Finite Group Databasebeta

← #85629

Subgroups — #85629

Order 384.

334 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 82 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#5945C211yesmaximal, characteristic
2#568⋊C3C213yesmaximal, complemented
3C2×#131⋊C3C213yesmaximal, complemented
4#3573C311yesmaximal, characteristic, complemented, Sylow
52−1+4⋊C3C413yes—
6#994C411yescharacteristic
7C2×#41⋊C3C2213yes—
8C3×#131—41nomaximal
9#131⋊C3C2211yescharacteristic, complemented
10#131⋊C3C2213yescomplemented
11#585C2×C311yescharacteristic
12#568—33no—
13#568—31no—
14#568C2×C313yescomplemented
15C2×#131—31no—
16C2×#131—33no—
17C2×#131C2×C313yescomplemented
18C3×#41—41no—
19#41⋊C3C2×C413yes—
20#41⋊C3C2311yescharacteristic
21C2×SL(2,𝔽3)C2×C413yes—
22C3×M16—43no—
23C2×C3×C8—43no—
242−1+4—31no—
252−1+4C3×C413yes—
26H(5,𝔽2)—33no—
27H(5,𝔽2)C3×C411yescharacteristic
28C2×#41—33no—
29C2×#41—31no—
30C2×#41C22×C313yes—
31#131—36no—
32#131—31no—
33#131—33no—
34#131—36no—
35#131—33no—
36#131—33no—
37#131—33no—
38#131A411yescharacteristic
39#131C22×C311yescharacteristic
40#131C22×C313yes—
41C2×M16—36no—
42C2×M16—33no—
43C2×M16—33no—
44C2×M16—33no—
45C22×C8—32no—
46C22×C8—33no—
47C3×Q8—41no—
48C3×D4—43no—
49C2×C3×C4—43no—
50SL(2,𝔽3)C22×C411yescharacteristic
51C3×C8—43no—
52C3×C8—41no—
53#41—36no—
54#41—31no—
55#41—33no—
56#41—36no—
57#41—33no—
58#41—33no—
59#41—33no—
60#41C2×C3×C413yes—
61#41C2×A411yescharacteristic
62#41C23×C311yescharacteristic
63C2×Q8—33no—
64C2×Q8—31no—
65C2×Q8C2×C3×C413yes—
66C2×D4—33no—
67C2×D4—36no—
68C2×D4—33no—
69C2×D4—33no—
70C22×C4—32no—
71C22×C4—33no—
72M16—36no—
73M16—33no—
74M16—36no—
75M16—33no—
76M16—31no—
77M16C2×A413yes—
78C2×C8—36no—
79C2×C8—33no—
80C2×C8—33no—
81C2×C8—33no—
82C2×C8—33no—
83C2×C8—31no—
84C2×C8C2×A413yes—
85C22×C3—43no—
86C3×C4—43no—
87C3×C4—41no—
88C23—32no—
89C23—33no—
90Q8—33no—
91Q8—33no—
92Q8C4×A411yescharacteristic
93Q8C22×C3×C411yescharacteristic
94D4—33no—
95D4—31no—
96D4—36no—
97D4—36no—
98D4—33no—
99D4C4×A413yes—
100C2×C4—36no—
101C2×C4—33no—
102C2×C4—33no—
103C2×C4—33no—
104C2×C4—33no—
105C2×C4—31no—
106C2×C4C22×A413yes—
107C8—33no—
108C8—31no—
109C8C22×A413yes—
110C8C22×A411yescharacteristic
111C2×C3—41no—
112C2×C3—83no—
113C22—33no—
114C22—31no—
115C22—123no—
116C22—122no—
117C22C2×C4×A413yes—
118C4—33no—
119C4—31no—
120C4C2×C4×A413yes—
121C4C23×A411yescharacteristic
122C3—41noSylow
123C2—23no—
124C2—61no—
125C2—63no—
126C2C22×C4×A411yescharacteristic
subgroupG/H
centerC8C22×A4
commutator, hypocenterQ8C22×C3×C4
FrattiniC4C23×A4
Fitting#3573C3
socleC2C22×C4×A4
prime radical#5945C2
cosocle#41C23×C3
hypercenter#131A4
LCSG ▷ Q8
UCSC1 ◁ C8 ◁ #131

Complements

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

NH ≅ G/Ntype
2. #568⋊C3C2semidirect
3. C2×#131⋊C3C2semidirect
4. #3573C3semidirect
9. #131⋊C3C22semidirect
10. #131⋊C3C22semidirect
14. #568C2×C3semidirect
17. C2×#131C2×C3semidirect

Maximal subgroups

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

subgroupindex
#59452
#568⋊C32
C2×#131⋊C32
#35733
C3×#1314