Finite Group Databasebeta

← #384⋊H(3,𝔽3) #13141234

Subgroups — #384⋊H(3,𝔽3) #13141234

Order 1728.

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

normal subgroups only

This group has 267 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×#384C311yesmaximal, characteristic, complemented
3C3×#384⋊C3C313yesmaximal, complemented
4C4×C22⋊H(3,𝔽3)—41nomaximal
5C32⋊(C42⋊C3)C411yescharacteristic, complemented
6C2×C32×C4⋊C4—31no—
7C2×C32×C4⋊C4—31no—
8C2×C32×C42C2×C311yescharacteristic
9C2×C3×C42⋊C3C2×C313yes—
10C2×C22⋊H(3,𝔽3)—41no—
11C3×#384—31no—
12C3×#384C3211yescharacteristic
13#384⋊C3—33no—
14C22×C32×C4—31no—
15C22×C32×C4—31no—
16C22×C32×C4A411yescharacteristic
17C3×C4×A4—43no—
18C32×C4⋊C4—62no—
19C32×C4⋊C4—62no—
20C32×C42—31no—
21C32×C42C3×C411yescharacteristic, complemented
22C3×C42⋊C3C3×C413yescomplemented
23C22⋊H(3,𝔽3)—41no—
24C4×H(3,𝔽3)—161no—
25C2×C3×C4⋊C4—31no—
26C2×C3×C4⋊C4—33no—
27C2×C3×C4⋊C4—33no—
28C2×C3×C4⋊C4—31no—
29C2×C3×C42—31no—
30C2×C3×C42C2×C3211yescharacteristic
31C2×C42⋊C3—33no—
32C23×C32C2×A411yescharacteristic
33C2×C3×A4—43no—
34C2×C32×C4—32no—
35C2×C32×C4—32no—
36C2×C32×C4—31no—
37C2×C32×C4—31no—
38C2×C32×C4—61no—
39C2×C32×C4—61no—
40C2×C32×C4—62no—
41#384H(3,𝔽3)11yescharacteristic, complemented, Sylow
42C2×H(3,𝔽3)—161no—
43C22×C3×C4—33no—
44C22×C3×C4—31no—
45C22×C3×C4—33no—
46C22×C3×C4—31no—
47C22×C3×C4—31no—
48C22×C3×C4C3×A411yescharacteristic
49C4×A4—123no—
50C3×C4⋊C4—62no—
51C3×C4⋊C4—62no—
52C3×C4⋊C4—66no—
53C3×C4⋊C4—66no—
54C3×C42—31no—
55C3×C42—33no—
56C3×C42—31no—
57C3×C42C32×C411yescharacteristic
58C42⋊C3—33no—
59C22×C32—31no—
60C22×C32—31no—
61C22×C32C4×A411yescharacteristic
62C3×A4—43no—
63C32×C4—32no—
64C32×C4—32no—
65C32×C4—41no—
66C32×C4—121no—
67C32×C4—163no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—31no—
70C2×C42C2×H(3,𝔽3)11yescharacteristic
71H(3,𝔽3)—161noSylow
72C23×C3—31no—
73C23×C3C2×C3×A411yescharacteristic
74C2×A4—123no—
75C2×C3×C4—33no—
76C2×C3×C4—33no—
77C2×C3×C4—36no—
78C2×C3×C4—32no—
79C2×C3×C4—32no—
80C2×C3×C4—36no—
81C2×C3×C4—31no—
82C2×C3×C4—31no—
83C2×C3×C4—61no—
84C2×C3×C4—63no—
85C2×C3×C4—63no—
86C2×C3×C4—66no—
87C2×C3×C4—61no—
88C2×C3×C4—62no—
89C2×C32—31no—
90C2×C32—31no—
91C2×C32—163no—
92C2×C32#127⋊C311yescharacteristic
93C22×C4—31no—
94C22×C4—31no—
95C22×C4C22⋊H(3,𝔽3)11yescharacteristic
96C4⋊C4—62no—
97C4⋊C4—62no—
98C42—31no—
99C42C4×H(3,𝔽3)11yescharacteristic, complemented
100C22×C3—31no—
101C22×C3—33no—
102C22×C3—33no—
103C22×C3—31no—
104C22×C3—31no—
105C22×C3C3×C4×A411yescharacteristic
106A4—123no—
107C3×C4—36no—
108C3×C4—36no—
109C3×C4—32no—
110C3×C4—32no—
111C3×C4—41no—
112C3×C4—121no—
113C3×C4—123no—
114C3×C4—121no—
115C3×C4—483no—
116C32—163no—
117C32#384⋊C311yescharacteristic, complemented
118C23C2×C22⋊H(3,𝔽3)11yescharacteristic
119C2×C4—32no—
120C2×C4—32no—
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—61no—
124C2×C4—61no—
125C2×C4—62no—
126C2×C3—33no—
127C2×C3—33no—
128C2×C3—31no—
129C2×C3—31no—
130C2×C3—31no—
131C2×C3—483no—
132C2×C3C3×#127⋊C311yescharacteristic
133C22—31no—
134C22—31no—
135C22C4×C22⋊H(3,𝔽3)11yescharacteristic
136C4—32no—
137C4—32no—
138C4—41no—
139C4—121no—
140C3—31no—
141C3—483no—
142C3C3×#384⋊C311yescharacteristic
143C2—31no—
144C2—31no—
145C2#127⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#127⋊C3
commutatorC3×C42C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC32×#384C3
prime radicalC2×C32⋊(C42⋊C3)C2
cosocleC2×C3×C42C2×C32
hypercenterC2×C32#127⋊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×#384C3semidirect
3. C3×#384⋊C3C3semidirect
5. C32⋊(C42⋊C3)C4semidirect
21. C32×C42C3×C4semidirect
22. C3×C42⋊C3C3×C4semidirect
41. #384H(3,𝔽3)semidirect
99. C42C4×H(3,𝔽3)semidirect
117. C32#384⋊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×#3843
C3×#384⋊C33
C4×C22⋊H(3,𝔽3)4