Subgroups — C₂×C₉×#496 #11767869

Order 1152.

1002 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₂²×C₉×D₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#496	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#496	C₃	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₉	—	2	2	no	—
10	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₉	C₂²	1	2	yes	—
12	C₂×C₉×SD₁₆	—	2	8	no	—
13	C₂×C₉×SD₁₆	C₂²	1	8	yes	complemented
14	C₂×C₉×D₈	—	2	8	no	—
15	C₂×C₉×D₈	C₂²	1	8	yes	complemented
16	C₂²×C₈×C₉	C₂²	1	2	yes	—
17	C₉×#127	—	2	2	no	—
18	C₉×#127	C₂²	1	4	yes	complemented
19	C₉×#124	—	2	2	no	—
20	C₉×#124	C₂²	1	4	yes	complemented
21	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×#97	C₂²	1	4	yes	complemented
24	C₂²×C₃×SD₁₆	C₂×C₃	1	2	yes	—
25	C₂²×C₃×D₈	C₂×C₃	1	2	yes	—
26	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
28	C₃×#496	C₂×C₃	1	8	yes	—
29	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
30	C₂⁴×C₉	—	2	1	no	—
31	C₂⁴×C₉	—	2	2	no	—
32	C₂×Q₈×C₉	—	2	1	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	C₂³	1	2	yes	—
35	C₂×D₄×C₉	—	2	4	no	—
36	C₂×D₄×C₉	—	2	4	no	—
37	C₂×D₄×C₉	—	2	4	no	—
38	C₂×D₄×C₉	—	2	1	no	—
39	C₂×D₄×C₉	—	2	2	no	—
40	C₂×D₄×C₉	—	2	8	no	—
41	C₂×D₄×C₉	—	4	8	no	—
42	C₂×D₄×C₉	C₂³	1	2	yes	—
43	C₂×D₄×C₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	D₄	1	2	yes	—
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×SD₁₆	—	2	16	no	—
48	C₉×D₈	—	2	16	no	—
49	C₂×C₈×C₉	D₄	1	8	yes	complemented
50	C₂×C₈×C₉	C₂³	1	4	yes	—
51	C₉×C₂²⋊C₄	—	2	8	no	—
52	C₄²×C₉	D₄	1	2	yes	—
53	C₄²×C₉	C₂³	1	2	yes	—
54	C₂×#496	C₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₃×D₄	—	2	2	no	—
57	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
59	C₂×C₃×SD₁₆	—	2	8	no	—
60	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	—
61	C₂×C₃×D₈	—	2	8	no	—
62	C₂×C₃×D₈	C₂²×C₃	1	8	yes	—
63	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
64	C₃×#127	—	2	2	no	—
65	C₃×#127	C₂²×C₃	1	4	yes	—
66	C₃×#124	—	2	2	no	—
67	C₃×#124	C₂²×C₃	1	4	yes	—
68	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
69	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
70	C₃×#97	C₂²×C₃	1	4	yes	—
71	C₂³×C₉	—	2	2	no	—
72	C₂³×C₉	—	2	4	no	—
73	C₂³×C₉	—	4	2	no	—
74	C₂³×C₉	—	4	4	no	—
75	C₂³×C₉	—	4	8	no	—
76	C₂³×C₉	—	4	4	no	—
77	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₉	—	2	4	no	—
79	Q₈×C₉	—	4	2	no	—
80	D₄×C₉	—	2	4	no	—
81	D₄×C₉	—	2	4	no	—
82	D₄×C₉	—	2	4	no	—
83	D₄×C₉	—	4	16	no	—
84	D₄×C₉	—	4	2	no	—
85	D₄×C₉	—	4	4	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	4	2	no	—
89	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
93	C₈×C₉	C₂×D₄	1	8	yes	complemented
94	C₂²×SD₁₆	C₂×C₉	1	2	yes	complemented
95	C₂²×D₈	C₂×C₉	1	2	yes	complemented
96	C₂×#127	C₂×C₉	1	1	yes	characteristic, complemented
97	C₂×#124	C₂×C₉	1	1	yes	characteristic, complemented
98	#496	C₂×C₉	1	8	yes	direct product factor
99	C₂×#97	C₂×C₉	1	1	yes	characteristic, complemented
100	C₂⁴×C₃	—	2	1	no	—
101	C₂⁴×C₃	—	2	2	no	—
102	C₂×C₃×Q₈	—	2	1	no	—
103	C₂×C₃×Q₈	—	2	4	no	—
104	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
105	C₂×C₃×D₄	—	2	4	no	—
106	C₂×C₃×D₄	—	2	4	no	—
107	C₂×C₃×D₄	—	2	4	no	—
108	C₂×C₃×D₄	—	2	1	no	—
109	C₂×C₃×D₄	—	2	2	no	—
110	C₂×C₃×D₄	—	2	8	no	—
111	C₂×C₃×D₄	—	4	8	no	—
112	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
113	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
116	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₃×SD₁₆	—	2	16	no	—
118	C₃×D₈	—	2	16	no	—
119	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
120	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
121	C₃×C₂²⋊C₄	—	2	8	no	—
122	C₃×C₄²	C₃×D₄	1	2	yes	—
123	C₃×C₄²	C₂³×C₃	1	2	yes	—
124	C₂²×C₉	—	4	2	no	—
125	C₂²×C₉	—	4	4	no	—
126	C₂²×C₉	—	4	8	no	—
127	C₂²×C₉	—	4	4	no	—
128	C₂²×C₉	—	8	8	no	—
129	C₂²×C₉	—	8	4	no	—
130	C₂²×C₉	#127	1	2	yes	—
131	C₂²×C₉	AGL(1,ℤ/8)	1	4	yes	—
132	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₉	—	2	4	no	—
134	C₄×C₉	—	4	2	no	—
135	C₄×C₉	#127	1	2	yes	—
136	C₄×C₉	C₂²×D₄	1	2	yes	—
137	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
138	C₂²×D₄	—	2	2	no	—
139	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
140	C₂²×D₄	C₂²×C₉	1	2	yes	—
141	C₂×SD₁₆	—	2	8	no	—
142	C₂×SD₁₆	C₂²×C₉	1	8	yes	complemented
143	C₂×D₈	—	2	8	no	—
144	C₂×D₈	C₂²×C₉	1	8	yes	complemented
145	C₂²×C₈	C₂²×C₉	1	2	yes	—
146	#127	—	2	2	no	—
147	#127	C₂²×C₉	1	4	yes	complemented
148	#124	—	2	2	no	—
149	#124	C₂²×C₉	1	4	yes	complemented
150	C₂×C₂²⋊C₄	—	2	2	no	—
151	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
152	#97	C₂²×C₉	1	4	yes	complemented
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	—	2	4	no	—
155	C₂³×C₃	—	4	2	no	—
156	C₂³×C₃	—	4	4	no	—
157	C₂³×C₃	—	4	8	no	—
158	C₂³×C₃	—	4	4	no	—
159	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
160	C₃×Q₈	—	2	4	no	—
161	C₃×Q₈	—	4	2	no	—
162	C₃×D₄	—	2	4	no	—
163	C₃×D₄	—	2	4	no	—
164	C₃×D₄	—	2	4	no	—
165	C₃×D₄	—	4	16	no	—
166	C₃×D₄	—	4	2	no	—
167	C₃×D₄	—	4	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
174	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
175	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
176	C₂×C₉	—	8	4	no	—
177	C₂×C₉	—	8	2	no	—
178	C₂×C₉	#496	1	4	yes	direct product factor
179	C₂×C₉	C₂×#127	1	1	yes	characteristic
180	C₂×C₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
181	C₂⁴	—	2	1	no	—
182	C₂⁴	—	2	2	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	C₂³×C₉	1	2	yes	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	8	no	—
192	C₂×D₄	—	4	8	no	—
193	C₂×D₄	C₂³×C₉	1	2	yes	—
194	C₂×D₄	C₂³×C₉	1	4	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	D₄×C₉	1	2	yes	—
197	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
198	SD₁₆	—	2	16	no	—
199	D₈	—	2	16	no	—
200	C₂×C₈	D₄×C₉	1	8	yes	complemented
201	C₂×C₈	C₂³×C₉	1	4	yes	—
202	C₂²⋊C₄	—	2	8	no	—
203	C₄²	D₄×C₉	1	2	yes	—
204	C₄²	C₂³×C₉	1	2	yes	—
205	C₂²×C₃	—	4	8	no	—
206	C₂²×C₃	—	4	4	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	4	4	no	—
209	C₂²×C₃	—	8	8	no	—
210	C₂²×C₃	—	8	4	no	—
211	C₂²×C₃	C₃×#127	1	2	yes	—
212	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	4	yes	—
213	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	4	2	no	—
216	C₃×C₄	C₃×#127	1	2	yes	—
217	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
218	C₉	C₂×#496	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	8	no	—
224	C₂³	—	4	4	no	—
225	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	4	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	4	no	—
231	D₄	—	4	16	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
238	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
240	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
241	C₈	C₂×D₄×C₉	1	8	yes	complemented
242	C₂×C₃	—	8	4	no	—
243	C₂×C₃	—	8	2	no	—
244	C₂×C₃	C₃×#496	1	4	yes	—
245	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
246	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	8	8	no	—
252	C₂²	—	8	4	no	—
253	C₂²	C₉×#127	1	2	yes	—
254	C₂²	C₉×AGL(1,ℤ/8)	1	4	yes	—
255	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	C₉×#127	1	2	yes	—
259	C₄	C₂²×D₄×C₉	1	2	yes	—
260	C₃	C₂×C₃×#496	1	1	yes	characteristic
261	C₂	—	8	4	no	—
262	C₂	—	8	2	no	—
263	C₂	C₉×#496	1	4	yes	direct product factor
264	C₂	C₂×C₉×#127	1	1	yes	characteristic
265	C₂	C₂×C₉×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#496	C₃
LCS	G ▷ C₂×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
1. C₂²×C₉×SD₁₆	C₂	semidirect
2. C₂²×C₉×D₈	C₂	semidirect
3. C₂×C₉×#127	C₂	semidirect
4. C₂×C₉×#124	C₂	semidirect
5. C₉×#496	C₂	direct (N × H)
6. C₂×C₉×#97	C₂	semidirect
13. C₂×C₉×SD₁₆	C₂²	semidirect
15. C₂×C₉×D₈	C₂²	semidirect
18. C₉×#127	C₂²	semidirect
20. C₉×#124	C₂²	semidirect
23. C₉×#97	C₂²	semidirect
49. C₂×C₈×C₉	D₄	semidirect
54. C₂×#496	C₉	direct (N × H)
93. C₈×C₉	C₂×D₄	semidirect
94. C₂²×SD₁₆	C₂×C₉	semidirect
95. C₂²×D₈	C₂×C₉	semidirect
96. C₂×#127	C₂×C₉	semidirect
97. C₂×#124	C₂×C₉	semidirect
98. #496	C₂×C₉	direct (N × H)
99. C₂×#97	C₂×C₉	semidirect
142. C₂×SD₁₆	C₂²×C₉	semidirect
144. C₂×D₈	C₂²×C₉	semidirect
147. #127	C₂²×C₉	semidirect
149. #124	C₂²×C₉	semidirect
152. #97	C₂²×C₉	semidirect
178. C₂×C₉	#496	direct (N × H)
200. C₂×C₈	D₄×C₉	semidirect
218. C₉	C₂×#496	direct (N × H)
241. C₈	C₂×D₄×C₉	semidirect
263. C₂	C₉×#496	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×SD₁₆	2
C₂²×C₉×D₈	2
C₂×C₉×#127	2
C₂×C₉×#124	2
C₉×#496	2
C₂×C₉×#97	2
C₂×C₃×#496	3

Subgroups — C2×C9×#496 #11767869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#496 #11767869