Finite Group Databasebeta

← #13135923

Subgroups — #13135923

Order 1728.

330 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 73 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic
2#11720584C212yesmaximal
3C3×#4548—34nomaximal
4C2×Q8×H(3,𝔽3)C411yescharacteristic
5C2×Q8×H(3,𝔽3)C2211yescharacteristic
6C4⋊C4×H(3,𝔽3)—22no—
7C42×H(3,𝔽3)C411yescharacteristic
8#88538—22no—
9C32×C4⋊Q8D314yes—
10C3×#836—38no—
11Q8×H(3,𝔽3)—41no—
12Q8×H(3,𝔽3)—41no—
13C2×C4×H(3,𝔽3)—21no—
14C2×C4×H(3,𝔽3)—21no—
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)D412yes—
17#6180—42no—
18C3×#356—91no—
19#4548—34no—
20C2×C32×Q8Dic314yes—
21C2×C32×Q8C2×D314yes—
22C32×C4⋊C4—28no—
23C32×C42Dic314yes—
24C3×#208—68no—
25C22×H(3,𝔽3)C22⋊C411yescharacteristic
26C4×H(3,𝔽3)—22no—
27C4×H(3,𝔽3)—22no—
28C4×H(3,𝔽3)—21no—
29C4×H(3,𝔽3)—41no—
30C3×C4⋊Q8—34no—
31C3×C4⋊Q8C3⋊D311yescharacteristic
32C3×#101—92no—
33#836—38no—
34C32×Q8—44no—
35C32×Q8—44no—
36C2×C32×C4—24no—
37C2×C32×C4—24no—
38C2×C32×C4C2×Dic314yes—
39C2×C32×C4#6718yes—
40C3×C3⋊C8—128no—
41#356—91noSylow
42C2×H(3,𝔽3)—21no—
43C2×H(3,𝔽3)#9911yescharacteristic
44C2×C3×Q8—34no—
45C2×C3×Q8—34no—
46C2×C3×Q8C3⋊Dic311yescharacteristic
47C2×C3×Q8C2×C3⋊D311yescharacteristic
48C3×M16—182no—
49C3×C4⋊C4—22no—
50C3×C4⋊C4—68no—
51C3×C42—34no—
52C3×C42C3⋊Dic311yescharacteristic
53#208—68no—
54C22×C32#21714yes—
55C32×C4—28no—
56C32×C4—28no—
57C32×C4—24no—
58C32×C4—44no—
59C4⋊Q8#27211yescharacteristic
60#101—92no—
61H(3,𝔽3)#35611yescharacteristic, complemented, Sylow
62C3×Q8—41no—
63C3×Q8—41no—
64C3×Q8—124no—
65C3×Q8—124no—
66C2×C3×C4—21no—
67C2×C3×C4—21no—
68C2×C3×C4—34no—
69C2×C3×C4—38no—
70C2×C3×C4—64no—
71C2×C3×C4—64no—
72C2×C3×C4C2×C3⋊Dic311yescharacteristic
73C2×C3×C4#63812yes—
74C3×C8—362no—
75C3⋊C8—128no—
76C2×C32—24no—
77C2×C32#83414yes—
78C2×Q8#108411yescharacteristic
79C2×Q8C2×#27211yescharacteristic
80M16—182no—
81C4⋊C4—22no—
82C42#108411yescharacteristic
83C22×C3—34no—
84C22×C3#375711yescharacteristic
85C3×C4—22no—
86C3×C4—22no—
87C3×C4—21no—
88C3×C4—41no—
89C3×C4—68no—
90C3×C4—64no—
91C3×C4—68no—
92C3×C4—124no—
93C32#454814yescomplemented
94Q8—41no—
95Q8—41no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4C2×#108411yescharacteristic
99C2×C4#623512yes—
100C8—362no—
101C2×C3—21no—
102C2×C3—34no—
103C2×C3—64no—
104C2×C3#6366111yescharacteristic
105C22#8855211yescharacteristic
106C4—22no—
107C4—22no—
108C4—21no—
109C4—41no—
110C3—34no—
111C3#1058941411yescharacteristic
112C2—21no—
113C2#1172058011yescharacteristic
subgroupG/H
center, socleC2×C3#63661
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×Q8C2×C3⋊D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC2×Q8×H(3,𝔽3)C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#356
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
61. H(3,𝔽3)#356semidirect
93. C32#4548semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#117205842
C3×#45483