Finite Group Databasebeta

← #44559633

Subgroups — #44559633

Order 32128.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22548740C211yesmaximal, characteristic, complemented
2C22×#16860795C211yesmaximal, characteristic, complemented
3#30632251C211yesmaximal, characteristic, complemented
4#30631832C214yesmaximal, complemented
5#30631821C214yesmaximal, complemented
6#30631692C211yesmaximal, characteristic, complemented
7#30631683C211yesmaximal, characteristic, complemented
8#30631654C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic, complemented
11#22548740—24no—
12C2×#16860797C2212yescomplemented
13C2×#16860795—24no—
14C2×#16860795C2214yescomplemented
15C2×#16860795C2212yescomplemented
16C2×#16860794C2211yescharacteristic, complemented
17C22×C251⋊C8C2211yescharacteristic, complemented
18#22548711—21no—
19#22548704—21no—
20#22548689C2214yescomplemented
21#22548683—22no—
22#22548674C2211yescharacteristic, complemented
23#22548639C2212yescomplemented
24#22548636C2212yescomplemented
25#22548636C2214yescomplemented
26C22⋊(C251⋊C8)C2214yescomplemented
27H(5,𝔽2)×C251—24no—
28H(5,𝔽2)×C251—24no—
29C2×#41×C251—21no—
30C2×#41×C251C2211yescharacteristic
31C2×#41×C251C2211yescharacteristic
32C2×#41×C251C2212yescomplemented
33C22×D4×C251—21no—
34C22×D4×C251—22no—
35C22×D4×C251C2211yescharacteristic
36C22×D4×C251C2212yescomplemented
37C23×D251—21no—
38C2×D1004—22no—
39C2×D1004—22no—
40C2×D1004—21no—
41C2×D1004C2312yes—
42C2×C22⋊D251—24no—
43#16860797—24no—
44#16860795—24no—
45#16860795—28no—
46#16860794—22no—
47C2×C251⋊C8—22no—
48C2×C251⋊C8C2312yes—
49C2×C251⋊C8C2312yes—
50C4⋊Dic251C2312yes—
51C22⋊Dic251—21no—
52(C2×D251)⋊C4—24no—
53C4×Dic251—21no—
54C24×C251—21no—
55C24×C251—22no—
56#41×C251—28no—
57#41×C251—24no—
58#41×C251—24no—
59#41×C251—22no—
60#41×C251—22no—
61#41×C251—22no—
62#41×C251D414yescomplemented
63C2×Q8×C251D411yescharacteristic
64C2×Q8×C251C2311yescharacteristic
65C2×D4×C251—216no—
66C2×D4×C251—28no—
67C2×D4×C251—21no—
68C2×D4×C251—22no—
69C2×D4×C251—24no—
70C2×D4×C251—24no—
71C2×D4×C251—21no—
72C2×D4×C251—22no—
73C2×D4×C251—22no—
74C2×D4×C251D412yes—
75C2×D4×C251D411yescharacteristic
76C2×D4×C251D414yescomplemented
77C2×D4×C251C2312yes—
78C2×D4×C251C2311yescharacteristic
79C22×C4×C251—21no—
80C22×C4×C251—22no—
81C22×C4×C251C2312yes—
82C22×C4×C251C2311yescharacteristic
83C22×D251—22no—
84C22×D251—42no—
85C22×D251—82no—
86D1004—22no—
87D1004—22no—
88D1004—42no—
89C22⋊D251—84no—
90C2×Dic251—22no—
91C251⋊C8—22no—
92C251⋊C8—22no—
93C23×C251—28no—
94C23×C251—21no—
95C23×C251—22no—
96C23×C251—24no—
97C23×C251—82no—
98C23×C251—84no—
99C23×C251C2×D412yes—
100C23×C251C2×D411yescharacteristic
101Q8×C251—22no—
102Q8×C251—21no—
103Q8×C251C2×D412yes—
104D4×C251—28no—
105D4×C251—28no—
106D4×C251—24no—
107D4×C251—24no—
108D4×C251—22no—
109D4×C251—24no—
110D4×C251—22no—
111D4×C251—21no—
112D4×C251C2×D412yes—
113D4×C251C2×D414yes—
114C2×C4×C251—28no—
115C2×C4×C251—21no—
116C2×C4×C251—24no—
117C2×C4×C251—22no—
118C2×C4×C251—22no—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2×D412yes—
122C2×C4×C251C2×D412yes—
123C2×C4×C251C2411yescharacteristic
124C22×C251—22no—
125C22×C251—22no—
126C22×C251—24no—
127C22×C251—22no—
128C22×C251—24no—
129C22×C251—44no—
130C22×C251—42no—
131C22×C251—84no—
132C22×C251—88no—
133C22×C251#12012yes—
134C22×C251C22×D411yescharacteristic
135C2×D251—42no—
136C2×D251—44no—
137C2×D251—84no—
138C4×C251—22no—
139C4×C251—22no—
140C4×C251#12012yes—
141C4×C251C22×D411yescharacteristic
142C4×C251C22×D411yescharacteristic
143Dic251—42no—
144C2×C251—22no—
145C2×C251—42no—
146C2×C251—44no—
147C2×C251—42no—
148C2×C251C2×#12011yescharacteristic
149C2×C251#57612yes—
150D251—82no—
151C251#300811yescharacteristic, complemented, Sylow
152#3008—2511nomaximal, Sylow
153C2×H(5,𝔽2)D25111yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—2511no—
155C22×D8—2511no—
156#535—2511no—
157#449—2514no—
158#447—2514no—
159#418—2511no—
160#416—2511no—
161#408—2511no—
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D25111yescharacteristic
166C2×#41C2×D25111yescharacteristic
167C2×#41C2×D25112yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—2511no—
171C22×D4C2×D25111yescharacteristic
172C22×D4C2×D25112yescomplemented
173AGL(1,ℤ/8)—5024no—
174C2×SD16—2512no—
175C2×D8—5024no—
176C2×D8—2514no—
177C2×D8—2512no—
178C2×M16—2511no—
179C22×C8—2511no—
180#127—5021no—
181#124—5021no—
182#121—2514no—
183#120—5022no—
184#117—2511no—
185#103—2512no—
186#102—2512no—
187#102—2514no—
188C22⋊C8—2514no—
189C24—21no—
190C24—22no—
191C24—5021no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D25114yescomplemented
199C2×Q8C22⋊D25111yescharacteristic
200C2×Q8C22×D25111yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—5024no—
211C2×D4—5021no—
212C2×D4—5022no—
213C2×D4—5022no—
214C2×D4—2512no—
215C2×D4C22⋊D25112yes—
216C2×D4C22⋊D25111yescharacteristic
217C2×D4C22⋊D25114yescomplemented
218C2×D4C22×D25112yes—
219C2×D4C22×D25111yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D25112yes—
223C22×C4C22×D25111yescharacteristic
224SD16—5024no—
225D8—5028no—
226D8—5024no—
227M16—5022no—
228C2×C8—5022no—
229C2×C8—2512no—
230C2×C8—2512no—
231C4⋊C4—2512no—
232C22⋊C4—5021no—
233C22⋊C4—5024no—
234C42—5021no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—20082no—
242C23—10042no—
243C23—5022no—
244C23C2×C22⋊D25112yes—
245C23C2×C22⋊D25111yescharacteristic
246Q8—21no—
247Q8—22no—
248Q8C2×C22⋊D25112yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—24no—
257D4—20084no—
258D4—10042no—
259D4—5022no—
260D4—5022no—
261D4C2×C22⋊D25112yes—
262D4C2×C22⋊D25114yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—5022no—
269C2×C4C2×C22⋊D25112yes—
270C2×C4C2×C22⋊D25112yes—
271C2×C4C2×C22⋊D25112yes—
272C2×C4C2×C22⋊D25111yescharacteristic
273C2×C4C23×D25111yescharacteristic
274C8—5022no—
275C8—5022no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—24no—
280C22—22no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—20084no—
286C22—10044no—
287C22—10042no—
288C22#2254868412yes—
289C22C22×C22⋊D25111yescharacteristic
290C4—22no—
291C4—22no—
292C4—10042no—
293C4#2254868412yes—
294C4C22×C22⋊D25111yescharacteristic
295C4C22×C22⋊D25111yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—20082no—
301C2C2×#2254868411yescharacteristic
302C2#3063249012yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×H(5,𝔽2)×C251C2
socleC22×C251C22×D4
hypercenterC2×H(5,𝔽2)D251
hypocenterC251#3008
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#22548740C2semidirect
2. C22×#16860795C2semidirect
3. #30632251C2semidirect
4. #30631832C2semidirect
5. #30631821C2semidirect
6. #30631692C2semidirect
7. #30631683C2semidirect
8. #30631654C2semidirect
9. C2×H(5,𝔽2)×C251C2semidirect
10. C22×D1004C22semidirect
12. C2×#16860797C22semidirect
14. C2×#16860795C22semidirect
15. C2×#16860795C22semidirect
16. C2×#16860794C22semidirect
17. C22×C251⋊C8C22semidirect
20. #22548689C22semidirect
22. #22548674C22semidirect
23. #22548639C22semidirect
24. #22548636C22semidirect
25. #22548636C22semidirect
26. C22⋊(C251⋊C8)C22semidirect
32. C2×#41×C251C22semidirect
36. C22×D4×C251C22semidirect
62. #41×C251D4semidirect
76. C2×D4×C251D4semidirect
151. C251#3008semidirect
153. C2×H(5,𝔽2)D251semidirect
167. C2×#41C2×D251semidirect
172. C22×D4C2×D251semidirect
198. #41C22⋊D251semidirect
217. C2×D4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487402
C22×#168607952
#306322512
#306318322
#306318212
#306316922
#306316832
#306316542
C2×H(5,𝔽2)×C2512
#3008251