Finite Group Databasebeta

← #44565146

Subgroups — #44565146

Order 32128.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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×#22548761C211yesmaximal, characteristic, complemented
3#30632499C218yesmaximal, complemented
4#30632492C218yesmaximal, complemented
5#30632487C216yesmaximal, complemented
6C2×#22548743C216yesmaximal, complemented
7#568×C251C211yesmaximal, characteristic, complemented
8#22548763—21no—
9#22548762C2214yescomplemented
10#22548761—24no—
11#22548761C2218yescomplemented
12#22548761C2214yescomplemented
13#22548760—23no—
14C2×#16860811—21no—
15C2×#16860810C2213yescomplemented
16C2×#16860810C2216yescomplemented
17#41×D251—24no—
18C22×D1004C2213yescomplemented
19C2×D4×D251—23no—
20#22548748C2214yescomplemented
21#22548748C22112yescomplemented
22#22548743C22112yescomplemented
23#22548743C22124yescomplemented
24#22548743C22112yescomplemented
25#22548737C22124yescomplemented
26C2×#16860799C22112yescomplemented
27C2×D2008C22112yescomplemented
28C2×#41×C251C2211yescharacteristic
29#131×C251C2218yescomplemented
30C2×M16×C251C2216yescomplemented
31C23×D251—23no—
32#16860811—24no—
33#16860811—28no—
34#16860810—26no—
35#16860810—26no—
36#16860810—26no—
37#16860810C23112yes—
38#16860810C2314yes—
39#16860810C23112yescomplemented
40#16860809—212no—
41C2×Dic502C2311yescharacteristic
42C2×Dic502C2316yescomplemented
43Q8×D251—24no—
44C2×D1004—23no—
45C2×D1004—26no—
46C2×D1004C2313yes—
47C2×D1004C23112yes—
48C2×D1004C2316yescomplemented
49C2×C22⋊D251—23no—
50C2×C22⋊D251—26no—
51D4×D251—224no—
52D4×D251—212no—
53C2×C4×D251—23no—
54C2×C4×D251—26no—
55Dic1004C2318yescomplemented
56#16860799C2318yescomplemented
57#16860799C23124yescomplemented
58D2008C23124yescomplemented
59#41×C251D414yes—
60#41×C251C2314yes—
61C2×Q8×C251D411yescharacteristic
62C2×D4×C251D413yes—
63C22×C4×C251C2313yes—
64M16×C251C2314yes—
65M16×C251C23112yescomplemented
66C2×C8×C251C23112yescomplemented
67C22×D251—23no—
68C22×D251—26no—
69C22×D251—212no—
70C22×D251—86no—
71Dic502—26no—
72Dic502—21no—
73Dic502C2414yes—
74Dic502C2412yes—
75D1004—212no—
76D1004—26no—
77D1004—23no—
78D1004C2414yes—
79D1004C2416yes—
80C22⋊D251—212no—
81C22⋊D251—212no—
82C22⋊D251—24no—
83C2×Dic251—21no—
84C2×Dic251—26no—
85C4×D251—24no—
86C4×D251—212no—
87C4×D251—212no—
88C23×C251C2×D413yes—
89Q8×C251C2×D414yes—
90D4×C251C2×D4112yes—
91C2×C4×C251C2×D413yes—
92C2×C4×C251C2×D416yes—
93C2×C4×C251C2416yes—
94C2×C4×C251C2411yescharacteristic
95C8×C251C2418yes—
96C22×C251—43no—
97C22×C251C22×D411yescharacteristic
98C22×C251C22×D416yes—
99C2×D251—24no—
100C2×D251—26no—
101C2×D251—46no—
102C2×D251—86no—
103C2×D251—812no—
104C4×C251C22×D411yescharacteristic
105C4×C251C22×D416yes—
106C4×C251C2511yescharacteristic
107Dic251—22no—
108Dic251—24no—
109C2×C251—21no—
110C2×C251—26no—
111C2×C251C23×D411yescharacteristic
112D251—46no—
113D251—44no—
114C251#358511yescharacteristic, complemented, Sylow
115#3585—2511nomaximal, Sylow
116#585—2511no—
117C2×H(5,𝔽2)—2511no—
118#577—2518no—
119#576—2518no—
120#575—2516no—
121C2×AGL(1,ℤ/8)—2516no—
122#568D25111yescharacteristic, complemented
1232−1+4—5021no—
1242−1+4—2514no—
125H(5,𝔽2)—5024no—
126H(5,𝔽2)—5023no—
127H(5,𝔽2)—2514no—
128H(5,𝔽2)—2518no—
129C2×#41—5024no—
130C2×#41—5021no—
131C2×#41—2516no—
132C2×#41—2513no—
133C2×#41C2×D25111yescharacteristic
134C22×D4—5023no—
135C22×D4—2513no—
136#137—2514no—
137#137—25112no—
138AGL(1,ℤ/8)—25112no—
139AGL(1,ℤ/8)—25124no—
140AGL(1,ℤ/8)—25112no—
141#135—25124no—
142C2×SD16—25112no—
143C2×D8—25112no—
144#131C2×D25118yescomplemented
145C2×M16C2×D25116yescomplemented
146C24—5023no—
147#41—5028no—
148#41—5024no—
149#41—50212no—
150#41—5026no—
151#41—5026no—
152#41—5026no—
153#41—25112no—
154#41—25112no—
155#41—2514no—
156#41D100414yes—
157#41C22×D25114yes—
158C2×Q8—5024no—
159C2×Q8—2516no—
160C2×Q8—2511no—
161C2×Q8D100411yescharacteristic
162C2×D4—5023no—
163C2×D4—5026no—
164C2×D4—50224no—
165C2×D4—50212no—
166C2×D4—5023no—
167C2×D4—5026no—
168C2×D4—2516no—
169C2×D4—25112no—
170C2×D4—2513no—
171C2×D4D100413yes—
172C22×C4—5023no—
173C22×C4—5026no—
174C22×C4C22×D25113yes—
175Q16—2518no—
176SD16—25124no—
177SD16—2518no—
178D8—25124no—
179M16C22×D25114yes—
180M16C22×D251112yescomplemented
181C2×C8C22×D251112yescomplemented
182C23—20086no—
183C23—5026no—
184C23—50212no—
185C23—5023no—
186C23C2×D100413yes—
187Q8—5026no—
188Q8—5021no—
189Q8—2514no—
190Q8—2512no—
191Q8C2×D100414yes—
192D4—50212no—
193D4—5024no—
194D4—50212no—
195D4—5023no—
196D4—50212no—
197D4—5026no—
198D4—2514no—
199D4—2516no—
200D4C2×D1004112yes—
201C2×C4—5021no—
202C2×C4—5026no—
203C2×C4—5024no—
204C2×C4—50212no—
205C2×C4—50212no—
206C2×C4C2×D100413yes—
207C2×C4C2×D100416yes—
208C2×C4C23×D25116yes—
209C2×C4C23×D25111yescharacteristic
210C8C23×D25118yes—
211C22—43no—
212C22—200812no—
213C22—20086no—
214C22—10046no—
215C22—5024no—
216C22—5026no—
217C22C22×D100411yescharacteristic
218C22C22×D100416yes—
219C4—5022no—
220C4—5024no—
221C4C22×D100411yescharacteristic
222C4C22×D100416yes—
223C4C24×D25111yescharacteristic
224C2—21no—
225C2—26no—
226C2—10046no—
227C2—10044no—
228C2C23×D100411yescharacteristic
subgroupG/H
centerC2C23×D1004
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
Fitting#568×C251C2
socleC2×C251C23×D4
hypercenter#568D251
hypocenterC251#3585
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #30632523C2semidirect
2. C2×#22548761C2semidirect
3. #30632499C2semidirect
4. #30632492C2semidirect
5. #30632487C2semidirect
6. C2×#22548743C2semidirect
7. #568×C251C2semidirect
9. #22548762C22semidirect
11. #22548761C22semidirect
12. #22548761C22semidirect
15. C2×#16860810C22semidirect
16. C2×#16860810C22semidirect
18. C22×D1004C22semidirect
20. #22548748C22semidirect
21. #22548748C22semidirect
22. #22548743C22semidirect
23. #22548743C22semidirect
24. #22548743C22semidirect
25. #22548737C22semidirect
26. C2×#16860799C22semidirect
27. C2×D2008C22semidirect
29. #131×C251C22semidirect
30. C2×M16×C251C22semidirect
39. #16860810C23semidirect
42. C2×Dic502C23semidirect
48. C2×D1004C23semidirect
55. Dic1004C23semidirect
56. #16860799C23semidirect
57. #16860799C23semidirect
58. D2008C23semidirect
65. M16×C251C23semidirect
66. C2×C8×C251C23semidirect
114. C251#3585semidirect
122. #568D251semidirect
144. #131C2×D251semidirect
145. C2×M16C2×D251semidirect
180. M16C22×D251semidirect
181. C2×C8C22×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
C2×#225487612
#306324992
#306324922
#306324872
C2×#225487432
#568×C2512
#3585251