Subgroups — C₉×#1993 #11766714

Order 1152.

414 conjugacy classes of subgroups.

The subgroup lattice has 195 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×#97	C₂	1	1	yes	maximal, characteristic
3	C₉×#382	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	4	yes	maximal
5	C₃×#1993	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	C₂²	1	1	yes	characteristic
14	C₉×#97	—	2	2	no	—
15	C₉×#95	—	2	1	no	—
16	C₉×#95	—	2	1	no	—
17	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
19	C₃×#382	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	2	no	—
30	C₂×C₈×C₉	Q₈	1	4	yes	complemented
31	C₉×C₄⋊C₄	—	2	4	no	—
32	C₉×C₄⋊C₄	—	2	2	no	—
33	C₉×C₄⋊C₄	—	2	4	no	—
34	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₄²×C₉	C₂×C₄	1	2	yes	—
36	C₄²×C₉	D₄	1	2	yes	—
37	#1993	C₉	1	1	yes	characteristic, direct product factor, Sylow
38	C₂²×C₃×Q₈	—	2	1	no	—
39	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
40	C₃×C₄⋊Q₈	—	2	2	no	—
41	C₃×C₄⋊Q₈	C₃×C₄	1	4	yes	—
42	C₂×C₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
44	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
45	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₃×#97	—	2	2	no	—
47	C₃×#95	—	2	1	no	—
48	C₃×#95	—	2	1	no	—
49	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₉	—	4	8	no	—
51	C₂×C₄×C₉	—	2	2	no	—
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	—	2	4	no	—
54	C₂×C₄×C₉	—	4	4	no	—
55	C₂×C₄×C₉	—	4	2	no	—
56	C₂×C₄×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	4	2	no	—
58	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₉	#41	1	2	yes	—
64	C₈×C₉	—	2	4	no	—
65	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
66	C₂×#97	C₂×C₉	1	1	yes	characteristic
67	#382	C₂×C₉	1	1	yes	characteristic
68	#340	C₂×C₉	1	4	yes	—
69	C₂×C₃×Q₈	—	2	4	no	—
70	C₂×C₃×Q₈	—	4	4	no	—
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	—	2	2	no	—
73	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₃×C₈	—	2	2	no	—
77	C₂×C₃×C₈	—	2	2	no	—
78	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
79	C₃×C₄⋊C₄	—	2	4	no	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	4	no	—
82	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
83	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
84	C₃×C₄²	C₃×D₄	1	2	yes	—
85	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₉	#124	1	1	yes	characteristic
87	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
89	C₂²×C₉	#137	1	2	yes	—
90	C₄×C₉	—	2	2	no	—
91	C₄×C₉	—	2	2	no	—
92	C₄×C₉	—	4	4	no	—
93	C₄×C₉	—	8	2	no	—
94	C₄×C₉	C₄×Q₈	1	2	yes	—
95	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
96	C₂²×Q₈	—	2	1	no	—
97	C₂²×C₈	C₂²×C₉	1	2	yes	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄⋊Q₈	C₄×C₉	1	4	yes	complemented
100	C₂×C₄⋊C₄	—	2	1	no	—
101	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
103	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
104	#97	—	2	2	no	—
105	#95	—	2	1	no	—
106	#95	—	2	1	no	—
107	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
108	C₃×Q₈	—	4	8	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	4	no	—
112	C₂×C₃×C₄	—	4	4	no	—
113	C₂×C₃×C₄	—	4	2	no	—
114	C₂×C₃×C₄	—	4	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
117	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
118	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
121	C₂×C₃×C₄	C₃×#41	1	2	yes	—
122	C₃×C₈	—	2	4	no	—
123	C₂×C₉	#391	1	1	yes	characteristic
124	C₂×C₉	#418	1	1	yes	characteristic
125	C₂×C₉	#419	1	1	yes	characteristic
126	C₂×C₉	#489	1	2	yes	—
127	C₂×C₉	#501	1	2	yes	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	4	4	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
133	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	Q₈×C₉	1	4	yes	complemented
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
142	C₄²	C₂×C₄×C₉	1	2	yes	—
143	C₄²	D₄×C₉	1	2	yes	—
144	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₃	C₃×#124	1	1	yes	characteristic
146	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
148	C₂²×C₃	C₃×#137	1	2	yes	—
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	2	2	no	—
151	C₃×C₄	—	4	4	no	—
152	C₃×C₄	—	8	2	no	—
153	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
154	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
155	C₉	#1993	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
157	Q₈	—	4	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
166	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
167	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
168	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
169	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
170	C₂×C₄	C₉×#41	1	2	yes	—
171	C₈	—	2	4	no	—
172	C₂×C₃	C₃×#391	1	1	yes	characteristic
173	C₂×C₃	C₃×#418	1	1	yes	characteristic
174	C₂×C₃	C₃×#419	1	1	yes	characteristic
175	C₂×C₃	C₃×#489	1	2	yes	—
176	C₂×C₃	C₃×#501	1	2	yes	—
177	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²	C₉×#124	1	1	yes	characteristic
179	C₂²	C₉×C₄⋊Q₈	1	1	yes	characteristic
180	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
181	C₂²	C₉×#137	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	8	2	no	—
186	C₄	C₄×Q₈×C₉	1	2	yes	—
187	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
188	C₃	C₃×#1993	1	1	yes	characteristic
189	C₂	C₉×#391	1	1	yes	characteristic
190	C₂	C₉×#418	1	1	yes	characteristic
191	C₂	C₉×#419	1	1	yes	characteristic
192	C₂	C₉×#489	1	2	yes	—
193	C₂	C₉×#501	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
30. C₂×C₈×C₉	Q₈	semidirect
37. #1993	C₉	direct (N × H)
99. C₄⋊Q₈	C₄×C₉	semidirect
137. C₂×C₈	Q₈×C₉	semidirect
155. C₉	#1993	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×#97	2
C₉×#382	2
C₉×#340	2
C₃×#1993	3

Subgroups — C9×#1993 #11766714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1993 #11766714