Finite Group Databasebeta

← #16844827

Subgroups — #16844827

Order 3968.

314 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 65 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#14521421C211yesmaximal, characteristic, complemented
2#14521339C212yesmaximal, complemented
3#14520793C211yesmaximal, characteristic, complemented
4D4⋊(C31⋊C8)C212yesmaximal, complemented
5D42×C31C211yesmaximal, characteristic
6C22×D4×C31—22no—
7C31×#127C2211yescharacteristic
8C31×#121—22no—
9C31×#120—22no—
10C4×D4×C31C2212yes—
11C2×#92480—22no—
12#11758446C2211yescharacteristic, complemented
13#11758420—22no—
14#11758396—22no—
15C4⋊(C31⋊C8)C2212yescomplemented
16C4×C31⋊C8C2211yescharacteristic, complemented
17C24×C31—22no—
18C2×D4×C31—24no—
19C2×D4×C31—22no—
20C2×D4×C31—44no—
21C2×D4×C31—44no—
22C2×D4×C31D412yes—
23C2×D4×C31D412yes—
24C22×C4×C31—22no—
25C2×Dic62—21no—
26C4⋊C4×C31D412yes—
27C22⋊C4×C31—22no—
28C22⋊C4×C31—22no—
29C42×C31C2311yescharacteristic
30#92480—44no—
31C4⋊Dic31—22no—
32C2×C31⋊C8—21no—
33C2×C31⋊C8—22no—
34C23×C31—22no—
35C23×C31—22no—
36C23×C31—48no—
37C23×C31—82no—
38D4×C31—22no—
39D4×C31—42no—
40D4×C31—48no—
41D4×C31—42no—
42D4×C31—42no—
43D4×C31SD1614yescomplemented
44C2×C4×C31—22no—
45C2×C4×C31—42no—
46C2×C4×C31—42no—
47C2×C4×C31C2×D411yescharacteristic
48C2×C4×C31C2×D412yes—
49C2×Dic31—41no—
50Dic62—42no—
51C31⋊C8—22no—
52C31⋊C8—42no—
53#1629—311nomaximal, Sylow
54C22×C31—24no—
55C22×C31—42no—
56C22×C31—42no—
57C22×C31—44no—
58C22×C31—42no—
59C22×C31—88no—
60C22×C31#12011yescharacteristic
61C4×C31—21no—
62C4×C31—42no—
63C4×C31C2×SD1612yes—
64C4×C31AGL(1,ℤ/8)12yes—
65Dic31—81no—
66D42D3111yescharacteristic
67#492—311no—
68#476—312no—
69#331—311no—
70D4⋊C8—312no—
71C2×C31—44no—
72C2×C31—82no—
73C2×C31#45012yes—
74C2×C31#45311yescharacteristic
75C22×D4—22no—
76C2×SD16—622no—
77C4⋊Q8—311no—
78#127C2×D3111yescharacteristic
79#121—22no—
80#120—22no—
81C4×D4C2×D3112yes—
82#106—622no—
83C4⋊C8—312no—
84#102—622no—
85C4×C8—311no—
86C31#162911yescharacteristic, complemented, Sylow
87C24—22no—
88C2×Q8—621no—
89C2×D4—24no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4C22⋊D3112yes—
94C2×D4C22⋊D3112yes—
95C22×C4—22no—
96SD16—1244no—
97C2×C8—621no—
98C2×C8—622no—
99C4⋊C4—622no—
100C4⋊C4C22⋊D3112yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42C22×D3111yescharacteristic
104C23—22no—
105C23—22no—
106C23—48no—
107C23—82no—
108Q8—1242no—
109D4—22no—
110D4—42no—
111D4—48no—
112D4—42no—
113D4—42no—
114D4#9248014yescomplemented
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—1241no—
119C2×C4C2×C22⋊D3111yescharacteristic
120C2×C4C2×C22⋊D3112yes—
121C8—622no—
122C8—1242no—
123C22—24no—
124C22—42no—
125C22—42no—
126C22—44no—
127C22—42no—
128C22—88no—
129C22#1175853011yescharacteristic
130C4—21no—
131C4—42no—
132C4—2481no—
133C4#1175850912yes—
134C4C2×#9248012yes—
135C2—44no—
136C2—82no—
137C2#1452120012yes—
138C2#1452121711yescharacteristic
subgroupG/H
centerC22#11758530
commutator, cosocleC42×C31C23
FrattiniC42C22×D31
Fitting, prime radicalD42×C31C2
socleC22×C31#120
hypercenterD42D31
hypocenterC31#1629
LCSG ▷ C42×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #14521421C2semidirect
2. #14521339C2semidirect
3. #14520793C2semidirect
4. D4⋊(C31⋊C8)C2semidirect
12. #11758446C22semidirect
15. C4⋊(C31⋊C8)C22semidirect
16. C4×C31⋊C8C22semidirect
43. D4×C31SD16semidirect
86. C31#1629semidirect
114. D4#92480semidirect

Maximal subgroups

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

subgroupindex
#145214212
#145213392
#145207932
D4⋊(C31⋊C8)2
D42×C312
#162931