Finite Group Databasebeta

← #37545412

Subgroups — #37545412

Order 23168.

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

normal subgroups only

This group has 546 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×SD16×D181C212yesmaximal, complemented
2Q8×D724C211yesmaximal, characteristic, complemented
3#26041914C211yesmaximal, characteristic
4#26041651C211yesmaximal, characteristic, complemented
5#26041507C212yesmaximal, complemented
6#26041485C211yesmaximal, characteristic, complemented
7#26041431C212yesmaximal, complemented
8#26041416C212yesmaximal, complemented
9C4×#15668436C211yesmaximal, characteristic, complemented
10#26041344C211yesmaximal, characteristic, complemented
11#460×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181C2212yescomplemented
13C2×D4×D181—21no—
14C2×D4×D181C2212yes—
15C2×#15668438C2212yes—
16C2×#15668437C2212yescomplemented
17C2×#15668437C2212yescomplemented
18C2×#15668436C2211yescharacteristic
19SD16×D181—28no—
20C2×C8×D181C2212yescomplemented
21#19153816—21no—
22#19153816C2211yescharacteristic, complemented
23#19153796—22no—
24#19153796C2212yescomplemented
25#19153787—21no—
26#19153787C2212yes—
27#19153784—22no—
28C4×D724—21no—
29C4×D724C2211yescharacteristic
30C4×D724C2211yescharacteristic
31#19153747C2211yescharacteristic, complemented
32#19153741C2212yescomplemented
33#19153740C2211yescharacteristic, complemented
34#19153736C2211yescharacteristic
35#19153728C2212yescomplemented
36C4×#13089678C2211yescharacteristic, complemented
37C2×SD16×C181C2212yescomplemented
38#127×C181C2211yescharacteristic
39C4×Q8×C181C2211yescharacteristic, complemented
40C4⋊C8×C181C2211yescharacteristic, complemented
41#102×C181C2212yescomplemented
42C23×D181—22no—
43C2×Dic362—22no—
44C2×Dic362C2312yes—
45Q8×D181—24no—
46Q8×D181—42no—
47C2×D724D411yescharacteristic
48C2×D724C2311yescharacteristic
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51D4×D181—24no—
52D4×D181—42no—
53D4×D181—44no—
54C2×C4×D181—22no—
55C2×C4×D181—21no—
56C2×C4×D181C2312yes—
57#15668438—24no—
58#15668437—24no—
59#15668437—42no—
60#15668436D414yescomplemented
61C2×#13089678C2311yescharacteristic
62C2×#13089678C2311yescharacteristic
63C8×D181—24no—
64C4⋊Dic181—22no—
65C4⋊Dic181—21no—
66C4⋊Dic181—21no—
67C4⋊Dic181D411yescharacteristic
68C4⋊Dic181C2311yescharacteristic
69C22⋊Dic181—22no—
70#15668426—22no—
71#15668426—21no—
72#15668426D412yes—
73C2×Q8×C181C2311yescharacteristic
74C2×D4×C181—21no—
75C2×D4×C181—21no—
76C2×D4×C181C2312yes—
77SD16×C181—24no—
78C2×C8×C181C2312yes—
79C4⋊C4×C181—21no—
80C4⋊C4×C181C2311yescharacteristic
81C42×C181—21no—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—44no—
85C22×D181—44no—
86C22×D181—82no—
87C22×D181C2×D412yes—
88Dic362—22no—
89Dic362—22no—
90Dic362—44no—
91D724—21no—
92D724SD1614yescomplemented
93D724C2×D412yes—
94C22⋊D181—44no—
95C22⋊D181—44no—
96C2×Dic181—21no—
97C2×Dic181—22no—
98C2×Dic181C2×D412yes—
99C4×D181—22no—
100C4×D181—22no—
101C4×D181—44no—
102C4×D181—42no—
103#13089678—21no—
104#13089678C2×D412yescomplemented
105C23×C181—22no—
106Q8×C181—21no—
107Q8×C181C2×D412yescomplemented
108D4×C181—22no—
109D4×C181—22no—
110D4×C181—42no—
111D4×C181—81no—
112C2×C4×C181—21no—
113C2×C4×C181—21no—
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D411yescharacteristic
116C2×C4×C181C2411yescharacteristic
117C8×C181—22no—
118C22×C181—42no—
119C22×C181—42no—
120C22×C181—42no—
121C22×C181C22×D411yescharacteristic
122C2×D181—22no—
123C2×D181—22no—
124C2×D181—22no—
125C2×D181—41no—
126C2×D181—41no—
127C2×D181—88no—
128C2×D181C2×SD1614yescomplemented
129C4×C181—22no—
130C4×C181—21no—
131C4×C181—41no—
132C4×C181C2×SD1612yescomplemented
133C4×C181C22×D411yescharacteristic
134C4×C181H(5,𝔽2)11yescharacteristic
135Dic181—22no—
136Dic181—42no—
137Dic181—41no—
138C2×C181—82no—
139C2×C181D4211yescharacteristic
140C2×C181C22×SD1611yescharacteristic
141C2×C181#57711yescharacteristic
142D181—24no—
143D181—42no—
144C181D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1811nomaximal, Sylow
146C22×SD16—1812no—
147D4×Q8—1811no—
148D42—1811no—
149#492—1811no—
150#465—1812no—
151#461—1811no—
152#460D18111yescharacteristic, complemented
153#450—1812no—
154#448—1812no—
155C4×SD16—1811no—
156C8×D4—1811no—
157C22×Q8—1812no—
158C22×D4—3621no—
159C22×D4—1812no—
160C2×SD16—3628no—
161C2×SD16—1812no—
162C2×SD16—1811no—
163C2×SD16—1812no—
164C2×SD16—1812no—
165C2×SD16C2×D18112yescomplemented
166C22×C8—1812no—
167C4⋊Q8—3621no—
168C4⋊Q8—1811no—
169#127C2×D18111yescharacteristic
170C22⋊Q8—3622no—
171C22⋊Q8—1812no—
172#121—3621no—
173#121—1812no—
174#120—3622no—
175C4×Q8C2×D18111yescharacteristic, complemented
176C4×D4—3621no—
177C4×D4—1811no—
178C4×D4—1811no—
179#106—1811no—
180C4⋊C8C2×D18111yescharacteristic, complemented
181#103—1811no—
182#103—1812no—
183#102—1811no—
184#102C2×D18112yescomplemented
185C22⋊C8—1812no—
186C4×C8—1811no—
187C24—3622no—
188C2×Q8—3622no—
189C2×Q8—3624no—
190C2×Q8—1812no—
191C2×Q8—7242no—
192C2×Q8C22×D18111yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—3624no—
196C2×D4—3622no—
197C2×D4—3622no—
198C2×D4—1811no—
199C2×D4—1811no—
200C2×D4—7242no—
201C2×D4—7244no—
202C2×D4C22×D18112yes—
203C22×C4—3622no—
204C22×C4—3621no—
205C22×C4—1812no—
206SD16—24no—
207SD16—3624no—
208SD16—3624no—
209SD16—1814no—
210SD16—7242no—
211C2×C8—3624no—
212C2×C8—1811no—
213C2×C8—1811no—
214C2×C8C22×D18112yes—
215C4⋊C4—21no—
216C4⋊C4—3622no—
217C4⋊C4—3621no—
218C4⋊C4—3621no—
219C4⋊C4—1811no—
220C4⋊C4—1811no—
221C4⋊C4C22×D18111yescharacteristic
222C22⋊C4—3622no—
223C22⋊C4—3622no—
224C22⋊C4—3621no—
225C22⋊C4—1812no—
226C42—21no—
227C42C22×D18111yescharacteristic
228C23—22no—
229C23—3621no—
230C23—14482no—
231C23—1812no—
232C23—7244no—
233C23—7244no—
234Q8—21no—
235Q8—3622no—
236Q8—3622no—
237Q8—7244no—
238Q8D4×D18112yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—3621no—
244D4—1814no—
245D4—1812no—
246D4—7244no—
247D4—7244no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—3622no—
251C2×C4—3622no—
252C2×C4—3621no—
253C2×C4—3622no—
254C2×C4—1812no—
255C2×C4—7244no—
256C2×C4—7242no—
257C2×C4D4×D18111yescharacteristic
258C2×C4D4×D18111yescharacteristic
259C2×C4C23×D18111yescharacteristic
260C8—22no—
261C8—3621no—
262C8—1812no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—3622no—
267C22—3622no—
268C22—3622no—
269C22—14488no—
270C22—1814no—
271C22—7241no—
272C22—7241no—
273C22C2×D4×D18111yescharacteristic
274C4—22no—
275C4—21no—
276C4—41no—
277C4—3622no—
278C4—7242no—
279C4—7241no—
280C4SD16×D18112yescomplemented
281C4C2×D4×D18111yescharacteristic
282C4#1915386211yescharacteristic
283C2—82no—
284C2—3624no—
285C2—7242no—
286C2#2604191311yescharacteristic
287C2C2×SD16×D18111yescharacteristic
288C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#460×C181C2
socleC22×C181C22×D4
prime radical#26041914C2
hypercenter#460D181
hypocenterC181D4×SD16
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D181C2semidirect
2. Q8×D724C2semidirect
4. #26041651C2semidirect
5. #26041507C2semidirect
6. #26041485C2semidirect
7. #26041431C2semidirect
8. #26041416C2semidirect
9. C4×#15668436C2semidirect
10. #26041344C2semidirect
11. #460×C181C2semidirect
12. C2×Q8×D181C22semidirect
16. C2×#15668437C22semidirect
17. C2×#15668437C22semidirect
20. C2×C8×D181C22semidirect
22. #19153816C22semidirect
24. #19153796C22semidirect
31. #19153747C22semidirect
32. #19153741C22semidirect
33. #19153740C22semidirect
35. #19153728C22semidirect
36. C4×#13089678C22semidirect
37. C2×SD16×C181C22semidirect
39. C4×Q8×C181C22semidirect
40. C4⋊C8×C181C22semidirect
41. #102×C181C22semidirect
60. #15668436D4semidirect
92. D724SD16semidirect
104. #13089678C2×D4semidirect
107. Q8×C181C2×D4semidirect
128. C2×D181C2×SD16semidirect
132. C4×C181C2×SD16semidirect
144. C181D4×SD16semidirect
152. #460D181semidirect
165. C2×SD16C2×D181semidirect
175. C4×Q8C2×D181semidirect
180. C4⋊C8C2×D181semidirect
184. #102C2×D181semidirect
238. Q8D4×D181semidirect
280. C4SD16×D181semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1812
Q8×D7242
#260419142
#260416512
#260415072
#260414852
#260414312
#260414162
C4×#156684362
#260413442
#460×C1812
D4×SD16181