Finite Group Databasebeta

← #16847724

Subgroups — #16847724

Order 3968.

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

normal subgroups only

This group has 226 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#14521238C212yesmaximal
2#14521226C211yesmaximal, characteristic
3#14520844C211yesmaximal, characteristic, complemented
4#14520839C212yesmaximal, complemented
5C31×#563C211yesmaximal, characteristic, complemented
6#11758561C2211yescharacteristic
7C31×C4⋊Q8—21no—
8C31×#126—22no—
9C31×#125—21no—
10C31×#124C2211yescharacteristic
11C31×#123C2212yes—
12C31×C22⋊Q8—22no—
13#11758519—21no—
14#11758506—22no—
15#11758422C2212yes—
16#11758421C2211yescharacteristic
17#11758394—22no—
18#11758393—21no—
19C2×Q8×C31—21no—
20C2×Q8×C31D411yescharacteristic
21C2×D4×C31C2311yescharacteristic
22C22×C4×C31D412yes—
23C2×C22⋊D31—22no—
24#92497—41no—
25D4×D31—41no—
26#92495—21no—
27C4⋊C4×C31—21no—
28C4⋊C4×C31—22no—
29C4⋊C4×C31—22no—
30C4⋊C4×C31—21no—
31C4⋊C4×C31—22no—
32C22⋊C4×C31—22no—
33C22⋊C4×C31—22no—
34C22⋊C4×C31D412yes—
35C42×C31—21no—
36C42×C31D411yescharacteristic
37C22⋊Dic31—22no—
38#92482—41no—
39#92481—41no—
40(C2×D31)⋊C4—42no—
41Dic31⋊C4—42no—
42#92474—21no—
43C23×C31C2×D412yes—
44C22×D31—22no—
45Q8×C31—41no—
46Q8×C31—81no—
47D4×C31—41no—
48C2×C4×C31—21no—
49C2×C4×C31—22no—
50C2×C4×C31—21no—
51C2×C4×C31—22no—
52C2×C4×C31—42no—
53C2×C4×C31C2×D411yescharacteristic
54C22⋊D31—42no—
55C22⋊D31—41no—
56C2×Dic31—22no—
57C2×Dic31—42no—
58D124—21no—
59C4×D31—41no—
60Dic62—21no—
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—42no—
70C4×C31—41no—
71C4×C31—42no—
72C4×C31—41no—
73Dic31—41no—
74Dic31—82no—
75#563D3111yescharacteristic, complemented
76#458—312no—
77#455—311no—
78#354—311no—
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—21no—
87#126—22no—
88#125—21no—
89#124C2×D3111yescharacteristic
90#123—622no—
91#123C2×D3112yes—
92C22⋊Q8—22no—
93#104—621no—
94#100—311no—
95#99—312no—
96#99—622no—
97C31#220111yescharacteristic, complemented, Sylow
98#41—621no—
99#41—1241no—
100C2×Q8—21no—
101C2×Q8C22⋊D3111yescharacteristic
102C2×D4—622no—
103C2×D4—1241no—
104C2×D4C22×D3111yescharacteristic
105C22×C4C22⋊D3112yes—
106Q16—1241no—
107SD16—1241no—
108M16—621no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—1242no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—622no—
118C22⋊C4—1242no—
119C22⋊C4C22⋊D3112yes—
120C42—21no—
121C42C22⋊D3111yescharacteristic
122C23—622no—
123C23C2×C22⋊D3112yes—
124Q8—41no—
125Q8—81no—
126Q8—621no—
127D4—41no—
128D4—621no—
129D4—1242no—
130D4—1241no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—622no—
137C2×C4—1241no—
138C2×C4—1242no—
139C2×C4C2×C22⋊D3111yescharacteristic
140C8—1241no—
141C22—22no—
142C22—42no—
143C22—1241no—
144C22—2482no—
145C22#1175853011yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—1241no—
152C4—2482no—
153C2—21no—
154C2—42no—
155C2—2481no—
156C2#1452123611yescharacteristic
subgroupG/H
centerC2#14521236
commutator, cosocleC2×D4×C31C23
FrattiniC2×D4C22×D31
FittingC31×#563C2
socleC2×C31B(4,𝔽2)
prime radical#11758561C22
hypercenter#563D31
hypocenterC31#2201
LCSG ▷ C2×D4×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #14520844C2semidirect
4. #14520839C2semidirect
5. C31×#563C2semidirect
75. #563D31semidirect
97. C31#2201semidirect

Maximal subgroups

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

subgroupindex
#145212382
#145212262
#145208442
#145208392
C31×#5632
#220131