Subgroups — C₉×#2502 #11767223

Order 1152.

816 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₉×#398	C₂	1	4	yes	maximal, complemented
4	C₉×#391	C₂	1	2	yes	maximal, complemented
5	C₉×#385	C₂	1	4	yes	maximal
6	C₉×#384	C₂	1	2	yes	maximal, complemented
7	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₃×#2502	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₉	C₂²	1	2	yes	—
12	C₉×C₂²⋊Q₈	—	2	4	no	—
13	C₂×C₉×C₄⋊C₄	—	2	4	no	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	—
19	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₉	C₂²	1	2	yes	—
22	C₉×#95	C₂²	1	2	yes	—
23	C₉×#95	C₂²	1	8	yes	—
24	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₃×#398	C₂×C₃	1	4	yes	—
27	C₃×#391	C₂×C₃	1	2	yes	—
28	C₃×#385	C₂×C₃	1	4	yes	—
29	C₃×#384	C₂×C₃	1	2	yes	—
30	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
31	C₂⁴×C₉	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₉	—	2	2	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂²×C₄×C₉	—	2	4	no	—
35	C₂²×C₄×C₉	—	2	4	no	—
36	C₂²×C₄×C₉	—	4	4	no	—
37	C₂²×C₄×C₉	D₄	1	4	yes	—
38	C₂²×C₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	4	yes	—
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×C₄⋊C₄	—	2	4	no	—
45	C₉×C₄⋊C₄	—	2	4	no	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	4	no	—
48	C₉×C₄⋊C₄	—	4	8	no	—
49	C₉×C₂²⋊C₄	—	2	4	no	—
50	C₉×C₂²⋊C₄	Q₈	1	8	yes	complemented
51	C₄²×C₉	—	2	4	no	—
52	#2502	C₉	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
56	C₃×C₂²⋊Q₈	—	2	4	no	—
57	C₂×C₃×C₄⋊C₄	—	2	4	no	—
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
63	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
66	C₃×#95	C₂²×C₃	1	2	yes	—
67	C₃×#95	C₂²×C₃	1	8	yes	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	—	2	2	no	—
70	C₂³×C₉	C₂×D₄	1	2	yes	—
71	C₂³×C₉	C₂×Q₈	1	4	yes	—
72	C₂³×C₉	C₂⁴	1	1	yes	characteristic
73	Q₈×C₉	—	4	4	no	—
74	C₂×C₄×C₉	—	2	4	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	8	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	4	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	4	no	—
83	C₂×C₄×C₉	—	2	8	no	—
84	C₂×C₄×C₉	—	4	8	no	—
85	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₉	C₂×Q₈	1	8	yes	—
88	C₂×C₄×C₉	#41	1	4	yes	—
89	C₂×C₂²⋊Q₈	C₂×C₉	1	1	yes	characteristic
90	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
91	#398	C₂×C₉	1	4	yes	complemented
92	#391	C₂×C₉	1	2	yes	complemented
93	#385	C₂×C₉	1	4	yes	—
94	#384	C₂×C₉	1	2	yes	complemented
95	C₄×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
96	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₃×Q₈	—	2	2	no	—
98	C₂×C₃×Q₈	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	4	no	—
100	C₂²×C₃×C₄	—	2	4	no	—
101	C₂²×C₃×C₄	—	4	4	no	—
102	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
103	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
105	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
106	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	4	no	—
110	C₃×C₄⋊C₄	—	2	4	no	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	4	8	no	—
114	C₃×C₂²⋊C₄	—	2	4	no	—
115	C₃×C₂²⋊C₄	C₃×Q₈	1	8	yes	—
116	C₃×C₄²	—	2	4	no	—
117	C₂²×C₉	—	2	2	no	—
118	C₂²×C₉	—	2	2	no	—
119	C₂²×C₉	—	2	8	no	—
120	C₂²×C₉	C₄⋊Q₈	1	4	yes	—
121	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₉	C₂²×Q₈	1	2	yes	—
123	C₂²×C₉	C₂×#41	1	2	yes	—
124	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
125	C₄×C₉	—	2	2	no	—
126	C₄×C₉	—	2	8	no	—
127	C₄×C₉	—	4	2	no	—
128	C₄×C₉	—	4	2	no	—
129	C₄×C₉	—	4	4	no	—
130	C₄×C₉	#123	1	4	yes	—
131	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
132	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₉	1	2	yes	—
134	C₂²⋊Q₈	—	2	4	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
143	C₂×C₄²	C₂²×C₉	1	2	yes	—
144	#95	C₂²×C₉	1	2	yes	—
145	#95	C₂²×C₉	1	8	yes	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	—	2	2	no	—
148	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
149	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
150	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
151	C₃×Q₈	—	4	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	4	8	no	—
163	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
164	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
165	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
166	C₂×C₃×C₄	C₃×#41	1	4	yes	—
167	C₂×C₉	—	2	4	no	—
168	C₂×C₉	C₂×#123	1	1	yes	characteristic
169	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
170	C₂×C₉	#535	1	1	yes	characteristic
171	C₂×C₉	#554	1	4	yes	—
172	C₂⁴	C₂³×C₉	1	1	yes	characteristic
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	D₄×C₉	1	4	yes	—
179	C₂²×C₄	C₂³×C₉	1	2	yes	—
180	C₂²×C₄	C₂³×C₉	1	4	yes	—
181	C₂²×C₄	C₂³×C₉	1	2	yes	—
182	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₉	1	4	yes	—
184	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	8	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	Q₈×C₉	1	8	yes	complemented
192	C₄²	—	2	4	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	8	no	—
196	C₂²×C₃	C₃×C₄⋊Q₈	1	4	yes	—
197	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₂²×C₃	C₂²×C₃×Q₈	1	2	yes	—
199	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
200	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	8	no	—
203	C₃×C₄	—	4	2	no	—
204	C₃×C₄	—	4	4	no	—
205	C₃×C₄	—	4	2	no	—
206	C₃×C₄	C₃×#123	1	4	yes	—
207	C₉	#2502	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂×D₄×C₉	1	2	yes	—
211	C₂³	C₂×Q₈×C₉	1	4	yes	—
212	C₂³	C₂⁴×C₉	1	1	yes	characteristic
213	Q₈	—	4	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
227	C₂×C₄	C₂×Q₈×C₉	1	8	yes	—
228	C₂×C₄	C₉×#41	1	4	yes	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	C₂×C₃×#123	1	1	yes	characteristic
231	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
232	C₂×C₃	C₃×#535	1	1	yes	characteristic
233	C₂×C₃	C₃×#554	1	4	yes	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	8	no	—
237	C₂²	C₉×C₄⋊Q₈	1	4	yes	—
238	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
239	C₂²	C₂²×Q₈×C₉	1	2	yes	—
240	C₂²	C₂×C₉×#41	1	2	yes	—
241	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	2	8	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	4	no	—
246	C₄	—	4	2	no	—
247	C₄	C₉×#123	1	4	yes	—
248	C₃	C₃×#2502	1	1	yes	characteristic
249	C₂	—	2	4	no	—
250	C₂	C₂×C₉×#123	1	1	yes	characteristic
251	C₂	C₂×C₉×C₄⋊Q₈	1	1	yes	characteristic
252	C₂	C₉×#535	1	1	yes	characteristic
253	C₂	C₉×#554	1	4	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#398	C₂	semidirect
4. C₉×#391	C₂	semidirect
6. C₉×#384	C₂	semidirect
50. C₉×C₂²⋊C₄	Q₈	semidirect
52. #2502	C₉	direct (N × H)
91. #398	C₂×C₉	semidirect
92. #391	C₂×C₉	semidirect
94. #384	C₂×C₉	semidirect
191. C₂²⋊C₄	Q₈×C₉	semidirect
207. C₉	#2502	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×C₉×C₂²⋊Q₈	2
C₂²×C₉×C₄⋊C₄	2
C₉×#398	2
C₉×#391	2
C₉×#385	2
C₉×#384	2
C₄×C₉×C₂²⋊C₄	2
C₃×#2502	3

Subgroups — C9×#2502 #11767223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2502 #11767223