Subgroups — C₉×#2547 #11767268

Order 1152.

906 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#124	C₂	1	2	yes	maximal
3	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
4	C₉×#397	C₂	1	4	yes	maximal, complemented
5	C₉×#391	C₂	1	2	yes	maximal, complemented
6	C₉×#380	C₂	1	4	yes	maximal, complemented
7	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2547	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₉	C₂²	1	2	yes	—
12	C₉×C₄⋊Q₈	—	2	4	no	—
13	C₉×#124	—	2	8	no	—
14	C₄×D₄×C₉	—	2	4	no	—
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	1	yes	characteristic
19	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
20	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×#95	C₂²	1	2	yes	—
24	C₉×#95	C₂²	1	8	yes	—
25	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×#124	C₂×C₃	1	2	yes	—
27	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
28	C₃×#397	C₂×C₃	1	4	yes	—
29	C₃×#391	C₂×C₃	1	2	yes	—
30	C₃×#380	C₂×C₃	1	4	yes	—
31	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
32	C₂⁴×C₉	C₂³	1	2	yes	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	—	2	4	no	—
35	C₂×Q₈×C₉	—	2	4	no	—
36	C₂×D₄×C₉	—	2	2	no	—
37	C₂×D₄×C₉	—	2	2	no	—
38	C₂×D₄×C₉	D₄	1	4	yes	—
39	C₂²×C₄×C₉	—	2	4	no	—
40	C₂²×C₄×C₉	—	2	8	no	—
41	C₂²×C₄×C₉	C₂³	1	4	yes	—
42	C₂²×C₄×C₉	C₂³	1	4	yes	—
43	C₂²×C₄×C₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	2	4	no	—
49	C₉×C₄⋊C₄	—	2	4	no	—
50	C₉×C₄⋊C₄	—	2	4	no	—
51	C₉×C₂²⋊C₄	—	2	16	no	—
52	C₉×C₂²⋊C₄	—	2	4	no	—
53	C₄²×C₉	—	2	4	no	—
54	C₄²×C₉	D₄	1	4	yes	—
55	#2547	C₉	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
57	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
59	C₃×C₄⋊Q₈	—	2	4	no	—
60	C₃×#124	—	2	8	no	—
61	C₃×C₄×D₄	—	2	4	no	—
62	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
63	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
67	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
69	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₃×#95	C₂²×C₃	1	2	yes	—
71	C₃×#95	C₂²×C₃	1	8	yes	—
72	C₂³×C₉	—	2	2	no	—
73	C₂³×C₉	—	2	4	no	—
74	C₂³×C₉	—	2	2	no	—
75	C₂³×C₉	—	2	4	no	—
76	C₂³×C₉	C₂×D₄	1	4	yes	—
77	C₂³×C₉	C₂⁴	1	1	yes	characteristic
78	Q₈×C₉	—	4	8	no	—
79	D₄×C₉	—	2	8	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	4	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	4	8	no	—
90	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₉	#41	1	2	yes	—
95	C₂×C₄×C₉	#41	1	4	yes	—
96	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
97	C₂×#124	C₂×C₉	1	2	yes	—
98	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
99	#397	C₂×C₉	1	4	yes	complemented
100	#391	C₂×C₉	1	2	yes	complemented
101	#380	C₂×C₉	1	4	yes	complemented
102	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
104	C₂×C₃×Q₈	—	2	4	no	—
105	C₂×C₃×Q₈	—	2	4	no	—
106	C₂×C₃×Q₈	—	2	4	no	—
107	C₂×C₃×D₄	—	2	2	no	—
108	C₂×C₃×D₄	—	2	2	no	—
109	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	2	8	no	—
112	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
114	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
115	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	2	2	no	—
119	C₃×C₄⋊C₄	—	2	4	no	—
120	C₃×C₄⋊C₄	—	2	4	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₂²⋊C₄	—	2	16	no	—
123	C₃×C₂²⋊C₄	—	2	4	no	—
124	C₃×C₄²	—	2	4	no	—
125	C₃×C₄²	C₃×D₄	1	4	yes	—
126	C₂²×C₉	—	2	4	no	—
127	C₂²×C₉	—	2	8	no	—
128	C₂²×C₉	—	4	2	no	—
129	C₂²×C₉	—	4	2	no	—
130	C₂²×C₉	—	4	4	no	—
131	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
133	C₂²×C₉	C₂×#41	1	1	yes	characteristic
134	C₂²×C₉	C₂×#41	1	2	yes	—
135	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
136	C₄×C₉	—	2	2	no	—
137	C₄×C₉	—	4	4	no	—
138	C₄×C₉	—	4	2	no	—
139	C₄×C₉	—	4	4	no	—
140	C₄×C₉	#121	1	4	yes	—
141	C₄×C₉	#124	1	4	yes	complemented
142	C₂²×Q₈	C₂²×C₉	1	2	yes	—
143	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
144	C₂³×C₄	C₂²×C₉	1	2	yes	—
145	C₄⋊Q₈	—	2	4	no	—
146	#124	—	2	8	no	—
147	C₄×D₄	—	2	4	no	—
148	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
154	C₂×C₄²	C₂²×C₉	1	2	yes	—
155	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
156	#95	C₂²×C₉	1	2	yes	—
157	#95	C₂²×C₉	1	8	yes	—
158	C₂³×C₃	—	2	2	no	—
159	C₂³×C₃	—	2	4	no	—
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	2	4	no	—
162	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
163	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
164	C₃×Q₈	—	4	8	no	—
165	C₃×D₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	4	8	no	—
176	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
178	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₃×C₄	C₃×#41	1	2	yes	—
181	C₂×C₃×C₄	C₃×#41	1	4	yes	—
182	C₂×C₉	—	4	4	no	—
183	C₂×C₉	C₂×#121	1	1	yes	characteristic
184	C₂×C₉	C₂×#124	1	1	yes	characteristic
185	C₂×C₉	#536	1	1	yes	characteristic
186	C₂×C₉	#547	1	2	yes	—
187	C₂×C₉	#556	1	2	yes	—
188	C₂⁴	C₂³×C₉	1	2	yes	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	D₄×C₉	1	4	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	C₂³×C₉	1	2	yes	—
198	C₂²×C₄	C₂³×C₉	1	4	yes	—
199	C₂²×C₄	C₂³×C₉	1	4	yes	—
200	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	16	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₄²	—	2	4	no	—
210	C₄²	D₄×C₉	1	4	yes	—
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	2	8	no	—
213	C₂²×C₃	—	4	2	no	—
214	C₂²×C₃	—	4	2	no	—
215	C₂²×C₃	—	4	4	no	—
216	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
217	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
218	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
220	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	4	4	no	—
223	C₃×C₄	—	4	2	no	—
224	C₃×C₄	—	4	4	no	—
225	C₃×C₄	C₃×#121	1	4	yes	—
226	C₃×C₄	C₃×#124	1	4	yes	—
227	C₉	#2547	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	—	2	2	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×D₄×C₉	1	4	yes	—
233	C₂³	C₂⁴×C₉	1	1	yes	characteristic
234	Q₈	—	4	8	no	—
235	D₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
247	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
248	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
249	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
250	C₂×C₄	C₉×#41	1	2	yes	—
251	C₂×C₄	C₉×#41	1	4	yes	—
252	C₂×C₃	—	4	4	no	—
253	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
254	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
255	C₂×C₃	C₃×#536	1	1	yes	characteristic
256	C₂×C₃	C₃×#547	1	2	yes	—
257	C₂×C₃	C₃×#556	1	2	yes	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
264	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
265	C₂²	C₂×C₉×#41	1	1	yes	characteristic
266	C₂²	C₂×C₉×#41	1	2	yes	—
267	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	4	no	—
272	C₄	C₉×#121	1	4	yes	—
273	C₄	C₉×#124	1	4	yes	complemented
274	C₃	C₃×#2547	1	1	yes	characteristic
275	C₂	—	4	4	no	—
276	C₂	C₂×C₉×#121	1	1	yes	characteristic
277	C₂	C₂×C₉×#124	1	1	yes	characteristic
278	C₂	C₉×#536	1	1	yes	characteristic
279	C₂	C₉×#547	1	2	yes	—
280	C₂	C₉×#556	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²×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
1. C₂×C₉×C₄⋊Q₈	C₂	semidirect
4. C₉×#397	C₂	semidirect
5. C₉×#391	C₂	semidirect
6. C₉×#380	C₂	semidirect
7. C₄×C₉×C₄⋊C₄	C₂	semidirect
55. #2547	C₉	direct (N × H)
96. C₂×C₄⋊Q₈	C₂×C₉	semidirect
99. #397	C₂×C₉	semidirect
100. #391	C₂×C₉	semidirect
101. #380	C₂×C₉	semidirect
102. C₄×C₄⋊C₄	C₂×C₉	semidirect
141. C₄×C₉	#124	semidirect
227. C₉	#2547	direct (N × H)
273. C₄	C₉×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#124	2
C₂×C₄×D₄×C₉	2
C₉×#397	2
C₉×#391	2
C₉×#380	2
C₄×C₉×C₄⋊C₄	2
C₃×#2547	3

Subgroups — C9×#2547 #11767268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2547 #11767268