Subgroups — C₉×#1997 #11766718

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 219 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	C₉×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₉×#343	C₂	1	4	yes	maximal
5	C₃×#1997	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	C₄	1	2	yes	—
7	C₂×C₉×M₁₆	C₂²	1	2	yes	—
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	C₉×C₄⋊Q₈	—	2	1	no	—
10	C₉×C₄⋊Q₈	—	2	1	no	—
11	C₉×C₄⋊Q₈	—	2	1	no	—
12	C₉×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₉×C₄⋊C₄	C₄	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊C₈	—	2	1	no	—
17	C₉×C₄⋊C₈	—	2	1	no	—
18	C₉×#97	—	2	1	no	—
19	C₄×C₈×C₉	—	2	1	no	—
20	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
21	C₃×#423	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
23	C₃×#343	C₂×C₃	1	4	yes	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	—	4	4	no	—
27	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₉×M₁₆	—	2	2	no	—
32	C₉×M₁₆	—	2	4	no	—
33	C₉×M₁₆	Q₈	1	4	yes	complemented
34	C₂×C₈×C₉	—	2	2	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₄⋊C₄	—	2	2	no	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₄²×C₉	D₄	1	1	yes	characteristic
41	C₄²×C₉	D₄	1	1	yes	characteristic
42	C₄²×C₉	D₄	1	1	yes	characteristic
43	C₄²×C₉	D₄	1	1	yes	characteristic
44	#1997	C₉	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₃×Q₈	C₃×C₄	1	2	yes	—
46	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
47	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
48	C₃×C₄⋊Q₈	—	2	1	no	—
49	C₃×C₄⋊Q₈	—	2	1	no	—
50	C₃×C₄⋊Q₈	—	2	1	no	—
51	C₃×C₄⋊Q₈	—	2	1	no	—
52	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
53	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	—	2	1	no	—
56	C₃×C₄⋊C₈	—	2	1	no	—
57	C₃×#97	—	2	1	no	—
58	C₃×C₄×C₈	—	2	1	no	—
59	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
60	Q₈×C₉	—	4	4	no	—
61	Q₈×C₉	—	4	4	no	—
62	C₂×C₄×C₉	—	2	2	no	—
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	4	no	—
65	C₂×C₄×C₉	—	4	4	no	—
66	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₉	#41	1	2	yes	—
74	C₈×C₉	—	2	4	no	—
75	C₈×C₉	—	4	2	no	—
76	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
77	#423	C₂×C₉	1	1	yes	characteristic
78	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic
79	#343	C₂×C₉	1	4	yes	—
80	C₂×C₃×Q₈	—	2	2	no	—
81	C₂×C₃×Q₈	—	2	2	no	—
82	C₂×C₃×Q₈	—	4	4	no	—
83	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
86	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
87	C₃×M₁₆	—	2	2	no	—
88	C₃×M₁₆	—	2	4	no	—
89	C₃×M₁₆	C₃×Q₈	1	4	yes	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₃×C₄⋊C₄	—	2	2	no	—
93	C₃×C₄⋊C₄	—	2	2	no	—
94	C₃×C₄⋊C₄	—	2	2	no	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
97	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
98	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
99	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
100	C₂²×C₉	—	2	1	no	—
101	C₂²×C₉	—	2	1	no	—
102	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₉	C₄×Q₈	1	2	yes	—
104	C₄×C₉	—	2	1	no	—
105	C₄×C₉	—	2	1	no	—
106	C₄×C₉	—	4	4	no	—
107	C₄×C₉	#100	1	2	yes	—
108	C₄×C₉	#101	1	2	yes	—
109	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
110	C₄×C₉	#124	1	1	yes	characteristic
111	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×Q₈	C₄×C₉	1	2	yes	—
113	C₂×M₁₆	C₂²×C₉	1	2	yes	—
114	C₂×M₁₆	C₂²×C₉	1	2	yes	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	—	2	1	no	—
119	C₂×C₄⋊C₄	C₄×C₉	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
121	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
122	C₄⋊C₈	—	2	1	no	—
123	C₄⋊C₈	—	2	1	no	—
124	#97	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₃×Q₈	—	4	4	no	—
128	C₃×Q₈	—	4	4	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	4	4	no	—
133	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
140	C₂×C₃×C₄	C₃×#41	1	2	yes	—
141	C₃×C₈	—	2	4	no	—
142	C₃×C₈	—	4	2	no	—
143	C₂×C₉	—	2	2	no	—
144	C₂×C₉	#391	1	1	yes	characteristic
145	C₂×C₉	C₂×#100	1	1	yes	characteristic
146	C₂×C₉	C₂×#101	1	1	yes	characteristic
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
153	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	4	no	—
156	M₁₆	Q₈×C₉	1	4	yes	complemented
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄²	D₄×C₉	1	1	yes	characteristic
164	C₄²	D₄×C₉	1	1	yes	characteristic
165	C₄²	D₄×C₉	1	1	yes	characteristic
166	C₄²	D₄×C₉	1	1	yes	characteristic
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	4	4	no	—
174	C₃×C₄	C₃×#100	1	2	yes	—
175	C₃×C₄	C₃×#101	1	2	yes	—
176	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
177	C₃×C₄	C₃×#124	1	1	yes	characteristic
178	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
179	C₉	#1997	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	Q₈	—	4	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
192	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
193	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
194	C₂×C₄	C₉×#41	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	4	2	no	—
197	C₂×C₃	—	2	2	no	—
198	C₂×C₃	C₃×#391	1	1	yes	characteristic
199	C₂×C₃	C₂×C₃×#100	1	1	yes	characteristic
200	C₂×C₃	C₂×C₃×#101	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²	C₄×Q₈×C₉	1	2	yes	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	4	no	—
208	C₄	C₉×#100	1	2	yes	—
209	C₄	C₉×#101	1	2	yes	—
210	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
211	C₄	C₉×#124	1	1	yes	characteristic
212	C₄	C₉×C₄⋊Q₈	1	1	yes	characteristic
213	C₃	C₃×#1997	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₉×#391	1	1	yes	characteristic
216	C₂	C₂×C₉×#100	1	1	yes	characteristic
217	C₂	C₂×C₉×#101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²×C₃×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
33. C₉×M₁₆	Q₈	semidirect
44. #1997	C₉	direct (N × H)
156. M₁₆	Q₈×C₉	semidirect
179. C₉	#1997	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
C₉×#423	2
C₄×C₉×M₁₆	2
C₉×#343	2
C₃×#1997	3

Subgroups — C9×#1997 #11766718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1997 #11766718