Finite Group Databasebeta

← #44559605

Subgroups — #44559605

Order 32128.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#16860810C211yesmaximal, characteristic, complemented
2C2×#22548745C211yesmaximal, characteristic, complemented
3C2×#22548740C211yesmaximal, characteristic, complemented
4#30631847C214yesmaximal, complemented
5#30631830C214yesmaximal, complemented
6#30631695C211yesmaximal, characteristic, complemented
7#30631691C211yesmaximal, characteristic, complemented
8#30631651C211yesmaximal, characteristic, complemented
9#525×C251C211yesmaximal, characteristic, complemented
10C2×#16860810—24no—
11C2×#16860810—24no—
12C2×#16860810—22no—
13C2×#16860810C2214yes—
14C22×Dic502C2211yescharacteristic
15C22×D1004C2211yescharacteristic
16C22×C22⋊D251—21no—
17C22×C4×D251—21no—
18#22548745—24no—
19#22548740—24no—
20C2×#16860800C2212yescomplemented
21C2×#16860798C2212yescomplemented
22C2×#16860797C2212yescomplemented
23C2×#16860795C2212yescomplemented
24C2×#16860794C2211yescharacteristic
25C2×#16860794C2211yescharacteristic
26#22548638C2214yescomplemented
27#22548635C2214yescomplemented
28C22⋊(C251⋊C8)C2214yescomplemented
29C2×#41×C251C2211yescharacteristic
30C23×C4×C251C2211yescharacteristic
31#123×C251—21no—
32C22⋊Q8×C251C2212yescomplemented
33#121×C251C2212yescomplemented
34#120×C251—21no—
35C4×D4×C251—22no—
36#117×C251C2211yescharacteristic
37C23×D251—21no—
38#16860810—216no—
39#16860810—28no—
40#16860810—28no—
41C2×Dic502—22no—
42C2×Dic502—21no—
43C2×Dic502D414yescomplemented
44C2×Dic502C2312yes—
45C2×D1004—22no—
46C2×D1004—21no—
47C2×D1004D414yescomplemented
48C2×D1004C2312yes—
49C2×C22⋊D251—28no—
50C2×C22⋊D251—24no—
51C22×Dic251—21no—
52C2×C4×D251—28no—
53C2×C4×D251—24no—
54#16860800—24no—
55#16860798—24no—
56#16860797—24no—
57#16860795—24no—
58#16860794—22no—
59#16860794—22no—
60C2×C251⋊C8C2312yes—
61C2×C251⋊C8C2312yes—
62C24×C251D411yescharacteristic
63#41×C251—22no—
64#41×C251—22no—
65C2×Q8×C251C2311yescharacteristic
66C2×D4×C251—22no—
67C2×D4×C251—21no—
68C2×D4×C251C2311yescharacteristic
69C22×C4×C251—22no—
70C22×C4×C251—22no—
71C22×C4×C251—21no—
72C22×C4×C251D411yescharacteristic
73C22×C4×C251D412yes—
74C22×C4×C251C2312yes—
75C22×C4×C251C2311yescharacteristic
76C4⋊C4×C251—22no—
77C4⋊C4×C251C2312yes—
78C22⋊C4×C251—21no—
79C22⋊C4×C251—22no—
80C22⋊C4×C251—22no—
81C42×C251—21no—
82C22×D251—24no—
83C22×D251—22no—
84C22×D251—44no—
85Dic502—24no—
86Dic502—22no—
87Dic502C2×D414yes—
88D1004—24no—
89D1004—22no—
90D1004C2×D414yes—
91C22⋊D251—216no—
92C2×Dic251—24no—
93C2×Dic251—22no—
94C4×D251—216no—
95C251⋊C8—22no—
96C251⋊C8—22no—
97C23×C251—22no—
98C23×C251—21no—
99C23×C251—42no—
100C23×C251C2×D412yes—
101C23×C251C2×D411yescharacteristic
102Q8×C251—21no—
103Q8×C251—21no—
104D4×C251—21no—
105D4×C251—21no—
106D4×C251—44no—
107D4×C251—42no—
108C2×C4×C251—22no—
109C2×C4×C251—24no—
110C2×C4×C251—22no—
111C2×C4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251—21no—
114C2×C4×C251—42no—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2411yescharacteristic
120C22×C251—22no—
121C22×C251—22no—
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—44no—
125C22×C251—41no—
126C22×C251—42no—
127C22×C251#12012yes—
128C22×C251C22×D411yescharacteristic
129C2×D251—24no—
130C2×D251—48no—
131C2×D251—44no—
132C4×C251—22no—
133C4×C251—41no—
134C4×C251—42no—
135C4×C251#12012yes—
136C4×C251C22×D411yescharacteristic
137C4×C251C22×D411yescharacteristic
138Dic251—24no—
139C2×C251—22no—
140C2×C251—42no—
141C2×C251—81no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57512yes—
144D251—44no—
145C251#300511yescharacteristic, complemented, Sylow
146#3005—2511nomaximal, Sylow
147C22×#41—2511no—
148C2×#137—2511no—
149C2×AGL(1,ℤ/8)—2511no—
150#525D25111yescharacteristic, complemented
151#452—2514no—
152#449—2514no—
153#419—2511no—
154#418—2511no—
155#407—2511no—
156C2×#41—5024no—
157C2×#41—5024no—
158C2×#41—5022no—
159C2×#41—2514no—
160C2×#41C2×D25111yescharacteristic
161C22×Q8—2511no—
162C22×D4—5021no—
163C22×D4—2511no—
164C23×C4—5021no—
165C23×C4C2×D25111yescharacteristic
166#137—5024no—
167AGL(1,ℤ/8)—5024no—
168C2×Q16—2512no—
169C2×SD16—2512no—
170C2×SD16—2512no—
171C2×D8—2512no—
172C2×M16—2511no—
173C2×M16—2511no—
174#123—21no—
175C22⋊Q8C2×D25112yescomplemented
176#121C2×D25112yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D25111yescharacteristic
180#103—2514no—
181#102—2514no—
182C22⋊C8—2514no—
183C24—5021no—
184C24C22⋊D25111yescharacteristic
185#41—22no—
186#41—22no—
187#41—50216no—
188#41—5028no—
189#41—5028no—
190C2×Q8—5022no—
191C2×Q8—5021no—
192C2×Q8—2514no—
193C2×Q8—2512no—
194C2×Q8C22×D25111yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—5028no—
198C2×D4—5024no—
199C2×D4—5022no—
200C2×D4—5021no—
201C2×D4—2514no—
202C2×D4—2512no—
203C2×D4C22×D25111yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—5028no—
208C22×C4—5021no—
209C22×C4—5024no—
210C22×C4C22⋊D25112yes—
211C22×C4C22⋊D25111yescharacteristic
212C22×C4C22×D25112yes—
213C22×C4C22×D25111yescharacteristic
214Q16—5024no—
215SD16—5024no—
216SD16—5024no—
217D8—5024no—
218M16—5022no—
219M16—5022no—
220C2×C8—2512no—
221C2×C8—2512no—
222C4⋊C4—22no—
223C4⋊C4C22×D25112yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—10044no—
232C23—5022no—
233C23—5024no—
234C23C2×C22⋊D25112yes—
235C23C2×C22⋊D25111yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—5022no—
239Q8—5024no—
240Q8—2514no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—50216no—
246D4—5022no—
247D4—5024no—
248D4—2514no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—50216no—
257C2×C4—5022no—
258C2×C4—5024no—
259C2×C4D4×D25112yes—
260C2×C4D4×D25112yes—
261C2×C4C2×C22⋊D25112yes—
262C2×C4C2×C22⋊D25111yescharacteristic
263C2×C4C23×D25111yescharacteristic
264C8—5022no—
265C8—5022no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—42no—
272C22—41no—
273C22—10048no—
274C22—10044no—
275C22—5024no—
276C22#2254868312yes—
277C22C22×C22⋊D25111yescharacteristic
278C4—22no—
279C4—42no—
280C4—41no—
281C4—5024no—
282C4#2254868312yes—
283C4C2×D4×D25111yescharacteristic
284C4C2×D4×D25111yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—10044no—
289C2C2×#2254868311yescharacteristic
290C2#3063248512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#525×C251C2
socleC22×C251C22×D4
hypercenter#525D251
hypocenterC251#3005
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#16860810C2semidirect
2. C2×#22548745C2semidirect
3. C2×#22548740C2semidirect
4. #30631847C2semidirect
5. #30631830C2semidirect
6. #30631695C2semidirect
7. #30631691C2semidirect
8. #30631651C2semidirect
9. #525×C251C2semidirect
20. C2×#16860800C22semidirect
21. C2×#16860798C22semidirect
22. C2×#16860797C22semidirect
23. C2×#16860795C22semidirect
26. #22548638C22semidirect
27. #22548635C22semidirect
28. C22⋊(C251⋊C8)C22semidirect
32. C22⋊Q8×C251C22semidirect
33. #121×C251C22semidirect
43. C2×Dic502D4semidirect
47. C2×D1004D4semidirect
145. C251#3005semidirect
150. #525D251semidirect
175. C22⋊Q8C2×D251semidirect
176. #121C2×D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168608102
C2×#225487452
C2×#225487402
#306318472
#306318302
#306316952
#306316912
#306316512
#525×C2512
#3005251