Finite Group Databasebeta

← #13154751

Subgroups — #13154751

Order 1728.

156 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 34 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#11723095C211yesmaximal, characteristic
2C3×#63768C211yesmaximal, characteristic
3#11722002C211yesmaximal, characteristic
4#10590137—31nomaximal
5#88989—21no—
6C3×C4⋊#157—21no—
7C3×C8×C3⋊D3C2211yescharacteristic
8C3⋊(C3⋊(C3⋊C16))C411yescharacteristic, complemented
9C32×C3⋊C16C411yescharacteristic, complemented
10#63770—31no—
11#63768D311yescharacteristic
12#63625—31no—
13#63543—62no—
14C2×C3×#157—41no—
15C3×C4×C3⋊D3D411yescharacteristic
16C3⋊#622—41no—
17C33×C8C2×C411yescharacteristic
18C3×C3⋊(C3⋊C8)Q811yescharacteristic
19#4487—91nomaximal
20C4⋊#157—21no—
21#3788—61no—
22C8×C3⋊D3C2×D311yescharacteristic
23C3×D3×C8—62no—
24C32×C16—31no—
25C3⋊(C3⋊C16)—31no—
26C3⋊(C3⋊C16)—62no—
27C3⋊(C3⋊C16)—122no—
28C3×C3⋊C16—22no—
29C2×C3×C3⋊D3SD1611yescharacteristic
30C3×#157—81no—
31C33×C4C4⋊C411yescharacteristic
32C3×C3⋊Dic3SD1611yescharacteristic
33C3×#106—91no—
34#819—91no—
35#812—91no—
36#798—182no—
37C2×#157—41no—
38C4×C3⋊D3#6711yescharacteristic
39C3×C4×D3—62no—
40#622—121no—
41C32×C8—22no—
42C32×C8—42no—
43C32×C8C4×D311yescharacteristic
44C3⋊(C3⋊C8)Dic611yescharacteristic
45C3×C3⋊C8—62no—
46#365—271noSylow
47C2×C33#10611yescharacteristic
48C3×C3⋊D3—21no—
49C2×C3×C8—91no—
50C3×C4⋊C4—181no—
51#208—181no—
52D3×C8—62no—
53C3×C16—62no—
54C3⋊C16—91no—
55C3⋊C16—182no—
56C3⋊C16—362no—
57C3⋊C16#15711yescharacteristic, complemented
58C2×C3⋊D3#21411yescharacteristic
59C2×C3×D3—62no—
60#157—81no—
61C32×C4—22no—
62C32×C4—42no—
63C32×C4Dic3⋊C411yescharacteristic
64C3⋊Dic3#21511yescharacteristic
65C3×Dic3—62no—
66M32—271no—
67#108—271no—
68#106—91no—
69C33#36511yescharacteristic, complemented, Sylow
70C2×C3×C4—91no—
71C2×C3×C4—361no—
72C4×D3—62no—
73C3×C8—22no—
74C3×C8—42no—
75C3×C8—91no—
76C3×C8C2×#15711yescharacteristic
77C3⋊C8—62no—
78C3⋊C8—361no—
79C2×C32—22no—
80C2×C32—42no—
81C2×C32#80811yescharacteristic
82C3⋊D3—21no—
83C3×D3—122no—
84M16—541no—
85C2×C8—91no—
86C4⋊C4—181no—
87C16—31no—
88C16—271no—
89C22×C3—91no—
90C2×D3—62no—
91C3×C4—22no—
92C3×C4—42no—
93C3×C4—91no—
94C3×C4—721no—
95C3×C4C4⋊#15711yescharacteristic
96Dic3—62no—
97C32—22no—
98C32—42no—
99C32#448711yescharacteristic, complemented
100C2×C4—91no—
101C2×C4—361no—
102C8—91no—
103C8—1081no—
104C8D3×#15711yescharacteristic
105C2×C3—22no—
106C2×C3—42no—
107C2×C3—181no—
108C2×C3#6376811yescharacteristic
109D3—122no—
110C22—91no—
111C4—91no—
112C4—721no—
113C4Dic3⋊#15711yescharacteristic
114C3—22no—
115C3—42no—
116C3#1059013711yescharacteristic, complemented
117C2—181no—
118C2#1172271411yescharacteristic
subgroupG/H
centerC2#11722714
commutator, FittingC33×C8C2×C4
Frattini, hypercenterC8D3×#157
socleC2×C33#106
prime radicalC2×C3×C3⋊D3SD16
cosocleC3×C8×C3⋊D3C22
hypocenterC33#365
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
8. C3⋊(C3⋊(C3⋊C16))C4semidirect
9. C32×C3⋊C16C4semidirect
57. C3⋊C16#157semidirect
69. C33#365semidirect
99. C32#4487semidirect
116. C3#10590137semidirect

Maximal subgroups

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

subgroupindex
#117230952
C3×#637682
#117220022
#105901373
#44879