Finite Group Databasebeta

← #11720325

Subgroups — #11720325

Order 864.

215 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C2×C4⋊#272C211yesmaximal, characteristic
2#3748⋊C3C212yesmaximal, complemented
3#63558—32nomaximal
4C22×#272C412yes—
5C2×C4×H(3,𝔽3)C2211yescharacteristic
6C4⋊#272—22no—
7(C3⋊(C3⋊C8))⋊C3—22no—
8C2×C3×D12—32no—
9C2×C3×D12—61no—
10#3748D312yescomplemented
11C3×#208—32no—
12C22×H(3,𝔽3)C2×C411yescharacteristic
13C2×#272—22no—
14C2×#272—42no—
15C4×H(3,𝔽3)D412yes—
16#833—91no—
17#823—92no—
18C22×C3×D3—34no—
19C22×C3×D3—62no—
20C2×C32×C4—21no—
21C2×C32×C4C2×D312yes—
22C3×D12—64no—
23C3×D12—62no—
24C3×D12—62no—
25C3⋊(C3⋊C8)—22no—
26C3×C3⋊C8—62no—
27C2×H(3,𝔽3)—21no—
28C2×H(3,𝔽3)C22⋊C411yescharacteristic
29#272—42no—
30C2×C3×D4—91no—
31C2×D12—32no—
32C2×D12—61no—
33C3×M16—92no—
34#208—32no—
35#208—182no—
36C22×C32—21no—
37C22×C32C4×D312yes—
38C2×C3×D3—64no—
39C2×C3×D3—64no—
40C2×C3×D3—124no—
41C2×C3×D3—124no—
42C32×C4—22no—
43C32×C4D1212yes—
44C32×C4#6712yes—
45#100—271noSylow
46H(3,𝔽3)#10011yescharacteristic, complemented, Sylow
47C23×C3—92no—
48C22×D3—34no—
49C22×D3—62no—
50C3×D4—182no—
51C2×C3×C4—62no—
52C2×C3×C4—121no—
53C2×C3×C4D3211yescharacteristic
54D12—62no—
55D12—62no—
56D12—64no—
57C3×C8—182no—
58C3⋊C8—62no—
59C3⋊C8—182no—
60C2×C32—22no—
61C2×C32—22no—
62C2×C32—21no—
63C2×C32(C2×D3)⋊C412yes—
64C3×D3—124no—
65C3×D3—124no—
66C2×D4—91no—
67M16—92no—
68C22×C3—62no—
69C22×C3—121no—
70C22×C3—182no—
71C22×C3—362no—
72C22×C3#62411yescharacteristic
73C2×D3—64no—
74C2×D3—64no—
75C2×D3—124no—
76C2×D3—124no—
77C3×C4—62no—
78C3×C4—62no—
79C3×C4—122no—
80C3×C4#62612yes—
81C32—21no—
82C32#82312yescomplemented
83C23—92no—
84D4—182no—
85C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
86C8—182no—
87C2×C3—21no—
88C2×C3—62no—
89C2×C3—122no—
90C2×C3—122no—
91C2×C3—121no—
92C2×C3—362no—
93C2×C3#372211yescharacteristic
94D3—124no—
95D3—124no—
96C22—182no—
97C22—362no—
98C22#619911yescharacteristic
99C4#620012yes—
100C3—62no—
101C3—121no—
102C3#6355911yescharacteristic
103C2—21no—
104C2—362no—
105C2#8845111yescharacteristic
subgroupG/H
centerC2#88451
commutatorC22×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4D32
Fitting, cosocleC2×C4×H(3,𝔽3)C22
socleC2×C3#3722
prime radicalC2×C4⋊#272C2
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#100
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #3748⋊C3C2semidirect
10. #3748D3semidirect
46. H(3,𝔽3)#100semidirect
82. C32#823semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#2722
#3748⋊C32
#635583