Finite Group Databasebeta

← #44559653

Subgroups — #44559653

Order 32128.

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

normal subgroups only

This group has 732 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
1C2×#22548762C211yesmaximal, characteristic, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C22×#16860799C211yesmaximal, characteristic, complemented
4#30632251C211yesmaximal, characteristic, complemented
5#30631834C214yesmaximal, complemented
6#30631828C214yesmaximal, complemented
7#30631697C211yesmaximal, characteristic, complemented
8#30631684C211yesmaximal, characteristic, complemented
9#408×C251C211yesmaximal, characteristic, complemented
10#22548762—24no—
11#22548762—24no—
12C2×#16860810C2211yescharacteristic
13C2×#16860810C2212yescomplemented
14C2×#16860809—22no—
15C2×#16860809—21no—
16C22×Dic502C2211yescharacteristic
17C22×Dic502C2212yescomplemented
18C2×Q8×D251—21no—
19C22×D1004C2211yescharacteristic
20#22548743—24no—
21C2×#16860799—24no—
22C2×#16860799C2214yes—
23C2×#16860799C2212yes—
24C2×D2008C2212yes—
25#22548711—21no—
26#22548704—21no—
27#22548689C2214yes—
28#22548683—22no—
29#22548674C2211yescharacteristic
30#22548640C2212yescomplemented
31#22548640C2214yescomplemented
32#22548637C2212yescomplemented
33C2×#41×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic
35C22×C8×C251C2211yescharacteristic
36C22⋊C8×C251C2214yescomplemented
37C23×D251—21no—
38#16860810—24no—
39#16860810—24no—
40#16860810—22no—
41#16860810D414yescomplemented
42#16860809—216no—
43#16860809—28no—
44C2×Dic502—24no—
45C2×Dic502—24no—
46C2×Dic502—21no—
47C2×Dic502—22no—
48C2×Dic502—22no—
49C2×Dic502D414yescomplemented
50C2×Dic502C2312yes—
51C2×Dic502C2311yescharacteristic
52Q8×D251—28no—
53C2×D1004—22no—
54C2×D1004—22no—
55C2×D1004—21no—
56C2×D1004C2312yes—
57C2×D1004C2311yescharacteristic
58C2×C22⋊D251—21no—
59C2×C22⋊D251—22no—
60C2×C22⋊D251—24no—
61C22×Dic251—21no—
62C22×Dic251—22no—
63C2×C4×D251—21no—
64C2×C4×D251—22no—
65#16860799—24no—
66#16860799—28no—
67D2008—24no—
68C4⋊Dic251C2312yes—
69C22⋊Dic251—21no—
70(C2×D251)⋊C4—24no—
71C4×Dic251—21no—
72#41×C251—22no—
73#41×C251—22no—
74C2×Q8×C251D411yescharacteristic
75C2×D4×C251D412yes—
76C2×D4×C251D411yescharacteristic
77C22×C4×C251C2312yes—
78C22×C4×C251C2311yescharacteristic
79M16×C251—22no—
80C2×C8×C251—22no—
81C2×C8×C251C2312yes—
82C2×C8×C251C2312yes—
83C22×D251—21no—
84C22×D251—22no—
85C22×D251—42no—
86C22×D251—82no—
87Dic502—28no—
88Dic502—24no—
89Dic502—21no—
90Dic502—22no—
91Dic502C2×D412yes—
92Dic502C2×D414yes—
93D1004—21no—
94D1004—22no—
95D1004—22no—
96D1004—42no—
97D1004C2×D412yes—
98C22⋊D251—28no—
99C22⋊D251—24no—
100C22⋊D251—84no—
101C2×Dic251—28no—
102C2×Dic251—24no—
103C2×Dic251—21no—
104C2×Dic251—22no—
105C2×Dic251—22no—
106C4×D251—28no—
107C4×D251—24no—
108C23×C251C2×D412yes—
109C23×C251C2×D411yescharacteristic
110Q8×C251—22no—
111D4×C251—24no—
112D4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2411yescharacteristic
120C8×C251—22no—
121C8×C251—22no—
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—24no—
125C22×C251#12012yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—22no—
128C2×D251—42no—
129C2×D251—42no—
130C2×D251—44no—
131C2×D251—84no—
132C4×C251—22no—
133C4×C251#12012yes—
134C4×C251C22×D411yescharacteristic
135C4×C251C22×D411yescharacteristic
136Dic251—24no—
137Dic251—22no—
138Dic251—42no—
139C2×C251—22no—
140C2×C251—42no—
141C2×C251C2×#12011yescharacteristic
142C2×C251#57712yes—
143D251—42no—
144D251—82no—
145C251#301011yescharacteristic, complemented, Sylow
146#3010—2511nomaximal, Sylow
147C2×2−1+4—2511no—
148C2×AGL(1,ℤ/8)—2511no—
149C22×SD16—2511no—
150#535—2511no—
151#449—2514no—
152#448—2514no—
153#419—2511no—
154#416—2511no—
155#408D25111yescharacteristic, complemented
1562−1+4—5024no—
1572−1+4—5024no—
158C2×#41—5022no—
159C2×#41—5021no—
160C2×#41—2512no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8—5021no—
164C22×Q8—2512no—
165C22×Q8—2511no—
166C22×D4—2511no—
167AGL(1,ℤ/8)—5024no—
168C2×SD16—5024no—
169C2×SD16—2514no—
170C2×SD16—2512no—
171C2×D8—2512no—
172C2×M16C2×D25111yescharacteristic
173C22×C8C2×D25111yescharacteristic
174#127—5021no—
175#124—5021no—
176#121—2514no—
177#120—5022no—
178#117—2511no—
179#103—2512no—
180#103—2514no—
181#102—2512no—
182C22⋊C8C2×D25114yescomplemented
183C24—5021no—
184#41—22no—
185#41—22no—
186#41—5024no—
187#41—5024no—
188#41—50216no—
189#41—5028no—
190#41—5022no—
191#41—2514no—
192C2×Q8—5024no—
193C2×Q8—5024no—
194C2×Q8—5028no—
195C2×Q8—5022no—
196C2×Q8—5022no—
197C2×Q8—5021no—
198C2×Q8—2514no—
199C2×Q8—2511no—
200C2×Q8—2512no—
201C2×Q8D100411yescharacteristic
202C2×D4—5024no—
203C2×D4—5022no—
204C2×D4—5021no—
205C2×D4—5022no—
206C2×D4—5022no—
207C2×D4—5021no—
208C2×D4—2512no—
209C2×D4—2511no—
210C2×D4D100412yes—
211C2×D4D100411yescharacteristic
212C22×C4—5022no—
213C22×C4—5021no—
214C22×C4—5022no—
215C22×C4—5021no—
216C22×C4C22×D25112yes—
217C22×C4C22×D25111yescharacteristic
218SD16—5028no—
219SD16—5024no—
220D8—5024no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D25112yes—
224C2×C8C22×D25112yes—
225C4⋊C4—2512no—
226C22⋊C4—5021no—
227C22⋊C4—5024no—
228C42—5021no—
229C23—20082no—
230C23—10042no—
231C23—5022no—
232C23—5021no—
233C23C2×D100412yes—
234C23C2×D100411yescharacteristic
235Q8—22no—
236Q8—5028no—
237Q8—5021no—
238Q8—5022no—
239Q8—5024no—
240Q8—2512no—
241Q8—2514no—
242D4—24no—
243D4—22no—
244D4—20084no—
245D4—10042no—
246D4—5028no—
247D4—5024no—
248D4—5022no—
249D4—5022no—
250D4—5021no—
251D4—2512no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—5028no—
255C2×C4—5022no—
256C2×C4—5028no—
257C2×C4—5022no—
258C2×C4—5021no—
259C2×C4—5024no—
260C2×C4—5024no—
261C2×C4D4×D25112yes—
262C2×C4D4×D25112yes—
263C2×C4C2×D100412yes—
264C2×C4C2×D100411yescharacteristic
265C2×C4C23×D25111yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—20084no—
272C22—10044no—
273C22—10042no—
274C22—10042no—
275C22—5022no—
276C22#2254868512yes—
277C22C22×D100411yescharacteristic
278C4—22no—
279C4—10042no—
280C4—5024no—
281C4—5022no—
282C4#2254868512yes—
283C4C2×D4×D25111yescharacteristic
284C4C2×D4×D25111yescharacteristic
285C2—22no—
286C2—42no—
287C2—20082no—
288C2—10042no—
289C2C2×#2254868511yescharacteristic
290C2#3063249912yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#408×C251C2
socleC22×C251C22×D4
hypercenter#408D251
hypocenterC251#3010
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#22548762C2semidirect
2. C2×#22548743C2semidirect
3. C22×#16860799C2semidirect
4. #30632251C2semidirect
5. #30631834C2semidirect
6. #30631828C2semidirect
7. #30631697C2semidirect
8. #30631684C2semidirect
9. #408×C251C2semidirect
13. C2×#16860810C22semidirect
17. C22×Dic502C22semidirect
30. #22548640C22semidirect
31. #22548640C22semidirect
32. #22548637C22semidirect
36. C22⋊C8×C251C22semidirect
41. #16860810D4semidirect
49. C2×Dic502D4semidirect
145. C251#3010semidirect
155. #408D251semidirect
182. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487622
C2×#225487432
C22×#168607992
#306322512
#306318342
#306318282
#306316972
#306316842
#408×C2512
#3010251