Finite Group Databasebeta

← #16855117

Subgroups — #16855117

Order 3968.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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#14521620C213yesmaximal, complemented
2#14521239C218yesmaximal, complemented
3#14521023C213yesmaximal, complemented
4C31×#585C211yesmaximal, characteristic
5C31×2−1+4—23no—
6C31×H(5,𝔽2)—23no—
7C31×H(5,𝔽2)C2214yes—
8C2×#41×C31—26no—
9C2×#41×C31C2213yes—
10C22×Dic62C2211yescharacteristic, complemented
11#11758521—23no—
12#11758513—23no—
13#11758512C22112yescomplemented
14#11758503C2213yescomplemented
15#11758501—26no—
16#11758422C22112yescomplemented
17#41×C31—24no—
18#41×C31—212no—
19#41×C31—26no—
20#41×C31—212no—
21#41×C31—26no—
22C2×Q8×C31—26no—
23C2×Q8×C31D413yes—
24C2×D4×C31—26no—
25C2×D4×C31—212no—
26C2×D4×C31D413yes—
27C2×D4×C31C2316yes—
28C22×C4×C31—24no—
29C22×C4×C31D416yes—
30C22×C4×C31C2311yescharacteristic
31C22×Dic31C2312yes—
32C2×Dic62—23no—
33C2×Dic62—23no—
34C22⋊Dic31—212no—
35C22⋊Dic31C2316yes—
36C4⋊Dic31—23no—
37Dic31⋊C4—212no—
38C4×Dic31—23no—
39C23×C31—24no—
40C23×C31C2×D416yes—
41C23×C31C2411yescharacteristic
42Q8×C31—24no—
43Q8×C31—26no—
44D4×C31—212no—
45D4×C31—212no—
46D4×C31—26no—
47C2×C4×C31—212no—
48C2×C4×C31—26no—
49C2×C4×C31—26no—
50C2×C4×C31C2×D416yes—
51C2×C4×C31C2×D413yes—
52C2×C4×C31C2×D413yes—
53C2×Dic31—26no—
54C2×Dic31—26no—
55Dic62—41no—
56Dic62—43no—
57#3028—311nomaximal, Sylow
58C22×C31—26no—
59C22×C31—41no—
60C22×C31—46no—
61C22×C31—84no—
62C22×C31C22×D413yes—
63C4×C31—26no—
64C4×C31#12011yescharacteristic
65C4×C31#12013yes—
66Dic31—46no—
67Dic31—42no—
68#585D3111yescharacteristic
69#536—313no—
70#458—318no—
71#410—313no—
72C2×C31—23no—
73C2×C31—46no—
74C2×C31C2×#12011yescharacteristic
752−1+4—23no—
76H(5,𝔽2)—23no—
77H(5,𝔽2)C2×D3114yes—
78C2×#41—26no—
79C2×#41C2×D3113yes—
80C22×Q8—311no—
81C4⋊Q8—623no—
82#124—623no—
83#123—3112no—
84C22⋊Q8—626no—
85#117—313no—
86#99—3112no—
87C31#302811yescharacteristic, complemented, Sylow
88#41—24no—
89#41—212no—
90#41—26no—
91#41—212no—
92#41—26no—
93C2×Q8—26no—
94C2×Q8—623no—
95C2×Q8—623no—
96C2×Q8C22⋊D3113yes—
97C2×D4—26no—
98C2×D4—212no—
99C2×D4C22⋊D3113yes—
100C2×D4C22×D3116yes—
101C22×C4—24no—
102C22×C4—312no—
103C22×C4C22⋊D3116yes—
104C22×C4C22×D3111yescharacteristic
105C4⋊C4—6212no—
106C4⋊C4—623no—
107C22⋊C4—316no—
108C22⋊C4—6212no—
109C42—623no—
110C23—24no—
111C23C2×C22⋊D3116yes—
112C23C23×D3111yescharacteristic
113Q8—24no—
114Q8—26no—
115Q8—1241no—
116Q8—1243no—
117D4—212no—
118D4—212no—
119D4—26no—
120C2×C4—212no—
121C2×C4—26no—
122C2×C4—26no—
123C2×C4—626no—
124C2×C4—626no—
125C2×C4C2×C22⋊D3116yes—
126C2×C4C2×C22⋊D3113yes—
127C2×C4C2×C22⋊D3113yes—
128C22—26no—
129C22—41no—
130C22—46no—
131C22—84no—
132C22C22×C22⋊D3113yes—
133C4—26no—
134C4—1246no—
135C4—1242no—
136C4#1175853011yescharacteristic
137C4#1175853013yes—
138C2—23no—
139C2—46no—
140C2C2×#1175853011yescharacteristic
subgroupG/H
centerC2C2×#11758530
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
Fitting, prime radicalC31×#585C2
socleC2×C31C2×#120
hypercenter#585D31
hypocenterC31#3028
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #14521620C2semidirect
2. #14521239C2semidirect
3. #14521023C2semidirect
10. C22×Dic62C22semidirect
13. #11758512C22semidirect
14. #11758503C22semidirect
16. #11758422C22semidirect
87. C31#3028semidirect

Maximal subgroups

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

subgroupindex
#145216202
#145212392
#145210232
C31×#5852
#302831