Subgroups — C₉×#2503 #11767224

Order 1152.

768 conjugacy classes of subgroups.

The subgroup lattice has 252 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×#400	C₂	1	4	yes	maximal, complemented
4	C₉×#391	C₂	1	2	yes	maximal, complemented
5	C₉×#382	C₂	1	2	yes	maximal, complemented
6	C₉×#381	C₂	1	4	yes	maximal
7	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₃×#2503	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₉	C₂²	1	2	yes	—
11	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄²×C₉	C₂²	1	2	yes	—
21	C₉×#95	C₂²	1	2	yes	—
22	C₉×#95	C₂²	1	8	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×#400	C₂×C₃	1	4	yes	—
27	C₃×#391	C₂×C₃	1	2	yes	—
28	C₃×#382	C₂×C₃	1	2	yes	—
29	C₃×#381	C₂×C₃	1	4	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	—	2	8	no	—
37	C₂²×C₄×C₉	D₄	1	4	yes	—
38	C₂²×C₄×C₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₉	C₂³	1	4	yes	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄	—	2	4	no	—
49	C₉×C₂²⋊C₄	—	2	4	no	—
50	C₄²×C₉	—	2	4	no	—
51	#2503	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₂²⋊Q₈	—	2	4	no	—
56	C₂×C₃×C₄⋊C₄	—	2	4	no	—
57	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
63	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
64	C₃×#95	C₂²×C₃	1	2	yes	—
65	C₃×#95	C₂²×C₃	1	8	yes	—
66	C₃×#95	C₂²×C₃	1	8	yes	—
67	C₂³×C₉	—	2	2	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	C₂×D₄	1	2	yes	—
70	C₂³×C₉	#41	1	4	yes	—
71	C₂³×C₉	C₂⁴	1	1	yes	characteristic
72	Q₈×C₉	—	4	4	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	8	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	4	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	2	8	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	4	no	—
83	C₂×C₄×C₉	—	2	4	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	#41	1	4	yes	—
88	C₂×C₂²⋊Q₈	C₂×C₉	1	1	yes	characteristic
89	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
90	#400	C₂×C₉	1	4	yes	complemented
91	#391	C₂×C₉	1	2	yes	complemented
92	#382	C₂×C₉	1	2	yes	complemented
93	#381	C₂×C₉	1	4	yes	—
94	C₄×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
95	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₃×Q₈	—	2	2	no	—
97	C₂×C₃×Q₈	—	2	4	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	4	no	—
100	C₂²×C₃×C₄	—	2	8	no	—
101	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
102	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
103	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
105	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
106	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	4	no	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	4	no	—
112	C₃×C₂²⋊C₄	—	2	4	no	—
113	C₃×C₂²⋊C₄	—	2	4	no	—
114	C₃×C₄²	—	2	4	no	—
115	C₂²×C₉	—	2	2	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	2	8	no	—
118	C₂²×C₉	#124	1	4	yes	—
119	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₉	C₂×#41	1	2	yes	—
121	C₂²×C₉	C₂×#41	1	2	yes	—
122	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
123	C₄×C₉	—	2	2	no	—
124	C₄×C₉	—	4	2	no	—
125	C₄×C₉	—	4	4	no	—
126	C₄×C₉	—	4	2	no	—
127	C₄×C₉	—	4	4	no	—
128	C₄×C₉	#123	1	4	yes	—
129	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₉	1	2	yes	—
131	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
132	C₂²⋊Q₈	—	2	4	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
135	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
140	C₂×C₄²	C₂²×C₉	1	2	yes	—
141	#95	C₂²×C₉	1	2	yes	—
142	#95	C₂²×C₉	1	8	yes	—
143	#95	C₂²×C₉	1	8	yes	—
144	C₂³×C₃	—	2	2	no	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
147	C₂³×C₃	C₃×#41	1	4	yes	—
148	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
149	C₃×Q₈	—	4	4	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	8	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	8	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	—	4	8	no	—
162	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
164	C₂×C₃×C₄	C₃×#41	1	4	yes	—
165	C₂×C₉	—	2	4	no	—
166	C₂×C₉	C₂×#123	1	1	yes	characteristic
167	C₂×C₉	C₂×#124	1	1	yes	characteristic
168	C₂×C₉	#536	1	1	yes	characteristic
169	C₂×C₉	#552	1	4	yes	—
170	C₂⁴	C₂³×C₉	1	1	yes	characteristic
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	D₄×C₉	1	4	yes	—
177	C₂²×C₄	C₂³×C₉	1	4	yes	—
178	C₂²×C₄	C₂³×C₉	1	2	yes	—
179	C₂²×C₄	C₂³×C₉	1	2	yes	—
180	C₂²×C₄	C₂³×C₉	1	4	yes	—
181	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₄²	—	2	4	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	8	no	—
193	C₂²×C₃	C₃×#124	1	4	yes	—
194	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
195	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
196	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
197	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	4	2	no	—
200	C₃×C₄	—	4	4	no	—
201	C₃×C₄	—	4	2	no	—
202	C₃×C₄	—	4	4	no	—
203	C₃×C₄	C₃×#123	1	4	yes	—
204	C₉	#2503	1	1	yes	characteristic, direct product factor, Sylow
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	C₂×D₄×C₉	1	2	yes	—
208	C₂³	C₉×#41	1	4	yes	—
209	C₂³	C₂⁴×C₉	1	1	yes	characteristic
210	Q₈	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
225	C₂×C₄	C₉×#41	1	4	yes	—
226	C₂×C₃	—	2	4	no	—
227	C₂×C₃	C₂×C₃×#123	1	1	yes	characteristic
228	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
229	C₂×C₃	C₃×#536	1	1	yes	characteristic
230	C₂×C₃	C₃×#552	1	4	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	8	no	—
234	C₂²	C₉×#124	1	4	yes	—
235	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
236	C₂²	C₂×C₉×#41	1	2	yes	—
237	C₂²	C₂×C₉×#41	1	2	yes	—
238	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	4	no	—
244	C₄	C₉×#123	1	4	yes	—
245	C₃	C₃×#2503	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	C₂×C₉×#123	1	1	yes	characteristic
248	C₂	C₂×C₉×#124	1	1	yes	characteristic
249	C₂	C₉×#536	1	1	yes	characteristic
250	C₂	C₉×#552	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×#400	C₂	semidirect
4. C₉×#391	C₂	semidirect
5. C₉×#382	C₂	semidirect
51. #2503	C₉	direct (N × H)
90. #400	C₂×C₉	semidirect
91. #391	C₂×C₉	semidirect
92. #382	C₂×C₉	semidirect
204. C₉	#2503	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×#400	2
C₉×#391	2
C₉×#382	2
C₉×#381	2
C₄×C₉×C₂²⋊C₄	2
C₃×#2503	3

Subgroups — C9×#2503 #11767224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2503 #11767224