Finite Group Databasebeta

← #44559607

Subgroups — #44559607

Order 32128.

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

normal subgroups only

This group has 758 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×#22548746C211yesmaximal, characteristic, complemented
2C2×#22548741C211yesmaximal, characteristic, complemented
3#30632198C211yesmaximal, characteristic, complemented
4#30631849C214yesmaximal, complemented
5#30631832C214yesmaximal, complemented
6#30631696C211yesmaximal, characteristic, complemented
7#30631692C211yesmaximal, characteristic, complemented
8#30631651C211yesmaximal, characteristic, complemented
9C22×#41×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11#22548746—24no—
12#22548741—24no—
13C2×#16860800C2212yescomplemented
14C2×#16860798C2212yescomplemented
15C2×#16860797C2212yescomplemented
16C2×#16860795C2212yescomplemented
17C2×#16860794C2211yescharacteristic
18C2×#16860794C2211yescharacteristic
19#22548698—21no—
20#22548694C2212yescomplemented
21#22548689C2212yescomplemented
22#22548684—21no—
23C4×C22⋊D251—22no—
24#22548674C2211yescharacteristic
25#22548639C2214yescomplemented
26#22548636C2214yescomplemented
27C22⋊(C251⋊C8)C2214yescomplemented
28C2×#41×C251—24no—
29C2×#41×C251—24no—
30C2×#41×C251—22no—
31C2×#41×C251C2214yes—
32C22×Q8×C251C2211yescharacteristic
33C22×D4×C251—21no—
34C22×D4×C251C2211yescharacteristic
35C23×C4×C251—21no—
36C23×C4×C251C2211yescharacteristic
37#16860810—22no—
38#16860810—22no—
39C2×Dic502C2311yescharacteristic
40C2×D1004C2311yescharacteristic
41C2×C22⋊D251—21no—
42C2×C22⋊D251—22no—
43C2×C4×D251—21no—
44#16860800—24no—
45#16860798—24no—
46#16860797—24no—
47#16860795—24no—
48#16860794—22no—
49#16860794—22no—
50C2×C251⋊C8C2312yes—
51C2×C251⋊C8C2312yes—
52C4⋊Dic251C2312yes—
53Dic251⋊C4—22no—
54C22⋊Dic251—21no—
55C22⋊Dic251—22no—
56(C2×D251)⋊C4—22no—
57C4×Dic251—21no—
58C24×C251—21no—
59C24×C251D411yescharacteristic
60#41×C251—216no—
61#41×C251—28no—
62#41×C251—28no—
63C2×Q8×C251—22no—
64C2×Q8×C251—21no—
65C2×Q8×C251D414yescomplemented
66C2×Q8×C251C2312yes—
67C2×D4×C251—28no—
68C2×D4×C251—24no—
69C2×D4×C251—21no—
70C2×D4×C251—22no—
71C2×D4×C251D414yescomplemented
72C2×D4×C251C2312yes—
73C22×C4×C251—28no—
74C22×C4×C251—21no—
75C22×C4×C251—24no—
76C22×C4×C251—22no—
77C22×C4×C251—22no—
78C22×C4×C251D411yescharacteristic
79C22×C4×C251D412yes—
80C22×C4×C251C2312yes—
81C22×C4×C251C2311yescharacteristic
82C22×D251—21no—
83Dic502—21no—
84Dic502—21no—
85D1004—21no—
86D1004—21no—
87C22⋊D251—44no—
88C22⋊D251—42no—
89C2×Dic251—22no—
90C2×Dic251—21no—
91C4×D251—42no—
92C251⋊C8—22no—
93C251⋊C8—22no—
94C23×C251—22no—
95C23×C251—24no—
96C23×C251—22no—
97C23×C251—42no—
98C23×C251—44no—
99C23×C251C2×D412yes—
100C23×C251C2×D411yescharacteristic
101Q8×C251—24no—
102Q8×C251—22no—
103Q8×C251C2×D414yes—
104D4×C251—216no—
105D4×C251—24no—
106D4×C251—22no—
107D4×C251C2×D414yes—
108C2×C4×C251—216no—
109C2×C4×C251—22no—
110C2×C4×C251—24no—
111C2×C4×C251—22no—
112C2×C4×C251—24no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2411yescharacteristic
120C22×C251—22no—
121C22×C251—22no—
122C22×C251—24no—
123C22×C251—22no—
124C22×C251—22no—
125C22×C251—44no—
126C22×C251—48no—
127C22×C251—44no—
128C22×C251#12012yes—
129C22×C251C22×D411yescharacteristic
130C2×D251—41no—
131C2×D251—42no—
132C4×C251—22no—
133C4×C251—24no—
134C4×C251#12012yes—
135C4×C251C22×D411yescharacteristic
136C4×C251C22×D411yescharacteristic
137Dic251—41no—
138Dic251—42no—
139C2×C251—22no—
140C2×C251—42no—
141C2×C251—44no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57512yes—
144D251—81no—
145C251#300511yescharacteristic, complemented, Sylow
146#3005—2511nomaximal, Sylow
147C22×#41D25111yescharacteristic, complemented
148C2×#137—2511no—
149C2×AGL(1,ℤ/8)—2511no—
150#525—2511no—
151#452—2514no—
152#449—2514no—
153#419—2511no—
154#418—2511no—
155#407—2511no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—2511no—
160C2×#41C2×D25114yes—
161C22×Q8C2×D25111yescharacteristic
162C22×D4—21no—
163C22×D4C2×D25111yescharacteristic
164C23×C4—21no—
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—5021no—
175C22⋊Q8—2512no—
176#121—2512no—
177#120—5021no—
178C4×D4—5022no—
179#117—2511no—
180#103—2514no—
181#102—2514no—
182C22⋊C8—2514no—
183C24—21no—
184C24C22⋊D25111yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—5022no—
189#41—5022no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—2511no—
193C2×Q8C22⋊D25114yescomplemented
194C2×Q8C22×D25112yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—5022no—
200C2×D4—5021no—
201C2×D4—2511no—
202C2×D4C22⋊D25114yescomplemented
203C2×D4C22×D25112yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—5021no—
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—5022no—
223C4⋊C4—2512no—
224C22⋊C4—5021no—
225C22⋊C4—5022no—
226C22⋊C4—5022no—
227C42—5021no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—42no—
232C23—44no—
233C23—5021no—
234C23C2×C22⋊D25112yes—
235C23C2×C22⋊D25111yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—5021no—
239Q8—5021no—
240Q8C2×C22⋊D25114yes—
241D4—216no—
242D4—22no—
243D4—24no—
244D4—10044no—
245D4—10042no—
246D4—5021no—
247D4—5021no—
248D4C2×C22⋊D25114yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—10042no—
257C2×C4—5022no—
258C2×C4—5021no—
259C2×C4C2×C22⋊D25112yes—
260C2×C4C2×C22⋊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—24no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—10042no—
275C22—10041no—
276C22#2254868412yes—
277C22C22×C22⋊D25111yescharacteristic
278C4—22no—
279C4—24no—
280C4—10042no—
281C4—10041no—
282C4#2254868412yes—
283C4C22×C22⋊D25111yescharacteristic
284C4C22×C22⋊D25111yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—20081no—
289C2C2×#2254868411yescharacteristic
290C2#3063248512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×#41×C251C2
socleC22×C251C22×D4
hypercenterC22×#41D251
hypocenterC251#3005
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
2. C2×#22548741C2semidirect
3. #30632198C2semidirect
4. #30631849C2semidirect
5. #30631832C2semidirect
6. #30631696C2semidirect
7. #30631692C2semidirect
8. #30631651C2semidirect
9. C22×#41×C251C2semidirect
13. C2×#16860800C22semidirect
14. C2×#16860798C22semidirect
15. C2×#16860797C22semidirect
16. C2×#16860795C22semidirect
20. #22548694C22semidirect
21. #22548689C22semidirect
25. #22548639C22semidirect
26. #22548636C22semidirect
27. C22⋊(C251⋊C8)C22semidirect
65. C2×Q8×C251D4semidirect
71. C2×D4×C251D4semidirect
145. C251#3005semidirect
147. C22×#41D251semidirect
193. C2×Q8C22⋊D251semidirect
202. C2×D4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C2×#225487412
#306321982
#306318492
#306318322
#306316962
#306316922
#306316512
C22×#41×C2512
#3005251