Finite Group Databasebeta

← #37545406

Subgroups — #37545406

Order 23168.

526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19153835C211yesmaximal, characteristic, complemented
2#26041923C212yesmaximal, complemented
3#26041662C211yesmaximal, characteristic, complemented
4#26041520C211yesmaximal, characteristic, complemented
5#26041512C211yesmaximal, characteristic, complemented
6#26041492C212yesmaximal, complemented
7#26041430C212yesmaximal, complemented
8#26041421C212yesmaximal, complemented
9D8×Dic181C211yesmaximal, characteristic, complemented
10C8×C22⋊D181C211yesmaximal, characteristic, complemented
11C22×D8×C181C211yesmaximal, characteristic, complemented
12C2×#15668448C2212yescomplemented
13#19153835—24no—
14C2×Dic724C2211yescharacteristic, complemented
15C2×#15668437C2212yescomplemented
16C2×#15668437C2212yescomplemented
17C2×C8×D181C2211yescharacteristic, complemented
18#19153803C2212yescomplemented
19#19153797—22no—
20#19153795C2212yes—
21#19153788—22no—
22#19153787C2212yescomplemented
23#19153785—22no—
24D4×Dic181C2212yescomplemented
25C4×C22⋊D181C2211yescharacteristic
26C2×C22⋊Dic181—22no—
27#19153750C2211yescharacteristic, complemented
28#19153745C2211yescharacteristic, complemented
29#19153741C2212yescomplemented
30#19153737C2212yescomplemented
31#19153737C2212yescomplemented
32C22⋊#13089678C2211yescharacteristic
33#19153728C2211yescharacteristic, complemented
34C8×Dic181C2211yescharacteristic, complemented
35C22×D4×C181C2212yes—
36C2×D8×C181—24no—
37C2×D8×C181C2211yescharacteristic
38C2×D8×C181C2212yes—
39C2×D8×C181C2211yescharacteristic
40C22×C8×C181C2211yescharacteristic
41#15668448—24no—
42#15668448—42no—
43C2×Dic362C2312yes—
44C2×C22⋊D181—22no—
45C2×C22⋊D181—22no—
46C2×C22⋊D181D411yescharacteristic
47C22×Dic181—22no—
48C22×Dic181—22no—
49C2×C4×D181C2311yescharacteristic
50Dic724—22no—
51#15668437—24no—
52#15668437—42no—
53C2×#13089678C2311yescharacteristic
54C2×#13089678C2311yescharacteristic
55C8×D181—22no—
56C4⋊Dic181C2312yes—
57#15668428—22no—
58#15668428D411yescharacteristic
59C22⋊Dic181—22no—
60C22⋊Dic181—22no—
61C22⋊Dic181—22no—
62C22⋊Dic181—22no—
63C22⋊Dic181—22no—
64C22⋊Dic181D411yescharacteristic
65#15668426D411yescharacteristic
66C4×Dic181C2311yescharacteristic
67C24×C181—22no—
68C2×D4×C181—24no—
69C2×D4×C181—22no—
70C2×D4×C181—42no—
71C2×D4×C181C2312yes—
72C2×D4×C181C2312yes—
73C22×C4×C181C2311yescharacteristic
74D8×C181—24no—
75D8×C181—22no—
76D8×C181D414yescomplemented
77C2×C8×C181—22no—
78C2×C8×C181C2311yescharacteristic
79C2×C8×C181C2311yescharacteristic
80C22×D181C2×D411yescharacteristic
81Dic362—22no—
82Dic362—22no—
83C22⋊D181—22no—
84C22⋊D181—44no—
85C22⋊D181—82no—
86C2×Dic181—22no—
87C2×Dic181—22no—
88C2×Dic181—44no—
89C2×Dic181—82no—
90C2×Dic181C2×D411yescharacteristic
91C2×Dic181C2×D411yescharacteristic
92C2×Dic181C2×D411yescharacteristic
93C4×D181—21no—
94C4×D181—21no—
95#13089678—21no—
96#13089678—21no—
97C23×C181—22no—
98C23×C181—22no—
99C23×C181—44no—
100C23×C181—44no—
101C23×C181—82no—
102C23×C181C2×D411yescharacteristic
103D4×C181—22no—
104D4×C181—22no—
105D4×C181—22no—
106D4×C181—44no—
107D4×C181C2×D414yes—
108C2×C4×C181—21no—
109C2×C4×C181—21no—
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2411yescharacteristic
112C8×C181—21no—
113C8×C181C2×D412yes—
114C22×C181—21no—
115C22×C181—21no—
116C22×C181—24no—
117C22×C181—42no—
118C22×C181—42no—
119C22×C181—42no—
120C22×C181—42no—
121C22×C181—42no—
122C22×C181—88no—
123C22×C181#13512yescomplemented
124C22×C181C22×D411yescharacteristic
125C2×D181—21no—
126C2×D181—21no—
127C2×D181—21no—
128C4×C181—21no—
129C4×C181C22×D411yescharacteristic
130C4×C181H(5,𝔽2)11yescharacteristic
131Dic181—21no—
132Dic181—21no—
133Dic181—21no—
134Dic181—42no—
135Dic181—42no—
136C2×C181—22no—
137C2×C181—44no—
138C2×C181—82no—
139C2×C181D4211yescharacteristic
140C2×C181C2×#13511yescharacteristic
141C2×C181#57611yescharacteristic
142D181—41no—
143C181#328011yescharacteristic, complemented, Sylow
144#3280—1811nomaximal, Sylow
145C2×#135—1811no—
146C22×D8D18111yescharacteristic, complemented
147#546—1812no—
148#495—1811no—
149#467—1811no—
150#466—1811no—
151#463—1812no—
152#450—1812no—
153#449—1812no—
154C4×D8—1811no—
155C8×D4—1811no—
156C2×#41—1812no—
157C22×D4C2×D18112yes—
158#135—3624no—
159C2×Q16—1811no—
160C2×SD16—1812no—
161C2×SD16—1812no—
162C2×D8—24no—
163C2×D8C2×D18112yes—
164C2×D8C2×D18111yescharacteristic
165C2×D8C2×D18111yescharacteristic
166C22×C8—1811no—
167C22×C8C2×D18111yescharacteristic
168#124—1812no—
169#123—3622no—
170C22⋊Q8—1812no—
171#121—3622no—
172#121—1812no—
173#120—3622no—
174C4×D4—1812no—
175C4×D4—1811no—
176C2×C22⋊C4—3622no—
177#107—1811no—
178C4⋊C8—1811no—
179#103—1812no—
180#102—1812no—
181#102—1812no—
182C22⋊C8—1811no—
183C22⋊C8—1811no—
184C4×C8—1811no—
185C24—22no—
186#41—3624no—
187#41—7242no—
188C2×Q8—1812no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—3622no—
193C2×D4—3622no—
194C2×D4—1811no—
195C2×D4C22×D18112yes—
196C2×D4C22×D18112yes—
197C22×C4—3622no—
198C22×C4—3622no—
199C22×C4—1811no—
200C22×C4C22×D18111yescharacteristic
201Q16—3622no—
202SD16—3624no—
203SD16—7242no—
204D8—24no—
205D8—22no—
206D8C22⋊D18114yescomplemented
207C2×C8—22no—
208C2×C8—3622no—
209C2×C8—1811no—
210C2×C8—1811no—
211C2×C8C22×D18111yescharacteristic
212C2×C8C22×D18111yescharacteristic
213C4⋊C4—3622no—
214C4⋊C4—1812no—
215C4⋊C4—1811no—
216C22⋊C4—3622no—
217C22⋊C4—3622no—
218C22⋊C4—3622no—
219C22⋊C4—3622no—
220C22⋊C4—3622no—
221C22⋊C4—1811no—
222C22⋊C4—1811no—
223C42—1811no—
224C23—22no—
225C23—22no—
226C23—44no—
227C23—44no—
228C23—82no—
229C23—1811no—
230C23D4×D18111yescharacteristic
231Q8—3622no—
232Q8—3622no—
233D4—22no—
234D4—22no—
235D4—22no—
236D4—44no—
237D4—3622no—
238D4—14482no—
239D4—7244no—
240D4C2×C22⋊D18114yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—3621no—
244C2×C4—3621no—
245C2×C4—3622no—
246C2×C4—3622no—
247C2×C4—14482no—
248C2×C4—1811no—
249C2×C4—1811no—
250C2×C4—1811no—
251C2×C4—7244no—
252C2×C4D4×D18111yescharacteristic
253C2×C4C23×D18111yescharacteristic
254C8—21no—
255C8—3621no—
256C8—3621no—
257C8C2×C22⋊D18112yes—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—88no—
267C22—3621no—
268C22—3621no—
269C22—3621no—
270C22#1915383512yescomplemented
271C22C2×D4×D18111yescharacteristic
272C4—21no—
273C4—3621no—
274C4—3621no—
275C4—3621no—
276C4—7242no—
277C4—7242no—
278C4C22×C22⋊D18111yescharacteristic
279C4#1915386111yescharacteristic
280C2—22no—
281C2—44no—
282C2—82no—
283C2—7241no—
284C2D4×C22⋊D18111yescharacteristic
285C2C2×#1915383511yescharacteristic
286C2#2604208011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC22×D8×C181C2
socleC22×C181C22×D4
hypercenterC22×D8D181
hypocenterC181#3280
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19153835C2semidirect
2. #26041923C2semidirect
3. #26041662C2semidirect
4. #26041520C2semidirect
5. #26041512C2semidirect
6. #26041492C2semidirect
7. #26041430C2semidirect
8. #26041421C2semidirect
9. D8×Dic181C2semidirect
10. C8×C22⋊D181C2semidirect
11. C22×D8×C181C2semidirect
12. C2×#15668448C22semidirect
14. C2×Dic724C22semidirect
15. C2×#15668437C22semidirect
16. C2×#15668437C22semidirect
17. C2×C8×D181C22semidirect
18. #19153803C22semidirect
22. #19153787C22semidirect
24. D4×Dic181C22semidirect
27. #19153750C22semidirect
28. #19153745C22semidirect
29. #19153741C22semidirect
30. #19153737C22semidirect
31. #19153737C22semidirect
33. #19153728C22semidirect
34. C8×Dic181C22semidirect
76. D8×C181D4semidirect
123. C22×C181#135semidirect
143. C181#3280semidirect
146. C22×D8D181semidirect
206. D8C22⋊D181semidirect
270. C22#19153835semidirect

Maximal subgroups

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

subgroupindex
C2×#191538352
#260419232
#260416622
#260415202
#260415122
#260414922
#260414302
#260414212
D8×Dic1812
C8×C22⋊D1812
C22×D8×C1812
#3280181