Finite Group Databasebeta

← #16847726

Subgroups — #16847726

Order 3968.

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

normal subgroups only

This group has 226 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#14521799C211yesmaximal, characteristic, complemented
2#14521240C212yesmaximal
3#14521226C211yesmaximal, characteristic
4#14520838C212yesmaximal, complemented
5C31×#354C211yesmaximal, characteristic, complemented
6#11758561C2211yescharacteristic
7C31×#124C2211yescharacteristic
8#11758521—21no—
9#11758519—21no—
10#11758512C2212yes—
11#11758463—22no—
12#11758456—22no—
13#11758455—22no—
14#11758447—21no—
15C31×#99C2212yes—
16#11758421C2211yescharacteristic
17#11758394—22no—
18#11758393—21no—
19C2×Q8×C31D411yescharacteristic
20C2×D4×C31C2311yescharacteristic
21C2×C22⋊D31—22no—
22C22×Dic31D412yes—
23#92497—41no—
24D4×D31—41no—
25#92495—21no—
26C2×Dic62—21no—
27C22⋊C4×C31—22no—
28C22⋊C4×C31—22no—
29C42×C31D411yescharacteristic
30C22⋊Dic31—22no—
31C22⋊Dic31D412yes—
32#92482—41no—
33#92481—41no—
34(C2×D31)⋊C4—42no—
35C4⋊Dic31—21no—
36C4⋊Dic31—21no—
37C4⋊Dic31—42no—
38Dic31⋊C4—22no—
39Dic31⋊C4—22no—
40Dic31⋊C4—22no—
41C4×Dic31—21no—
42#92474—21no—
43C23×C31C2×D412yes—
44C22×D31—22no—
45Q8×C31—41no—
46D4×C31—41no—
47C2×C4×C31—21no—
48C2×C4×C31—21no—
49C2×C4×C31—42no—
50C2×C4×C31C2×D411yescharacteristic
51C22⋊D31—42no—
52C22⋊D31—41no—
53C2×Dic31—22no—
54C2×Dic31—22no—
55C2×Dic31—22no—
56C2×Dic31—42no—
57D124—21no—
58C4×D31—41no—
59Dic62—21no—
60Dic62—81no—
61C31⋊C8—41no—
62#2201—311nomaximal, Sylow
63C22×C31—22no—
64C22×C31—42no—
65C22×C31#12011yescharacteristic
66C2×D31—41no—
67C2×D31—82no—
68C4×C31—21no—
69C4×C31—41no—
70C4×C31—41no—
71C4×C31—82no—
72Dic31—42no—
73Dic31—41no—
74Dic31—42no—
75#563—311no—
76#458—312no—
77#455—311no—
78#354D3111yescharacteristic, complemented
79#352—312no—
80C2×C31—21no—
81C2×C31—42no—
82C2×C31B(4,𝔽2)11yescharacteristic
83D31—81no—
84H(5,𝔽2)—311no—
85#137—621no—
86C4⋊Q8—621no—
87#126—622no—
88#125—621no—
89#124C2×D3111yescharacteristic
90#123—312no—
91#123—622no—
92C22⋊Q8—622no—
93#104—621no—
94#100—311no—
95#99—622no—
96#99C2×D3112yes—
97C31#220111yescharacteristic, complemented, Sylow
98#41—621no—
99#41—1241no—
100C2×Q8—621no—
101C2×Q8D12411yescharacteristic
102C2×D4—622no—
103C2×D4—1241no—
104C2×D4C22×D3111yescharacteristic
105C22×C4—312no—
106Q16—1241no—
107SD16—1241no—
108M16—621no—
109C4⋊C4—622no—
110C4⋊C4—622no—
111C4⋊C4—621no—
112C4⋊C4—621no—
113C4⋊C4—622no—
114C4⋊C4—1242no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—312no—
118C22⋊C4—622no—
119C22⋊C4—1242no—
120C42—621no—
121C42D12411yescharacteristic
122C23—622no—
123C23D4×D3112yes—
124Q8—41no—
125Q8—621no—
126Q8—2481no—
127D4—41no—
128D4—621no—
129D4—1242no—
130D4—1241no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—622no—
135C2×C4—622no—
136C2×C4—622no—
137C2×C4—1241no—
138C2×C4—1242no—
139C2×C4C2×D12411yescharacteristic
140C8—1241no—
141C22—22no—
142C22—42no—
143C22—1241no—
144C22—2482no—
145C22#1175845911yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—1242no—
151C4—1241no—
152C4—1242no—
153C2—21no—
154C2—42no—
155C2—2481no—
156C2#1452123711yescharacteristic
subgroupG/H
centerC2#14521237
commutator, cosocleC2×D4×C31C23
FrattiniC2×D4C22×D31
FittingC31×#354C2
socleC2×C31B(4,𝔽2)
prime radical#11758561C22
hypercenter#354D31
hypocenterC31#2201
LCSG ▷ C2×D4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #14521799C2semidirect
4. #14520838C2semidirect
5. C31×#354C2semidirect
78. #354D31semidirect
97. C31#2201semidirect

Maximal subgroups

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

subgroupindex
#145217992
#145212402
#145212262
#145208382
C31×#3542
#220131