Finite Group Databasebeta

← #37545321

Subgroups — #37545321

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×#19153841C212yesmaximal, complemented
2D4×D724C211yesmaximal, characteristic, complemented
3#26041653C211yesmaximal, characteristic, complemented
4#26041527C212yesmaximal, complemented
5#26041481C211yesmaximal, characteristic, complemented
6#26041474C211yesmaximal, characteristic, complemented
7#26041429C212yesmaximal, complemented
8#26041424C212yesmaximal, complemented
9C4×#15668434C211yesmaximal, characteristic, complemented
10#26041357C211yesmaximal, characteristic, complemented
11#547×C181C211yesmaximal, characteristic, complemented
12C22×D724C2212yes—
13C2×D4×D181—21no—
14#19153841—28no—
15C2×#15668436C2212yes—
16C2×#15668436C2212yescomplemented
17C2×#15668436C2212yescomplemented
18C2×#15668434C2211yescharacteristic
19C2×#15668434C2212yescomplemented
20C2×#15668433C2212yescomplemented
21#19153813C2211yescharacteristic
22#19153791—21no—
23#19153790C2212yes—
24#19153786—22no—
25C4×D724C2211yescharacteristic
26#19153749C2211yescharacteristic, complemented
27C4⋊#13089678C2211yescharacteristic, complemented
28#19153740C2212yescomplemented
29#19153737C2211yescharacteristic, complemented
30#19153736C2211yescharacteristic
31#19153736C2212yescomplemented
32C22⋊#13089678C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
34C2×#41×C181C2212yescomplemented
35C4⋊Q8×C181C2211yescharacteristic, complemented
36#123×C181—22no—
37C22⋊Q8×C181C2212yescomplemented
38#121×C181—21no—
39C4×D4×C181C2211yescharacteristic
40C4×D4×C181C2211yescharacteristic, complemented
41C2×C4⋊C4×C181—21no—
42C23×D181—22no—
43C2×D724—24no—
44C2×D724—22no—
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—42no—
48C2×D724C2312yes—
49C2×D724C2311yescharacteristic
50C2×C22⋊D181—22no—
51D4×D181—44no—
52C2×C4×D181—21no—
53#15668436—24no—
54#15668436—24no—
55#15668436—42no—
56#15668434—24no—
57#15668434D414yescomplemented
58#15668433—24no—
59C2×#13089678C2311yescharacteristic
60C2×#13089678C2312yes—
61C2×#13089678C2311yescharacteristic
62C4⋊Dic181C2311yescharacteristic
63#15668426—21no—
64#15668426—22no—
65#41×C181—24no—
66#41×C181—42no—
67C2×Q8×C181C2312yes—
68C2×D4×C181—22no—
69C2×D4×C181D411yescharacteristic
70C2×D4×C181C2311yescharacteristic
71C22×C4×C181—21no—
72C22×C4×C181—22no—
73C22×C4×C181C2312yes—
74C4⋊C4×C181—21no—
75C4⋊C4×C181—22no—
76C4⋊C4×C181—21no—
77C4⋊C4×C181D411yescharacteristic
78C4⋊C4×C181C2311yescharacteristic
79C22⋊C4×C181—22no—
80C22⋊C4×C181—21no—
81C22⋊C4×C181D412yes—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—22no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88D724—22no—
89D724—21no—
90D724—22no—
91D724—44no—
92D724—42no—
93D724—81no—
94D724C2×D412yes—
95C22⋊D181—44no—
96C2×Dic181—21no—
97C4×D181—42no—
98#13089678—22no—
99#13089678—21no—
100#13089678C2×D412yescomplemented
101C23×C181—21no—
102C23×C181C2×D412yes—
103Q8×C181—22no—
104Q8×C181—22no—
105D4×C181—22no—
106D4×C181—21no—
107D4×C181—44no—
108D4×C181C2×D412yescomplemented
109C2×C4×C181—22no—
110C2×C4×C181—22no—
111C2×C4×C181—21no—
112C2×C4×C181—22no—
113C2×C4×C181—44no—
114C2×C4×C181—42no—
115C2×C4×C181C2×D411yescharacteristic
116C2×C4×C181C2×D412yes—
117C2×C4×C181C2×D411yescharacteristic
118C2×C4×C181C2411yescharacteristic
119C22×C181—22no—
120C22×C181—22no—
121C22×C181—22no—
122C22×C181—22no—
123C22×C181—41no—
124C22×C181—41no—
125C22×C181C22×D411yescharacteristic
126C2×D181—22no—
127C2×D181—41no—
128C2×D181—41no—
129C2×D181—42no—
130C2×D181—42no—
131C2×D181—42no—
132C2×D181—88no—
133C4×C181—22no—
134C4×C181—21no—
135C4×C181—42no—
136C4×C181—41no—
137C4×C181AGL(1,ℤ/8)12yescomplemented
138C4×C181C22×D411yescharacteristic
139C4×C181H(5,𝔽2)11yescharacteristic
140Dic181—41no—
141C2×C181—42no—
142C2×C181—42no—
143C2×C181D4211yescharacteristic
144C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C181#57711yescharacteristic
146D181—42no—
147D181—82no—
148C181#327211yescharacteristic, complemented, Sylow
149#3272—1811nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1812no—
151#547D18111yescharacteristic, complemented
152D42—1811no—
153#492—1811no—
154#468—1812no—
155#461—1811no—
156#459—1811no—
157#450—1812no—
158#449—1812no—
159C4×D8—1811no—
160#436—1811no—
161C2×#41C2×D18112yescomplemented
162C22×D4—3621no—
163C22×D4—1812no—
164AGL(1,ℤ/8)—3628no—
165C2×SD16—1812no—
166C2×SD16—1812no—
167C2×SD16—1812no—
168C2×D8—1812no—
169C2×D8—1811no—
170C2×M16—1812no—
171C4⋊Q8C2×D18111yescharacteristic, complemented
172#127—1811no—
173#123—22no—
174C22⋊Q8C2×D18112yescomplemented
175#121—21no—
176#121—3621no—
177#121—1812no—
178#120—3622no—
179C4×D4—1811no—
180C4×D4C2×D18111yescharacteristic
181C4×D4C2×D18111yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1811no—
184C4⋊C8—1811no—
185#103—1812no—
186#102—1811no—
187#102—1811no—
188#102—1812no—
189C22⋊C8—1812no—
190C4×C8—1811no—
191C24—3622no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D18112yes—
195C2×D4—22no—
196C2×D4—3624no—
197C2×D4—3622no—
198C2×D4—3621no—
199C2×D4—3622no—
200C2×D4—3621no—
201C2×D4—1812no—
202C2×D4—1811no—
203C2×D4—7242no—
204C2×D4—7244no—
205C2×D4C22⋊D18111yescharacteristic
206C2×D4C22×D18111yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—3621no—
210C22×C4C22×D18112yes—
211SD16—3624no—
212SD16—3624no—
213SD16—7242no—
214D8—3624no—
215D8—1814no—
216M16—3624no—
217C2×C8—1811no—
218C2×C8—1812no—
219C2×C8—1811no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1811no—
224C4⋊C4C22⋊D18111yescharacteristic
225C4⋊C4C22×D18111yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—3621no—
229C22⋊C4—3622no—
230C22⋊C4C22⋊D18112yes—
231C42C22×D18111yescharacteristic
232C23—21no—
233C23—3621no—
234C23—3622no—
235C23—14482no—
236C23—7244no—
237C23—7244no—
238C23C2×C22⋊D18112yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—3622no—
245D4—3621no—
246D4—3622no—
247D4—14481no—
248D4—1812no—
249D4—7244no—
250D4—7244no—
251D4—7242no—
252D4D4×D18112yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—3621no—
260C2×C4—7242no—
261C2×C4C2×C22⋊D18111yescharacteristic
262C2×C4C2×C22⋊D18112yes—
263C2×C4C2×C22⋊D18111yescharacteristic
264C2×C4C23×D18111yescharacteristic
265C8—3622no—
266C8—3621no—
267C8—1812no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—3622no—
275C22—14488no—
276C22—7242no—
277C22—7242no—
278C22—7242no—
279C22—7241no—
280C22—7241no—
281C22C22×C22⋊D18111yescharacteristic
282C4—22no—
283C4—21no—
284C4—42no—
285C4—41no—
286C4—7241no—
287C4#1915384112yescomplemented
288C4C2×D4×D18111yescharacteristic
289C4#1915386111yescharacteristic
290C2—42no—
291C2—42no—
292C2—14482no—
293C2—7242no—
294C2D4×C22⋊D18111yescharacteristic
295C2C2×#1915384111yescharacteristic
296C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#547×C181C2
socleC22×C181C22×D4
hypercenter#547D181
hypocenterC181#3272
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19153841C2semidirect
2. D4×D724C2semidirect
3. #26041653C2semidirect
4. #26041527C2semidirect
5. #26041481C2semidirect
6. #26041474C2semidirect
7. #26041429C2semidirect
8. #26041424C2semidirect
9. C4×#15668434C2semidirect
10. #26041357C2semidirect
11. #547×C181C2semidirect
16. C2×#15668436C22semidirect
17. C2×#15668436C22semidirect
19. C2×#15668434C22semidirect
20. C2×#15668433C22semidirect
26. #19153749C22semidirect
27. C4⋊#13089678C22semidirect
28. #19153740C22semidirect
29. #19153737C22semidirect
31. #19153736C22semidirect
32. C22⋊#13089678C22semidirect
33. C4×#13089678C22semidirect
34. C2×#41×C181C22semidirect
35. C4⋊Q8×C181C22semidirect
37. C22⋊Q8×C181C22semidirect
40. C4×D4×C181C22semidirect
57. #15668434D4semidirect
100. #13089678C2×D4semidirect
108. D4×C181C2×D4semidirect
137. C4×C181AGL(1,ℤ/8)semidirect
148. C181#3272semidirect
151. #547D181semidirect
161. C2×#41C2×D181semidirect
171. C4⋊Q8C2×D181semidirect
174. C22⋊Q8C2×D181semidirect
181. C4×D4C2×D181semidirect
252. D4D4×D181semidirect
287. C4#19153841semidirect

Maximal subgroups

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

subgroupindex
C2×#191538412
D4×D7242
#260416532
#260415272
#260414812
#260414742
#260414292
#260414242
C4×#156684342
#260413572
#547×C1812
#3272181