Subgroups — C₂×C₉×#486 #11767849

Order 1152.

792 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
3	C₉×#486	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₉×#102	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#486	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	—	2	1	no	—
8	C₂²×D₄×C₉	—	2	1	no	—
9	C₂²×D₄×C₉	—	2	1	no	—
10	C₂²×D₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₉	C₂²	1	2	yes	—
12	C₉×C₄⋊Q₈	—	2	2	no	—
13	C₉×C₄⋊Q₈	C₂²	1	4	yes	complemented
14	C₉×#127	—	2	2	no	—
15	C₉×#127	C₂²	1	4	yes	—
16	C₂×C₉×C₄⋊C₄	—	2	1	no	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₉×#102	C₂²	1	16	yes	complemented
20	C₄×C₈×C₉	C₂²	1	4	yes	complemented
21	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
23	C₃×#486	C₂×C₃	1	8	yes	—
24	C₂×C₃×#102	C₂×C₃	1	4	yes	—
25	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
26	C₂⁴×C₉	—	2	2	no	—
27	C₂×Q₈×C₉	—	2	4	no	—
28	C₂×Q₈×C₉	—	4	4	no	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	4	no	—
31	C₂×D₄×C₉	—	2	4	no	—
32	C₂×D₄×C₉	—	2	8	no	—
33	C₂×D₄×C₉	—	4	4	no	—
34	C₂×D₄×C₉	—	4	4	no	—
35	C₂×D₄×C₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₉	—	2	4	no	—
41	C₂×C₈×C₉	C₂³	1	4	yes	—
42	C₉×C₄⋊C₄	—	2	4	no	—
43	C₉×C₄⋊C₄	—	2	2	no	—
44	C₉×C₄⋊C₄	C₂³	1	4	yes	—
45	C₄²×C₉	D₄	1	2	yes	—
46	C₄²×C₉	C₂³	1	2	yes	—
47	C₂×#486	C₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₃×Q₈	—	2	1	no	—
49	C₂²×C₃×D₄	—	2	1	no	—
50	C₂²×C₃×D₄	—	2	1	no	—
51	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
52	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
53	C₃×C₄⋊Q₈	—	2	2	no	—
54	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
55	C₃×#127	—	2	2	no	—
56	C₃×#127	C₂²×C₃	1	4	yes	—
57	C₂×C₃×C₄⋊C₄	—	2	1	no	—
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
60	C₃×#102	C₂²×C₃	1	16	yes	—
61	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
62	C₂³×C₉	—	2	4	no	—
63	C₂³×C₉	—	4	4	no	—
64	C₂³×C₉	—	4	4	no	—
65	C₂³×C₉	—	4	4	no	—
66	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₉	—	4	8	no	—
68	D₄×C₉	—	2	8	no	—
69	D₄×C₉	—	4	4	no	—
70	D₄×C₉	—	4	8	no	—
71	D₄×C₉	—	4	8	no	—
72	C₂×C₄×C₉	—	2	4	no	—
73	C₂×C₄×C₉	—	4	4	no	—
74	C₂×C₄×C₉	D₈	1	4	yes	—
75	C₂×C₄×C₉	SD₁₆	1	4	yes	—
76	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₉	#41	1	4	yes	—
80	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
81	C₈×C₉	—	2	4	no	—
82	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
83	C₂×#127	C₂×C₉	1	1	yes	characteristic
84	#486	C₂×C₉	1	8	yes	direct product factor
85	C₂×#102	C₂×C₉	1	4	yes	complemented
86	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
87	C₂⁴×C₃	—	2	2	no	—
88	C₂×C₃×Q₈	—	2	4	no	—
89	C₂×C₃×Q₈	—	4	4	no	—
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂×C₃×D₄	—	2	4	no	—
92	C₂×C₃×D₄	—	2	4	no	—
93	C₂×C₃×D₄	—	2	8	no	—
94	C₂×C₃×D₄	—	4	4	no	—
95	C₂×C₃×D₄	—	4	4	no	—
96	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
101	C₂×C₃×C₈	—	2	4	no	—
102	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
103	C₃×C₄⋊C₄	—	2	4	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
106	C₃×C₄²	C₃×D₄	1	2	yes	—
107	C₃×C₄²	C₂³×C₃	1	2	yes	—
108	C₂²×C₉	—	4	4	no	—
109	C₂²×C₉	—	4	4	no	—
110	C₂²×C₉	—	4	4	no	—
111	C₂²×C₉	—	8	8	no	—
112	C₂²×C₉	#124	1	2	yes	—
113	C₂²×C₉	C₂×D₈	1	2	yes	—
114	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
115	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₉	—	4	4	no	—
117	C₄×C₉	#124	1	2	yes	—
118	C₄×C₉	C₂×D₈	1	4	yes	—
119	C₄×C₉	C₂×SD₁₆	1	4	yes	—
120	C₄×C₉	C₂×#41	1	2	yes	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂²×D₄	—	2	1	no	—
123	C₂²×D₄	—	2	1	no	—
124	C₂²×D₄	C₂²×C₉	1	2	yes	—
125	C₂²×C₈	C₂²×C₉	1	2	yes	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂²×C₉	1	4	yes	complemented
128	#127	—	2	2	no	—
129	#127	C₂²×C₉	1	4	yes	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
132	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
133	#102	C₂²×C₉	1	16	yes	complemented
134	C₄×C₈	C₂²×C₉	1	4	yes	complemented
135	C₂³×C₃	—	2	4	no	—
136	C₂³×C₃	—	4	4	no	—
137	C₂³×C₃	—	4	4	no	—
138	C₂³×C₃	—	4	4	no	—
139	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
140	C₃×Q₈	—	4	8	no	—
141	C₃×D₄	—	2	8	no	—
142	C₃×D₄	—	4	4	no	—
143	C₃×D₄	—	4	8	no	—
144	C₃×D₄	—	4	8	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
148	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
149	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
151	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
152	C₂×C₃×C₄	C₃×#41	1	4	yes	—
153	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
154	C₃×C₈	—	2	4	no	—
155	C₂×C₉	—	8	4	no	—
156	C₂×C₉	#486	1	4	yes	direct product factor
157	C₂×C₉	C₂×#124	1	1	yes	characteristic
158	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
159	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂⁴	—	2	2	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	8	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	C₂³×C₉	1	4	yes	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
172	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	C₂³×C₉	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	C₂³×C₉	1	4	yes	—
179	C₄²	D₄×C₉	1	2	yes	—
180	C₄²	C₂³×C₉	1	2	yes	—
181	C₂²×C₃	—	4	4	no	—
182	C₂²×C₃	—	4	4	no	—
183	C₂²×C₃	—	4	4	no	—
184	C₂²×C₃	—	8	8	no	—
185	C₂²×C₃	C₃×#124	1	2	yes	—
186	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
187	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
188	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₃×C₄	—	4	4	no	—
190	C₃×C₄	C₃×#124	1	2	yes	—
191	C₃×C₄	C₂×C₃×D₈	1	4	yes	—
192	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
193	C₃×C₄	C₂×C₃×#41	1	2	yes	—
194	C₉	C₂×#486	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	—	2	4	no	—
196	C₂³	—	4	4	no	—
197	C₂³	—	4	4	no	—
198	C₂³	—	4	4	no	—
199	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
200	Q₈	—	4	8	no	—
201	D₄	—	2	8	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	8	no	—
204	D₄	—	4	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	C₉×D₈	1	4	yes	—
208	C₂×C₄	C₉×SD₁₆	1	4	yes	—
209	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
211	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
212	C₂×C₄	C₉×#41	1	4	yes	—
213	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₂×C₃	—	8	4	no	—
216	C₂×C₃	C₃×#486	1	4	yes	—
217	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
218	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
219	C₂×C₃	C₂²×C₃×SD₁₆	1	1	yes	characteristic
220	C₂²	—	4	4	no	—
221	C₂²	—	4	4	no	—
222	C₂²	—	4	4	no	—
223	C₂²	—	8	8	no	—
224	C₂²	C₉×#124	1	2	yes	—
225	C₂²	C₂×C₉×D₈	1	2	yes	—
226	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
227	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
228	C₄	—	4	4	no	—
229	C₄	C₉×#124	1	2	yes	—
230	C₄	C₂×C₉×D₈	1	4	yes	—
231	C₄	C₂×C₉×SD₁₆	1	4	yes	—
232	C₄	C₂×C₉×#41	1	2	yes	—
233	C₃	C₂×C₃×#486	1	1	yes	characteristic
234	C₂	—	8	4	no	—
235	C₂	C₉×#486	1	4	yes	direct product factor
236	C₂	C₂×C₉×#124	1	1	yes	characteristic
237	C₂	C₂²×C₉×D₈	1	1	yes	characteristic
238	C₂	C₂²×C₉×SD₁₆	1	1	yes	characteristic

	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×#127	C₂×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. C₂×C₉×C₄⋊Q₈	C₂	semidirect
3. C₉×#486	C₂	direct (N × H)
4. C₂×C₉×#102	C₂	semidirect
5. C₂×C₄×C₈×C₉	C₂	semidirect
13. C₉×C₄⋊Q₈	C₂²	semidirect
19. C₉×#102	C₂²	semidirect
20. C₄×C₈×C₉	C₂²	semidirect
47. C₂×#486	C₉	direct (N × H)
82. C₂×C₄⋊Q₈	C₂×C₉	semidirect
84. #486	C₂×C₉	direct (N × H)
85. C₂×#102	C₂×C₉	semidirect
86. C₂×C₄×C₈	C₂×C₉	semidirect
127. C₄⋊Q₈	C₂²×C₉	semidirect
133. #102	C₂²×C₉	semidirect
134. C₄×C₈	C₂²×C₉	semidirect
156. C₂×C₉	#486	direct (N × H)
194. C₉	C₂×#486	direct (N × H)
235. C₂	C₉×#486	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#127	2
C₉×#486	2
C₂×C₉×#102	2
C₂×C₄×C₈×C₉	2
C₂×C₃×#486	3

Subgroups — C2×C9×#486 #11767849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#486 #11767849