Subgroups — C₂×C₄×C₉×D₈ #11767657

Order 1152.

924 conjugacy classes of subgroups.

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

	subgroup	G/H
center	C₂²×C₄×C₉	D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂²×C₃×D₈	C₂×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄×C₉	C₂	semidirect
3. C₄×C₉×D₈	C₂	direct (N × H)
4. C₂×C₉×#107	C₂	semidirect
5. C₂×C₉×#102	C₂	semidirect
6. C₂×C₄×C₈×C₉	C₂	semidirect
10. C₂×C₉×D₈	C₄	direct (N × H)
15. C₄×D₄×C₉	C₂²	semidirect
19. C₉×#107	C₂²	semidirect
20. C₉×#102	C₂²	semidirect
21. C₄×C₈×C₉	C₂²	semidirect
31. C₂×D₄×C₉	C₂×C₄	semidirect
39. C₉×D₈	C₂×C₄	direct (N × H)
41. C₂×C₈×C₉	C₂×C₄	semidirect
49. C₂×C₄×D₈	C₉	direct (N × H)
71. D₄×C₉	C₂²×C₄	semidirect
76. C₂×C₄×C₉	D₈	direct (N × H)
84. C₈×C₉	C₂²×C₄	semidirect
86. C₂×C₄×D₄	C₂×C₉	semidirect
87. C₄×D₈	C₂×C₉	direct (N × H)
88. C₂×#107	C₂×C₉	semidirect
89. C₂×#102	C₂×C₉	semidirect
90. C₂×C₄×C₈	C₂×C₉	semidirect
123. C₄×C₉	C₂×D₈	direct (N × H)
128. C₂×D₈	C₄×C₉	direct (N × H)
133. C₄×D₄	C₂²×C₉	semidirect
137. #107	C₂²×C₉	semidirect
138. #102	C₂²×C₉	semidirect
139. C₄×C₈	C₂²×C₉	semidirect
162. C₂×C₉	C₄×D₈	direct (N × H)
169. C₂×D₄	C₂×C₄×C₉	semidirect
177. D₈	C₂×C₄×C₉	direct (N × H)
179. C₂×C₈	C₂×C₄×C₉	semidirect
201. C₉	C₂×C₄×D₈	direct (N × H)
209. D₄	C₂²×C₄×C₉	semidirect
214. C₂×C₄	C₉×D₈	direct (N × H)
222. C₈	C₂²×C₄×C₉	semidirect
239. C₄	C₂×C₉×D₈	direct (N × H)
244. C₂	C₄×C₉×D₈	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×C₉×D₈	2
C₂×C₄×D₄×C₉	2
C₄×C₉×D₈	2
C₂×C₉×#107	2
C₂×C₉×#102	2
C₂×C₄×C₈×C₉	2
C₂×C₃×C₄×D₈	3

Subgroups — C2×C4×C9×D8 #11767657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₉×D₈ #11767657