Finite Group Databasebeta

← #44559627

Subgroups — #44559627

Order 32128.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C2×#22548748C212yesmaximal, complemented
3C2×#22548694C211yesmaximal, characteristic, complemented
4#30631851C214yesmaximal, complemented
5#30631845C212yesmaximal, complemented
6#30631840C212yesmaximal, complemented
7#30631689C212yesmaximal, complemented
8#407×C251C211yesmaximal, characteristic, complemented
9C2×#16860810—28no—
10C2×#16860810—21no—
11C2×#16860810—21no—
12C2×#16860810C2212yes—
13C2×#16860810C2212yes—
14C22×Dic502C2211yescharacteristic
15C22×Dic502C2211yescharacteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×C22⋊D251—21no—
18C22×C4×D251—21no—
19#22548748—28no—
20C2×Dic1004C2214yescomplemented
21C2×#16860799C2214yescomplemented
22#22548694—21no—
23#22548694—21no—
24#22548694C2212yes—
25#22548694C2212yescomplemented
26C2×C4⋊Dic251C2211yescharacteristic, complemented
27C2×Dic251⋊C4—21no—
28C2×C22⋊Dic251—21no—
29#22548640C2214yescomplemented
30#22548637C2214yescomplemented
31C23×C4×C251C2211yescharacteristic
32C2×M16×C251C2212yescomplemented
33C22⋊C8×C251C2212yes—
34C22⋊C8×C251C2212yescomplemented
35C23×D251—21no—
36#16860810—28no—
37#16860810—28no—
38#16860810—24no—
39#16860810—24no—
40#16860810—24no—
41#16860810D418yes—
42C2×Dic502—24no—
43C2×Dic502—21no—
44C2×Dic502—21no—
45C2×Dic502—24no—
46C2×Dic502C2312yes—
47C2×Dic502C2312yes—
48C2×D1004—24no—
49C2×D1004—21no—
50C2×D1004C2312yes—
51C2×C22⋊D251—28no—
52C2×C22⋊D251—22no—
53C2×C22⋊D251—22no—
54C22×Dic251—21no—
55C22×Dic251—21no—
56C22×Dic251—21no—
57C2×C4×D251—28no—
58C2×C4×D251—22no—
59C2×C4×D251—22no—
60Dic1004—28no—
61#16860799—28no—
62C4⋊Dic251—21no—
63C4⋊Dic251C2312yes—
64Dic251⋊C4—22no—
65Dic251⋊C4—22no—
66C22⋊Dic251—22no—
67C22⋊Dic251—22no—
68C24×C251D411yescharacteristic
69C22×C4×C251—24no—
70C22×C4×C251D411yescharacteristic
71C22×C4×C251D411yescharacteristic
72C22×C4×C251D411yescharacteristic
73C22×C4×C251C2311yescharacteristic
74C22×C4×C251C2311yescharacteristic
75C22×C4×C251C2311yescharacteristic
76M16×C251—24no—
77C2×C8×C251C2314yes—
78C22×D251—24no—
79C22×D251—22no—
80C22×D251—42no—
81C22×D251—42no—
82Dic502—24no—
83Dic502—22no—
84Dic502—24no—
85Dic502—42no—
86Dic502C2×D414yes—
87D1004—24no—
88D1004—22no—
89D1004C2×D414yes—
90C22⋊D251—28no—
91C22⋊D251—28no—
92C2×Dic251—24no—
93C2×Dic251—22no—
94C2×Dic251—22no—
95C2×Dic251—22no—
96C2×Dic251—42no—
97C2×Dic251—42no—
98C4×D251—28no—
99C4×D251—28no—
100C23×C251—22no—
101C23×C251—22no—
102C23×C251—41no—
103C23×C251C2×D411yescharacteristic
104C23×C251C2×D411yescharacteristic
105C23×C251C2×D411yescharacteristic
106C2×C4×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251C2×D414yes—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2411yescharacteristic
116C8×C251—24no—
117C22×C251—21no—
118C22×C251—21no—
119C22×C251—21no—
120C22×C251—21no—
121C22×C251—21no—
122C22×C251—21no—
123C22×C251—21no—
124C22×C251—44no—
125C22×C251#12012yes—
126C22×C251#13712yescomplemented
127C22×C251C22×D411yescharacteristic
128C2×D251—24no—
129C2×D251—48no—
130C2×D251—42no—
131C2×D251—42no—
132C4×C251—21no—
133C4×C251—21no—
134C4×C251#12012yes—
135C4×C251C22×D412yes—
136Dic251—24no—
137Dic251—42no—
138Dic251—42no—
139C2×C251—22no—
140C2×C251—22no—
141C2×C251—41no—
142C2×C251C2×#12011yescharacteristic
143C2×C251C2×#13711yescharacteristic
144C2×C251#57511yescharacteristic
145D251—44no—
146C251#300711yescharacteristic, complemented, Sylow
147#3007—2511nomaximal, Sylow
148C22×#41—2511no—
149C2×#137—2512no—
150C2×C22⋊Q8—2511no—
151#452—2514no—
152#451—2512no—
153#450—2512no—
154#417—2512no—
155#407D25111yescharacteristic, complemented
156C2×#41—5028no—
157C2×#41—5021no—
158C2×#41—5021no—
159C2×#41—2512no—
160C2×#41—2512no—
161C22×Q8—2511no—
162C22×Q8—2511no—
163C22×D4—5021no—
164C22×D4—2511no—
165C23×C4—5021no—
166C23×C4C2×D25111yescharacteristic
167#137—5028no—
168C2×Q16—2514no—
169C2×SD16—2514no—
170C2×M16C2×D25112yescomplemented
171C22⋊Q8—5021no—
172C22⋊Q8—5021no—
173C22⋊Q8—2512no—
174C22⋊Q8—2512no—
175C2×C4⋊C4—5021no—
176C2×C4⋊C4—2511no—
177C2×C22⋊C4—5021no—
178#103—2514no—
179#102—2514no—
180C22⋊C8C2×D25112yes—
181C22⋊C8C2×D25112yescomplemented
182C24—5021no—
183C24D100411yescharacteristic
184#41—5028no—
185#41—5028no—
186#41—5024no—
187#41—5024no—
188#41—5024no—
189#41—2518no—
190C2×Q8—5024no—
191C2×Q8—5024no—
192C2×Q8—5021no—
193C2×Q8—5021no—
194C2×Q8—2512no—
195C2×Q8—2512no—
196C2×D4—5024no—
197C2×D4—5028no—
198C2×D4—5022no—
199C2×D4—5022no—
200C2×D4—5021no—
201C2×D4—2512no—
202C22×C4—24no—
203C22×C4—5028no—
204C22×C4—5021no—
205C22×C4—5021no—
206C22×C4—5021no—
207C22×C4—5022no—
208C22×C4—5022no—
209C22×C4D100411yescharacteristic
210C22×C4D100411yescharacteristic
211C22×C4D100411yescharacteristic
212C22×C4C22×D25111yescharacteristic
213C22×C4C22×D25111yescharacteristic
214C22×C4C22×D25111yescharacteristic
215Q16—5028no—
216SD16—5028no—
217M16—24no—
218C2×C8C22×D25114yes—
219C4⋊C4—5022no—
220C4⋊C4—5022no—
221C4⋊C4—5021no—
222C4⋊C4—2512no—
223C22⋊C4—5022no—
224C22⋊C4—5022no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—10042no—
229C23—10042no—
230C23—5024no—
231C23—5022no—
232C23C2×D100411yescharacteristic
233C23C2×D100411yescharacteristic
234C23C2×D100411yescharacteristic
235Q8—10042no—
236Q8—5024no—
237Q8—5024no—
238Q8—5022no—
239Q8—2514no—
240D4—5024no—
241D4—5028no—
242D4—5028no—
243D4—5022no—
244D4—2514no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—10042no—
251C2×C4—10042no—
252C2×C4—5028no—
253C2×C4—5022no—
254C2×C4—5022no—
255C2×C4—5024no—
256C2×C4—5022no—
257C2×C4—5028no—
258C2×C4D4×D25114yes—
259C2×C4C2×D100411yescharacteristic
260C2×C4C2×D100411yescharacteristic
261C2×C4C2×D100411yescharacteristic
262C2×C4C23×D25111yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—10048no—
273C22—10042no—
274C22—10042no—
275C22—5024no—
276C22#2254868512yes—
277C22#2254874812yescomplemented
278C22C22×D100411yescharacteristic
279C4—21no—
280C4—21no—
281C4—10042no—
282C4—10042no—
283C4—5024no—
284C4#2254868512yes—
285C4C2×D4×D25112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—10044no—
290C2C2×#2254868511yescharacteristic
291C2C2×#2254874811yescharacteristic
292C2#3063248711yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
prime radicalC22×#16860810C2
hypercenter#407D251
hypocenterC251#3007
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#22548748C2semidirect
3. C2×#22548694C2semidirect
4. #30631851C2semidirect
5. #30631845C2semidirect
6. #30631840C2semidirect
7. #30631689C2semidirect
8. #407×C251C2semidirect
15. C22×Dic502C22semidirect
20. C2×Dic1004C22semidirect
21. C2×#16860799C22semidirect
25. #22548694C22semidirect
26. C2×C4⋊Dic251C22semidirect
29. #22548640C22semidirect
30. #22548637C22semidirect
32. C2×M16×C251C22semidirect
34. C22⋊C8×C251C22semidirect
126. C22×C251#137semidirect
146. C251#3007semidirect
155. #407D251semidirect
170. C2×M16C2×D251semidirect
181. C22⋊C8C2×D251semidirect
277. C22#22548748semidirect

Maximal subgroups

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

subgroupindex
C22×#168608102
C2×#225487482
C2×#225486942
#306318512
#306318452
#306318402
#306316892
#407×C2512
#3007251