Finite Group Databasebeta

← #16856113

Subgroups — #16856113

Order 3968.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#14521600C211yesmaximal, characteristic
2#125×D31C211yesmaximal, characteristic, complemented
3#14521417C211yesmaximal, characteristic, complemented
4#14521415C211yesmaximal, characteristic, complemented
5#14521414C211yesmaximal, characteristic, complemented
6#14521405C214yesmaximal, complemented
7#14521049C214yesmaximal, complemented
8#14521001C211yesmaximal, characteristic, complemented
9C31×#491C211yesmaximal, characteristic, complemented
10C2×#92499C2212yes—
11C31×C4⋊Q8C2211yescharacteristic
12C31×#125C2211yescharacteristic
13#11758524C2212yes—
14C2×#92469C2212yes—
15#11758472—22no—
16#11758470—22no—
17C4⋊C4×D31—22no—
18C4⋊C4×D31C2212yes—
19#11758467C2212yes—
20#11758466—22no—
21#11758451C2211yescharacteristic
22C42×D31C2211yescharacteristic
23#11758447C2211yescharacteristic
24C31×#103C2214yes—
25C31×#97C2211yescharacteristic
26#11758423C2214yes—
27#11758409C2214yes—
28C8×Dic31C2212yes—
29#11758397C2214yes—
30#11758391C2211yescharacteristic
31C2×Q8×C31C2312yes—
32#92499—28no—
33C2×D124—22no—
34C2×D124—22no—
35C2×D124C2312yes—
36C2×C4×D31—22no—
37C2×C4×D31—22no—
38C2×C4×D31D412yes—
39C2×C4×D31C2311yescharacteristic
40C2×C8×C31C2312yes—
41C4⋊C4×C31—22no—
42C4⋊C4×C31—22no—
43C4⋊C4×C31C2312yes—
44C42×C31C2311yescharacteristic
45(C2×D31)⋊C4—24no—
46C4⋊Dic31—22no—
47C4⋊Dic31C2312yes—
48Dic31⋊C4—22no—
49Dic31⋊C4—22no—
50Dic31⋊C4—22no—
51C4×Dic31D412yes—
52C4×Dic31C2311yescharacteristic
53C2×C31⋊C8C2312yes—
54#92469—24no—
55C22×D31—22no—
56C22×D31C2×D411yescharacteristic
57Q8×C31—24no—
58C2×C4×C31—22no—
59C2×C4×C31—22no—
60C2×C4×C31C2×D412yes—
61C2×C4×C31C2411yescharacteristic
62C2×Dic31—22no—
63C2×Dic31C2×D412yes—
64C2×Dic31C2×D411yescharacteristic
65D124—24no—
66D124—44no—
67D124—44no—
68C4×D31—42no—
69C4×D31—44no—
70C4×D31—44no—
71C4×D31#4114yes—
72C8×C31—22no—
73C31⋊C8—22no—
74#3142—311nomaximal, Sylow
75C22×C31C22×D411yescharacteristic
76C2×D31—22no—
77C2×D31—42no—
78C2×D31—44no—
79C2×D31#12412yes—
80C4×C31—22no—
81C4×C31—42no—
82C4×C31—42no—
83C4×C31C2×#4112yes—
84Dic31—22no—
85Dic31—42no—
86Dic31#12412yes—
87#532—311no—
88C2×#125—311no—
89#491—311no—
90#491D3111yescharacteristic, complemented
91#490—311no—
92#490—311no—
93#488—314no—
94#417—314no—
95C4×M16—311no—
96C2×C31C2×#12411yescharacteristic
97C2×C31#57512yes—
98D31—22no—
99D31—82no—
100C2×#41—312no—
101C2×M16—312no—
102C4⋊Q8C2×D3111yescharacteristic
103#127—311no—
104#125—312no—
105#125—311no—
106#125—622no—
107#125C2×D3111yescharacteristic
108#124—312no—
109#121—622no—
110C4×D4—622no—
111C2×C4⋊C4—312no—
112C2×C4⋊C4—622no—
113C2×C42—311no—
114#103—314no—
115#103C2×D3114yes—
116#102—314no—
117#102—314no—
118#97—311no—
119#97C2×D3111yescharacteristic
120C4×C8—312no—
121C31#314211yescharacteristic, complemented, Sylow
122#41—628no—
123C2×Q8C22×D3112yes—
124C2×D4—312no—
125C2×D4—622no—
126C2×D4—622no—
127C22×C4—311no—
128C22×C4—312no—
129C22×C4—622no—
130C22×C4—622no—
131M16—624no—
132C2×C8—312no—
133C2×C8C22×D3112yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—312no—
137C4⋊C4—622no—
138C4⋊C4—622no—
139C4⋊C4—622no—
140C4⋊C4—622no—
141C4⋊C4C22×D3112yes—
142C22⋊C4—624no—
143C42—312no—
144C42—311no—
145C42C22×D3111yescharacteristic
146C23—311no—
147C23—622no—
148Q8—24no—
149D4—624no—
150D4—1244no—
151D4—1244no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—311no—
155C2×C4—314no—
156C2×C4—312no—
157C2×C4—622no—
158C2×C4—1242no—
159C2×C4—1244no—
160C2×C4—1244no—
161C2×C4D4×D3112yes—
162C2×C4C23×D3111yescharacteristic
163C8—22no—
164C8—622no—
165C22—312no—
166C22—622no—
167C22—1242no—
168C22—1244no—
169C22C2×D4×D3111yescharacteristic
170C4—22no—
171C4—42no—
172C4—42no—
173C4—312no—
174C4—622no—
175C4—1242no—
176C4#41×D3112yes—
177C2—622no—
178C2—2482no—
179C2#124×D3111yescharacteristic
180C2#1452184812yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#491C2
socleC22×C31C22×D4
prime radical#14521600C2
hypercenter#491D31
hypocenterC31#3142
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #125×D31C2semidirect
3. #14521417C2semidirect
4. #14521415C2semidirect
5. #14521414C2semidirect
6. #14521405C2semidirect
7. #14521049C2semidirect
8. #14521001C2semidirect
9. C31×#491C2semidirect
90. #491D31semidirect
121. C31#3142semidirect

Maximal subgroups

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

subgroupindex
#145216002
#125×D312
#145214172
#145214152
#145214142
#145214052
#145210492
#145210012
C31×#4912
#314231