Finite Group Databasebeta

← #13141233

Subgroups — #13141233

Order 1728.

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

normal subgroups only

This group has 240 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×#384C311yesmaximal, characteristic
3#384⋊C9C313yesmaximal, complemented
4C4×#1094—41nomaximal
5#88455C411yescharacteristic, complemented
6C2×C32×C4⋊C4—31no—
7C2×C32×C4⋊C4—31no—
8C2×C32×C42C2×C311yescharacteristic
9C2×C42⋊C9C2×C313yes—
10C2×#1094—41no—
11C3×#384—31no—
12C3×#384C3211yescharacteristic
13C22×C32×C4—31no—
14C22×C32×C4—31no—
15C22×C32×C4A411yescharacteristic
16C32×C4⋊C4—62no—
17C32×C4⋊C4—62no—
18C32×C42—31no—
19C32×C42C3×C411yescharacteristic
20C4×C22⋊C9—43no—
21C42⋊C9C3×C413yescomplemented
22#1094—41no—
23C4×3−1+2—161no—
24C2×C3×C4⋊C4—31no—
25C2×C3×C4⋊C4—33no—
26C2×C3×C4⋊C4—33no—
27C2×C3×C4⋊C4—31no—
28C2×C3×C42—31no—
29C2×C3×C42C2×C3211yescharacteristic
30C23×C32C2×A411yescharacteristic
31C2×C32×C4—32no—
32C2×C32×C4—32no—
33C2×C32×C4—31no—
34C2×C32×C4—31no—
35C2×C32×C4—61no—
36C2×C32×C4—61no—
37C2×C32×C4—62no—
38C2×C22⋊C9—43no—
39#3843−1+211yescharacteristic, complemented, Sylow
40C2×3−1+2—161no—
41C22×C3×C4—33no—
42C22×C3×C4—31no—
43C22×C3×C4—33no—
44C22×C3×C4—31no—
45C22×C3×C4—31no—
46C22×C3×C4C3×A411yescharacteristic
47C3×C4⋊C4—62no—
48C3×C4⋊C4—62no—
49C3×C4⋊C4—66no—
50C3×C4⋊C4—66no—
51C3×C42—31no—
52C3×C42—33no—
53C3×C42—31no—
54C3×C42C32×C411yescharacteristic
55C22×C32—31no—
56C22×C32—31no—
57C22×C32C4×A411yescharacteristic
58C32×C4—32no—
59C32×C4—32no—
60C32×C4—41no—
61C32×C4—121no—
62C22⋊C9—43no—
63C4×C9—163no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C42C2×3−1+211yescharacteristic
673−1+2—161noSylow
68C23×C3—31no—
69C23×C3C2×C3×A411yescharacteristic
70C2×C3×C4—33no—
71C2×C3×C4—33no—
72C2×C3×C4—36no—
73C2×C3×C4—32no—
74C2×C3×C4—32no—
75C2×C3×C4—36no—
76C2×C3×C4—31no—
77C2×C3×C4—31no—
78C2×C3×C4—61no—
79C2×C3×C4—63no—
80C2×C3×C4—63no—
81C2×C3×C4—66no—
82C2×C3×C4—61no—
83C2×C3×C4—62no—
84C2×C32—31no—
85C2×C32—31no—
86C2×C32#127⋊C311yescharacteristic
87C2×C9—163no—
88C22×C4—31no—
89C22×C4—31no—
90C22×C4#109411yescharacteristic
91C4⋊C4—62no—
92C4⋊C4—62no—
93C42—31no—
94C42C4×3−1+211yescharacteristic, complemented
95C22×C3—31no—
96C22×C3—33no—
97C22×C3—33no—
98C22×C3—31no—
99C22×C3—31no—
100C22×C3C3×C4×A411yescharacteristic
101C3×C4—36no—
102C3×C4—36no—
103C3×C4—32no—
104C3×C4—32no—
105C3×C4—41no—
106C3×C4—121no—
107C3×C4—123no—
108C3×C4—121no—
109C32#384⋊C311yescharacteristic
110C9—163no—
111C23C2×#109411yescharacteristic
112C2×C4—32no—
113C2×C4—32no—
114C2×C4—31no—
115C2×C4—31no—
116C2×C4—61no—
117C2×C4—61no—
118C2×C4—62no—
119C2×C3—33no—
120C2×C3—33no—
121C2×C3—31no—
122C2×C3—31no—
123C2×C3—31no—
124C2×C3C3×#127⋊C311yescharacteristic
125C22—31no—
126C22—31no—
127C22C4×#109411yescharacteristic
128C4—32no—
129C4—32no—
130C4—41no—
131C4—121no—
132C3—31no—
133C3C3×#384⋊C311yescharacteristic
134C2—31no—
135C2—31no—
136C2#1172135311yescharacteristic
subgroupG/H
centerC2×C3C3×#127⋊C3
commutatorC3×C42C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC32×#384C3
prime radicalC23×C32C2×A4
cosocleC2×C3×C42C2×C32
hypercenterC2×C32#127⋊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. #384⋊C9C3semidirect
5. #88455C4semidirect
21. C42⋊C9C3×C4semidirect
39. #3843−1+2semidirect
94. 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×#3843
#384⋊C93
C4×#10944