Subgroups — C₂×C₉×#497 #11767870

Order 1152.

846 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×Q₁₆	C₂	1	2	yes	maximal, complemented
2	C₂²×C₉×SD₁₆	C₂	1	2	yes	maximal
3	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic
5	C₉×#497	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#497	C₃	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×Q₁₆	—	2	8	no	—
12	C₂×C₉×Q₁₆	C₂²	1	8	yes	complemented
13	C₂×C₉×SD₁₆	—	2	8	no	—
14	C₂×C₉×SD₁₆	C₂²	1	8	yes	—
15	C₂²×C₈×C₉	C₂²	1	2	yes	—
16	C₉×C₄⋊Q₈	—	2	2	no	—
17	C₉×C₄⋊Q₈	C₂²	1	4	yes	complemented
18	C₉×#124	—	2	2	no	—
19	C₉×#124	C₂²	1	4	yes	—
20	C₂×C₉×C₄⋊C₄	—	2	2	no	—
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×Q₁₆	C₂×C₃	1	2	yes	—
25	C₂²×C₃×SD₁₆	C₂×C₃	1	2	yes	—
26	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
28	C₃×#497	C₂×C₃	1	8	yes	—
29	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
30	C₂⁴×C₉	—	2	1	no	—
31	C₂×Q₈×C₉	—	2	1	no	—
32	C₂×Q₈×C₉	—	2	2	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	—	2	4	no	—
35	C₂×Q₈×C₉	—	2	4	no	—
36	C₂×Q₈×C₉	C₂³	1	2	yes	—
37	C₂×Q₈×C₉	C₂³	1	4	yes	—
38	C₂×D₄×C₉	—	2	4	no	—
39	C₂×D₄×C₉	—	2	1	no	—
40	C₂×D₄×C₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	—	2	2	no	—
43	C₂²×C₄×C₉	D₄	1	2	yes	—
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₉×Q₁₆	—	2	16	no	—
46	C₉×SD₁₆	—	2	16	no	—
47	C₂×C₈×C₉	D₄	1	8	yes	—
48	C₂×C₈×C₉	C₂³	1	4	yes	—
49	C₉×C₄⋊C₄	—	2	8	no	—
50	C₉×C₂²⋊C₄	—	2	8	no	—
51	C₄²×C₉	D₄	1	2	yes	—
52	C₄²×C₉	C₂³	1	2	yes	—
53	C₂×#497	C₉	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
55	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×Q₁₆	—	2	8	no	—
58	C₂×C₃×Q₁₆	C₂²×C₃	1	8	yes	—
59	C₂×C₃×SD₁₆	—	2	8	no	—
60	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	—
61	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
62	C₃×C₄⋊Q₈	—	2	2	no	—
63	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
64	C₃×#124	—	2	2	no	—
65	C₃×#124	C₂²×C₃	1	4	yes	—
66	C₂×C₃×C₄⋊C₄	—	2	2	no	—
67	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
68	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
69	C₃×#97	C₂²×C₃	1	4	yes	—
70	C₂³×C₉	—	2	2	no	—
71	C₂³×C₉	—	4	2	no	—
72	C₂³×C₉	—	4	4	no	—
73	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₉	—	2	4	no	—
75	Q₈×C₉	—	2	4	no	—
76	Q₈×C₉	—	2	4	no	—
77	Q₈×C₉	—	4	2	no	—
78	Q₈×C₉	—	4	4	no	—
79	D₄×C₉	—	2	4	no	—
80	D₄×C₉	—	4	2	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	4	no	—
84	C₂×C₄×C₉	—	4	2	no	—
85	C₂×C₄×C₉	—	4	4	no	—
86	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
89	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
90	C₈×C₉	C₂×D₄	1	8	yes	—
91	C₂²×Q₁₆	C₂×C₉	1	2	yes	complemented
92	C₂²×SD₁₆	C₂×C₉	1	2	yes	—
93	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
94	C₂×#124	C₂×C₉	1	1	yes	characteristic
95	#497	C₂×C₉	1	8	yes	direct product factor
96	C₂×#97	C₂×C₉	1	1	yes	characteristic, complemented
97	C₂⁴×C₃	—	2	1	no	—
98	C₂×C₃×Q₈	—	2	1	no	—
99	C₂×C₃×Q₈	—	2	2	no	—
100	C₂×C₃×Q₈	—	2	4	no	—
101	C₂×C₃×Q₈	—	2	4	no	—
102	C₂×C₃×Q₈	—	2	4	no	—
103	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
104	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
105	C₂×C₃×D₄	—	2	4	no	—
106	C₂×C₃×D₄	—	2	1	no	—
107	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₃×Q₁₆	—	2	16	no	—
113	C₃×SD₁₆	—	2	16	no	—
114	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
115	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
116	C₃×C₄⋊C₄	—	2	8	no	—
117	C₃×C₂²⋊C₄	—	2	8	no	—
118	C₃×C₄²	C₃×D₄	1	2	yes	—
119	C₃×C₄²	C₂³×C₃	1	2	yes	—
120	C₂²×C₉	—	4	2	no	—
121	C₂²×C₉	—	4	4	no	—
122	C₂²×C₉	—	8	4	no	—
123	C₂²×C₉	#127	1	2	yes	—
124	C₂²×C₉	#137	1	4	yes	—
125	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₉	—	2	4	no	—
127	C₄×C₉	—	4	2	no	—
128	C₄×C₉	—	4	4	no	—
129	C₄×C₉	#127	1	2	yes	—
130	C₄×C₉	C₂²×D₄	1	2	yes	—
131	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
132	C₂²×Q₈	C₂²×C₉	1	2	yes	—
133	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
134	C₂×Q₁₆	—	2	8	no	—
135	C₂×Q₁₆	C₂²×C₉	1	8	yes	complemented
136	C₂×SD₁₆	—	2	8	no	—
137	C₂×SD₁₆	C₂²×C₉	1	8	yes	—
138	C₂²×C₈	C₂²×C₉	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	C₂²×C₉	1	4	yes	complemented
141	#124	—	2	2	no	—
142	#124	C₂²×C₉	1	4	yes	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₂²⋊C₄	—	2	2	no	—
145	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
146	#97	C₂²×C₉	1	4	yes	complemented
147	C₂³×C₃	—	2	2	no	—
148	C₂³×C₃	—	4	2	no	—
149	C₂³×C₃	—	4	4	no	—
150	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
151	C₃×Q₈	—	2	4	no	—
152	C₃×Q₈	—	2	4	no	—
153	C₃×Q₈	—	2	4	no	—
154	C₃×Q₈	—	4	2	no	—
155	C₃×Q₈	—	4	4	no	—
156	C₃×D₄	—	2	4	no	—
157	C₃×D₄	—	4	2	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	4	no	—
163	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
164	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
166	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
167	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
168	C₂×C₉	—	8	2	no	—
169	C₂×C₉	#497	1	4	yes	direct product factor
170	C₂×C₉	C₂×#127	1	1	yes	characteristic
171	C₂×C₉	C₂×#137	1	2	yes	—
172	C₂⁴	—	2	1	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	C₂³×C₉	1	2	yes	—
179	C₂×Q₈	C₂³×C₉	1	4	yes	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	C₂³×C₉	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	D₄×C₉	1	2	yes	—
186	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
187	Q₁₆	—	2	16	no	—
188	SD₁₆	—	2	16	no	—
189	C₂×C₈	D₄×C₉	1	8	yes	—
190	C₂×C₈	C₂³×C₉	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₂²⋊C₄	—	2	8	no	—
193	C₄²	D₄×C₉	1	2	yes	—
194	C₄²	C₂³×C₉	1	2	yes	—
195	C₂²×C₃	—	4	4	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	—	8	4	no	—
198	C₂²×C₃	C₃×#127	1	2	yes	—
199	C₂²×C₃	C₃×#137	1	4	yes	—
200	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	—	4	4	no	—
204	C₃×C₄	C₃×#127	1	2	yes	—
205	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
206	C₉	C₂×#497	1	1	yes	characteristic, direct product factor, Sylow
207	C₂³	—	2	2	no	—
208	C₂³	—	4	2	no	—
209	C₂³	—	4	4	no	—
210	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
211	Q₈	—	2	4	no	—
212	Q₈	—	2	4	no	—
213	Q₈	—	2	4	no	—
214	Q₈	—	4	2	no	—
215	Q₈	—	4	4	no	—
216	D₄	—	2	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
224	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
226	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
227	C₈	C₂×D₄×C₉	1	8	yes	—
228	C₂×C₃	—	8	2	no	—
229	C₂×C₃	C₃×#497	1	4	yes	—
230	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
231	C₂×C₃	C₂×C₃×#137	1	2	yes	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	C₉×#127	1	2	yes	—
236	C₂²	C₉×#137	1	4	yes	—
237	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	4	no	—
241	C₄	C₉×#127	1	2	yes	—
242	C₄	C₂²×D₄×C₉	1	2	yes	—
243	C₃	C₂×C₃×#497	1	1	yes	characteristic
244	C₂	—	8	2	no	—
245	C₂	C₉×#497	1	4	yes	direct product factor
246	C₂	C₂×C₉×#127	1	1	yes	characteristic
247	C₂	C₂×C₉×#137	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	C₂²×C₃×D₄	C₂²×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×Q₁₆	C₂	semidirect
3. C₂×C₉×C₄⋊Q₈	C₂	semidirect
5. C₉×#497	C₂	direct (N × H)
6. C₂×C₉×#97	C₂	semidirect
12. C₂×C₉×Q₁₆	C₂²	semidirect
17. C₉×C₄⋊Q₈	C₂²	semidirect
23. C₉×#97	C₂²	semidirect
53. C₂×#497	C₉	direct (N × H)
91. C₂²×Q₁₆	C₂×C₉	semidirect
93. C₂×C₄⋊Q₈	C₂×C₉	semidirect
95. #497	C₂×C₉	direct (N × H)
96. C₂×#97	C₂×C₉	semidirect
135. C₂×Q₁₆	C₂²×C₉	semidirect
140. C₄⋊Q₈	C₂²×C₉	semidirect
146. #97	C₂²×C₉	semidirect
169. C₂×C₉	#497	direct (N × H)
206. C₉	C₂×#497	direct (N × H)
245. C₂	C₉×#497	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×Q₁₆	2
C₂²×C₉×SD₁₆	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#124	2
C₉×#497	2
C₂×C₉×#97	2
C₂×C₃×#497	3

Subgroups — C2×C9×#497 #11767870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#497 #11767870