Finite Group Databasebeta

← #383⋊H(3,𝔽3) #13141231

Subgroups — #383⋊H(3,𝔽3) #13141231

Order 1728.

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

normal subgroups only

This group has 219 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×C32⋊(C42⋊C3)C211yesmaximal, characteristic
2C32×#383C311yesmaximal, characteristic, complemented
3C3×#383⋊C3C313yesmaximal, complemented
4C4×C22⋊H(3,𝔽3)—41nomaximal
5C32⋊(C42⋊C3)C411yescharacteristic, complemented
6C2×C32×C42C2×C311yescharacteristic
7C2×C3×C42⋊C3C2×C313yes—
8C32×#95—31no—
9C32×#95—31no—
10C2×C22⋊H(3,𝔽3)—41no—
11C3×#383—31no—
12C3×#383C3211yescharacteristic
13#383⋊C3—33no—
14C22×C32×C4—31no—
15C22×C32×C4—31no—
16C22×C32×C4A411yescharacteristic
17C3×C4×A4—43no—
18C32×C42—31no—
19C32×C42C3×C411yescharacteristic, complemented
20C3×C42⋊C3C3×C413yescomplemented
21C22⋊H(3,𝔽3)—41no—
22C4×H(3,𝔽3)—161no—
23C2×C3×C42—31no—
24C2×C3×C42C2×C3211yescharacteristic
25C2×C42⋊C3—33no—
26C3×#95—31no—
27C3×#95—33no—
28C3×#95—33no—
29C3×#95—31no—
30C23×C32C2×A411yescharacteristic
31C2×C3×A4—43no—
32C2×C32×C4—31no—
33C2×C32×C4—31no—
34C2×C32×C4—61no—
35C2×C32×C4—61no—
36C2×C32×C4—61no—
37C2×C32×C4—62no—
38C2×C32×C4—61no—
39#383H(3,𝔽3)11yescharacteristic, complemented, Sylow
40C2×H(3,𝔽3)—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
47C4×A4—123no—
48C3×C42—31no—
49C3×C42—33no—
50C3×C42—31no—
51C3×C42C32×C411yescharacteristic
52C42⋊C3—33no—
53C22×C32—31no—
54C22×C32—31no—
55C22×C32C4×A411yescharacteristic
56C3×A4—43no—
57C32×C4—41no—
58C32×C4—61no—
59C32×C4—61no—
60C32×C4—121no—
61C32×C4—163no—
62C2×C42C2×H(3,𝔽3)11yescharacteristic
63#95—31no—
64#95—31no—
65H(3,𝔽3)—161noSylow
66C23×C3—31no—
67C23×C3C2×C3×A411yescharacteristic
68C2×A4—123no—
69C2×C3×C4—33no—
70C2×C3×C4—33no—
71C2×C3×C4—31no—
72C2×C3×C4—31no—
73C2×C3×C4—61no—
74C2×C3×C4—61no—
75C2×C3×C4—63no—
76C2×C3×C4—63no—
77C2×C3×C4—61no—
78C2×C3×C4—63no—
79C2×C3×C4—62no—
80C2×C3×C4—66no—
81C2×C3×C4—63no—
82C2×C3×C4—61no—
83C2×C32—31no—
84C2×C32—31no—
85C2×C32—163no—
86C2×C32#126⋊C311yescharacteristic
87C22×C4—31no—
88C22×C4—31no—
89C22×C4C22⋊H(3,𝔽3)11yescharacteristic
90C42—31no—
91C42C4×H(3,𝔽3)11yescharacteristic, complemented
92C22×C3—31no—
93C22×C3—33no—
94C22×C3—33no—
95C22×C3—31no—
96C22×C3—31no—
97C22×C3C3×C4×A411yescharacteristic
98A4—123no—
99C3×C4—41no—
100C3×C4—63no—
101C3×C4—63no—
102C3×C4—61no—
103C3×C4—61no—
104C3×C4—121no—
105C3×C4—123no—
106C3×C4—121no—
107C3×C4—483no—
108C32—163no—
109C32#383⋊C311yescharacteristic, complemented
110C23C2×C22⋊H(3,𝔽3)11yescharacteristic
111C2×C4—31no—
112C2×C4—31no—
113C2×C4—61no—
114C2×C4—61no—
115C2×C4—61no—
116C2×C4—62no—
117C2×C4—61no—
118C2×C3—33no—
119C2×C3—33no—
120C2×C3—31no—
121C2×C3—31no—
122C2×C3—31no—
123C2×C3—483no—
124C2×C3C3×#126⋊C311yescharacteristic
125C22—31no—
126C22—31no—
127C22C4×C22⋊H(3,𝔽3)11yescharacteristic
128C4—41no—
129C4—61no—
130C4—61no—
131C4—121no—
132C3—31no—
133C3—483no—
134C3C3×#383⋊C311yescharacteristic
135C2—31no—
136C2—31no—
137C2#126⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#126⋊C3
commutatorC3×C42C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC32×#383C3
prime radicalC2×C32⋊(C42⋊C3)C2
cosocleC2×C3×C42C2×C32
hypercenterC2×C32#126⋊C3
hypocenterC42C4×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C32×#383C3semidirect
3. C3×#383⋊C3C3semidirect
5. C32⋊(C42⋊C3)C4semidirect
19. C32×C42C3×C4semidirect
20. C3×C42⋊C3C3×C4semidirect
39. #383H(3,𝔽3)semidirect
91. C42C4×H(3,𝔽3)semidirect
109. C32#383⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32⋊(C42⋊C3)2
C32×#3833
C3×#383⋊C33
C4×C22⋊H(3,𝔽3)4