Subgroups — C₉×#2546 #11767267

Order 1152.

810 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×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
4	C₉×#398	C₂	1	4	yes	maximal
5	C₉×#391	C₂	1	2	yes	maximal
6	C₉×#389	C₂	1	4	yes	maximal
7	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₃×#2546	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×C₄⋊Q₈	—	2	8	no	—
12	C₉×C₄⋊Q₈	—	2	4	no	—
13	C₄×Q₈×C₉	—	2	4	no	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄²×C₉	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	2	yes	—
26	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
28	C₃×#398	C₂×C₃	1	4	yes	—
29	C₃×#391	C₂×C₃	1	2	yes	—
30	C₃×#389	C₂×C₃	1	4	yes	—
31	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
32	C₂×Q₈×C₉	—	2	2	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	—	2	4	no	—
35	C₂×Q₈×C₉	—	2	2	no	—
36	C₂×Q₈×C₉	—	2	4	no	—
37	C₂×Q₈×C₉	D₄	1	4	yes	—
38	C₂²×C₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₉	C₂³	1	2	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	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₉×C₄⋊C₄	—	2	4	no	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	16	no	—
48	C₉×C₄⋊C₄	—	2	4	no	—
49	C₉×C₄⋊C₄	—	2	4	no	—
50	C₉×C₄⋊C₄	Q₈	1	8	yes	complemented
51	C₄²×C₉	—	2	4	no	—
52	C₄²×C₉	—	2	4	no	—
53	C₄²×C₉	D₄	1	4	yes	—
54	#2546	C₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
56	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₃×C₄⋊Q₈	—	2	8	no	—
58	C₃×C₄⋊Q₈	—	2	4	no	—
59	C₃×C₄×Q₈	—	2	4	no	—
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
62	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
67	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
68	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
69	C₃×#95	C₂²×C₃	1	2	yes	—
70	C₃×#95	C₂²×C₃	1	8	yes	—
71	C₂³×C₉	C₂⁴	1	1	yes	characteristic
72	Q₈×C₉	—	2	8	no	—
73	Q₈×C₉	—	4	8	no	—
74	C₂×C₄×C₉	—	2	4	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	4	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	4	no	—
84	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₉	C₂×Q₈	1	8	yes	—
90	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
91	C₂×C₄×C₉	#41	1	2	yes	—
92	C₂×C₄⋊Q₈	C₂×C₉	1	2	yes	—
93	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
94	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
95	#398	C₂×C₉	1	4	yes	—
96	#391	C₂×C₉	1	2	yes	—
97	#389	C₂×C₉	1	4	yes	—
98	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
99	C₂×C₃×Q₈	—	2	2	no	—
100	C₂×C₃×Q₈	—	2	4	no	—
101	C₂×C₃×Q₈	—	2	4	no	—
102	C₂×C₃×Q₈	—	2	2	no	—
103	C₂×C₃×Q₈	—	2	4	no	—
104	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
105	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
106	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
109	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
110	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	2	2	no	—
114	C₃×C₄⋊C₄	—	2	16	no	—
115	C₃×C₄⋊C₄	—	2	4	no	—
116	C₃×C₄⋊C₄	—	2	4	no	—
117	C₃×C₄⋊C₄	C₃×Q₈	1	8	yes	—
118	C₃×C₄²	—	2	4	no	—
119	C₃×C₄²	—	2	4	no	—
120	C₃×C₄²	C₃×D₄	1	4	yes	—
121	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₉	C₂²×Q₈	1	2	yes	—
124	C₂²×C₉	C₂×#41	1	1	yes	characteristic
125	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
126	C₄×C₉	—	2	4	no	—
127	C₄×C₉	—	2	2	no	—
128	C₄×C₉	—	2	8	no	—
129	C₄×C₉	—	4	4	no	—
130	C₄×C₉	—	4	2	no	—
131	C₄×C₉	#121	1	4	yes	—
132	C₄×C₉	C₄⋊Q₈	1	4	yes	complemented
133	C₂²×Q₈	C₂²×C₉	1	2	yes	—
134	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	8	no	—
136	C₄⋊Q₈	—	2	4	no	—
137	C₄×Q₈	—	2	4	no	—
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	2	yes	—
141	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	—
142	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
143	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
144	C₂×C₄²	C₂²×C₉	1	2	yes	—
145	C₂×C₄²	C₂²×C₉	1	2	yes	—
146	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
147	#95	C₂²×C₉	1	2	yes	—
148	#95	C₂²×C₉	1	8	yes	—
149	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
150	C₃×Q₈	—	2	8	no	—
151	C₃×Q₈	—	4	8	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	4	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	4	no	—
162	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
164	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
165	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
168	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
169	C₂×C₃×C₄	C₃×#41	1	2	yes	—
170	C₂×C₉	C₂×#121	1	1	yes	characteristic
171	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
172	C₂×C₉	#536	1	1	yes	characteristic
173	C₂×C₉	D₄×Q₈	1	2	yes	—
174	C₂×C₉	#557	1	2	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	D₄×C₉	1	4	yes	—
181	C₂²×C₄	C₂³×C₉	1	2	yes	—
182	C₂²×C₄	C₂³×C₉	1	4	yes	—
183	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₉	1	2	yes	—
185	C₂²×C₄	C₂³×C₉	1	4	yes	—
186	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	16	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	Q₈×C₉	1	8	yes	complemented
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	D₄×C₉	1	4	yes	—
197	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
199	C₂²×C₃	C₂²×C₃×Q₈	1	2	yes	—
200	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
201	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	8	no	—
205	C₃×C₄	—	4	4	no	—
206	C₃×C₄	—	4	2	no	—
207	C₃×C₄	C₃×#121	1	4	yes	—
208	C₃×C₄	C₃×C₄⋊Q₈	1	4	yes	—
209	C₉	#2546	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³	C₂⁴×C₉	1	1	yes	characteristic
211	Q₈	—	2	8	no	—
212	Q₈	—	4	8	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
225	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
227	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
228	C₂×C₄	C₂×Q₈×C₉	1	8	yes	—
229	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
230	C₂×C₄	C₉×#41	1	2	yes	—
231	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
232	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
233	C₂×C₃	C₃×#536	1	1	yes	characteristic
234	C₂×C₃	C₃×D₄×Q₈	1	2	yes	—
235	C₂×C₃	C₃×#557	1	2	yes	—
236	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
237	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
238	C₂²	C₂²×Q₈×C₉	1	2	yes	—
239	C₂²	C₂×C₉×#41	1	1	yes	characteristic
240	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	8	no	—
244	C₄	—	4	4	no	—
245	C₄	—	4	2	no	—
246	C₄	C₉×#121	1	4	yes	—
247	C₄	C₉×C₄⋊Q₈	1	4	yes	complemented
248	C₃	C₃×#2546	1	1	yes	characteristic
249	C₂	C₂×C₉×#121	1	1	yes	characteristic
250	C₂	C₂×C₉×C₄⋊Q₈	1	1	yes	characteristic
251	C₂	C₉×#536	1	1	yes	characteristic
252	C₂	D₄×Q₈×C₉	1	2	yes	—
253	C₂	C₉×#557	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, prime radical, cosocle	C₂³×C₃	C₂⁴×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
50. C₉×C₄⋊C₄	Q₈	semidirect
54. #2546	C₉	direct (N × H)
132. C₄×C₉	C₄⋊Q₈	semidirect
193. C₄⋊C₄	Q₈×C₉	semidirect
209. C₉	#2546	direct (N × H)
247. C₄	C₉×C₄⋊Q₈	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×C₄×Q₈×C₉	2
C₉×#398	2
C₉×#391	2
C₉×#389	2
C₄×C₉×C₄⋊C₄	2
C₃×#2546	3

Subgroups — C9×#2546 #11767267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2546 #11767267