Finite Group Databasebeta

← #401⋊H(3,𝔽3) #13141252

Subgroups — #401⋊H(3,𝔽3) #13141252

Order 1728.

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

normal subgroups only

This group has 190 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#95⋊H(3,𝔽3)C211yesmaximal, characteristic
2C32×#401C311yesmaximal, characteristic, complemented
3C3×#401⋊C3C313yesmaximal, complemented
4C4×C22⋊H(3,𝔽3)—41nomaximal
5C32×#95—31no—
6C32×#95—31no—
7C32×#95C2×C311yescharacteristic
8C3×#95⋊C3C2×C313yes—
9C2×C22⋊H(3,𝔽3)—41no—
10C3×#401—31no—
11C3×#401C3211yescharacteristic
12#401⋊C3—33no—
13C22×C32×C4—31no—
14C22×C32×C4—31no—
15C22×C32×C4A411yescharacteristic
16C3×C4×A4—43no—
17C22⋊H(3,𝔽3)—41no—
18C4×H(3,𝔽3)—161no—
19C3×#95—31no—
20C3×#95—33no—
21C3×#95—33no—
22C3×#95—31no—
23C3×#95—31no—
24C3×#95C2×C3211yescharacteristic
25#95⋊C3—33no—
26C23×C32C2×A411yescharacteristic
27C2×C3×A4—43no—
28C2×C32×C4—61no—
29C2×C32×C4—61no—
30C2×C32×C4—62no—
31C2×C32×C4—61no—
32C2×C32×C4—61no—
33C2×C32×C4—61no—
34#401H(3,𝔽3)11yescharacteristic, complemented, Sylow
35C2×H(3,𝔽3)—161no—
36C22×C3×C4—33no—
37C22×C3×C4—33no—
38C22×C3×C4—31no—
39C22×C3×C4—31no—
40C22×C3×C4—31no—
41C22×C3×C4C3×A411yescharacteristic
42C4×A4—123no—
43C22×C32—31no—
44C22×C32—31no—
45C22×C32#41⋊C311yescharacteristic
46C3×A4—43no—
47C32×C4—41no—
48C32×C4—121no—
49C32×C4—121no—
50C32×C4—163no—
51#95—31no—
52#95—31no—
53#95C2×H(3,𝔽3)11yescharacteristic
54H(3,𝔽3)—161noSylow
55C23×C3—31no—
56C23×C3C2×C3×A411yescharacteristic
57C2×A4—123no—
58C2×C3×C4—61no—
59C2×C3×C4—63no—
60C2×C3×C4—62no—
61C2×C3×C4—61no—
62C2×C3×C4—66no—
63C2×C3×C4—63no—
64C2×C3×C4—63no—
65C2×C3×C4—63no—
66C2×C3×C4—61no—
67C2×C3×C4—61no—
68C2×C3×C4—63no—
69C2×C3×C4—61no—
70C2×C32—31no—
71C2×C32—31no—
72C2×C32—163no—
73C2×C32#126⋊C311yescharacteristic
74C22×C4—31no—
75C22×C4—31no—
76C22×C4C22⋊H(3,𝔽3)11yescharacteristic
77C22×C3—31no—
78C22×C3—33no—
79C22×C3—33no—
80C22×C3—31no—
81C22×C3—31no—
82C22×C3C3×#41⋊C311yescharacteristic
83A4—123no—
84C3×C4—41no—
85C3×C4—121no—
86C3×C4—123no—
87C3×C4—123no—
88C3×C4—121no—
89C3×C4—121no—
90C3×C4—483no—
91C32—163no—
92C32#401⋊C311yescharacteristic, complemented
93C23C2×C22⋊H(3,𝔽3)11yescharacteristic
94C2×C4—61no—
95C2×C4—62no—
96C2×C4—61no—
97C2×C4—61no—
98C2×C4—61no—
99C2×C4—61no—
100C2×C3—33no—
101C2×C3—33no—
102C2×C3—31no—
103C2×C3—31no—
104C2×C3—31no—
105C2×C3—483no—
106C2×C3C3×#126⋊C311yescharacteristic
107C22—31no—
108C22—31no—
109C22#41⋊H(3,𝔽3)11yescharacteristic
110C4—41no—
111C4—121no—
112C4—121no—
113C3—31no—
114C3—483no—
115C3C3×#401⋊C311yescharacteristic
116C2—31no—
117C2—31no—
118C2#126⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#126⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#401C3
prime radical#95⋊H(3,𝔽3)C2
hypercenterC2×C32#126⋊C3
hypocenter#95C2×H(3,𝔽3)
LCSG ▷ C3×#95 ▷ #95
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C32×#401C3semidirect
3. C3×#401⋊C3C3semidirect
34. #401H(3,𝔽3)semidirect
92. C32#401⋊C3semidirect

Maximal subgroups

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

subgroupindex
#95⋊H(3,𝔽3)2
C32×#4013
C3×#401⋊C33
C4×C22⋊H(3,𝔽3)4