Finite Group Databasebeta

← #72220228

Subgroups — #72220228

Order 2592.

318 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 23 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#13049411C212yesmaximal
2#13049399C211yesmaximal, characteristic
3C2×#66472—22no—
4#10620150C2211yescharacteristic
5C3⋊(C3⋊#622)C411yescharacteristic, complemented
6C32×#622C411yescharacteristic, complemented
7#88924—62no—
8C2×C3⋊(C3⋊(C3⋊D3))D411yescharacteristic
9#66472—42no—
10C32×C3⋊Dic3C2×C411yescharacteristic
11C32×C3⋊Dic3Q811yescharacteristic
12#63768—91nomaximal
13#63745—91nomaximal
14#6289—62no—
15#6289—121no—
16C3⋊#622—62no—
17C3×#622—22no—
18C2×C34C4⋊C411yescharacteristic
19C3⋊(C3⋊(C3⋊D3))SD1612yes—
20C4⋊#157—92no—
21C4⋊#157—92no—
22#3787—91no—
23C8×C3⋊D3—91no—
24C2×C3⋊(C3⋊D3)—62no—
25C2×C3⋊(C3⋊D3)—121no—
26C2×C3⋊(C3⋊D3)—244no—
27C3×C3⋊Dic3—22no—
28C3×C3⋊Dic3—41no—
29C32×Dic3—62no—
30C32×Dic3—121no—
31C34#10611yescharacteristic, complemented, Sylow
32C2×#157—182no—
33C2×#157—182no—
34C2×#157—368no—
35C4×C3⋊D3—91no—
36C4×C3⋊D3—91no—
37#624—364no—
38#622—91no—
39#622#15711yescharacteristic, complemented
40C32×C8—91no—
41C3⋊(C3⋊C8)—91no—
42C2×C33—22no—
43C2×C33—41no—
44C2×C33—84no—
45C3⋊(C3⋊D3)—64no—
46C3⋊(C3⋊D3)—122no—
47C3⋊(C3⋊D3)—248no—
48D3×C8—542no—
49C2×C3⋊D3—91no—
50C2×C3⋊D3—91no—
51C2×C3⋊D3—364no—
52C2×C3⋊D3—368no—
53C2×C3⋊D3—724no—
54C2×C3⋊D3—724no—
55C2×C3⋊D3—722no—
56#157—362no—
57#157—362no—
58#157—3616no—
59C32×C4—91no—
60C32×C4—91no—
61C3⋊Dic3PSU(3,𝔽2)11yescharacteristic
62C3⋊Dic3C2×#15711yescharacteristic
63C3×Dic3—124no—
64C3×Dic3—124no—
65#106—811noSylow
66C33—22no—
67C33—41no—
68C33—84no—
69C4×D3—542no—
70C4×D3—1081no—
71C3×C8—182no—
72C3⋊C8—542no—
73C2×C32—44no—
74C2×C32—48no—
75C2×C32—84no—
76C2×C32—84no—
77C2×C32—82no—
78C2×C32#377711yescharacteristic
79C2×C32C4⋊#15711yescharacteristic
80C3⋊D3—92no—
81C3⋊D3—92no—
82C3⋊D3—368no—
83C3⋊D3—368no—
84C3⋊D3—368no—
85C3⋊D3—728no—
86C3⋊D3—728no—
87C3⋊D3—724no—
88C2×C8—811no—
89C4⋊C4—812no—
90C2×D3—542no—
91C2×D3—1081no—
92C2×D3—2164no—
93C3×C4—182no—
94C3×C4—361no—
95Dic3—62no—
96Dic3—121no—
97C32—44no—
98C32—48no—
99C32—84no—
100C32—84no—
101C32—82no—
102C32#6374511yescharacteristic, complemented
103C32#6376811yescharacteristic, complemented
104C2×C4—811no—
105C2×C4—1622no—
106C8—91no—
107C8—811no—
108C2×C3—22no—
109C2×C3—41no—
110C2×C3—84no—
111D3—544no—
112D3—1082no—
113D3—2168no—
114C22—811no—
115C4—91no—
116C4—91no—
117C4—3242no—
118C3—22no—
119C3—41no—
120C3—84no—
121C2—812no—
122C2#1304978311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#13049783
commutatorC32×C3⋊Dic3C2×C4
Fitting, socleC2×C34C4⋊C4
prime radicalC2×C3⋊(C3⋊(C3⋊D3))D4
cosocle#10620150C22
hypocenterC34#106
LCSG ▷ C32×C3⋊Dic3 ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
5. C3⋊(C3⋊#622)C4semidirect
6. C32×#622C4semidirect
31. C34#106semidirect
39. #622#157semidirect
102. C32#63745semidirect
103. C32#63768semidirect

Maximal subgroups

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

subgroupindex
#130494112
#130493992
#637689
#637459