Subgroups — C₉×#1988 #11766709

Order 1152.

438 conjugacy classes of subgroups.

The subgroup lattice has 210 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×#1988	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	2	no	—
23	C₂×Q₈×C₉	—	4	2	no	—
24	C₂²×C₄×C₉	—	2	2	no	—
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₉	—	2	2	no	—
30	C₂×C₈×C₉	—	2	4	no	—
31	C₉×C₄⋊C₄	—	2	4	no	—
32	C₉×C₄⋊C₄	—	2	2	no	—
33	C₉×C₄⋊C₄	—	2	2	no	—
34	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₉×C₄⋊C₄	Q₈	1	4	yes	complemented
36	C₄²×C₉	—	4	2	no	—
37	C₄²×C₉	C₂×C₄	1	2	yes	—
38	C₄²×C₉	D₄	1	2	yes	—
39	#1988	C₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₃×Q₈	—	2	1	no	—
41	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
42	C₃×C₄⋊Q₈	—	2	2	no	—
43	C₃×C₄⋊Q₈	C₃×C₄	1	4	yes	—
44	C₂×C₃×C₄⋊C₄	—	2	1	no	—
45	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃×C₄²	—	2	1	no	—
48	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₄⋊C₈	—	2	2	no	—
50	C₃×#95	—	2	1	no	—
51	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
52	Q₈×C₉	—	4	8	no	—
53	C₂×C₄×C₉	—	2	2	no	—
54	C₂×C₄×C₉	—	2	4	no	—
55	C₂×C₄×C₉	—	2	4	no	—
56	C₂×C₄×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	4	2	no	—
58	C₂×C₄×C₉	—	4	2	no	—
59	C₂×C₄×C₉	—	4	2	no	—
60	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₉	D₈	1	2	yes	—
63	C₂×C₄×C₉	SD₁₆	1	2	yes	—
64	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₉	#41	1	2	yes	—
68	C₈×C₉	—	4	2	no	—
69	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
70	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
71	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
72	#340	C₂×C₉	1	4	yes	—
73	C₂×C₃×Q₈	—	2	4	no	—
74	C₂×C₃×Q₈	—	4	2	no	—
75	C₂×C₃×Q₈	—	4	2	no	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	2	no	—
78	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	4	no	—
83	C₃×C₄⋊C₄	—	2	4	no	—
84	C₃×C₄⋊C₄	—	2	2	no	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
87	C₃×C₄⋊C₄	C₃×Q₈	1	4	yes	—
88	C₃×C₄²	—	4	2	no	—
89	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
90	C₃×C₄²	C₃×D₄	1	2	yes	—
91	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
93	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
94	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₂²×C₉	#137	1	2	yes	—
96	C₄×C₉	—	2	2	no	—
97	C₄×C₉	—	4	4	no	—
98	C₄×C₉	—	4	4	no	—
99	C₄×C₉	#102	1	4	yes	—
100	C₄×C₉	C₄×Q₈	1	2	yes	—
101	C₄×C₉	#124	1	1	yes	characteristic
102	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
103	C₂²×Q₈	—	2	1	no	—
104	C₂²×C₈	C₂²×C₉	1	2	yes	—
105	C₄⋊Q₈	—	2	2	no	—
106	C₄⋊Q₈	C₄×C₉	1	4	yes	complemented
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	#95	—	2	1	no	—
114	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
115	C₃×Q₈	—	4	8	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	—	4	2	no	—
121	C₂×C₃×C₄	—	4	2	no	—
122	C₂×C₃×C₄	—	4	2	no	—
123	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
124	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	C₃×D₈	1	2	yes	—
126	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
130	C₂×C₃×C₄	C₃×#41	1	2	yes	—
131	C₃×C₈	—	4	2	no	—
132	C₂×C₉	#391	1	1	yes	characteristic
133	C₂×C₉	C₂×#102	1	1	yes	characteristic
134	C₂×C₉	#419	1	1	yes	characteristic
135	C₂×C₉	#474	1	2	yes	—
136	C₂×C₉	#475	1	2	yes	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
143	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
151	C₄⋊C₄	Q₈×C₉	1	4	yes	complemented
152	C₄²	—	4	2	no	—
153	C₄²	C₂×C₄×C₉	1	2	yes	—
154	C₄²	D₄×C₉	1	2	yes	—
155	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
157	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
158	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
159	C₂²×C₃	C₃×#137	1	2	yes	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	4	4	no	—
162	C₃×C₄	—	4	4	no	—
163	C₃×C₄	C₃×#102	1	4	yes	—
164	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
165	C₃×C₄	C₃×#124	1	1	yes	characteristic
166	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
167	C₉	#1988	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄	C₉×D₈	1	2	yes	—
180	C₂×C₄	C₉×SD₁₆	1	2	yes	—
181	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
183	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
184	C₂×C₄	C₉×#41	1	2	yes	—
185	C₈	—	4	2	no	—
186	C₂×C₃	C₃×#391	1	1	yes	characteristic
187	C₂×C₃	C₂×C₃×#102	1	1	yes	characteristic
188	C₂×C₃	C₃×#419	1	1	yes	characteristic
189	C₂×C₃	C₃×#474	1	2	yes	—
190	C₂×C₃	C₃×#475	1	2	yes	—
191	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
192	C₂²	C₉×C₂²⋊Q₈	1	2	yes	—
193	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
194	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
195	C₂²	C₉×#137	1	2	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	4	no	—
199	C₄	C₉×#102	1	4	yes	—
200	C₄	C₄×Q₈×C₉	1	2	yes	—
201	C₄	C₉×#124	1	1	yes	characteristic
202	C₄	C₉×C₄⋊Q₈	1	1	yes	characteristic
203	C₃	C₃×#1988	1	1	yes	characteristic
204	C₂	C₉×#391	1	1	yes	characteristic
205	C₂	C₂×C₉×#102	1	1	yes	characteristic
206	C₂	C₉×#419	1	1	yes	characteristic
207	C₂	C₉×#474	1	2	yes	—
208	C₂	C₉×#475	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
35. C₉×C₄⋊C₄	Q₈	semidirect
39. #1988	C₉	direct (N × H)
106. C₄⋊Q₈	C₄×C₉	semidirect
151. C₄⋊C₄	Q₈×C₉	semidirect
167. C₉	#1988	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×#1988	3

Subgroups — C9×#1988 #11766709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1988 #11766709