Finite Group Databasebeta

← #16847737

Subgroups — #16847737

Order 3968.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#14521231C211yesmaximal, characteristic, complemented
2#14521221C212yesmaximal, complemented
3#14520848C211yesmaximal, characteristic, complemented
4#14520846C212yesmaximal, complemented
5C31×#555C211yesmaximal, characteristic, complemented
6#11758564C2211yescharacteristic
7C31×C4⋊Q8C2211yescharacteristic
8C31×#124—21no—
9C31×#124C2212yes—
10C31×#121—22no—
11C4×D4×C31—21no—
12C31×#117—22no—
13#11758519—21no—
14#11758509—22no—
15#11758424C2211yescharacteristic
16#11758424C2212yes—
17#11758393—22no—
18#11758393—21no—
19C2×Q8×C31D411yescharacteristic
20C2×Q8×C31C2311yescharacteristic
21C2×D4×C31—21no—
22C2×D4×C31—21no—
23C2×D4×C31D412yes—
24C22×C4×C31—22no—
25#92499—41no—
26Q8×D31—41no—
27#92495—22no—
28#92495—21no—
29C4⋊C4×C31—21no—
30C4⋊C4×C31—22no—
31C22⋊C4×C31—22no—
32C22⋊C4×C31—22no—
33C22⋊C4×C31—22no—
34C42×C31—21no—
35C42×C31D412yes—
36C42×C31D411yescharacteristic
37#92482—41no—
38#92481—41no—
39#92480—42no—
40#92479—42no—
41#92474—22no—
42#92474—21no—
43C23×C31—22no—
44Q8×C31—41no—
45Q8×C31—41no—
46D4×C31—42no—
47D4×C31—42no—
48D4×C31—81no—
49C2×C4×C31—21no—
50C2×C4×C31—22no—
51C2×C4×C31—21no—
52C2×C4×C31—44no—
53C2×C4×C31C2×D411yescharacteristic
54C2×C4×C31C2×D412yes—
55C22⋊D31—41no—
56D124—22no—
57D124—21no—
58C4×D31—42no—
59C4×D31—41no—
60Dic62—22no—
61Dic62—21no—
62C31⋊C8—42no—
63C31⋊C8—41no—
64#2204—311nomaximal, Sylow
65C22×C31—42no—
66C22×C31—42no—
67C22×C31—42no—
68C22×C31#12011yescharacteristic
69C2×D31—41no—
70C4×C31—22no—
71C4×C31—22no—
72C4×C31—21no—
73C4×C31—42no—
74C4×C31—41no—
75Dic31—41no—
76#555D3111yescharacteristic, complemented
77#456—311no—
78#454—312no—
79#356—311no—
80#355—312no—
81C2×C31—21no—
82C2×C31—82no—
83C2×C31B(4,𝔽2)11yescharacteristic
84D31—81no—
852−1+4—311no—
86#137—621no—
87AGL(1,ℤ/8)—622no—
88C4⋊Q8C2×D3111yescharacteristic
89#124—21no—
90#124C2×D3112yes—
91#121—22no—
92C4×D4—21no—
93#117—22no—
94#104—622no—
95#104—621no—
96#101—311no—
97#101—312no—
98C31#220411yescharacteristic, complemented, Sylow
99#41—622no—
100#41—621no—
101#41—1241no—
102C2×Q8—1241no—
103C2×Q8C22⋊D3111yescharacteristic
104C2×Q8C22×D3111yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D3112yes—
108C22×C4—22no—
109Q16—1241no—
110SD16—1242no—
111SD16—1241no—
112D8—1242no—
113M16—622no—
114M16—621no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D3112yes—
122C42C22⋊D3111yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—622no—
127Q8—621no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—622no—
132D4—621no—
133D4—1241no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—1242no—
139C2×C4—1241no—
140C2×C4C2×C22⋊D3111yescharacteristic
141C2×C4C2×C22⋊D3112yes—
142C8—1242no—
143C8—1241no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—1241no—
148C22#1175853011yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—1241no—
155C2—21no—
156C2—82no—
157C2—2481no—
158C2#1452123611yescharacteristic
subgroupG/H
centerC2#14521236
commutator, cosocleC2×Q8×C31C23
FrattiniC2×Q8C22×D31
FittingC31×#555C2
socleC2×C31B(4,𝔽2)
hypercenter#555D31
hypocenterC31#2204
LCSG ▷ C2×Q8×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #14521231C2semidirect
2. #14521221C2semidirect
3. #14520848C2semidirect
4. #14520846C2semidirect
5. C31×#555C2semidirect
76. #555D31semidirect
98. C31#2204semidirect

Maximal subgroups

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

subgroupindex
#145212312
#145212212
#145208482
#145208462
C31×#5552
#220431