Subgroups — C₉×#1824 #11766545

Order 1152.

516 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₉×#380	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	4	yes	maximal, complemented
5	C₃×#1824	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₉	—	2	1	no	—
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₂×C₉×C₄⋊C₄	—	2	1	no	—
10	C₂×C₉×C₄⋊C₄	—	2	2	no	—
11	C₂×C₉×C₄⋊C₄	C₄	1	4	yes	complemented
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
15	C₉×C₂²⋊C₈	—	2	2	no	—
16	C₉×#95	—	2	1	no	—
17	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×#380	C₂×C₃	1	1	yes	characteristic
20	C₃×#340	C₂×C₃	1	4	yes	—
21	C₂⁴×C₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₉	—	2	4	no	—
23	C₂²×C₄×C₉	—	2	4	no	—
24	C₂²×C₄×C₉	—	2	2	no	—
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	4	4	no	—
27	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	D₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₉	—	2	2	no	—
32	C₂×C₈×C₉	—	2	4	no	—
33	C₉×C₄⋊C₄	—	2	4	no	—
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₉×C₄⋊C₄	—	2	4	no	—
36	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₉×C₂²⋊C₄	—	4	2	no	—
38	#1824	C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂³×C₃×C₄	—	2	1	no	—
40	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
42	C₂×C₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₃×C₄⋊C₄	C₃×C₄	1	4	yes	—
45	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
46	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
48	C₃×C₂²⋊C₈	—	2	2	no	—
49	C₃×#95	—	2	1	no	—
50	C₂³×C₉	—	2	2	no	—
51	C₂³×C₉	—	2	2	no	—
52	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₉	SD₁₆	1	2	yes	—
54	C₂³×C₉	Q₁₆	1	2	yes	—
55	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₉	—	2	2	no	—
57	C₂×C₄×C₉	—	2	4	no	—
58	C₂×C₄×C₉	—	2	4	no	—
59	C₂×C₄×C₉	—	2	4	no	—
60	C₂×C₄×C₉	—	4	8	no	—
61	C₂×C₄×C₉	—	4	4	no	—
62	C₂×C₄×C₉	—	4	2	no	—
63	C₂×C₄×C₉	—	4	2	no	—
64	C₂×C₄×C₉	—	4	2	no	—
65	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₉	#41	1	4	yes	—
69	C₈×C₉	—	4	2	no	—
70	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
71	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
72	#380	C₂×C₉	1	1	yes	characteristic
73	#340	C₂×C₉	1	4	yes	complemented
74	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	—	2	4	no	—
76	C₂²×C₃×C₄	—	2	4	no	—
77	C₂²×C₃×C₄	—	2	2	no	—
78	C₂²×C₃×C₄	—	2	2	no	—
79	C₂²×C₃×C₄	—	4	4	no	—
80	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
81	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	—	2	4	no	—
86	C₃×C₄⋊C₄	—	2	4	no	—
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	2	4	no	—
89	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
90	C₃×C₂²⋊C₄	—	4	2	no	—
91	C₂²×C₉	—	2	2	no	—
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	—	2	4	no	—
94	C₂²×C₉	—	2	4	no	—
95	C₂²×C₉	#103	1	4	yes	—
96	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₉	#123	1	2	yes	—
98	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
99	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
100	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
101	C₄×C₉	—	2	2	no	—
102	C₄×C₉	—	4	4	no	—
103	C₄×C₉	—	8	2	no	—
104	C₄×C₉	#117	1	2	yes	—
105	C₄×C₉	#123	1	2	yes	—
106	C₂³×C₄	—	2	1	no	—
107	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₉	1	2	yes	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	C₄×C₉	1	4	yes	complemented
112	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
114	C₂×C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	#95	—	2	1	no	—
117	C₂³×C₃	—	2	2	no	—
118	C₂³×C₃	—	2	2	no	—
119	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
120	C₂³×C₃	C₃×SD₁₆	1	2	yes	—
121	C₂³×C₃	C₃×Q₁₆	1	2	yes	—
122	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	4	8	no	—
128	C₂×C₃×C₄	—	4	4	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	—	4	2	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
133	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₃×#41	1	4	yes	—
136	C₃×C₈	—	4	2	no	—
137	C₂×C₉	—	2	4	no	—
138	C₂×C₉	#381	1	1	yes	characteristic
139	C₂×C₉	C₂×#103	1	1	yes	characteristic
140	C₂×C₉	#418	1	1	yes	characteristic
141	C₂×C₉	#481	1	2	yes	—
142	C₂×C₉	#484	1	2	yes	—
143	C₂⁴	D₄×C₉	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
150	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
151	C₂²×C₄	D₄×C₉	1	2	yes	—
152	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₂²×C₃	—	2	2	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	4	no	—
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	C₃×#103	1	4	yes	—
165	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₃	C₃×#123	1	2	yes	—
167	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
168	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
169	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	4	4	no	—
172	C₃×C₄	—	8	2	no	—
173	C₃×C₄	C₃×#117	1	2	yes	—
174	C₃×C₄	C₃×#123	1	2	yes	—
175	C₉	#1824	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
179	C₂³	C₉×SD₁₆	1	2	yes	—
180	C₂³	C₉×Q₁₆	1	2	yes	—
181	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
192	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
193	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
194	C₂×C₄	C₉×#41	1	4	yes	—
195	C₈	—	4	2	no	—
196	C₂×C₃	—	2	4	no	—
197	C₂×C₃	C₃×#381	1	1	yes	characteristic
198	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
199	C₂×C₃	C₃×#418	1	1	yes	characteristic
200	C₂×C₃	C₃×#481	1	2	yes	—
201	C₂×C₃	C₃×#484	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	C₉×#103	1	4	yes	—
207	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²	C₉×#123	1	2	yes	—
209	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
210	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
211	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
212	C₄	—	2	2	no	—
213	C₄	—	4	4	no	—
214	C₄	—	8	2	no	—
215	C₄	C₉×#117	1	2	yes	—
216	C₄	C₉×#123	1	2	yes	—
217	C₃	C₃×#1824	1	1	yes	characteristic
218	C₂	—	2	4	no	—
219	C₂	C₉×#381	1	1	yes	characteristic
220	C₂	C₂×C₉×#103	1	1	yes	characteristic
221	C₂	C₉×#418	1	1	yes	characteristic
222	C₂	C₉×#481	1	2	yes	—
223	C₂	C₉×#484	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	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂⁴×C₃	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#340	C₂	semidirect
11. C₂×C₉×C₄⋊C₄	C₄	semidirect
38. #1824	C₉	direct (N × H)
73. #340	C₂×C₉	semidirect
111. C₂×C₄⋊C₄	C₄×C₉	semidirect
175. C₉	#1824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₂×C₉×C₂²⋊C₈	2
C₉×#380	2
C₉×#340	2
C₃×#1824	3

Subgroups — C9×#1824 #11766545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1824 #11766545