Subgroups — C₉×#2568 #11767289

Order 1152.

702 conjugacy classes of subgroups.

The subgroup lattice has 243 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
2	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#400	C₂	1	4	yes	maximal
4	C₉×#391	C₂	1	2	yes	maximal
5	C₉×#391	C₂	1	4	yes	maximal
6	C₉×#382	C₂	1	2	yes	maximal
7	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₃×#2568	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	4	no	—
12	C₄×Q₈×C₉	—	2	4	no	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
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×C₄²×C₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₉×#95	C₂²	1	2	yes	—
22	C₉×#95	C₂²	1	8	yes	—
23	C₉×#95	C₂²	1	8	yes	—
24	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
26	C₃×#400	C₂×C₃	1	4	yes	—
27	C₃×#391	C₂×C₃	1	4	yes	—
28	C₃×#391	C₂×C₃	1	2	yes	—
29	C₃×#382	C₂×C₃	1	2	yes	—
30	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
31	C₂×Q₈×C₉	—	2	2	no	—
32	C₂×Q₈×C₉	—	2	4	no	—
33	C₂×Q₈×C₉	—	2	2	no	—
34	C₂×Q₈×C₉	—	2	8	no	—
35	C₂×Q₈×C₉	Q₈	1	4	yes	complemented
36	C₂²×C₄×C₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₉×C₄⋊C₄	—	2	4	no	—
44	C₉×C₄⋊C₄	—	2	2	no	—
45	C₉×C₄⋊C₄	—	2	4	no	—
46	C₉×C₄⋊C₄	—	2	4	no	—
47	C₉×C₄⋊C₄	—	2	4	no	—
48	C₄²×C₉	—	2	4	no	—
49	C₄²×C₉	—	2	4	no	—
50	C₄²×C₉	D₄	1	4	yes	—
51	#2568	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
53	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₃×C₄⋊Q₈	—	2	4	no	—
55	C₃×C₄×Q₈	—	2	4	no	—
56	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
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×C₄²	C₂²×C₃	1	2	yes	—
62	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
63	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
68	Q₈×C₉	—	2	8	no	—
69	Q₈×C₉	—	4	8	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	4	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	8	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
84	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₉	#41	1	2	yes	—
86	C₂×C₄×C₉	#41	1	4	yes	—
87	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
88	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
89	#400	C₂×C₉	1	4	yes	—
90	#391	C₂×C₉	1	4	yes	—
91	#391	C₂×C₉	1	2	yes	—
92	#382	C₂×C₉	1	2	yes	—
93	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
94	C₂×C₃×Q₈	—	2	2	no	—
95	C₂×C₃×Q₈	—	2	4	no	—
96	C₂×C₃×Q₈	—	2	2	no	—
97	C₂×C₃×Q₈	—	2	8	no	—
98	C₂×C₃×Q₈	C₃×Q₈	1	4	yes	—
99	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
100	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
101	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
102	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	4	no	—
110	C₃×C₄⋊C₄	—	2	4	no	—
111	C₃×C₄²	—	2	4	no	—
112	C₃×C₄²	—	2	4	no	—
113	C₃×C₄²	C₃×D₄	1	4	yes	—
114	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₉	C₂×#41	1	1	yes	characteristic
117	C₂²×C₉	C₂×#41	1	2	yes	—
118	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
119	C₄×C₉	—	2	4	no	—
120	C₄×C₉	—	2	2	no	—
121	C₄×C₉	—	4	4	no	—
122	C₄×C₉	—	4	2	no	—
123	C₄×C₉	—	4	4	no	—
124	C₄×C₉	C₂²⋊Q₈	1	4	yes	—
125	C₄×C₉	#124	1	4	yes	—
126	C₂²×Q₈	C₂²×C₉	1	2	yes	—
127	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	4	no	—
129	C₄×Q₈	—	2	4	no	—
130	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
135	C₂×C₄²	C₂²×C₉	1	2	yes	—
136	C₂×C₄²	C₂²×C₉	1	2	yes	—
137	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
138	#95	C₂²×C₉	1	2	yes	—
139	#95	C₂²×C₉	1	8	yes	—
140	#95	C₂²×C₉	1	8	yes	—
141	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
142	C₃×Q₈	—	2	8	no	—
143	C₃×Q₈	—	4	8	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	8	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
158	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
159	C₂×C₃×C₄	C₃×#41	1	2	yes	—
160	C₂×C₃×C₄	C₃×#41	1	4	yes	—
161	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₉	C₂×#124	1	1	yes	characteristic
163	C₂×C₉	#536	1	1	yes	characteristic
164	C₂×C₉	#556	1	2	yes	—
165	C₂×C₉	#557	1	2	yes	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	8	no	—
170	C₂×Q₈	Q₈×C₉	1	4	yes	complemented
171	C₂²×C₄	C₂³×C₉	1	4	yes	—
172	C₂²×C₄	C₂³×C₉	1	2	yes	—
173	C₂²×C₄	C₂³×C₉	1	4	yes	—
174	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₉	1	2	yes	—
176	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	2	4	no	—
185	C₄²	D₄×C₉	1	4	yes	—
186	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
187	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
188	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
189	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
190	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
191	C₃×C₄	—	2	4	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	4	4	no	—
194	C₃×C₄	—	4	2	no	—
195	C₃×C₄	—	4	4	no	—
196	C₃×C₄	C₃×C₂²⋊Q₈	1	4	yes	—
197	C₃×C₄	C₃×#124	1	4	yes	—
198	C₉	#2568	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	C₂⁴×C₉	1	1	yes	characteristic
200	Q₈	—	2	8	no	—
201	Q₈	—	4	8	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
215	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
216	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
217	C₂×C₄	C₉×#41	1	2	yes	—
218	C₂×C₄	C₉×#41	1	4	yes	—
219	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
221	C₂×C₃	C₃×#536	1	1	yes	characteristic
222	C₂×C₃	C₃×#556	1	2	yes	—
223	C₂×C₃	C₃×#557	1	2	yes	—
224	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
225	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
226	C₂²	C₂×C₉×#41	1	1	yes	characteristic
227	C₂²	C₂×C₉×#41	1	2	yes	—
228	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	4	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	4	no	—
234	C₄	C₉×C₂²⋊Q₈	1	4	yes	—
235	C₄	C₉×#124	1	4	yes	—
236	C₃	C₃×#2568	1	1	yes	characteristic
237	C₂	C₂×C₉×C₂²⋊Q₈	1	1	yes	characteristic
238	C₂	C₂×C₉×#124	1	1	yes	characteristic
239	C₂	C₉×#536	1	1	yes	characteristic
240	C₂	C₉×#556	1	2	yes	—
241	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
35. C₂×Q₈×C₉	Q₈	semidirect
51. #2568	C₉	direct (N × H)
170. C₂×Q₈	Q₈×C₉	semidirect
198. C₉	#2568	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×C₄⋊Q₈	2
C₂×C₄×Q₈×C₉	2
C₉×#400	2
C₉×#391	2
C₉×#382	2
C₄×C₉×C₄⋊C₄	2
C₃×#2568	3

Subgroups — C9×#2568 #11767289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2568 #11767289