Finite Group Databasebeta

← #16847742

Subgroups — #16847742

Order 3968.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#14521779C211yesmaximal, characteristic, complemented
2#14521234C212yesmaximal
3#14521231C211yesmaximal, characteristic
4#14520847C212yesmaximal, complemented
5C31×#356C211yesmaximal, characteristic, complemented
6#11758564C2211yescharacteristic
7C31×C4⋊Q8C2211yescharacteristic
8#11758521—21no—
9#11758521C2212yes—
10#11758519—21no—
11#11758486—22no—
12#11758465—22no—
13#11758464—22no—
14C4×Dic62—21no—
15C31×#101C2212yes—
16#11758424C2211yescharacteristic
17#11758413—22no—
18#11758393—21no—
19C2×Q8×C31D411yescharacteristic
20C2×Q8×C31C2311yescharacteristic
21#92499—41no—
22Q8×D31—41no—
23#92495—21no—
24#92495—22no—
25C2×Dic62—21no—
26C2×Dic62—21no—
27C2×Dic62D412yes—
28M16×C31—22no—
29C4⋊C4×C31—22no—
30C42×C31D411yescharacteristic
31#92482—41no—
32#92481—41no—
33C4⋊Dic31—21no—
34Dic31⋊C4—22no—
35Dic31⋊C4—22no—
36Dic31⋊C4—22no—
37C4×Dic31—22no—
38C4×Dic31—21no—
39C4×Dic31D412yes—
40#92474—21no—
41Dic124—42no—
42#92470—42no—
43Q8×C31—41no—
44Q8×C31—41no—
45C2×C4×C31—21no—
46C2×C4×C31—21no—
47C2×C4×C31C2×D411yescharacteristic
48C2×C4×C31C2×D412yes—
49C22⋊D31—41no—
50C2×Dic31—22no—
51C2×Dic31—22no—
52D124—22no—
53D124—21no—
54C4×D31—41no—
55C4×D31—42no—
56Dic62—24no—
57Dic62—22no—
58Dic62—21no—
59Dic62—42no—
60Dic62—81no—
61C8×C31—42no—
62C31⋊C8—41no—
63#2205—311nomaximal, Sylow
64C22×C31#12011yescharacteristic
65C2×D31—41no—
66C4×C31—22no—
67C4×C31—21no—
68C4×C31—22no—
69C4×C31—41no—
70Dic31—24no—
71Dic31—41no—
72Dic31—42no—
73Q82—311no—
74#456—311no—
75#456—312no—
76#356—312no—
77#356D3111yescharacteristic, complemented
78C2×C31—21no—
79C2×C31B(4,𝔽2)11yescharacteristic
80D31—81no—
812−1+4—311no—
82#137—622no—
83#137—621no—
84C4⋊Q8—312no—
85C4⋊Q8—622no—
86C4⋊Q8—621no—
87C4⋊Q8C2×D3111yescharacteristic
88C4×Q8—622no—
89C4×Q8—621no—
90#104—622no—
91#104—621no—
92#101—311no—
93#101C2×D3112yes—
94C31#220511yescharacteristic, complemented, Sylow
95#41—621no—
96#41—622no—
97#41—1241no—
98C2×Q8—312no—
99C2×Q8—621no—
100C2×Q8—621no—
101C2×Q8—1241no—
102C2×Q8D12411yescharacteristic
103C2×Q8C22×D3111yescharacteristic
104Q16—1242no—
105Q16—1241no—
106SD16—1242no—
107SD16—1241no—
108M16—22no—
109M16—621no—
110C4⋊C4—22no—
111C4⋊C4—622no—
112C4⋊C4—621no—
113C4⋊C4—622no—
114C4⋊C4—622no—
115C42—312no—
116C42—622no—
117C42—621no—
118C42D12411yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—624no—
122Q8—622no—
123Q8—621no—
124Q8—1242no—
125Q8—2481no—
126D4—622no—
127D4—621no—
128D4—1241no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—622no—
132C2×C4—622no—
133C2×C4—1242no—
134C2×C4—1241no—
135C2×C4C2×D12411yescharacteristic
136C2×C4D4×D3112yes—
137C8—42no—
138C8—1241no—
139C22—1241no—
140C22#1175845911yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—624no—
146C4—1242no—
147C4—1241no—
148C2—21no—
149C2—2481no—
150C2#1452123711yescharacteristic
subgroupG/H
centerC2#14521237
commutator, cosocleC2×Q8×C31C23
FrattiniC2×Q8C22×D31
FittingC31×#356C2
socleC2×C31B(4,𝔽2)
prime radical#11758564C22
hypercenter#356D31
hypocenterC31#2205
LCSG ▷ C2×Q8×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #14521779C2semidirect
4. #14520847C2semidirect
5. C31×#356C2semidirect
77. #356D31semidirect
94. C31#2205semidirect

Maximal subgroups

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

subgroupindex
#145217792
#145212342
#145212312
#145208472
C31×#3562
#220531