Subgroups — C₉×#2312 #11767033

Order 1152.

1008 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₉×#388	C₂	1	8	yes	maximal, complemented
3	C₉×#380	C₂	1	2	yes	maximal, complemented
4	C₉×#379	C₂	1	2	yes	maximal
5	C₄×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×#2312	C₃	1	1	yes	maximal, characteristic
7	C₂⁵×C₉	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	complemented
11	C₂×C₉×C₂²⋊C₄	C₄	1	8	yes	complemented
12	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄²×C₉	C₂²	1	4	yes	complemented
17	C₉×#95	C₂²	1	4	yes	complemented
18	C₉×#95	C₂²	1	4	yes	complemented
19	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₃×#388	C₂×C₃	1	8	yes	—
21	C₃×#380	C₂×C₃	1	2	yes	—
22	C₃×#379	C₂×C₃	1	2	yes	—
23	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	2	yes	—
24	C₂⁴×C₉	—	2	4	no	—
25	C₂⁴×C₉	—	4	2	no	—
26	C₂⁴×C₉	—	4	2	no	—
27	C₂⁴×C₉	C₂×C₄	1	4	yes	—
28	C₂⁴×C₉	C₂³	1	2	yes	—
29	C₂⁴×C₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₉	—	4	2	no	—
31	C₂²×C₄×C₉	—	4	2	no	—
32	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₉	C₂³	1	8	yes	—
35	C₂²×C₄×C₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	2	yes	—
37	C₉×C₄⋊C₄	—	4	4	no	—
38	C₉×C₂²⋊C₄	—	2	8	no	—
39	C₉×C₂²⋊C₄	—	4	4	no	—
40	C₉×C₂²⋊C₄	—	4	4	no	—
41	C₉×C₂²⋊C₄	C₂×C₄	1	16	yes	—
42	C₄²×C₉	—	4	4	no	—
43	#2312	C₉	1	1	yes	characteristic, direct product factor, Sylow
44	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
45	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
48	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	8	yes	—
49	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
50	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
52	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
53	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
54	C₃×#95	C₂²×C₃	1	4	yes	—
55	C₃×#95	C₂²×C₃	1	4	yes	—
56	C₂³×C₉	—	2	2	no	—
57	C₂³×C₉	—	2	2	no	—
58	C₂³×C₉	—	2	8	no	—
59	C₂³×C₉	—	2	4	no	—
60	C₂³×C₉	—	2	4	no	—
61	C₂³×C₉	—	2	4	no	—
62	C₂³×C₉	—	4	4	no	—
63	C₂³×C₉	—	4	4	no	—
64	C₂³×C₉	—	4	16	no	—
65	C₂³×C₉	C₂²×C₄	1	4	yes	—
66	C₂³×C₉	C₂²×C₄	1	2	yes	—
67	C₂³×C₉	#41	1	4	yes	—
68	C₂³×C₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	8	no	—
72	C₂×C₄×C₉	—	2	8	no	—
73	C₂×C₄×C₉	—	2	8	no	—
74	C₂×C₄×C₉	—	4	4	no	—
75	C₂×C₄×C₉	—	4	4	no	—
76	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
78	C₂²×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
79	#388	C₂×C₉	1	8	yes	complemented
80	#380	C₂×C₉	1	2	yes	complemented
81	#379	C₂×C₉	1	2	yes	—
82	C₄×C₂²⋊C₄	C₂×C₉	1	2	yes	complemented
83	C₂⁴×C₃	—	2	4	no	—
84	C₂⁴×C₃	—	4	2	no	—
85	C₂⁴×C₃	—	4	2	no	—
86	C₂⁴×C₃	C₂×C₃×C₄	1	4	yes	—
87	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
88	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	—	4	2	no	—
90	C₂²×C₃×C₄	—	4	2	no	—
91	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
92	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
93	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
94	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
95	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
96	C₃×C₄⋊C₄	—	4	4	no	—
97	C₃×C₂²⋊C₄	—	2	8	no	—
98	C₃×C₂²⋊C₄	—	4	4	no	—
99	C₃×C₂²⋊C₄	—	4	4	no	—
100	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	—
101	C₃×C₄²	—	4	4	no	—
102	C₂²×C₉	—	2	2	no	—
103	C₂²×C₉	—	2	2	no	—
104	C₂²×C₉	—	2	8	no	—
105	C₂²×C₉	—	2	4	no	—
106	C₂²×C₉	—	2	4	no	—
107	C₂²×C₉	—	2	4	no	—
108	C₂²×C₉	—	4	16	no	—
109	C₂²×C₉	—	4	8	no	—
110	C₂²×C₉	#117	1	4	yes	—
111	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
112	C₂²×C₉	C₂×#41	1	2	yes	—
113	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
115	C₄×C₉	—	2	4	no	—
116	C₄×C₉	—	2	4	no	—
117	C₄×C₉	—	4	8	no	—
118	C₂⁵	C₂²×C₉	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
120	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	complemented
122	C₂×C₂²⋊C₄	C₄×C₉	1	8	yes	complemented
123	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
127	C₂×C₄²	C₂²×C₉	1	4	yes	complemented
128	#95	C₂²×C₉	1	4	yes	complemented
129	#95	C₂²×C₉	1	4	yes	complemented
130	C₂³×C₃	—	2	2	no	—
131	C₂³×C₃	—	2	2	no	—
132	C₂³×C₃	—	2	8	no	—
133	C₂³×C₃	—	2	4	no	—
134	C₂³×C₃	—	2	4	no	—
135	C₂³×C₃	—	2	4	no	—
136	C₂³×C₃	—	4	4	no	—
137	C₂³×C₃	—	4	4	no	—
138	C₂³×C₃	—	4	16	no	—
139	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
140	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
141	C₂³×C₃	C₃×#41	1	4	yes	—
142	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	8	no	—
146	C₂×C₃×C₄	—	2	8	no	—
147	C₂×C₃×C₄	—	2	8	no	—
148	C₂×C₃×C₄	—	4	4	no	—
149	C₂×C₃×C₄	—	4	4	no	—
150	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
151	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
152	C₂×C₉	—	2	4	no	—
153	C₂×C₉	—	4	4	no	—
154	C₂×C₉	C₂×#117	1	1	yes	characteristic
155	C₂×C₉	#518	1	1	yes	characteristic
156	C₂×C₉	#518	1	1	yes	characteristic
157	C₂×C₉	#538	1	4	yes	—
158	C₂⁴	—	2	4	no	—
159	C₂⁴	—	4	2	no	—
160	C₂⁴	—	4	2	no	—
161	C₂⁴	C₂×C₄×C₉	1	4	yes	—
162	C₂⁴	C₂³×C₉	1	2	yes	—
163	C₂⁴	C₂³×C₉	1	1	yes	characteristic
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
167	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
168	C₂²×C₄	C₂³×C₉	1	8	yes	—
169	C₂²×C₄	C₂³×C₉	1	2	yes	—
170	C₂²×C₄	C₂³×C₉	1	2	yes	—
171	C₄⋊C₄	—	4	4	no	—
172	C₂²⋊C₄	—	2	8	no	—
173	C₂²⋊C₄	—	4	4	no	—
174	C₂²⋊C₄	—	4	4	no	—
175	C₂²⋊C₄	C₂×C₄×C₉	1	16	yes	—
176	C₄²	—	4	4	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	8	no	—
180	C₂²×C₃	—	2	4	no	—
181	C₂²×C₃	—	2	4	no	—
182	C₂²×C₃	—	2	4	no	—
183	C₂²×C₃	—	4	16	no	—
184	C₂²×C₃	—	4	8	no	—
185	C₂²×C₃	C₃×#117	1	4	yes	—
186	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
187	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
188	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
189	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	4	no	—
192	C₃×C₄	—	4	8	no	—
193	C₉	#2312	1	1	yes	characteristic, direct product factor, Sylow
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	8	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	4	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	4	16	no	—
203	C₂³	C₂²×C₄×C₉	1	4	yes	—
204	C₂³	C₂²×C₄×C₉	1	2	yes	—
205	C₂³	C₉×#41	1	4	yes	—
206	C₂³	C₂⁴×C₉	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
215	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	—	4	4	no	—
218	C₂×C₃	C₂×C₃×#117	1	1	yes	characteristic
219	C₂×C₃	C₃×#518	1	1	yes	characteristic
220	C₂×C₃	C₃×#518	1	1	yes	characteristic
221	C₂×C₃	C₃×#538	1	4	yes	—
222	C₂²	—	2	8	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	16	no	—
229	C₂²	—	4	8	no	—
230	C₂²	C₉×#117	1	4	yes	—
231	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
232	C₂²	C₂×C₉×#41	1	2	yes	—
233	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
234	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	8	no	—
238	C₃	C₃×#2312	1	1	yes	characteristic
239	C₂	—	2	4	no	—
240	C₂	—	4	4	no	—
241	C₂	C₂×C₉×#117	1	1	yes	characteristic
242	C₂	C₉×#518	1	1	yes	characteristic
243	C₂	C₉×#518	1	1	yes	characteristic
244	C₂	C₉×#538	1	4	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#388	C₂	semidirect
3. C₉×#380	C₂	semidirect
5. C₄×C₉×C₂²⋊C₄	C₂	semidirect
10. C₂×C₉×C₄⋊C₄	C₂²	semidirect
11. C₂×C₉×C₂²⋊C₄	C₄	semidirect
16. C₂×C₄²×C₉	C₂²	semidirect
17. C₉×#95	C₂²	semidirect
18. C₉×#95	C₂²	semidirect
43. #2312	C₉	direct (N × H)
79. #388	C₂×C₉	semidirect
80. #380	C₂×C₉	semidirect
82. C₄×C₂²⋊C₄	C₂×C₉	semidirect
121. C₂×C₄⋊C₄	C₂²×C₉	semidirect
122. C₂×C₂²⋊C₄	C₄×C₉	semidirect
127. C₂×C₄²	C₂²×C₉	semidirect
128. #95	C₂²×C₉	semidirect
129. #95	C₂²×C₉	semidirect
193. C₉	#2312	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×C₂²⋊C₄	2
C₉×#388	2
C₉×#380	2
C₉×#379	2
C₄×C₉×C₂²⋊C₄	2
C₃×#2312	3

Subgroups — C9×#2312 #11767033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2312 #11767033