Subgroups — C₉×#3230 #11767951

Order 1152.

540 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	C₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₉×#500	C₂	1	2	yes	maximal, complemented
4	C₉×#487	C₂	1	2	yes	maximal, complemented
5	C₉×#484	C₂	1	4	yes	maximal
6	C₉×#477	C₂	1	4	yes	maximal, complemented
7	C₉×#431	C₂	1	1	yes	maximal, characteristic
8	C₃×#3230	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	—	2	1	no	—
10	C₉×C₄⋊Q₈	—	2	1	no	—
11	C₉×C₄⋊Q₈	—	2	1	no	—
12	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×#125	—	2	1	no	—
17	C₉×C₂²⋊Q₈	—	2	1	no	—
18	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₄×Q₈×C₉	—	2	1	no	—
20	C₄×Q₈×C₉	C₂²	1	2	yes	—
21	C₉×#117	—	2	1	no	—
22	C₂×C₉×C₄⋊C₄	—	2	1	no	—
23	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₉×#107	C₂²	1	8	yes	—
26	C₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₉×#103	C₂²	1	8	yes	—
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	—
30	C₃×#533	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	C₃×#500	C₂×C₃	1	2	yes	—
33	C₃×#487	C₂×C₃	1	2	yes	—
34	C₃×#484	C₂×C₃	1	4	yes	—
35	C₃×#477	C₂×C₃	1	4	yes	—
36	C₃×#431	C₂×C₃	1	1	yes	characteristic
37	C₂×Q₈×C₉	—	2	1	no	—
38	C₂×Q₈×C₉	—	2	2	no	—
39	C₂×Q₈×C₉	—	2	2	no	—
40	C₂×Q₈×C₉	—	4	4	no	—
41	C₂×Q₈×C₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₉	—	2	2	no	—
43	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₉	C₂³	1	4	yes	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₄⋊C₄	—	2	1	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	—	2	2	no	—
55	C₉×C₄⋊C₄	C₂³	1	4	yes	—
56	C₉×C₄⋊C₄	C₂³	1	2	yes	—
57	C₉×C₂²⋊C₄	—	2	2	no	—
58	C₄²×C₉	—	2	2	no	—
59	C₄²×C₉	D₄	1	1	yes	characteristic
60	C₄²×C₉	D₄	1	1	yes	characteristic
61	C₄²×C₉	C₂³	1	1	yes	characteristic
62	C₄²×C₉	C₂³	1	1	yes	characteristic
63	#3230	C₉	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²×C₃×Q₈	—	2	1	no	—
65	C₃×C₄⋊Q₈	—	2	1	no	—
66	C₃×C₄⋊Q₈	—	2	1	no	—
67	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
68	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
69	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
70	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₃×#125	—	2	1	no	—
72	C₃×C₂²⋊Q₈	—	2	1	no	—
73	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
74	C₃×C₄×Q₈	—	2	1	no	—
75	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
76	C₃×#117	—	2	1	no	—
77	C₂×C₃×C₄⋊C₄	—	2	1	no	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
80	C₃×#107	C₂²×C₃	1	8	yes	—
81	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
82	C₃×#103	C₂²×C₃	1	8	yes	—
83	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
85	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₉	—	2	4	no	—
87	Q₈×C₉	—	4	4	no	—
88	Q₈×C₉	—	4	2	no	—
89	Q₈×C₉	—	4	4	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	4	4	no	—
96	C₂×C₄×C₉	Q₁₆	1	4	yes	—
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
103	C₈×C₉	—	2	4	no	—
104	#533	C₂×C₉	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
106	#500	C₂×C₉	1	2	yes	complemented
107	#487	C₂×C₉	1	2	yes	complemented
108	#484	C₂×C₉	1	4	yes	—
109	#477	C₂×C₉	1	4	yes	complemented
110	#431	C₂×C₉	1	1	yes	characteristic
111	C₂×C₃×Q₈	—	2	1	no	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₂×C₃×Q₈	—	4	4	no	—
115	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	1	no	—
125	C₃×C₄⋊C₄	—	2	1	no	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	—	2	2	no	—
129	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
130	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
131	C₃×C₂²⋊C₄	—	2	2	no	—
132	C₃×C₄²	—	2	2	no	—
133	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
134	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
135	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
136	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₉	—	2	1	no	—
138	C₂²×C₉	—	2	1	no	—
139	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
140	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	—	4	4	no	—
145	C₄×C₉	—	4	2	no	—
146	C₄×C₉	C₂×Q₁₆	1	2	yes	—
147	C₄×C₉	C₂×Q₁₆	1	2	yes	—
148	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
149	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
150	C₂²×Q₈	—	2	1	no	—
151	C₄⋊Q₈	—	2	1	no	—
152	C₄⋊Q₈	—	2	1	no	—
153	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
154	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
155	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
156	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
157	#125	—	2	1	no	—
158	C₂²⋊Q₈	—	2	1	no	—
159	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	C₂²×C₉	1	2	yes	—
162	#117	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
165	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
166	#107	C₂²×C₉	1	8	yes	—
167	C₄⋊C₈	C₂²×C₉	1	2	yes	—
168	#103	C₂²×C₉	1	8	yes	—
169	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
170	C₄×C₈	C₂²×C₉	1	2	yes	—
171	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
172	C₃×Q₈	—	2	4	no	—
173	C₃×Q₈	—	4	4	no	—
174	C₃×Q₈	—	4	2	no	—
175	C₃×Q₈	—	4	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	4	4	no	—
182	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
183	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
189	C₃×C₈	—	2	4	no	—
190	C₂×C₉	—	2	2	no	—
191	C₂×C₉	#536	1	1	yes	characteristic
192	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
193	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	C₂³×C₉	1	2	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
201	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₉	1	4	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	C₂³×C₉	1	4	yes	—
213	C₄⋊C₄	C₂³×C₉	1	2	yes	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	D₄×C₉	1	1	yes	characteristic
217	C₄²	D₄×C₉	1	1	yes	characteristic
218	C₄²	C₂³×C₉	1	1	yes	characteristic
219	C₄²	C₂³×C₉	1	1	yes	characteristic
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
223	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	4	4	no	—
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	C₂×C₃×Q₁₆	1	2	yes	—
230	C₃×C₄	C₂×C₃×Q₁₆	1	2	yes	—
231	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
233	C₉	#3230	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
235	Q₈	—	2	4	no	—
236	Q₈	—	4	4	no	—
237	Q₈	—	4	4	no	—
238	Q₈	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	C₉×Q₁₆	1	4	yes	—
246	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
251	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₂×C₃	—	2	2	no	—
254	C₂×C₃	C₃×#536	1	1	yes	characteristic
255	C₂×C₃	C₂²×C₃×Q₁₆	1	1	yes	characteristic
256	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	C₂×C₉×Q₁₆	1	2	yes	—
260	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	4	no	—
265	C₄	—	4	2	no	—
266	C₄	C₂×C₉×Q₁₆	1	2	yes	—
267	C₄	C₂×C₉×Q₁₆	1	2	yes	—
268	C₄	C₉×AGL(1,ℤ/8)	1	2	yes	—
269	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
270	C₃	C₃×#3230	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	C₉×#536	1	1	yes	characteristic
273	C₂	C₂²×C₉×Q₁₆	1	1	yes	characteristic
274	C₂	C₂×C₉×AGL(1,ℤ/8)	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	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#500	C₂	semidirect
4. C₉×#487	C₂	semidirect
6. C₉×#477	C₂	semidirect
63. #3230	C₉	direct (N × H)
106. #500	C₂×C₉	semidirect
107. #487	C₂×C₉	semidirect
109. #477	C₂×C₉	semidirect
233. C₉	#3230	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₂×C₉×C₄⋊Q₈	2
C₉×#500	2
C₉×#487	2
C₉×#484	2
C₉×#477	2
C₉×#431	2
C₃×#3230	3

Subgroups — C9×#3230 #11767951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3230 #11767951