Finite Group Databasebeta

← #44559610

Subgroups — #44559610

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×#22548748C211yesmaximal, characteristic, complemented
3C2×#22548743C211yesmaximal, characteristic, complemented
4#30632198C211yesmaximal, characteristic, complemented
5#30631851C214yesmaximal, complemented
6#30631834C214yesmaximal, complemented
7#30631697C211yesmaximal, characteristic, complemented
8#30631693C211yesmaximal, characteristic, complemented
9#407×C251C211yesmaximal, characteristic, complemented
10C2×#16860810—24no—
11C2×#16860810—24no—
12C2×#16860810—22no—
13C2×#16860810C2211yescharacteristic
14C2×#16860810C2214yes—
15C22×Dic502C2211yescharacteristic
16C22×D1004C2211yescharacteristic
17C22×C22⋊D251—21no—
18C22×C4×D251—21no—
19#22548748—24no—
20#22548743—24no—
21C2×Dic1004C2212yescomplemented
22C2×#16860799C2212yescomplemented
23C2×#16860799C2212yescomplemented
24C2×D2008C2212yescomplemented
25#22548698—21no—
26#22548694C2212yescomplemented
27#22548689C2212yescomplemented
28#22548684—21no—
29C4×C22⋊D251—22no—
30#22548674C2211yescharacteristic
31#22548640C2214yescomplemented
32#22548637C2214yescomplemented
33C23×C4×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic
35C2×M16×C251C2211yescharacteristic
36C22⋊C8×C251C2214yescomplemented
37C23×D251—21no—
38#16860810—216no—
39#16860810—28no—
40#16860810—28no—
41#16860810—22no—
42#16860810—22no—
43C2×Dic502—22no—
44C2×Dic502—21no—
45C2×Dic502D414yescomplemented
46C2×Dic502C2311yescharacteristic
47C2×Dic502C2312yes—
48C2×D1004—22no—
49C2×D1004—21no—
50C2×D1004D414yescomplemented
51C2×D1004C2311yescharacteristic
52C2×D1004C2312yes—
53C2×C22⋊D251—28no—
54C2×C22⋊D251—24no—
55C2×C22⋊D251—21no—
56C2×C22⋊D251—22no—
57C22×Dic251—21no—
58C2×C4×D251—28no—
59C2×C4×D251—24no—
60C2×C4×D251—21no—
61Dic1004—24no—
62#16860799—24no—
63#16860799—24no—
64D2008—24no—
65C4⋊Dic251C2312yes—
66Dic251⋊C4—22no—
67C22⋊Dic251—21no—
68C22⋊Dic251—22no—
69(C2×D251)⋊C4—22no—
70C4×Dic251—21no—
71C24×C251D411yescharacteristic
72C22×C4×C251—22no—
73C22×C4×C251—22no—
74C22×C4×C251D411yescharacteristic
75C22×C4×C251D412yes—
76C22×C4×C251C2312yes—
77C22×C4×C251C2311yescharacteristic
78M16×C251—22no—
79M16×C251—22no—
80C2×C8×C251C2312yes—
81C2×C8×C251C2312yes—
82C22×D251—24no—
83C22×D251—22no—
84C22×D251—21no—
85C22×D251—44no—
86Dic502—24no—
87Dic502—22no—
88Dic502—21no—
89Dic502—21no—
90Dic502C2×D414yes—
91D1004—24no—
92D1004—22no—
93D1004—21no—
94D1004—21no—
95D1004C2×D414yes—
96C22⋊D251—216no—
97C22⋊D251—44no—
98C22⋊D251—42no—
99C2×Dic251—24no—
100C2×Dic251—22no—
101C2×Dic251—22no—
102C2×Dic251—21no—
103C4×D251—216no—
104C4×D251—42no—
105C23×C251—22no—
106C23×C251—42no—
107C23×C251C2×D412yes—
108C23×C251C2×D411yescharacteristic
109C2×C4×C251—22no—
110C2×C4×C251—24no—
111C2×C4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251C2×D412yes—
114C2×C4×C251C2×D412yes—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251C2411yescharacteristic
118C8×C251—22no—
119C8×C251—22no—
120C22×C251—22no—
121C22×C251—22no—
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—44no—
125C22×C251#12012yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—24no—
128C2×D251—48no—
129C2×D251—44no—
130C2×D251—41no—
131C2×D251—42no—
132C4×C251—22no—
133C4×C251#12012yes—
134C4×C251C22×D411yescharacteristic
135C4×C251C22×D411yescharacteristic
136Dic251—24no—
137Dic251—41no—
138Dic251—42no—
139C2×C251—22no—
140C2×C251—42no—
141C2×C251C2×#12011yescharacteristic
142C2×C251#57512yes—
143D251—44no—
144D251—81no—
145C251#300511yescharacteristic, complemented, Sylow
146#3005—2511nomaximal, Sylow
147C22×#41—2511no—
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#407D25111yescharacteristic, complemented
156C2×#41—5024no—
157C2×#41—5024no—
158C2×#41—5022no—
159C2×#41—2511no—
160C2×#41—2514no—
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×M16C2×D25111yescharacteristic
173C2×M16C2×D25111yescharacteristic
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⋊C8C2×D25114yescomplemented
183C24—5021no—
184C24D100411yescharacteristic
185#41—50216no—
186#41—5028no—
187#41—5028no—
188#41—5022no—
189#41—5022no—
190C2×Q8—5022no—
191C2×Q8—5021no—
192C2×Q8—2514no—
193C2×Q8—2511no—
194C2×Q8—2512no—
195C2×D4—5022no—
196C2×D4—5021no—
197C2×D4—5028no—
198C2×D4—5024no—
199C2×D4—5022no—
200C2×D4—5021no—
201C2×D4—2514no—
202C2×D4—2511no—
203C2×D4—2512no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—5028no—
207C22×C4—5021no—
208C22×C4—5021no—
209C22×C4—5024no—
210C22×C4D100412yes—
211C22×C4D100411yescharacteristic
212C22×C4C22×D25112yes—
213C22×C4C22×D25111yescharacteristic
214Q16—5024no—
215SD16—5024no—
216SD16—5024no—
217D8—5024no—
218M16—22no—
219M16—22no—
220C2×C8C22×D25112yes—
221C2×C8C22×D25112yes—
222C4⋊C4—5022no—
223C4⋊C4—2512no—
224C22⋊C4—5021no—
225C22⋊C4—5022no—
226C22⋊C4—5022no—
227C42—5021no—
228C23—22no—
229C23—42no—
230C23—10044no—
231C23—5021no—
232C23—5022no—
233C23—5024no—
234C23C2×D100412yes—
235C23C2×D100411yescharacteristic
236Q8—5021no—
237Q8—5021no—
238Q8—5022no—
239Q8—5024no—
240Q8—2514no—
241D4—10044no—
242D4—10042no—
243D4—50216no—
244D4—5021no—
245D4—5021no—
246D4—5022no—
247D4—5024no—
248D4—2514no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—10042no—
254C2×C4—50216no—
255C2×C4—5022no—
256C2×C4—5021no—
257C2×C4—5022no—
258C2×C4—5024no—
259C2×C4D4×D25112yes—
260C2×C4D4×D25112yes—
261C2×C4C2×D100412yes—
262C2×C4C2×D100411yescharacteristic
263C2×C4C23×D25111yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—10048no—
272C22—10042no—
273C22—10041no—
274C22—10044no—
275C22—5024no—
276C22#2254868512yes—
277C22C22×D100411yescharacteristic
278C4—22no—
279C4—10042no—
280C4—10041no—
281C4—5024no—
282C4#2254868512yes—
283C4C2×D4×D25111yescharacteristic
284C4C2×D4×D25111yescharacteristic
285C2—22no—
286C2—42no—
287C2—20081no—
288C2—10044no—
289C2C2×#2254868511yescharacteristic
290C2#3063248712yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
hypercenter#407D251
hypocenterC251#3005
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
1. C22×#16860810C2semidirect
2. C2×#22548748C2semidirect
3. C2×#22548743C2semidirect
4. #30632198C2semidirect
5. #30631851C2semidirect
6. #30631834C2semidirect
7. #30631697C2semidirect
8. #30631693C2semidirect
9. #407×C251C2semidirect
21. C2×Dic1004C22semidirect
22. C2×#16860799C22semidirect
23. C2×#16860799C22semidirect
24. C2×D2008C22semidirect
26. #22548694C22semidirect
27. #22548689C22semidirect
31. #22548640C22semidirect
32. #22548637C22semidirect
36. C22⋊C8×C251C22semidirect
45. C2×Dic502D4semidirect
50. C2×D1004D4semidirect
145. C251#3005semidirect
155. #407D251semidirect
182. C22⋊C8C2×D251semidirect

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×#225487432
#306321982
#306318512
#306318342
#306316972
#306316932
#407×C2512
#3005251