Subgroups — C₉×#1987 #11766708

Order 1152.

438 conjugacy classes of subgroups.

The subgroup lattice has 204 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
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	4	yes	maximal
5	C₃×#1987	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	—	2	1	no	—
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₉×C₄⋊Q₈	—	2	2	no	—
9	C₉×C₄⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₉	—	2	1	no	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊C₈	—	2	2	no	—
16	C₉×#95	—	2	1	no	—
17	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₃×#340	C₂×C₃	1	4	yes	—
21	C₂×Q₈×C₉	—	2	4	no	—
22	C₂×Q₈×C₉	—	4	4	no	—
23	C₂²×C₄×C₉	—	2	2	no	—
24	C₂²×C₄×C₉	—	2	2	no	—
25	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉	—	2	2	no	—
29	C₂×C₈×C₉	—	2	4	no	—
30	C₉×C₄⋊C₄	—	2	4	no	—
31	C₉×C₄⋊C₄	—	2	2	no	—
32	C₉×C₄⋊C₄	—	2	2	no	—
33	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₉×C₄⋊C₄	Q₈	1	4	yes	complemented
35	C₄²×C₉	—	4	2	no	—
36	C₄²×C₉	C₂×C₄	1	2	yes	—
37	C₄²×C₉	D₄	1	2	yes	—
38	#1987	C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃×Q₈	—	2	1	no	—
40	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
41	C₃×C₄⋊Q₈	—	2	2	no	—
42	C₃×C₄⋊Q₈	C₃×C₄	1	4	yes	—
43	C₂×C₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
45	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₂×C₃×C₄²	—	2	1	no	—
47	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₃×C₄⋊C₈	—	2	2	no	—
49	C₃×#95	—	2	1	no	—
50	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₉	—	4	8	no	—
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	—	2	4	no	—
54	C₂×C₄×C₉	—	2	4	no	—
55	C₂×C₄×C₉	—	4	2	no	—
56	C₂×C₄×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	4	4	no	—
58	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₉	SD₁₆	1	2	yes	—
61	C₂×C₄×C₉	Q₁₆	1	2	yes	—
62	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₉	#41	1	2	yes	—
66	C₈×C₉	—	4	2	no	—
67	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
68	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
69	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
70	#340	C₂×C₉	1	4	yes	—
71	C₂×C₃×Q₈	—	2	4	no	—
72	C₂×C₃×Q₈	—	4	4	no	—
73	C₂²×C₃×C₄	—	2	2	no	—
74	C₂²×C₃×C₄	—	2	2	no	—
75	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
78	C₂×C₃×C₈	—	2	2	no	—
79	C₂×C₃×C₈	—	2	4	no	—
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
84	C₃×C₄⋊C₄	C₃×Q₈	1	4	yes	—
85	C₃×C₄²	—	4	2	no	—
86	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
87	C₃×C₄²	C₃×D₄	1	2	yes	—
88	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
90	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
92	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
93	C₄×C₉	—	2	2	no	—
94	C₄×C₉	—	4	4	no	—
95	C₄×C₉	—	4	4	no	—
96	C₄×C₉	#103	1	4	yes	—
97	C₄×C₉	C₄×Q₈	1	2	yes	—
98	C₄×C₉	#124	1	1	yes	characteristic
99	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
100	C₂²×Q₈	—	2	1	no	—
101	C₂²×C₈	C₂²×C₉	1	2	yes	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	C₄×C₉	1	4	yes	complemented
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
107	C₂×C₄²	—	2	1	no	—
108	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
109	C₄⋊C₈	—	2	2	no	—
110	#95	—	2	1	no	—
111	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
112	C₃×Q₈	—	4	8	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	4	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	—	4	2	no	—
118	C₂×C₃×C₄	—	4	4	no	—
119	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
120	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
121	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
122	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
123	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
126	C₂×C₃×C₄	C₃×#41	1	2	yes	—
127	C₃×C₈	—	4	2	no	—
128	C₂×C₉	#391	1	1	yes	characteristic
129	C₂×C₉	C₂×#103	1	1	yes	characteristic
130	C₂×C₉	#418	1	1	yes	characteristic
131	C₂×C₉	#476	1	2	yes	—
132	C₂×C₉	#477	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
138	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
146	C₄⋊C₄	Q₈×C₉	1	4	yes	complemented
147	C₄²	—	4	2	no	—
148	C₄²	C₂×C₄×C₉	1	2	yes	—
149	C₄²	D₄×C₉	1	2	yes	—
150	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
152	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
153	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
154	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	4	4	no	—
157	C₃×C₄	—	4	4	no	—
158	C₃×C₄	C₃×#103	1	4	yes	—
159	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
160	C₃×C₄	C₃×#124	1	1	yes	characteristic
161	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
162	C₉	#1987	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
164	Q₈	—	4	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
172	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
173	C₂×C₄	C₉×SD₁₆	1	2	yes	—
174	C₂×C₄	C₉×Q₁₆	1	2	yes	—
175	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
176	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
177	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
178	C₂×C₄	C₉×#41	1	2	yes	—
179	C₈	—	4	2	no	—
180	C₂×C₃	C₃×#391	1	1	yes	characteristic
181	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
182	C₂×C₃	C₃×#418	1	1	yes	characteristic
183	C₂×C₃	C₃×#476	1	2	yes	—
184	C₂×C₃	C₃×#477	1	2	yes	—
185	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
186	C₂²	C₉×C₂²⋊Q₈	1	2	yes	—
187	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
188	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
189	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	4	4	no	—
193	C₄	C₉×#103	1	4	yes	—
194	C₄	C₄×Q₈×C₉	1	2	yes	—
195	C₄	C₉×#124	1	1	yes	characteristic
196	C₄	C₉×C₄⋊Q₈	1	1	yes	characteristic
197	C₃	C₃×#1987	1	1	yes	characteristic
198	C₂	C₉×#391	1	1	yes	characteristic
199	C₂	C₂×C₉×#103	1	1	yes	characteristic
200	C₂	C₉×#418	1	1	yes	characteristic
201	C₂	C₉×#476	1	2	yes	—
202	C₂	C₉×#477	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₃×D₄
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
9. C₉×C₄⋊Q₈	C₄	semidirect
34. C₉×C₄⋊C₄	Q₈	semidirect
38. #1987	C₉	direct (N × H)
103. C₄⋊Q₈	C₄×C₉	semidirect
146. C₄⋊C₄	Q₈×C₉	semidirect
162. C₉	#1987	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×C₄⋊C₈	2
C₄×C₉×C₄⋊C₄	2
C₉×#340	2
C₃×#1987	3

Subgroups — C9×#1987 #11766708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1987 #11766708