Finite Group Databasebeta

← #13141251

Subgroups — #13141251

Order 1728.

166 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 29 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#11720339C211yesmaximal, characteristic
2C32×#401C311yesmaximal, characteristic
3#401⋊C9C313yesmaximal, complemented
4C4×#1094—41nomaximal
5C32×#95—31no—
6C32×#95—31no—
7C32×#95C2×C311yescharacteristic
8#95⋊C9C2×C313yes—
9C2×#1094—41no—
10C3×#401—31no—
11C3×#401C3211yescharacteristic
12C22×C32×C4—31no—
13C22×C32×C4—31no—
14C22×C32×C4A411yescharacteristic
15C4×C22⋊C9—43no—
16#1094—41no—
17C4×3−1+2—161no—
18C3×#95—31no—
19C3×#95—33no—
20C3×#95—33no—
21C3×#95—31no—
22C3×#95—31no—
23C3×#95C2×C3211yescharacteristic
24C23×C32C2×A411yescharacteristic
25C2×C32×C4—61no—
26C2×C32×C4—61no—
27C2×C32×C4—62no—
28C2×C32×C4—61no—
29C2×C32×C4—61no—
30C2×C32×C4—61no—
31C2×C22⋊C9—43no—
32#4013−1+211yescharacteristic, complemented, Sylow
33C2×3−1+2—161no—
34C22×C3×C4—33no—
35C22×C3×C4—33no—
36C22×C3×C4—31no—
37C22×C3×C4—31no—
38C22×C3×C4—31no—
39C22×C3×C4C3×A411yescharacteristic
40C22×C32—31no—
41C22×C32—31no—
42C22×C32#41⋊C311yescharacteristic
43C32×C4—41no—
44C32×C4—121no—
45C32×C4—121no—
46C22⋊C9—43no—
47C4×C9—163no—
48#95—31no—
49#95—31no—
50#95C2×3−1+211yescharacteristic
513−1+2—161noSylow
52C23×C3—31no—
53C23×C3C2×C3×A411yescharacteristic
54C2×C3×C4—61no—
55C2×C3×C4—63no—
56C2×C3×C4—62no—
57C2×C3×C4—61no—
58C2×C3×C4—66no—
59C2×C3×C4—63no—
60C2×C3×C4—63no—
61C2×C3×C4—63no—
62C2×C3×C4—61no—
63C2×C3×C4—61no—
64C2×C3×C4—63no—
65C2×C3×C4—61no—
66C2×C32—31no—
67C2×C32—31no—
68C2×C32#126⋊C311yescharacteristic
69C2×C9—163no—
70C22×C4—31no—
71C22×C4—31no—
72C22×C4#109411yescharacteristic
73C22×C3—31no—
74C22×C3—33no—
75C22×C3—33no—
76C22×C3—31no—
77C22×C3—31no—
78C22×C3C3×#41⋊C311yescharacteristic
79C3×C4—41no—
80C3×C4—121no—
81C3×C4—123no—
82C3×C4—123no—
83C3×C4—121no—
84C3×C4—121no—
85C32#401⋊C311yescharacteristic
86C9—163no—
87C23C2×#109411yescharacteristic
88C2×C4—61no—
89C2×C4—62no—
90C2×C4—61no—
91C2×C4—61no—
92C2×C4—61no—
93C2×C4—61no—
94C2×C3—33no—
95C2×C3—33no—
96C2×C3—31no—
97C2×C3—31no—
98C2×C3—31no—
99C2×C3C3×#126⋊C311yescharacteristic
100C22—31no—
101C22—31no—
102C22#8869111yescharacteristic
103C4—41no—
104C4—121no—
105C4—121no—
106C3—31no—
107C3C3×#401⋊C311yescharacteristic
108C2—31no—
109C2—31no—
110C2#1172135011yescharacteristic
subgroupG/H
centerC2×C3C3×#126⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#401C3
prime radicalC23×C32C2×A4
hypercenterC2×C32#126⋊C3
hypocenter#95C2×3−1+2
LCSG ▷ C3×#95 ▷ #95
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
3. #401⋊C9C3semidirect
32. #4013−1+2semidirect

Maximal subgroups

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

subgroupindex
#117203392
C32×#4013
#401⋊C93
C4×#10944