Finite Group Databasebeta

← #16860492

Subgroups — #16860492

Order 3968.

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#14521885C211yesmaximal, characteristic, complemented
2C2×#11758566C211yesmaximal, characteristic, complemented
3#14521861C218yesmaximal, complemented
4#14521854C218yesmaximal, complemented
5#14521849C216yesmaximal, complemented
6C2×#11758485C216yesmaximal, complemented
7C31×#568C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9#11758567C2214yescomplemented
10#11758566—24no—
11#11758566C2218yescomplemented
12#11758566C2214yescomplemented
13#41×D31—24no—
14#11758564—21no—
15C2×#92499—21no—
16#11758561—23no—
17C2×D4×D31—23no—
18C2×#92495C2213yescomplemented
19C2×#92495C2216yescomplemented
20C22×D124C2213yescomplemented
21C31×#131C2218yescomplemented
22C2×M16×C31C2216yescomplemented
23#11758486C2214yescomplemented
24#11758486C22112yescomplemented
25#11758485C22112yescomplemented
26#11758485C22124yescomplemented
27#11758485C22112yescomplemented
28#11758481C22124yescomplemented
29C2×D248C22112yescomplemented
30C2×#92470C22112yescomplemented
31C23×D31—23no—
32#41×C31D414yes—
33#41×C31C2314yes—
34C2×Q8×C31D411yescharacteristic
35C2×D4×C31D413yes—
36C22×C4×C31C2313yes—
37C2×C22⋊D31—23no—
38C2×C22⋊D31—26no—
39#92499—28no—
40#92499—24no—
41Q8×D31—24no—
42#92497—212no—
43D4×D31—224no—
44D4×D31—212no—
45#92495—26no—
46#92495—26no—
47#92495—26no—
48#92495C23112yes—
49#92495C2314yes—
50#92495C23112yescomplemented
51C2×D124—23no—
52C2×D124—26no—
53C2×D124C23112yes—
54C2×D124C2313yes—
55C2×D124C2316yescomplemented
56C2×C4×D31—23no—
57C2×C4×D31—26no—
58C2×Dic62C2311yescharacteristic
59C2×Dic62C2316yescomplemented
60M16×C31C2314yes—
61M16×C31C23112yescomplemented
62C2×C8×C31C23112yescomplemented
63Dic124C2318yescomplemented
64D248C23124yescomplemented
65#92470C23124yescomplemented
66#92470C2318yescomplemented
67C23×C31C2×D413yes—
68C22×D31—23no—
69C22×D31—26no—
70C22×D31—212no—
71C22×D31—86no—
72Q8×C31C2×D414yes—
73D4×C31C2×D4112yes—
74C2×C4×C31C2×D413yes—
75C2×C4×C31C2×D416yes—
76C2×C4×C31C2416yes—
77C2×C4×C31C2411yescharacteristic
78C22⋊D31—24no—
79C22⋊D31—212no—
80C22⋊D31—212no—
81C2×Dic31—21no—
82C2×Dic31—26no—
83D124—23no—
84D124—212no—
85D124—26no—
86D124C2416yes—
87D124C2414yes—
88C4×D31—24no—
89C4×D31—212no—
90C4×D31—212no—
91Dic62—21no—
92Dic62—26no—
93Dic62C2414yes—
94Dic62C2412yes—
95C8×C31C2418yes—
96#3585—311nomaximal, Sylow
97C22×C31—43no—
98C22×C31C22×D411yescharacteristic
99C22×C31C22×D416yes—
100C2×D31—24no—
101C2×D31—26no—
102C2×D31—46no—
103C2×D31—86no—
104C2×D31—812no—
105C4×C31C22×D411yescharacteristic
106C4×C31C22×D416yes—
107C4×C31C2511yescharacteristic
108Dic31—24no—
109Dic31—22no—
110#585—311no—
111C2×H(5,𝔽2)—311no—
112#577—318no—
113#576—318no—
114#575—316no—
115C2×AGL(1,ℤ/8)—316no—
116#568D3111yescharacteristic, complemented
117C2×C31—21no—
118C2×C31—26no—
119C2×C31C23×D411yescharacteristic
120D31—46no—
121D31—44no—
1222−1+4—314no—
1232−1+4—621no—
124H(5,𝔽2)—314no—
125H(5,𝔽2)—318no—
126H(5,𝔽2)—624no—
127H(5,𝔽2)—623no—
128C2×#41—316no—
129C2×#41—313no—
130C2×#41—624no—
131C2×#41—621no—
132C2×#41C2×D3111yescharacteristic
133C22×D4—313no—
134C22×D4—623no—
135#137—314no—
136#137—3112no—
137AGL(1,ℤ/8)—3112no—
138AGL(1,ℤ/8)—3124no—
139AGL(1,ℤ/8)—3112no—
140#135—3124no—
141C2×SD16—3112no—
142C2×D8—3112no—
143#131C2×D3118yescomplemented
144C2×M16C2×D3116yescomplemented
145C31#358511yescharacteristic, complemented, Sylow
146C24—623no—
147#41—3112no—
148#41—314no—
149#41—3112no—
150#41—628no—
151#41—624no—
152#41—6212no—
153#41—626no—
154#41—626no—
155#41—626no—
156#41D12414yes—
157#41C22×D3114yes—
158C2×Q8—311no—
159C2×Q8—316no—
160C2×Q8—624no—
161C2×Q8D12411yescharacteristic
162C2×D4—3112no—
163C2×D4—313no—
164C2×D4—316no—
165C2×D4—623no—
166C2×D4—626no—
167C2×D4—6224no—
168C2×D4—6212no—
169C2×D4—623no—
170C2×D4—626no—
171C2×D4D12413yes—
172C22×C4—623no—
173C22×C4—626no—
174C22×C4C22×D3113yes—
175Q16—318no—
176SD16—3124no—
177SD16—318no—
178D8—3124no—
179M16C22×D3114yes—
180M16C22×D31112yescomplemented
181C2×C8C22×D31112yescomplemented
182C23—623no—
183C23—626no—
184C23—6212no—
185C23—2486no—
186C23C2×D12413yes—
187Q8—314no—
188Q8—312no—
189Q8—626no—
190Q8—621no—
191Q8C2×D12414yes—
192D4—314no—
193D4—316no—
194D4—6212no—
195D4—624no—
196D4—6212no—
197D4—623no—
198D4—6212no—
199D4—626no—
200D4C2×D124112yes—
201C2×C4—621no—
202C2×C4—626no—
203C2×C4—624no—
204C2×C4—6212no—
205C2×C4—6212no—
206C2×C4C2×D12413yes—
207C2×C4C2×D12416yes—
208C2×C4C23×D3116yes—
209C2×C4C23×D3111yescharacteristic
210C8C23×D3118yes—
211C22—43no—
212C22—624no—
213C22—626no—
214C22—1246no—
215C22—24812no—
216C22—2486no—
217C22C22×D12411yescharacteristic
218C22C22×D12416yes—
219C4—622no—
220C4—624no—
221C4C22×D12411yescharacteristic
222C4C22×D12416yes—
223C4C24×D3111yescharacteristic
224C2—21no—
225C2—26no—
226C2—1246no—
227C2—1244no—
228C2C23×D12411yescharacteristic
subgroupG/H
centerC2C23×D124
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC31×#568C2
socleC2×C31C23×D4
hypercenter#568D31
hypocenterC31#3585
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. C2×#11758566C2semidirect
3. #14521861C2semidirect
4. #14521854C2semidirect
5. #14521849C2semidirect
6. C2×#11758485C2semidirect
7. C31×#568C2semidirect
9. #11758567C22semidirect
11. #11758566C22semidirect
12. #11758566C22semidirect
18. C2×#92495C22semidirect
19. C2×#92495C22semidirect
20. C22×D124C22semidirect
21. C31×#131C22semidirect
22. C2×M16×C31C22semidirect
23. #11758486C22semidirect
24. #11758486C22semidirect
25. #11758485C22semidirect
26. #11758485C22semidirect
27. #11758485C22semidirect
28. #11758481C22semidirect
29. C2×D248C22semidirect
30. C2×#92470C22semidirect
50. #92495C23semidirect
55. C2×D124C23semidirect
59. C2×Dic62C23semidirect
61. M16×C31C23semidirect
62. C2×C8×C31C23semidirect
63. Dic124C23semidirect
64. D248C23semidirect
65. #92470C23semidirect
66. #92470C23semidirect
116. #568D31semidirect
143. #131C2×D31semidirect
144. C2×M16C2×D31semidirect
145. C31#3585semidirect
180. M16C22×D31semidirect
181. C2×C8C22×D31semidirect

Maximal subgroups

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

subgroupindex
#145218852
C2×#117585662
#145218612
#145218542
#145218492
C2×#117584852
C31×#5682
#358531