79	C₈×D₉	C₂²×C₃	1	4	yes	—
80	Dic₃₆	C₂²×C₃	1	4	yes	complemented
81	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
82	C₂×AGL(1,ℤ/9)	Q₁₆	1	4	yes	direct product factor
83	C₂×AGL(1,ℤ/9)	C₂×D₄	1	2	yes	—
84	C₄×3₋¹⁺²	—	2	4	no	—
85	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
86	C₄×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
87	3₋¹⁺²⋊C₄	—	2	4	no	—
88	3₋¹⁺²⋊C₄	C₂×D₄	1	2	yes	—
89	C₂²×C₃×Q₈	—	9	2	no	—
90	C₂×D₃×Q₈	—	3	2	no	—
91	C₂×C₃×Q₁₆	—	3	1	no	—
92	C₂×C₃×Q₁₆	—	9	1	no	—
93	C₂×C₃×Q₁₆	—	9	8	no	—
94	C₂×C₃×Q₁₆	—	9	2	no	—
95	C₂×C₃×Q₁₆	C₃×D₃	1	1	yes	characteristic
96	C₂²×C₃×C₈	—	9	1	no	—
97	C₂×#216	—	3	2	no	—
98	D₃×Q₁₆	—	3	8	no	—
99	C₂×Dic₁₂	—	3	1	no	—
100	C₂×D₃×C₈	—	3	1	no	—
101	C₂²×C₃×D₃	—	3	1	no	—
102	C₃²×Q₈	—	2	2	no	—
103	C₃²×Q₈	C₂²×D₃	1	4	yes	—
104	C₂×C₃²×C₄	—	2	2	no	—
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃×Dic₃	—	3	1	no	—
107	C₂×C₃×Dic₃	—	6	2	no	—
108	C₃×C₄×D₃	—	3	2	no	—
109	C₃×C₄×D₃	—	3	2	no	—
110	C₃×C₄×D₃	—	6	8	no	—
111	C₃×Dic₆	—	3	4	no	—
112	C₃×Dic₆	—	6	8	no	—
113	C₃×Dic₆	—	6	2	no	—
114	C₂²×D₉	C₃×D₄	1	1	yes	characteristic
115	C₃²×C₈	C₂²×D₃	1	2	yes	—
116	C₃×C₃⋊C₈	—	3	2	no	—
117	Q₈×C₉	—	2	2	no	—
118	Q₈×C₉	—	2	4	no	—
119	Q₈×C₉	—	4	2	no	—
120	Q₈×C₉	C₂³×C₃	1	4	yes	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	4	2	no	—
124	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
125	C₂×Dic₉	—	2	2	no	—
126	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
127	C₄×D₉	—	2	8	no	—
128	C₄×D₉	C₃×D₄	1	2	yes	—
129	C₄×D₉	C₂³×C₃	1	2	yes	—
130	Dic₁₈	—	2	2	no	—
131	Dic₁₈	—	2	8	no	—
132	Dic₁₈	C₂³×C₃	1	4	yes	—
133	C₈×C₉	—	2	2	no	—
134	C₈×C₉	C₂³×C₃	1	2	yes	—
135	C₉⋊C₈	C₂³×C₃	1	2	yes	—
136	C₂²×Q₁₆	—	9	1	no	Sylow
137	C₂×3₋¹⁺²	C₂×Q₁₆	1	2	yes	complemented
138	C₂×3₋¹⁺²	C₂²×D₄	1	1	yes	characteristic
139	AGL(1,ℤ/9)	C₂×Q₁₆	1	4	yes	direct product factor
140	C₂×C₃×Q₈	—	3	2	no	—
141	C₂×C₃×Q₈	—	9	2	no	—
142	C₂×C₃×Q₈	—	9	8	no	—
143	C₂×C₃×Q₈	—	18	2	no	—
144	C₂×C₃×Q₈	—	18	4	no	—
145	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
146	C₂²×C₃×C₄	—	9	1	no	—
147	C₂²×C₃×C₄	—	18	2	no	—
148	D₃×Q₈	—	3	8	no	—
149	D₃×Q₈	—	6	4	no	—
150	C₂×C₄×D₃	—	3	1	no	—
151	C₂×C₄×D₃	—	6	2	no	—
152	C₂×Dic₆	—	3	2	no	—
153	C₂×Dic₆	—	6	2	no	—
154	C₃×Q₁₆	—	3	4	no	—
155	C₃×Q₁₆	—	9	4	no	—
156	C₃×Q₁₆	—	9	8	no	—
157	C₃×Q₁₆	C₂×C₃×D₃	1	4	yes	—
158	C₂×C₃×C₈	—	3	1	no	—
159	C₂×C₃×C₈	—	9	4	no	—
160	C₂×C₃×C₈	—	9	1	no	—
161	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
162	#216	—	3	8	no	—
163	C₂×C₃⋊C₈	—	3	1	no	—
164	Dic₁₂	—	3	4	no	—
165	D₃×C₈	—	3	4	no	—
166	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
167	C₂×C₃×D₃	—	3	4	no	—
168	C₂×C₃×D₃	—	3	2	no	—
169	C₃²×C₄	—	2	4	no	—
170	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
171	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
172	C₃×Dic₃	—	3	2	no	—
173	C₃×Dic₃	—	6	4	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×D₉	C₃×Q₁₆	1	4	yes	complemented
177	C₂×D₉	C₂×C₃×D₄	1	2	yes	—
178	C₄×C₉	—	2	4	no	—
179	C₄×C₉	—	2	1	no	—
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	4	4	no	—
182	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
183	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
184	Dic₉	—	2	4	no	—
185	Dic₉	C₂×C₃×D₄	1	2	yes	—
186	C₂²×Q₈	—	9	2	no	—
187	C₂×Q₁₆	—	9	2	no	—
188	C₂×Q₁₆	—	9	8	no	—
189	C₂×Q₁₆	—	9	1	no	—
190	C₂×Q₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor
191	C₂²×C₈	—	9	1	no	—
192	3₋¹⁺²	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₃	—	9	1	no	—
194	C₂²×D₃	—	3	1	no	—
195	C₃×Q₈	—	2	2	no	—
196	C₃×Q₈	—	3	4	no	—
197	C₃×Q₈	—	6	2	no	—
198	C₃×Q₈	—	9	4	no	—
199	C₃×Q₈	—	18	2	no	—
200	C₃×Q₈	—	18	8	no	—
201	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	3	1	no	—
204	C₂×C₃×C₄	—	6	2	no	—
205	C₂×C₃×C₄	—	9	2	no	—
206	C₂×C₃×C₄	—	9	2	no	—
207	C₂×C₃×C₄	—	9	1	no	—
208	C₂×C₃×C₄	—	18	8	no	—
209	C₂×C₃×C₄	—	18	2	no	—
210	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
211	C₂×Dic₃	—	3	1	no	—
212	C₂×Dic₃	—	6	2	no	—
213	C₄×D₃	—	3	2	no	—
214	C₄×D₃	—	3	2	no	—
215	C₄×D₃	—	6	8	no	—
216	Dic₆	—	3	4	no	—
217	Dic₆	—	6	8	no	—
218	Dic₆	—	6	2	no	—
219	C₃×C₈	—	3	2	no	—
220	C₃×C₈	—	9	2	no	—
221	C₃×C₈	C₂²×C₃×D₃	1	2	yes	—
222	C₃⋊C₈	—	3	2	no	—
223	C₂×C₃²	D₃×Q₁₆	1	2	yes	—
224	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
225	C₃×D₃	—	3	4	no	—
226	C₂×C₉	—	2	2	no	—
227	C₂×C₉	—	2	1	no	—
228	C₂×C₉	C₂×C₃×Q₁₆	1	2	yes	complemented
229	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
230	D₉	C₂×C₃×Q₁₆	1	4	yes	complemented
231	C₂×Q₈	—	9	8	no	—
232	C₂×Q₈	—	9	2	no	—
233	C₂×Q₈	—	18	2	no	—
234	C₂×Q₈	—	18	4	no	—
235	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
236	C₂²×C₄	—	9	1	no	—
237	C₂²×C₄	—	18	2	no	—
238	Q₁₆	—	9	8	no	—
239	Q₁₆	—	9	4	no	—
240	Q₁₆	C₂×AGL(1,ℤ/9)	1	4	yes	direct product factor
241	C₂×C₈	—	9	1	no	—
242	C₂×C₈	—	9	4	no	—
243	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
244	C₂²×C₃	—	3	1	no	—
245	C₂²×C₃	—	9	2	no	—
246	C₂²×C₃	—	9	4	no	—
247	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
248	C₂×D₃	—	3	4	no	—
249	C₂×D₃	—	3	2	no	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	3	1	no	—
252	C₃×C₄	—	3	1	no	—
253	C₃×C₄	—	6	4	no	—
254	C₃×C₄	—	9	2	no	—
255	C₃×C₄	—	18	4	no	—
256	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
257	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
258	Dic₃	—	3	2	no	—
259	Dic₃	—	6	4	no	—
260	C₃²	C₂×D₃×Q₁₆	1	1	yes	characteristic
261	C₉	—	2	1	no	—
262	C₉	C₂²×C₃×Q₁₆	1	1	yes	characteristic, complemented
263	C₂³	—	9	1	no	—
264	Q₈	—	2	2	no	—
265	Q₈	—	9	4	no	—
266	Q₈	—	18	8	no	—
267	Q₈	—	18	2	no	—
268	Q₈	C₂²×AGL(1,ℤ/9)	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	9	1	no	—
271	C₂×C₄	—	9	2	no	—
272	C₂×C₄	—	9	2	no	—
273	C₂×C₄	—	18	8	no	—
274	C₂×C₄	—	18	2	no	—
275	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
276	C₈	—	9	2	no	—
277	C₈	C₂²×AGL(1,ℤ/9)	1	2	yes	—
278	C₂×C₃	—	3	2	no	—
279	C₂×C₃	—	3	1	no	—
280	C₂×C₃	—	9	4	no	—
281	C₂×C₃	C₃×D₃×Q₁₆	1	2	yes	—
282	C₂×C₃	C₂×C₃×D₃×D₄	1	1	yes	characteristic
283	D₃	—	3	4	no	—
284	C₂²	—	9	4	no	—
285	C₂²	—	9	2	no	—
286	C₂²	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	9	2	no	—
289	C₄	—	18	4	no	—
290	C₄	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
291	C₄	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
292	C₃	—	3	1	no	—
293	C₃	C₂×C₃×D₃×Q₁₆	1	1	yes	characteristic
294	C₂	—	9	4	no	—
295	C₂	Q₁₆×AGL(1,ℤ/9)	1	2	yes	direct product factor
296	C₂	C₂×D₄×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,ℤ/9)
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	C₂×Q₁₆×3₋¹⁺²	C₂
socle	C₂²×C₃	C₃×D₃×D₄
prime radical	C₂²×AGL(1,ℤ/9)	D₄
hypercenter	C₂×Q₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₂²×C₃×Q₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×3₋¹⁺²	C₂	semidirect
3. C₂×#3674⋊C₃	C₂	semidirect
4. Q₁₆×AGL(1,ℤ/9)	C₂	direct (N × H)
6. C₂×Dic₃₆⋊C₃	C₂	semidirect
8. C₂×Q₁₆×D₉	C₃	semidirect
16. Q₁₆×3₋¹⁺²	C₂²	semidirect
18. #3674⋊C₃	C₂²	semidirect
21. Dic₃₆⋊C₃	C₂²	semidirect
30. C₂×C₉×Q₁₆	C₂×C₃	semidirect
31. C₂×#3674	C₂×C₃	semidirect
32. Q₁₆×D₉	C₂×C₃	semidirect
34. C₂×Dic₃₆	C₂×C₃	semidirect
74. C₉×Q₁₆	C₂²×C₃	semidirect
77. #3674	C₂²×C₃	semidirect
80. Dic₃₆	C₂²×C₃	semidirect
82. C₂×AGL(1,ℤ/9)	Q₁₆	direct (N × H)
137. C₂×3₋¹⁺²	C₂×Q₁₆	semidirect
139. AGL(1,ℤ/9)	C₂×Q₁₆	direct (N × H)
176. C₂×D₉	C₃×Q₁₆	semidirect
190. C₂×Q₁₆	AGL(1,ℤ/9)	direct (N × H)
192. 3₋¹⁺²	C₂²×Q₁₆	semidirect
228. C₂×C₉	C₂×C₃×Q₁₆	semidirect
230. D₉	C₂×C₃×Q₁₆	semidirect
240. Q₁₆	C₂×AGL(1,ℤ/9)	direct (N × H)
262. C₉	C₂²×C₃×Q₁₆	semidirect
295. C₂	Q₁₆×AGL(1,ℤ/9)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×AGL(1,ℤ/9)	2
C₂×Q₁₆×3₋¹⁺²	2
C₂×#3674⋊C₃	2
Q₁₆×AGL(1,ℤ/9)	2
C₂×C₈×AGL(1,ℤ/9)	2
C₂×Dic₃₆⋊C₃	2
C₂×C₃×D₃×Q₁₆	3
C₂×Q₁₆×D₉	3

Subgroups — C2×Q16×AGL(1,ℤ/9) #13153467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₁₆×AGL(1,ℤ/9) #13153467