Finite Group Databasebeta

← #44559739

Subgroups — #44559739

Order 32128.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#30632523C211yesmaximal, characteristic, complemented
2C2×#22548743C212yesmaximal, complemented
3#30632251C211yesmaximal, characteristic, complemented
4#30631862C214yesmaximal, complemented
5#30631857C214yesmaximal, complemented
6#30631703C212yesmaximal, complemented
7#413×C251C211yesmaximal, characteristic, complemented
8#22548763C2212yescomplemented
9#22548762—22no—
10#22548761—22no—
11#22548760—22no—
12#22548760C2212yescomplemented
13C2×#16860810—22no—
14C2×#16860810C2212yes—
15#41×D251—24no—
16C22×D1004C2211yescharacteristic, complemented
17#22548743—24no—
18#22548743C2218yescomplemented
19C2×#16860799—22no—
20C2×D2008—22no—
21#22548711C2212yescomplemented
22#22548704C2212yescomplemented
23#22548689—22no—
24#22548683—22no—
25#22548674C2211yescharacteristic, complemented
26#22548641C2218yescomplemented
27C2×#41×C251C2211yescharacteristic
28C2×M16×C251C2212yescomplemented
29#101×C251C2212yescomplemented
30#100×C251C2212yescomplemented
31C23×D251—21no—
32#16860811—24no—
33#16860810—24no—
34#16860810—24no—
35#16860810—22no—
36#16860810—24no—
37#16860810—22no—
38#16860810D414yescomplemented
39#16860810C2314yes—
40#16860809—28no—
41#16860809—24no—
42C2×Dic502—22no—
43C2×Dic502D412yescomplemented
44Q8×D251—24no—
45C2×D1004—22no—
46C2×D1004—22no—
47C2×D1004—22no—
48C2×D1004—21no—
49C2×D1004D412yescomplemented
50C2×D1004C2312yes—
51C2×C22⋊D251—24no—
52C2×C22⋊D251—22no—
53C2×C22⋊D251—24no—
54D4×D251—28no—
55D4×D251—24no—
56C2×C4×D251—24no—
57C2×C4×D251—22no—
58#16860799—28no—
59D2008—28no—
60C4⋊Dic251—21no—
61C22⋊Dic251—21no—
62(C2×D251)⋊C4—24no—
63C4×Dic251C2312yes—
64#41×C251—22no—
65#41×C251—22no—
66C2×Q8×C251C2311yescharacteristic
67C2×D4×C251D412yes—
68C2×D4×C251C2311yescharacteristic
69C22×C4×C251D412yes—
70C22×C4×C251C2311yescharacteristic
71M16×C251—22no—
72M16×C251C2314yes—
73C2×C8×C251—22no—
74C22×D251—24no—
75C22×D251—22no—
76C22×D251—22no—
77C22×D251—42no—
78C22×D251—82no—
79Dic502—24no—
80Dic502—22no—
81Dic502C2×D414yes—
82D1004—24no—
83D1004—22no—
84D1004—24no—
85D1004—41no—
86D1004—41no—
87D1004C2×D414yes—
88C22⋊D251—28no—
89C22⋊D251—24no—
90C22⋊D251—24no—
91C22⋊D251—84no—
92C2×Dic251—24no—
93C2×Dic251—22no—
94C2×Dic251—22no—
95C4×D251—28no—
96C4×D251—24no—
97C4×D251—24no—
98C23×C251C2×D412yes—
99C23×C251C2×D411yescharacteristic
100Q8×C251—22no—
101D4×C251—24no—
102D4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251C2×D412yes—
106C2×C4×C251C2×D412yes—
107C2×C4×C251C2×D412yes—
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2411yescharacteristic
110C8×C251—24no—
111C22×C251—22no—
112C22×C251—41no—
113C22×C251—42no—
114C22×C251#12012yes—
115C22×C251C22×D411yescharacteristic
116C2×D251—24no—
117C2×D251—44no—
118C2×D251—42no—
119C2×D251—82no—
120C2×D251—84no—
121C2×D251—84no—
122C4×C251—22no—
123C4×C251#12011yescharacteristic
124C4×C251#12011yescharacteristic
125C4×C251C22×D412yes—
126Dic251—24no—
127Dic251—42no—
128C2×C251—21no—
129C2×C251—22no—
130C2×C251—42no—
131C2×C251C2×#12011yescharacteristic
132D251—44no—
133D251—82no—
134C251#302111yescharacteristic, complemented, Sylow
135#3021—2511nomaximal, Sylow
136#585—2511no—
137C2×AGL(1,ℤ/8)—2512no—
138#535—2511no—
139#454—2514no—
140#453—2514no—
141#421—2512no—
142#413D25111yescharacteristic, complemented
1432−1+4—5022no—
1442−1+4—2512no—
145H(5,𝔽2)—5022no—
146H(5,𝔽2)—5022no—
147H(5,𝔽2)—2512no—
148C2×#41—5024no—
149C2×#41—5022no—
150C2×#41—2512no—
151C2×#41C2×D25111yescharacteristic
152C22×D4—2511no—
153AGL(1,ℤ/8)—5024no—
154AGL(1,ℤ/8)—2518no—
155C2×SD16—5022no—
156C2×D8—5022no—
157C2×M16C2×D25112yescomplemented
158#127—2512no—
159#124—2512no—
160#121—5022no—
161#120—5022no—
162#117—2511no—
163#104—2518no—
164#101C2×D25112yescomplemented
165#100C2×D25112yescomplemented
166C24—5021no—
167#41—22no—
168#41—22no—
169#41—5024no—
170#41—5024no—
171#41—5028no—
172#41—5024no—
173#41—5024no—
174#41—5022no—
175#41—5022no—
176#41—5024no—
177#41—2514no—
178#41—2514no—
179C2×Q8—5022no—
180C2×Q8—5024no—
181C2×Q8—2512no—
182C2×Q8C22×D25111yescharacteristic
183C2×D4—5022no—
184C2×D4—5028no—
185C2×D4—5024no—
186C2×D4—5024no—
187C2×D4—5024no—
188C2×D4—5022no—
189C2×D4—5022no—
190C2×D4—5022no—
191C2×D4—5021no—
192C2×D4—2512no—
193C2×D4—2512no—
194C2×D4D100412yes—
195C2×D4C22×D25111yescharacteristic
196C22×C4—5024no—
197C22×C4—5022no—
198C22×C4D100412yes—
199C22×C4C22×D25111yescharacteristic
200SD16—5028no—
201D8—5028no—
202M16—22no—
203M16C22×D25114yes—
204C2×C8—22no—
205C4⋊C4—5021no—
206C22⋊C4—5021no—
207C22⋊C4—5024no—
208C42—2512no—
209C23—20082no—
210C23—10042no—
211C23—5024no—
212C23—5022no—
213C23—5022no—
214C23C2×D100412yes—
215C23C2×D100411yescharacteristic
216Q8—22no—
217Q8—5024no—
218Q8—5022no—
219Q8—2514no—
220D4—24no—
221D4—22no—
222D4—20084no—
223D4—10041no—
224D4—10041no—
225D4—5024no—
226D4—5028no—
227D4—5024no—
228D4—5024no—
229D4—5024no—
230D4—5022no—
231D4—2514no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—5028no—
235C2×C4—5022no—
236C2×C4—5024no—
237C2×C4—5022no—
238C2×C4—5024no—
239C2×C4—5024no—
240C2×C4D4×D25112yes—
241C2×C4D4×D25112yes—
242C2×C4C2×D100412yes—
243C2×C4C2×D100411yescharacteristic
244C2×C4C23×D25111yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—20084no—
250C22—20082no—
251C22—20084no—
252C22—10044no—
253C22—10042no—
254C22—5024no—
255C22#2254868512yes—
256C22C22×D100411yescharacteristic
257C4—22no—
258C4—10042no—
259C4—5024no—
260C4#2254868511yescharacteristic
261C4#2254868511yescharacteristic
262C4C2×D4×D25112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—20082no—
267C2—10044no—
268C2C2×#2254868511yescharacteristic
subgroupG/H
centerC2C2×#22548685
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#413×C251C2
socleC2×C251C2×#120
hypercenter#413D251
hypocenterC251#3021
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #30632523C2semidirect
2. C2×#22548743C2semidirect
3. #30632251C2semidirect
4. #30631862C2semidirect
5. #30631857C2semidirect
6. #30631703C2semidirect
7. #413×C251C2semidirect
8. #22548763C22semidirect
12. #22548760C22semidirect
16. C22×D1004C22semidirect
18. #22548743C22semidirect
21. #22548711C22semidirect
22. #22548704C22semidirect
25. #22548674C22semidirect
26. #22548641C22semidirect
28. C2×M16×C251C22semidirect
29. #101×C251C22semidirect
30. #100×C251C22semidirect
38. #16860810D4semidirect
43. C2×Dic502D4semidirect
49. C2×D1004D4semidirect
134. C251#3021semidirect
142. #413D251semidirect
157. C2×M16C2×D251semidirect
164. #101C2×D251semidirect
165. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
C2×#225487432
#306322512
#306318622
#306318572
#306317032
#413×C2512
#3021251