Subgroups — C₂×C₄×C₈×D₉ #11857094

Order 1152.

1140 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₉	C₂	1	2	yes	maximal
2	C₂×C₄²×D₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×Dic₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₉	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×D₃×C₈	—	3	1	no	maximal
8	C₂²×C₄×D₉	C₄	1	2	yes	—
9	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₉	C₄	1	2	yes	—
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×D₉	C₄	1	16	yes	direct product factor
16	C₂×C₈×D₉	C₂²	1	8	yes	—
17	C₄²×D₉	C₄	1	4	yes	—
18	C₄²×D₉	C₂²	1	4	yes	—
19	C₄×C₈×C₉	C₂²	1	4	yes	complemented
20	C₈×Dic₉	C₂²	1	8	yes	complemented
21	C₄×C₉⋊C₈	C₂²	1	4	yes	complemented
22	C₂²×D₃×C₈	—	3	2	no	—
23	C₂×C₄²×D₃	—	3	1	no	—
24	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
25	C₂×C₈×Dic₃	—	3	2	no	—
26	C₂×C₄×C₃⋊C₈	—	3	1	no	—
27	C₄×D₃×C₈	—	3	8	no	—
28	C₂³×D₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₉	C₈	1	16	yes	direct product factor
34	C₂×C₄×D₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₉	C₂×C₄	1	8	yes	—
36	C₂×C₄×D₉	C₂×C₄	1	8	yes	—
37	C₂×C₄×D₉	C₂³	1	2	yes	—
38	C₂×C₈×C₉	C₂×C₄	1	8	yes	complemented
39	C₂×C₈×C₉	C₂³	1	4	yes	—
40	C₄²×C₉	C₂×C₄	1	2	yes	—
41	C₄²×C₉	C₂³	1	2	yes	—
42	C₄×Dic₉	C₂×C₄	1	2	yes	—
43	C₄×Dic₉	C₂×C₄	1	8	yes	—
44	C₄×Dic₉	C₂³	1	2	yes	—
45	C₂×C₉⋊C₈	C₂×C₄	1	8	yes	complemented
46	C₂×C₉⋊C₈	C₂³	1	4	yes	—
47	C₈×D₉	C₂×C₄	1	32	yes	direct product factor
48	C₂²×C₄×C₈	—	9	1	no	Sylow
49	C₂²×C₄×D₃	—	3	2	no	—
50	C₂²×C₄×D₃	—	3	1	no	—
51	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
52	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
53	C₂×C₄×Dic₃	—	3	2	no	—
54	C₂×C₄×Dic₃	—	3	1	no	—
55	C₂²×C₃⋊C₈	—	3	2	no	—
56	C₂×D₃×C₈	—	3	16	no	—
57	C₂×D₃×C₈	—	3	8	no	—
58	C₄²×D₃	—	3	4	no	—
59	C₄²×D₃	—	3	4	no	—
60	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
61	C₈×Dic₃	—	3	8	no	—
62	C₄×C₃⋊C₈	—	3	4	no	—
63	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×D₉	C₄²	1	4	yes	—
65	C₂²×D₉	C₂×C₈	1	8	yes	—
66	C₂²×D₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂×C₈	1	8	yes	complemented
68	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₉	C₄²	1	4	yes	—
73	C₂×Dic₉	C₂×C₈	1	8	yes	complemented
74	C₂×Dic₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₉	C₂²×C₄	1	4	yes	—
76	C₄×D₉	C₄²	1	8	yes	—
77	C₄×D₉	C₂×C₈	1	32	yes	direct product factor
78	C₄×D₉	C₂²×C₄	1	8	yes	—
79	C₈×C₉	C₂²×C₄	1	8	yes	complemented
80	C₉⋊C₈	C₂²×C₄	1	8	yes	complemented
81	C₂³×C₈	—	9	2	no	—
82	C₂²×C₄²	—	9	1	no	—
83	C₂×C₄×C₈	—	9	2	no	—
84	C₂×C₄×C₈	—	9	1	no	—
85	C₂×C₄×C₈	—	9	8	no	—
86	C₂×C₄×C₈	D₉	1	1	yes	characteristic, direct product factor
87	C₂³×D₃	—	3	1	no	—
88	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
89	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	3	1	no	—
91	C₂²×Dic₃	—	3	2	no	—
92	C₂×C₄×D₃	—	3	16	no	—
93	C₂×C₄×D₃	—	3	2	no	—
94	C₂×C₄×D₃	—	3	8	no	—
95	C₂×C₄×D₃	—	3	8	no	—
96	C₂×C₄×D₃	—	3	2	no	—
97	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
98	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
99	C₃×C₄²	C₄×D₃	1	2	yes	—
100	C₃×C₄²	C₂²×D₃	1	2	yes	—
101	C₄×Dic₃	—	3	2	no	—
102	C₄×Dic₃	—	3	8	no	—
103	C₄×Dic₃	—	3	2	no	—
104	C₂×C₃⋊C₈	—	3	8	no	—
105	C₂×C₃⋊C₈	—	3	4	no	—
106	D₃×C₈	—	3	32	no	—
107	C₂²×C₉	C₂×C₄²	1	2	yes	—
108	C₂²×C₉	C₂²×C₈	1	4	yes	—
109	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×D₉	C₄×C₈	1	16	yes	direct product factor
111	C₂×D₉	C₂×C₄²	1	4	yes	—
112	C₂×D₉	C₂²×C₈	1	8	yes	—
113	C₄×C₉	C₂×C₄²	1	2	yes	—
114	C₄×C₉	C₂²×C₈	1	8	yes	complemented
115	C₄×C₉	C₂³×C₄	1	2	yes	—
116	Dic₉	C₂×C₄²	1	4	yes	—
117	Dic₉	C₂²×C₈	1	8	yes	complemented
118	C₂³×C₄	—	9	2	no	—
119	C₂³×C₄	—	9	1	no	—
120	C₂²×C₈	—	9	2	no	—
121	C₂²×C₈	—	9	16	no	—
122	C₂²×C₈	—	9	8	no	—
123	C₂²×C₈	C₂×D₉	1	2	yes	—
124	C₂×C₄²	—	9	2	no	—
125	C₂×C₄²	—	9	1	no	—
126	C₂×C₄²	—	9	4	no	—
127	C₂×C₄²	—	9	4	no	—
128	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
129	C₄×C₈	—	9	8	no	—
130	C₄×C₈	—	9	4	no	—
131	C₄×C₈	C₂×D₉	1	4	yes	direct product factor
132	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂²×D₃	—	3	4	no	—
134	C₂²×D₃	—	3	8	no	—
135	C₂²×D₃	—	3	2	no	—
136	C₂×C₃×C₄	D₃×C₈	1	8	yes	—
137	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
139	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
140	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
141	C₂×Dic₃	—	3	4	no	—
142	C₂×Dic₃	—	3	8	no	—
143	C₂×Dic₃	—	3	2	no	—
144	C₂×Dic₃	—	3	4	no	—
145	C₄×D₃	—	3	8	no	—
146	C₄×D₃	—	3	32	no	—
147	C₄×D₃	—	3	8	no	—
148	C₃×C₈	C₂×C₄×D₃	1	8	yes	—
149	C₃⋊C₈	—	3	8	no	—
150	C₂×C₉	C₂×C₄×C₈	1	4	yes	complemented
151	C₂×C₉	C₂²×C₄²	1	1	yes	characteristic
152	C₂×C₉	C₂³×C₈	1	2	yes	—
153	D₉	C₂×C₄×C₈	1	8	yes	direct product factor
154	C₂⁴	—	9	1	no	—
155	C₂²×C₄	—	9	1	no	—
156	C₂²×C₄	—	9	2	no	—
157	C₂²×C₄	—	9	16	no	—
158	C₂²×C₄	—	9	2	no	—
159	C₂²×C₄	—	9	8	no	—
160	C₂²×C₄	—	9	8	no	—
161	C₂²×C₄	—	9	2	no	—
162	C₂²×C₄	C₄×D₉	1	2	yes	—
163	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
164	C₂×C₈	—	9	8	no	—
165	C₂×C₈	—	9	4	no	—
166	C₂×C₈	—	9	32	no	—
167	C₂×C₈	C₄×D₉	1	8	yes	direct product factor
168	C₂×C₈	C₂²×D₉	1	4	yes	—
169	C₄²	—	9	2	no	—
170	C₄²	—	9	8	no	—
171	C₄²	—	9	2	no	—
172	C₄²	C₄×D₉	1	2	yes	—
173	C₄²	C₂²×D₉	1	2	yes	—
174	C₂²×C₃	C₄²×D₃	1	2	yes	—
175	C₂²×C₃	C₂×D₃×C₈	1	4	yes	—
176	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
177	C₂×D₃	—	3	16	no	—
178	C₂×D₃	—	3	4	no	—
179	C₂×D₃	—	3	8	no	—
180	C₃×C₄	C₄²×D₃	1	2	yes	—
181	C₃×C₄	C₂×D₃×C₈	1	8	yes	—
182	C₃×C₄	C₂²×C₄×D₃	1	2	yes	—
183	Dic₃	—	3	4	no	—
184	Dic₃	—	3	8	no	—
185	C₉	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
186	C₂³	—	9	4	no	—
187	C₂³	—	9	2	no	—
188	C₂³	—	9	8	no	—
189	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
190	C₂×C₄	—	9	4	no	—
191	C₂×C₄	—	9	2	no	—
192	C₂×C₄	—	9	4	no	—
193	C₂×C₄	—	9	8	no	—
194	C₂×C₄	—	9	8	no	—
195	C₂×C₄	—	9	8	no	—
196	C₂×C₄	—	9	32	no	—
197	C₂×C₄	C₈×D₉	1	8	yes	direct product factor
198	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₉	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₉	1	4	yes	—
201	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
202	C₈	—	9	8	no	—
203	C₈	C₂×C₄×D₉	1	8	yes	direct product factor
204	C₂×C₃	C₄×D₃×C₈	1	4	yes	—
205	C₂×C₃	C₂×C₄²×D₃	1	1	yes	characteristic
206	C₂×C₃	C₂²×D₃×C₈	1	2	yes	—
207	D₃	—	3	8	no	—
208	C₂²	—	9	4	no	—
209	C₂²	—	9	8	no	—
210	C₂²	—	9	16	no	—
211	C₂²	C₄²×D₉	1	2	yes	—
212	C₂²	C₂×C₈×D₉	1	4	yes	—
213	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
214	C₄	—	9	4	no	—
215	C₄	—	9	8	no	—
216	C₄	C₄²×D₉	1	2	yes	—
217	C₄	C₂×C₈×D₉	1	8	yes	direct product factor
218	C₄	C₂²×C₄×D₉	1	2	yes	—
219	C₃	C₂×C₄×D₃×C₈	1	1	yes	characteristic
220	C₂	—	9	8	no	—
221	C₂	C₄×C₈×D₉	1	4	yes	direct product factor
222	C₂	C₂×C₄²×D₉	1	1	yes	characteristic
223	C₂	C₂²×C₈×D₉	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	D₉
commutator, hypocenter	C₉	C₂²×C₄×C₈
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle	C₂³×C₃	C₂×C₄×D₃
prime radical	C₂³×D₉	C₂×C₄
cosocle	C₂×C₄×C₉	C₂⁴
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₈×C₉	C₂	semidirect
4. C₂×C₈×Dic₉	C₂	semidirect
5. C₂×C₄×C₉⋊C₈	C₂	semidirect
6. C₄×C₈×D₉	C₂	direct (N × H)
15. C₂×C₈×D₉	C₄	direct (N × H)
19. C₄×C₈×C₉	C₂²	semidirect
20. C₈×Dic₉	C₂²	semidirect
21. C₄×C₉⋊C₈	C₂²	semidirect
33. C₂×C₄×D₉	C₈	direct (N × H)
38. C₂×C₈×C₉	C₂×C₄	semidirect
45. C₂×C₉⋊C₈	C₂×C₄	semidirect
47. C₈×D₉	C₂×C₄	direct (N × H)
67. C₂×C₄×C₉	C₂×C₈	semidirect
73. C₂×Dic₉	C₂×C₈	semidirect
77. C₄×D₉	C₂×C₈	direct (N × H)
79. C₈×C₉	C₂²×C₄	semidirect
80. C₉⋊C₈	C₂²×C₄	semidirect
86. C₂×C₄×C₈	D₉	direct (N × H)
110. C₂×D₉	C₄×C₈	direct (N × H)
114. C₄×C₉	C₂²×C₈	semidirect
117. Dic₉	C₂²×C₈	semidirect
131. C₄×C₈	C₂×D₉	direct (N × H)
150. C₂×C₉	C₂×C₄×C₈	semidirect
153. D₉	C₂×C₄×C₈	direct (N × H)
167. C₂×C₈	C₄×D₉	direct (N × H)
185. C₉	C₂²×C₄×C₈	semidirect
197. C₂×C₄	C₈×D₉	direct (N × H)
203. C₈	C₂×C₄×D₉	direct (N × H)
217. C₄	C₂×C₈×D₉	direct (N × H)
221. C₂	C₄×C₈×D₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₉	2
C₂×C₄²×D₉	2
C₂×C₄×C₈×C₉	2
C₂×C₈×Dic₉	2
C₂×C₄×C₉⋊C₈	2
C₄×C₈×D₉	2
C₂×C₄×D₃×C₈	3

Subgroups — C2×C4×C8×D9 #11857094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₈×D₉ #11857094