Finite Group Databasebeta

← #13136933

Subgroups — #13136933

Order 1728.

201 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 25 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
1Q16×#272C211yesmaximal, characteristic
2#11720776C211yesmaximal, characteristic
3#11720771C211yesmaximal, characteristic
4Q8×#272—21no—
5#88627—21no—
6Q16×H(3,𝔽3)C411yescharacteristic
7#88550—21no—
8#88530C411yescharacteristic
9C8×#272C2211yescharacteristic
10#88410—21no—
11C3×D3×Q16—62no—
12Q8×H(3,𝔽3)—21no—
13C4×#272—41no—
14C4×#272D411yescharacteristic
15#6229—21no—
16#6229—41no—
17#6188—41no—
18C8×H(3,𝔽3)C2×C411yescharacteristic
19#6180D411yescharacteristic
20C3×#362—91nomaximal
21C3×D3×Q8—64no—
22C32×Q16—22no—
23C3×#216—64no—
24C3×Dic12—62no—
25C3×D3×C8—62no—
26C2×#272SD1611yescharacteristic
27C4×H(3,𝔽3)—41no—
28C4×H(3,𝔽3)C22⋊C411yescharacteristic
29#1084—41no—
30#1084D811yescharacteristic
31C2×C3×Q16—91no—
32D3×Q16—62no—
33C3×M32—91no—
34C3×#108—91no—
35C32×Q8—24no—
36C3×C4×D3—62no—
37C3×C4×D3—124no—
38C3×Dic6—64no—
39C3×Dic6—124no—
40C32×C8—22no—
41C3×C3⋊C8—62no—
42#362—91noSylow
43C2×H(3,𝔽3)#10211yescharacteristic
44#272—21no—
45C2×C3×Q8—181no—
46D3×Q8—64no—
47C3×Q16—62no—
48C3×Q16—91no—
49C3×Q16—181no—
50C3×Q16#15711yescharacteristic
51C3×M16—181no—
52C2×C3×C8—91no—
53#216—64no—
54Dic12—62no—
55D3×C8—62no—
56C3×C16—181no—
57C2×C3×D3—62no—
58C32×C4—22no—
59C32×C4—44no—
60C3×Dic3—62no—
61C3×Dic3—124no—
62C2×Q16—91no—
63M32—91no—
64#108—91no—
65H(3,𝔽3)#36211yescharacteristic, complemented, Sylow
66C3×Q8—21no—
67C3×Q8—64no—
68C3×Q8—181no—
69C3×Q8—361no—
70C2×C3×C4—91no—
71C2×C3×C4—361no—
72C4×D3—62no—
73C4×D3—124no—
74Dic6—64no—
75Dic6—124no—
76C3×C8—62no—
77C3×C8—91no—
78C3×C8—361no—
79C3×C8C2×#15711yescharacteristic
80C3⋊C8—62no—
81C2×C32—22no—
82C3×D3—64no—
83C2×Q8—181no—
84Q16—91no—
85Q16—181no—
86Q16#108811yescharacteristic
87M16—181no—
88C2×C8—91no—
89C16—181no—
90C22×C3—91no—
91C2×D3—62no—
92C3×C4—41no—
93C3×C4—62no—
94C3×C4—91no—
95C3×C4—124no—
96C3×C4—361no—
97C3×C4C22⋊#15711yescharacteristic
98Dic3—62no—
99Dic3—124no—
100C32—22no—
101Q8—21no—
102Q8—181no—
103Q8—361no—
104C2×C4—91no—
105C2×C4—361no—
106C8—91no—
107C8—361no—
108C8C2×#108811yescharacteristic
109C2×C3—62no—
110C2×C3—181no—
111C2×C3#6378211yescharacteristic
112D3—64no—
113C22—91no—
114C4—41no—
115C4—91no—
116C4—361no—
117C4C22⋊#108811yescharacteristic
118C3—62no—
119C3#1059019911yescharacteristic
120C2—181no—
121C2#1172078811yescharacteristic
subgroupG/H
center, socleC2×C3#63782
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC3×C8C2×#157
FittingQ16×H(3,𝔽3)C4
prime radicalC2×#272SD16
cosocleC8×#272C22
hypercenterC3×Q16#157
hypocenterH(3,𝔽3)#362
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×Q16

Complements

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

NH ≅ G/Ntype
65. H(3,𝔽3)#362semidirect

Maximal subgroups

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

subgroupindex
Q16×#2722
#117207762
#117207712
C3×#3629