Finite Group Databasebeta

← #19146217

Subgroups — #19146217

Order 5504.

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

normal subgroups only

This group has 556 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×#13063353C212yesmaximal, complemented
2#14583570C211yesmaximal, characteristic, complemented
3D4×D172C211yesmaximal, characteristic, complemented
4#14583276C211yesmaximal, characteristic, complemented
5#14583152C212yesmaximal, complemented
6#14583100C211yesmaximal, characteristic, complemented
7#14583095C211yesmaximal, characteristic, complemented
8#14583046C212yesmaximal, complemented
9#14583035C212yesmaximal, complemented
10C4×#10621873C211yesmaximal, characteristic, complemented
11C43×#436C211yesmaximal, characteristic, complemented
12C2×#10621898C2212yescomplemented
13C22×D172C2212yes—
14C2×D4×D43—21no—
15#13063353—28no—
16C2×#10621873C2211yescharacteristic
17C2×#10621873C2212yescomplemented
18C2×D344C2212yes—
19C2×D344C2212yescomplemented
20C2×D344C2212yescomplemented
21#13063322C2211yescharacteristic
22#13063322C2211yescharacteristic, complemented
23#13063321—21no—
24#13063300—21no—
25#13063299C2212yes—
26#13063299C2212yescomplemented
27#13063298—22no—
28#13063295—22no—
29C4×Dic86C2211yescharacteristic, complemented
30C4×D172C2211yescharacteristic
31#13063287—21no—
32#13063257C2211yescharacteristic, complemented
33#13063250C2211yescharacteristic, complemented
34#13063247C2211yescharacteristic
35#13063247C2212yescomplemented
36#13063247C2212yescomplemented
37C2×M16×C43C2212yescomplemented
38C4×D4×C43C2211yescharacteristic
39C4⋊C8×C43C2211yescharacteristic, complemented
40C22⋊C8×C43C2212yescomplemented
41C4×C8×C43C2211yescharacteristic, complemented
42C23×D43—22no—
43C2×D4×C43D411yescharacteristic
44C22×C4×C43C2312yes—
45C2×C22⋊D43—22no—
46C2×C22⋊D43—22no—
47C2×C22⋊D43—22no—
48D4×D43—44no—
49#10621898—24no—
50#10621898—42no—
51C2×D172—21no—
52C2×D172—22no—
53C2×D172—21no—
54C2×D172—21no—
55C2×D172—21no—
56C2×D172—24no—
57C2×D172—42no—
58C2×D172C2312yes—
59C2×D172C2312yes—
60C2×D172C2311yescharacteristic
61C2×C4×D43—22no—
62C2×C4×D43—21no—
63C2×Dic86C2311yescharacteristic
64M16×C43—24no—
65C2×C8×C43C2311yescharacteristic
66C2×C8×C43C2312yes—
67C2×C8×C43C2311yescharacteristic
68C4⋊C4×C43D411yescharacteristic
69C22⋊C4×C43D412yes—
70C42×C43C2311yescharacteristic
71(C2×D43)⋊C4—22no—
72(C2×D43)⋊C4—21no—
73(C2×D43)⋊C4—22no—
74C4⋊Dic43C2311yescharacteristic
75C4⋊Dic43C2311yescharacteristic
76Dic43⋊C4—21no—
77C4×Dic43—21no—
78D344—24no—
79D344—24no—
80D344—42no—
81#10621873—24no—
82#10621873D414yescomplemented
83C23×C43C2×D412yes—
84C22×D43—21no—
85C22×D43—22no—
86C22×D43—22no—
87C22×D43—44no—
88C22×D43—44no—
89C22×D43—82no—
90D4×C43—22no—
91C2×C4×C43—22no—
92C2×C4×C43—22no—
93C2×C4×C43C2×D411yescharacteristic
94C2×C4×C43C2×D412yes—
95C2×C4×C43C2×D411yescharacteristic
96C2×C4×C43C2411yescharacteristic
97C22⋊D43—44no—
98C22⋊D43—44no—
99C22⋊D43—82no—
100C2×Dic43—21no—
101C2×Dic43—21no—
102C2×Dic43—21no—
103D172—22no—
104D172—22no—
105D172—21no—
106D172—22no—
107D172—22no—
108D172—44no—
109D172—42no—
110D172—42no—
111D172—81no—
112D172—81no—
113D172C2×D412yes—
114C4×D43—44no—
115C4×D43—42no—
116Dic86—21no—
117Dic86C2×D412yescomplemented
118C8×C43—22no—
119C8×C43—21no—
120C8×C43C2×D412yescomplemented
121C22×C43—22no—
122C22×C43—22no—
123C22×C43—22no—
124C22×C43C22×D411yescharacteristic
125C2×D43—22no—
126C2×D43—41no—
127C2×D43—41no—
128C2×D43—42no—
129C2×D43—42no—
130C2×D43—42no—
131C2×D43—42no—
132C2×D43—42no—
133C2×D43—42no—
134C2×D43—88no—
135C4×C43—22no—
136C4×C43—21no—
137C4×C43AGL(1,ℤ/8)12yescomplemented
138C4×C43C22×D411yescharacteristic
139C4×C43H(5,𝔽2)11yescharacteristic
140Dic43—22no—
141Dic43—41no—
142Dic43—41no—
143#3274—431nomaximal, Sylow
144C2×C43—42no—
145C2×C43D4211yescharacteristic
146C2×C43C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C43#57611yescharacteristic
148D43—42no—
149D43—82no—
150D43—82no—
151C2×AGL(1,ℤ/8)—432no—
152#550—431no—
153D42—431no—
154#493—431no—
155#469—432no—
156#460—431no—
157#459—431no—
158#449—432no—
159#447—432no—
160C4×SD16—431no—
161#436D4311yescharacteristic, complemented
162C43#327411yescharacteristic, complemented, Sylow
163C2×#41—432no—
164C22×D4—432no—
165C22×D4—861no—
166AGL(1,ℤ/8)—868no—
167C2×SD16—431no—
168C2×SD16—432no—
169C2×D8—432no—
170C2×D8—432no—
171C2×D8—432no—
172C2×M16C2×D4312yescomplemented
173#127—431no—
174#127—431no—
175#127—861no—
176#121—432no—
177#121—432no—
178#121—861no—
179#121—862no—
180#120—862no—
181C4×Q8—431no—
182C4×D4—431no—
183C4×D4—861no—
184C4×D4C2×D4311yescharacteristic
185#106—431no—
186C4⋊C8C2×D4311yescharacteristic, complemented
187#103—431no—
188#102—431no—
189#102—432no—
190#102—432no—
191C22⋊C8C2×D4312yescomplemented
192C4×C8C2×D4311yescharacteristic, complemented
193C24—862no—
194#41—864no—
195#41—1722no—
196C2×Q8—431no—
197C2×D4—431no—
198C2×D4—432no—
199C2×D4—432no—
200C2×D4—864no—
201C2×D4—862no—
202C2×D4—862no—
203C2×D4—862no—
204C2×D4—861no—
205C2×D4—861no—
206C2×D4—861no—
207C2×D4—862no—
208C2×D4—861no—
209C2×D4—1722no—
210C2×D4—1724no—
211C2×D4D17211yescharacteristic
212C22×C4—861no—
213C22×C4—862no—
214C22×C4C22×D4312yes—
215SD16—434no—
216SD16—864no—
217D8—864no—
218D8—864no—
219D8—1722no—
220M16—24no—
221C2×C8C22×D4311yescharacteristic
222C2×C8C22×D4312yes—
223C2×C8C22×D4311yescharacteristic
224C4⋊C4—431no—
225C4⋊C4—431no—
226C4⋊C4—861no—
227C4⋊C4D17211yescharacteristic
228C22⋊C4—861no—
229C22⋊C4—862no—
230C22⋊C4—862no—
231C22⋊C4D17212yes—
232C42—861no—
233C42C22×D4311yescharacteristic
234C23—861no—
235C23—862no—
236C23—862no—
237C23—3442no—
238C23—1724no—
239C23—1724no—
240C23C2×D17212yes—
241Q8—432no—
242Q8—861no—
243D4—22no—
244D4—432no—
245D4—862no—
246D4—862no—
247D4—861no—
248D4—862no—
249D4—862no—
250D4—3441no—
251D4—3441no—
252D4—3442no—
253D4—1724no—
254D4—1724no—
255D4—1724no—
256D4—1722no—
257D4—1722no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—861no—
261C2×C4—861no—
262C2×C4—861no—
263C2×C4—1724no—
264C2×C4—1722no—
265C2×C4C2×D17211yescharacteristic
266C2×C4C2×D17212yes—
267C2×C4C2×D17211yescharacteristic
268C2×C4C23×D4311yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D4312yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—862no—
276C22—3448no—
277C22—1722no—
278C22—1722no—
279C22—1722no—
280C22—1722no—
281C22—1722no—
282C22—1722no—
283C22—1721no—
284C22—1721no—
285C22C22×D17211yescharacteristic
286C4—21no—
287C4—22no—
288C4—862no—
289C4—1721no—
290C4—1721no—
291C4#1306335312yescomplemented
292C4C2×D4×D4311yescharacteristic
293C4#1306337111yescharacteristic
294C2—42no—
295C2—3442no—
296C2—3442no—
297C2—1722no—
298C2D4×D17211yescharacteristic
299C2C2×#1306335311yescharacteristic
300C2#1458370411yescharacteristic
subgroupG/H
centerC22C22×D172
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#436C2
socleC22×C43C22×D4
hypercenter#436D43
hypocenterC43#3274
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#13063353C2semidirect
2. #14583570C2semidirect
3. D4×D172C2semidirect
4. #14583276C2semidirect
5. #14583152C2semidirect
6. #14583100C2semidirect
7. #14583095C2semidirect
8. #14583046C2semidirect
9. #14583035C2semidirect
10. C4×#10621873C2semidirect
11. C43×#436C2semidirect
12. C2×#10621898C22semidirect
17. C2×#10621873C22semidirect
19. C2×D344C22semidirect
20. C2×D344C22semidirect
22. #13063322C22semidirect
26. #13063299C22semidirect
29. C4×Dic86C22semidirect
32. #13063257C22semidirect
33. #13063250C22semidirect
35. #13063247C22semidirect
36. #13063247C22semidirect
37. C2×M16×C43C22semidirect
39. C4⋊C8×C43C22semidirect
40. C22⋊C8×C43C22semidirect
41. C4×C8×C43C22semidirect
82. #10621873D4semidirect
117. Dic86C2×D4semidirect
120. C8×C43C2×D4semidirect
137. C4×C43AGL(1,ℤ/8)semidirect
161. #436D43semidirect
162. C43#3274semidirect
172. C2×M16C2×D43semidirect
186. C4⋊C8C2×D43semidirect
191. C22⋊C8C2×D43semidirect
192. C4×C8C2×D43semidirect
271. C8D4×D43semidirect
291. C4#13063353semidirect