Subgroups — C₉×#2054 #11766775

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#102	C₂	1	2	yes	maximal, complemented
4	C₉×#390	C₂	1	2	yes	maximal, complemented
5	C₉×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2054	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	C₂²	1	2	yes	—
8	C₂×C₉×D₈	—	2	4	no	—
9	C₂×C₉×D₈	—	2	2	no	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
14	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊C₈	—	2	2	no	—
17	C₉×#102	—	2	4	no	—
18	C₂²×C₃×D₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#102	C₂×C₃	1	2	yes	—
21	C₃×#390	C₂×C₃	1	2	yes	—
22	C₃×#340	C₂×C₃	1	1	yes	characteristic
23	C₂⁴×C₉	—	2	2	no	—
24	C₂×D₄×C₉	—	2	2	no	—
25	C₂×D₄×C₉	—	2	2	no	—
26	C₂×D₄×C₉	—	2	2	no	—
27	C₂×D₄×C₉	—	2	2	no	—
28	C₂×D₄×C₉	—	2	2	no	—
29	C₂×D₄×C₉	D₄	1	4	yes	—
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₉×D₈	—	2	8	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	—	2	2	no	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₂×C₈×C₉	D₄	1	2	yes	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₉×C₄⋊C₄	—	2	2	no	—
41	C₉×C₂²⋊C₄	—	4	4	no	—
42	C₉×C₂²⋊C₄	—	4	4	no	—
43	C₄²×C₉	—	2	1	no	—
44	C₄²×C₉	—	2	1	no	—
45	#2054	C₉	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃×D₈	—	2	4	no	—
48	C₂×C₃×D₈	—	2	2	no	—
49	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
53	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	2	no	—
56	C₃×#102	—	2	4	no	—
57	C₂³×C₉	—	2	4	no	—
58	C₂³×C₉	—	4	2	no	—
59	C₂³×C₉	—	4	2	no	—
60	C₂³×C₉	—	4	2	no	—
61	C₂³×C₉	—	4	2	no	—
62	C₂³×C₉	—	4	2	no	—
63	C₂³×C₉	—	4	2	no	—
64	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
65	D₄×C₉	—	2	4	no	—
66	D₄×C₉	—	2	4	no	—
67	D₄×C₉	—	4	2	no	—
68	D₄×C₉	—	4	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	4	2	no	—
74	C₂×C₄×C₉	—	4	2	no	—
75	C₂×C₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	D₈	1	2	yes	complemented
77	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₉	#41	1	1	yes	characteristic
80	C₂×C₄×C₉	#41	1	2	yes	—
81	C₈×C₉	—	2	2	no	—
82	C₈×C₉	—	4	1	no	—
83	C₂²×D₈	C₂×C₉	1	1	yes	characteristic
84	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
85	C₂×#102	C₂×C₉	1	2	yes	complemented
86	#390	C₂×C₉	1	2	yes	complemented
87	#340	C₂×C₉	1	1	yes	characteristic, complemented
88	C₂⁴×C₃	—	2	2	no	—
89	C₂×C₃×D₄	—	2	2	no	—
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂×C₃×D₄	—	2	2	no	—
92	C₂×C₃×D₄	—	2	2	no	—
93	C₂×C₃×D₄	—	2	2	no	—
94	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
95	C₂²×C₃×C₄	—	2	2	no	—
96	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₃×D₈	—	2	8	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₂²⋊C₄	—	4	4	no	—
107	C₃×C₂²⋊C₄	—	4	4	no	—
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	—	2	1	no	—
110	C₂²×C₉	—	4	4	no	—
111	C₂²×C₉	—	4	4	no	—
112	C₂²×C₉	—	4	4	no	—
113	C₂²×C₉	—	8	2	no	—
114	C₂²×C₉	—	8	2	no	—
115	C₂²×C₉	—	8	2	no	—
116	C₂²×C₉	—	8	2	no	—
117	C₂²×C₉	#121	1	2	yes	—
118	C₂²×C₉	#121	1	1	yes	characteristic
119	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
120	C₂²×C₉	#135	1	1	yes	characteristic
121	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	—	4	1	no	—
124	C₄×C₉	—	8	2	no	—
125	C₄×C₉	#120	1	1	yes	characteristic
126	C₄×C₉	#123	1	2	yes	—
127	C₄×C₉	#124	1	1	yes	characteristic
128	C₂²×D₄	C₂²×C₉	1	2	yes	—
129	C₂×D₈	—	2	4	no	—
130	C₂×D₈	—	2	2	no	—
131	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
132	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
134	C₂×C₂²⋊C₄	—	2	2	no	—
135	C₂×C₂²⋊C₄	—	2	2	no	—
136	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	#102	—	2	4	no	—
139	C₂³×C₃	—	2	4	no	—
140	C₂³×C₃	—	4	2	no	—
141	C₂³×C₃	—	4	2	no	—
142	C₂³×C₃	—	4	2	no	—
143	C₂³×C₃	—	4	2	no	—
144	C₂³×C₃	—	4	2	no	—
145	C₂³×C₃	—	4	2	no	—
146	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
147	C₃×D₄	—	2	4	no	—
148	C₃×D₄	—	2	4	no	—
149	C₃×D₄	—	4	2	no	—
150	C₃×D₄	—	4	2	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	—	4	2	no	—
157	C₂×C₃×C₄	—	4	2	no	—
158	C₂×C₃×C₄	C₃×D₈	1	2	yes	—
159	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
162	C₂×C₃×C₄	C₃×#41	1	2	yes	—
163	C₃×C₈	—	2	2	no	—
164	C₃×C₈	—	4	1	no	—
165	C₂×C₉	—	8	4	no	—
166	C₂×C₉	#394	1	1	yes	characteristic
167	C₂×C₉	#459	1	2	yes	—
168	C₂×C₉	#463	1	2	yes	—
169	C₂×C₉	#466	1	1	yes	characteristic
170	C₂×C₉	#469	1	1	yes	characteristic
171	C₂⁴	—	2	2	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	D₄×C₉	1	4	yes	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
182	D₈	—	2	8	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	D₄×C₉	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	4	4	no	—
190	C₂²⋊C₄	—	4	4	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₂²×C₃	—	4	4	no	—
194	C₂²×C₃	—	4	4	no	—
195	C₂²×C₃	—	4	4	no	—
196	C₂²×C₃	—	8	2	no	—
197	C₂²×C₃	—	8	2	no	—
198	C₂²×C₃	—	8	2	no	—
199	C₂²×C₃	—	8	2	no	—
200	C₂²×C₃	C₃×#121	1	2	yes	—
201	C₂²×C₃	C₃×#121	1	1	yes	characteristic
202	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
203	C₂²×C₃	C₃×#135	1	1	yes	characteristic
204	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	4	1	no	—
207	C₃×C₄	—	8	2	no	—
208	C₃×C₄	C₃×#120	1	1	yes	characteristic
209	C₃×C₄	C₃×#123	1	2	yes	—
210	C₃×C₄	C₃×#124	1	1	yes	characteristic
211	C₉	#2054	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³	—	2	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
220	D₄	—	2	4	no	—
221	D₄	—	2	4	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₉×D₈	1	2	yes	complemented
232	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
233	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
234	C₂×C₄	C₉×#41	1	1	yes	characteristic
235	C₂×C₄	C₉×#41	1	2	yes	—
236	C₈	—	2	2	no	—
237	C₈	—	4	1	no	—
238	C₂×C₃	—	8	4	no	—
239	C₂×C₃	C₃×#394	1	1	yes	characteristic
240	C₂×C₃	C₃×#459	1	2	yes	—
241	C₂×C₃	C₃×#463	1	2	yes	—
242	C₂×C₃	C₃×#466	1	1	yes	characteristic
243	C₂×C₃	C₃×#469	1	1	yes	characteristic
244	C₂²	—	4	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	C₉×#121	1	2	yes	—
252	C₂²	C₉×#121	1	1	yes	characteristic
253	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
254	C₂²	C₉×#135	1	1	yes	characteristic
255	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	8	2	no	—
259	C₄	C₉×#120	1	1	yes	characteristic
260	C₄	C₉×#123	1	2	yes	—
261	C₄	C₉×#124	1	1	yes	characteristic
262	C₃	C₃×#2054	1	1	yes	characteristic
263	C₂	—	8	4	no	—
264	C₂	C₉×#394	1	1	yes	characteristic
265	C₂	C₉×#459	1	2	yes	—
266	C₂	C₉×#463	1	2	yes	—
267	C₂	C₉×#466	1	1	yes	characteristic
268	C₂	C₉×#469	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂²×C₃×D₈	C₂×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊C₈	C₂	semidirect
3. C₂×C₉×#102	C₂	semidirect
4. C₉×#390	C₂	semidirect
5. C₉×#340	C₂	semidirect
45. #2054	C₉	direct (N × H)
76. C₂×C₄×C₉	D₈	semidirect
84. C₂×C₄⋊C₈	C₂×C₉	semidirect
85. C₂×#102	C₂×C₉	semidirect
86. #390	C₂×C₉	semidirect
87. #340	C₂×C₉	semidirect
211. C₉	#2054	direct (N × H)
231. C₂×C₄	C₉×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×D₈	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#102	2
C₉×#390	2
C₉×#340	2
C₃×#2054	3

Subgroups — C9×#2054 #11766775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2054 #11766775