Finite Group Databasebeta

← #16845314

Subgroups — #16845314

Order 3968.

190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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#14521445C211yesmaximal, characteristic, complemented
2#14521423C211yesmaximal, characteristic
3#14521401C211yesmaximal, characteristic, complemented
4#14521397C211yesmaximal, characteristic
5#14521396C211yesmaximal, characteristic
6C8×C31⋊C8C211yesmaximal, characteristic, complemented
7C31×#487C211yesmaximal, characteristic, complemented
8C31×C4⋊Q8C2211yescharacteristic
9C31×C4⋊Q8C2211yescharacteristic
10C2×#92481—22no—
11#11758451C2211yescharacteristic
12#11758446C2211yescharacteristic
13C31×#103—22no—
14C4×C8×C31C2211yescharacteristic
15#11758409—22no—
16#11758398—22no—
17#11758397—22no—
18#11758396—22no—
19C4×C31⋊C8C2211yescharacteristic
20C4×C31⋊C8C2211yescharacteristic
21C2×Q8×C31—21no—
22C2×Q8×C31—21no—
23C2×D124—21no—
24C2×D124—21no—
25C2×D124—21no—
26C2×Dic62—21no—
27C2×C8×C31D412yes—
28C4⋊C4×C31—21no—
29C4⋊C4×C31—21no—
30C4⋊C4×C31—21no—
31C4⋊C4×C31—21no—
32C42×C31C2311yescharacteristic
33#92481—44no—
34C4⋊Dic31—21no—
35C4⋊Dic31—21no—
36C2×C31⋊C8D412yes—
37C2×C31⋊C8D412yes—
38C22×D31—41no—
39Q8×C31—42no—
40Q8×C31—42no—
41C2×C4×C31—41no—
42C2×C4×C31—41no—
43C2×C4×C31C2×D411yescharacteristic
44C2×C4×C31C2×D411yescharacteristic
45C2×C4×C31C2×D411yescharacteristic
46C2×Dic31—41no—
47D124—42no—
48D124—42no—
49D124—42no—
50Dic62—42no—
51C8×C31—22no—
52C31⋊C8—22no—
53C31⋊C8SD1614yes—
54#1708—311nomaximal, Sylow
55C22×C31#12711yescharacteristic
56C2×D31—81no—
57C2×D31—81no—
58C2×D31—81no—
59C4×C31—81no—
60C4×C31—81no—
61C4×C31C2×SD1612yes—
62C4×C31#13512yes—
63C4×C31#13512yes—
64Dic31—81no—
65#498—311no—
66#492—311no—
67#487—311no—
68#487D3111yescharacteristic, complemented
69#486—311no—
70#486—311no—
71C82—311no—
72C2×C31#49211yescharacteristic
73C2×C31#49511yescharacteristic
74C2×C31#49511yescharacteristic
75D31—161no—
76C2×SD16—622no—
77C4⋊Q8—311no—
78C4⋊Q8C2×D3111yescharacteristic
79C4⋊Q8C2×D3111yescharacteristic
80#127—311no—
81#106—622no—
82#103—22no—
83#103—622no—
84#102—622no—
85#102—622no—
86C4×C8—311no—
87C4×C8—311no—
88C4×C8C2×D3111yescharacteristic
89C31#170811yescharacteristic, complemented, Sylow
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—621no—
93C2×D4—621no—
94C2×D4—621no—
95C2×D4—621no—
96SD16—1244no—
97C2×C8—312no—
98C2×C8—312no—
99C2×C8C22⋊D3112yes—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—621no—
105C4⋊C4—621no—
106C42C22×D3111yescharacteristic
107C23—1241no—
108Q8—42no—
109Q8—42no—
110Q8—1242no—
111D4—1242no—
112D4—1242no—
113D4—1242no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—1241no—
117C2×C4D4×D3111yescharacteristic
118C2×C4D4×D3111yescharacteristic
119C2×C4C2×C22⋊D3111yescharacteristic
120C8—22no—
121C8—314no—
122C8—622no—
123C22—2481no—
124C22—2481no—
125C22—2481no—
126C22#1175851711yescharacteristic
127C4—81no—
128C4—81no—
129C4—2481no—
130C4SD16×D3112yes—
131C4#1175849612yes—
132C4#1175852712yes—
133C2—4961no—
134C2#1452142011yescharacteristic
135C2#1452143111yescharacteristic
136C2#1452143411yescharacteristic
subgroupG/H
centerC22#11758517
commutator, cosocleC42×C31C23
FrattiniC42C22×D31
FittingC31×#487C2
socleC22×C31#127
prime radical#11758451C22
hypercenter#487D31
hypocenterC31#1708
LCSG ▷ C42×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #14521445C2semidirect
3. #14521401C2semidirect
6. C8×C31⋊C8C2semidirect
7. C31×#487C2semidirect
68. #487D31semidirect
89. C31#1708semidirect

Maximal subgroups

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

subgroupindex
#145214452
#145214232
#145214012
#145213972
#145213962
C8×C31⋊C82
C31×#4872
#170831