Finite Group Databasebeta

← #13141230

Subgroups — #13141230

Order 1728.

192 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C2×#88455C211yesmaximal, characteristic
2C32×#383C311yesmaximal, characteristic
3#383⋊C9C313yesmaximal, complemented
4C4×#1094—41nomaximal
5#88455C411yescharacteristic, complemented
6C2×C32×C42C2×C311yescharacteristic
7C32×#95—31no—
8C32×#95—31no—
9C2×C42⋊C9C2×C313yes—
10C2×#1094—41no—
11C3×#383—31no—
12C3×#383C3211yescharacteristic
13C22×C32×C4—31no—
14C22×C32×C4—31no—
15C22×C32×C4A411yescharacteristic
16C32×C42—31no—
17C32×C42C3×C411yescharacteristic
18C4×C22⋊C9—43no—
19C42⋊C9C3×C413yescomplemented
20#1094—41no—
21C4×3−1+2—161no—
22C2×C3×C42—31no—
23C2×C3×C42C2×C3211yescharacteristic
24C3×#95—31no—
25C3×#95—33no—
26C3×#95—33no—
27C3×#95—31no—
28C23×C32C2×A411yescharacteristic
29C2×C32×C4—31no—
30C2×C32×C4—31no—
31C2×C32×C4—61no—
32C2×C32×C4—61no—
33C2×C32×C4—61no—
34C2×C32×C4—62no—
35C2×C32×C4—61no—
36C2×C22⋊C9—43no—
37#3833−1+211yescharacteristic, complemented, Sylow
38C2×3−1+2—161no—
39C22×C3×C4—33no—
40C22×C3×C4—31no—
41C22×C3×C4—33no—
42C22×C3×C4—31no—
43C22×C3×C4—31no—
44C22×C3×C4C3×A411yescharacteristic
45C3×C42—31no—
46C3×C42—33no—
47C3×C42—31no—
48C3×C42C32×C411yescharacteristic
49C22×C32—31no—
50C22×C32—31no—
51C22×C32C4×A411yescharacteristic
52C32×C4—41no—
53C32×C4—61no—
54C32×C4—61no—
55C32×C4—121no—
56C22⋊C9—43no—
57C4×C9—163no—
58C2×C42C2×3−1+211yescharacteristic
59#95—31no—
60#95—31no—
613−1+2—161noSylow
62C23×C3—31no—
63C23×C3C2×C3×A411yescharacteristic
64C2×C3×C4—33no—
65C2×C3×C4—33no—
66C2×C3×C4—31no—
67C2×C3×C4—31no—
68C2×C3×C4—61no—
69C2×C3×C4—61no—
70C2×C3×C4—63no—
71C2×C3×C4—63no—
72C2×C3×C4—61no—
73C2×C3×C4—63no—
74C2×C3×C4—62no—
75C2×C3×C4—66no—
76C2×C3×C4—63no—
77C2×C3×C4—61no—
78C2×C32—31no—
79C2×C32—31no—
80C2×C32#126⋊C311yescharacteristic
81C2×C9—163no—
82C22×C4—31no—
83C22×C4—31no—
84C22×C4#109411yescharacteristic
85C42—31no—
86C42C4×3−1+211yescharacteristic, complemented
87C22×C3—31no—
88C22×C3—33no—
89C22×C3—33no—
90C22×C3—31no—
91C22×C3—31no—
92C22×C3C3×C4×A411yescharacteristic
93C3×C4—41no—
94C3×C4—63no—
95C3×C4—63no—
96C3×C4—61no—
97C3×C4—61no—
98C3×C4—121no—
99C3×C4—123no—
100C3×C4—121no—
101C32#383⋊C311yescharacteristic
102C9—163no—
103C23C2×#109411yescharacteristic
104C2×C4—31no—
105C2×C4—31no—
106C2×C4—61no—
107C2×C4—61no—
108C2×C4—61no—
109C2×C4—62no—
110C2×C4—61no—
111C2×C3—33no—
112C2×C3—33no—
113C2×C3—31no—
114C2×C3—31no—
115C2×C3—31no—
116C2×C3C3×#126⋊C311yescharacteristic
117C22—31no—
118C22—31no—
119C22C4×#109411yescharacteristic
120C4—41no—
121C4—61no—
122C4—61no—
123C4—121no—
124C3—31no—
125C3C3×#383⋊C311yescharacteristic
126C2—31no—
127C2—31no—
128C2#1172135011yescharacteristic
subgroupG/H
centerC2×C3C3×#126⋊C3
commutatorC3×C42C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC32×#383C3
prime radicalC23×C32C2×A4
cosocleC2×C3×C42C2×C32
hypercenterC2×C32#126⋊C3
hypocenterC42C4×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
3. #383⋊C9C3semidirect
5. #88455C4semidirect
19. C42⋊C9C3×C4semidirect
37. #3833−1+2semidirect
86. C42C4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C2×#884552
C32×#3833
#383⋊C93
C4×#10944