Finite Group Databasebeta

← C9×#3026 #11767747

Subgroups — C9×#3026 #11767747

Order 1152.

1200 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 318 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C9×#534C213yesmaximal, complemented
3C9×#458C214yesmaximal, complemented
4C9×B(4,𝔽2)C214yesmaximal, complemented
5C2×C9×#99C213yesmaximal, complemented
6C3×#3026C311yesmaximal, characteristic
7C9×H(5,𝔽2)—26no—
8C9×H(5,𝔽2)C2214yes—
9C2×C9×#41—23no—
10C22×D4×C9—23no—
11C22×D4×C9C2213yes—
12C22×D4×C9C2211yescharacteristic, complemented
13C9×#123—23no—
14C9×#123C2216yescomplemented
15C9×#121—26no—
16C9×#120—23no—
17C9×#120C2216yescomplemented
18C2×C9×C22⋊C4C2213yescomplemented
19C9×#99C22112yescomplemented
20C2×C3×H(5,𝔽2)C2×C311yescharacteristic
21C3×#534C2×C313yes—
22C3×#458C2×C314yes—
23C3×B(4,𝔽2)C2×C314yes—
24C2×C3×#99C2×C313yes—
25C24×C9—21no—
26C24×C9D413yescomplemented
27C24×C9C2311yescharacteristic
28C24×C9C2311yescharacteristic, complemented
29C9×#41—212no—
30C9×#41—212no—
31C2×Q8×C9—21no—
32C2×D4×C9—23no—
33C2×D4×C9—212no—
34C2×D4×C9—26no—
35C2×D4×C9—212no—
36C2×D4×C9—26no—
37C2×D4×C9—23no—
38C2×D4×C9—23no—
39C2×D4×C9—26no—
40C2×D4×C9D416yescomplemented
41C2×D4×C9C2316yes—
42C22×C4×C9—23no—
43C22×C4×C9—23no—
44C22×C4×C9D413yescomplemented
45C22×C4×C9C2311yescharacteristic, complemented
46C9×C4⋊C4—26no—
47C9×C22⋊C4—23no—
48C9×C22⋊C4—26no—
49C9×C22⋊C4—26no—
50C9×C22⋊C4C2316yescomplemented
51#3026C911yescharacteristic, direct product factor, Sylow
52C3×H(5,𝔽2)—26no—
53C3×H(5,𝔽2)C22×C314yes—
54C2×C3×#41—23no—
55C22×C3×D4—23no—
56C22×C3×D4C22×C313yes—
57C22×C3×D4C22×C311yescharacteristic
58C3×#123—23no—
59C3×#123C22×C316yes—
60C3×#121—26no—
61C3×#120—23no—
62C3×#120C22×C316yes—
63C2×C3×C22⋊C4C22×C313yes—
64C3×#99C22×C3112yes—
65C23×C9—24no—
66C23×C9—23no—
67C23×C9—23no—
68C23×C9—23no—
69C23×C9—81no—
70C23×C9—81no—
71C23×C9—83no—
72C23×C9—82no—
73C23×C9C2×D416yes—
74C23×C9C2×D413yes—
75C23×C9C2×D413yes—
76C23×C9C2411yescharacteristic
77Q8×C9—24no—
78D4×C9—212no—
79D4×C9—212no—
80D4×C9—212no—
81D4×C9—41no—
82D4×C9—43no—
83D4×C9—86no—
84C2×C4×C9—212no—
85C2×C4×C9—23no—
86C2×C4×C9—23no—
87C2×C4×C9—26no—
88C2×C4×C9—23no—
89C2×C4×C9—83no—
90C2×C4×C9C2×D416yes—
91C2×H(5,𝔽2)C2×C911yescharacteristic, complemented
92#534C2×C913yescomplemented
93#458C2×C914yescomplemented
94B(4,𝔽2)C2×C914yescomplemented
95C2×#99C2×C913yescomplemented
96C24×C3—21no—
97C24×C3C3×D413yes—
98C24×C3C23×C311yescharacteristic
99C24×C3C23×C311yescharacteristic
100C3×#41—212no—
101C3×#41—212no—
102C2×C3×Q8—21no—
103C2×C3×D4—23no—
104C2×C3×D4—212no—
105C2×C3×D4—26no—
106C2×C3×D4—212no—
107C2×C3×D4—26no—
108C2×C3×D4—23no—
109C2×C3×D4—23no—
110C2×C3×D4—26no—
111C2×C3×D4C3×D416yes—
112C2×C3×D4C23×C316yes—
113C22×C3×C4—23no—
114C22×C3×C4—23no—
115C22×C3×C4C3×D413yes—
116C22×C3×C4C23×C311yescharacteristic
117C3×C4⋊C4—26no—
118C3×C22⋊C4—23no—
119C3×C22⋊C4—26no—
120C3×C22⋊C4—26no—
121C3×C22⋊C4C23×C316yes—
122C22×C9—26no—
123C22×C9—43no—
124C22×C9—43no—
125C22×C9—41no—
126C22×C9—46no—
127C22×C9—46no—
128C22×C9—46no—
129C22×C9—41no—
130C22×C9—83no—
131C22×C9—84no—
132C22×C9#12011yescharacteristic
133C22×C9#12013yes—
134C22×C9C22×D413yes—
135C4×C9—26no—
136C4×C9—46no—
137C4×C9—41no—
138H(5,𝔽2)—26no—
139H(5,𝔽2)C22×C914yes—
140C2×#41—23no—
141C22×D4—23no—
142C22×D4C22×C913yes—
143C22×D4C22×C911yescharacteristic, complemented
144#123—23no—
145#123C22×C916yescomplemented
146#121—26no—
147#120—23no—
148#120C22×C916yescomplemented
149C2×C22⋊C4C22×C913yescomplemented
150#99C22×C9112yescomplemented
151C23×C3—24no—
152C23×C3—23no—
153C23×C3—23no—
154C23×C3—23no—
155C23×C3—81no—
156C23×C3—81no—
157C23×C3—83no—
158C23×C3—82no—
159C23×C3C2×C3×D416yes—
160C23×C3C2×C3×D413yes—
161C23×C3C2×C3×D413yes—
162C23×C3C24×C311yescharacteristic
163C3×Q8—24no—
164C3×D4—212no—
165C3×D4—212no—
166C3×D4—212no—
167C3×D4—41no—
168C3×D4—43no—
169C3×D4—86no—
170C2×C3×C4—212no—
171C2×C3×C4—23no—
172C2×C3×C4—23no—
173C2×C3×C4—26no—
174C2×C3×C4—23no—
175C2×C3×C4—83no—
176C2×C3×C4C2×C3×D416yes—
177C2×C9—23no—
178C2×C9—21no—
179C2×C9—23no—
180C2×C9—46no—
181C2×C9—81no—
182C2×C9C2×#12011yescharacteristic
183C24—21no—
184C24D4×C913yescomplemented
185C24C23×C911yescharacteristic
186C24C23×C911yescharacteristic, complemented
187#41—212no—
188#41—212no—
189C2×Q8—21no—
190C2×D4—23no—
191C2×D4—212no—
192C2×D4—26no—
193C2×D4—212no—
194C2×D4—26no—
195C2×D4—23no—
196C2×D4—23no—
197C2×D4—26no—
198C2×D4D4×C916yescomplemented
199C2×D4C23×C916yes—
200C22×C4—23no—
201C22×C4—23no—
202C22×C4D4×C913yescomplemented
203C22×C4C23×C911yescharacteristic, complemented
204C4⋊C4—26no—
205C22⋊C4—23no—
206C22⋊C4—26no—
207C22⋊C4—26no—
208C22⋊C4C23×C916yescomplemented
209C22×C3—26no—
210C22×C3—43no—
211C22×C3—43no—
212C22×C3—41no—
213C22×C3—46no—
214C22×C3—46no—
215C22×C3—46no—
216C22×C3—41no—
217C22×C3—83no—
218C22×C3—84no—
219C22×C3C3×#12011yescharacteristic
220C22×C3C3×#12013yes—
221C22×C3C22×C3×D413yes—
222C3×C4—26no—
223C3×C4—46no—
224C3×C4—41no—
225C9#302611yescharacteristic, direct product factor, Sylow
226C23—24no—
227C23—23no—
228C23—23no—
229C23—23no—
230C23—81no—
231C23—81no—
232C23—83no—
233C23—82no—
234C23C2×D4×C916yes—
235C23C2×D4×C913yes—
236C23C2×D4×C913yes—
237C23C24×C911yescharacteristic
238Q8—24no—
239D4—212no—
240D4—212no—
241D4—212no—
242D4—41no—
243D4—43no—
244D4—86no—
245C2×C4—212no—
246C2×C4—23no—
247C2×C4—23no—
248C2×C4—26no—
249C2×C4—23no—
250C2×C4—83no—
251C2×C4C2×D4×C916yes—
252C2×C3—23no—
253C2×C3—21no—
254C2×C3—23no—
255C2×C3—46no—
256C2×C3—81no—
257C2×C3C2×C3×#12011yescharacteristic
258C22—26no—
259C22—43no—
260C22—43no—
261C22—41no—
262C22—46no—
263C22—46no—
264C22—46no—
265C22—41no—
266C22—83no—
267C22—84no—
268C22C9×#12011yescharacteristic
269C22C9×#12013yes—
270C22C22×D4×C913yes—
271C4—26no—
272C4—46no—
273C4—41no—
274C3C3×#302611yescharacteristic
275C2—23no—
276C2—21no—
277C2—23no—
278C2—46no—
279C2—81no—
280C2C2×C9×#12011yescharacteristic
subgroupG/H
centerC2×C9C2×#120
commutatorC23C24×C9
Frattini, cosocleC23×C3C24×C3
socleC2×C3C2×C3×#120
prime radicalC3×#3026C3
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C9 ◁ C24×C9 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C9×H(5,𝔽2)C2semidirect
2. C9×#534C2semidirect
3. C9×#458C2semidirect
4. C9×B(4,𝔽2)C2semidirect
5. C2×C9×#99C2semidirect
12. C22×D4×C9C22semidirect
14. C9×#123C22semidirect
17. C9×#120C22semidirect
18. C2×C9×C22⋊C4C22semidirect
19. C9×#99C22semidirect
26. C24×C9D4semidirect
28. C24×C9C23semidirect
40. C2×D4×C9D4semidirect
44. C22×C4×C9D4semidirect
45. C22×C4×C9C23semidirect
50. C9×C22⋊C4C23semidirect
51. #3026C9direct (N × H)
91. C2×H(5,𝔽2)C2×C9semidirect
92. #534C2×C9semidirect
93. #458C2×C9semidirect
94. B(4,𝔽2)C2×C9semidirect
95. C2×#99C2×C9semidirect
143. C22×D4C22×C9semidirect
145. #123C22×C9semidirect
148. #120C22×C9semidirect
149. C2×C22⋊C4C22×C9semidirect
150. #99C22×C9semidirect
184. C24D4×C9semidirect
186. C24C23×C9semidirect
198. C2×D4D4×C9semidirect
202. C22×C4D4×C9semidirect
203. C22×C4C23×C9semidirect
208. C22⋊C4C23×C9semidirect
225. C9#3026direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C9×H(5,𝔽2)2
C9×#5342
C9×#4582
C9×B(4,𝔽2)2
C2×C9×#992
C3×#30263