Finite Group Databasebeta

← #11920997

Subgroups — #11920997

Order 1152.

226 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 37 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#10593641C212yesmaximal, complemented
2C3×#504⋊C3C211yesmaximal, characteristic, complemented
3#73040—33nomaximal
4#71454—31nomaximal
5#64267—22no—
6C3×#131⋊C3C2211yescharacteristic
7#63626—41nomaximal
8#5407—36no—
9C3×#504D311yescharacteristic, complemented
10#5142—32no—
11C2×D48—31no—
12#504⋊C3D313yescomplemented
13#4516—32no—
14#4506—31no—
15C3×#41⋊C3D411yescharacteristic
16C3⋊GL(2,𝔽3)—42no—
17#3745—42no—
18C32×C16—41no—
19#2221—91noSylow
20#986—66no—
21C3×#131C2×D311yescharacteristic
22#949—62no—
23C2×D24—32no—
24#131⋊C3C2×D313yes—
25C3×M32—31no—
26C2×C3×C16—31no—
27#823—62no—
28#819—32no—
29D48—61no—
30D48—61no—
31D48—123no—
32C4⋊(C3⋊D3)—82no—
33C3×SL(2,𝔽3)D811yescharacteristic
34C32×C8—41no—
35C2×D16—91no—
36#504C3⋊D311yescharacteristic, complemented
37#472—92no—
38#368—91no—
39#361—92no—
40C3×#41D1211yescharacteristic
41C2×D12—62no—
42#41⋊C3D1213yes—
43GL(2,𝔽3)—126no—
44C3×M16—31no—
45C2×C3×C8—31no—
46#215—122no—
47#213—122no—
48#208—62no—
49D24—62no—
50D24—62no—
51D24—126no—
52C3×C16—31no—
53C3×C16—43no—
54C3×C16S411yescharacteristic
55C2×C3⋊D3—162no—
56C32×C4—41no—
57AGL(1,ℤ/8)—182no—
58C2×D8—92no—
59#131C2×C3⋊D311yescharacteristic
60D16—181no—
61D16—181no—
62M32—31no—
63C2×C16—31no—
64#108—92no—
65#100—182no—
66C22×D3—122no—
67C3×Q8D2411yescharacteristic
68C3×D4—31no—
69C2×C3×C4—31no—
70D12—122no—
71D12—122no—
72D12—246no—
73SL(2,𝔽3)D2413yes—
74C3×C8—31no—
75C3×C8—43no—
76C3×C8C2×S411yescharacteristic
77C3⋊C8—122no—
78C2×C32—41no—
79C3⋊D3—322no—
80#41C4⋊(C3⋊D3)11yescharacteristic
81C2×D4—182no—
82SD16—362no—
83D8—182no—
84D8—182no—
85D8—362no—
86M16—31no—
87M16—182no—
88C2×C8—31no—
89C16—31no—
90C16C3⋊S411yescharacteristic
91C22×C3—31no—
92C2×D3—242no—
93C2×D3—486no—
94C2×D3—482no—
95C3×C4—31no—
96C3×C4—43no—
97C3×C4C4⋊S411yescharacteristic
98C32—41noSylow
99C23—362no—
100Q8#374511yescharacteristic
101D4—31no—
102D4—362no—
103D4—362no—
104C2×C4—31no—
105C8—31no—
106C8—362no—
107C8C2×C3⋊S411yescharacteristic
108C2×C3—43no—
109C2×C3—61no—
110C2×C3#540211yescharacteristic
111D3—482no—
112D3—966no—
113C22—31no—
114C22—722no—
115C22—1442no—
116C4—31no—
117C4C4⋊(C3⋊S4)11yescharacteristic
118C3—43no—
119C3#7304011yescharacteristic, complemented
120C2—61no—
121C2—1442no—
122C2#1059363611yescharacteristic
subgroupG/H
centerC2#10593636
commutator, cosocleC3×#131⋊C3C22
FrattiniC8C2×C3⋊S4
FittingC3×#504D3
socleC2×C3#5402
hypercenterC16C3⋊S4
hypocenterC3×SL(2,𝔽3)D8
LCSG ▷ C3×#131⋊C3 ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
1. #10593641C2semidirect
2. C3×#504⋊C3C2semidirect
9. C3×#504D3semidirect
12. #504⋊C3D3semidirect
36. #504C3⋊D3semidirect
119. C3#73040semidirect

Maximal subgroups

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

subgroupindex
#105936412
C3×#504⋊C32
#730403
#714543
#636264