Finite Group Databasebeta

← #395⋊H(3,𝔽3) #13141243

Subgroups — #395⋊H(3,𝔽3) #13141243

Order 1728.

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

normal subgroups only

This group has 272 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#11722587C211yesmaximal, characteristic
2C32×#395C311yesmaximal, characteristic, complemented
3C3×#395⋊C3C313yesmaximal, complemented
4C4×C22⋊H(3,𝔽3)—41nomaximal
5C22×C32×Q8C2×C311yescharacteristic
6C3×C22⋊SL(2,𝔽3)C2×C313yes—
7C2×C32×C4⋊C4—31no—
8C32×#95—31no—
9C2×C22⋊H(3,𝔽3)—41no—
10Q8⋊H(3,𝔽3)—82no—
11C3×#395—31no—
12C3×#395C3211yescharacteristic
13#395⋊C3—33no—
14C2×C32×Q8—62no—
15C22×C32×C4—31no—
16C22×C32×C4—31no—
17C22×C32×C4A411yescharacteristic
18C3×C4×A4—43no—
19C32×C4⋊C4—62no—
20C22⋊H(3,𝔽3)—41no—
21C4×H(3,𝔽3)—161no—
22C22×C3×Q8—31no—
23C22×C3×Q8C2×C3211yescharacteristic
24C22⋊SL(2,𝔽3)—33no—
25C2×C3×C4⋊C4—33no—
26C2×C3×C4⋊C4—31no—
27C3×#95—31no—
28C3×#95—33no—
29C23×C32C2×A411yescharacteristic
30C2×C3×A4—43no—
31C32×Q8—22no—
32C32×Q8—62no—
33C2×C32×C4—32no—
34C2×C32×C4—32no—
35C2×C32×C4—61no—
36C2×C32×C4—61no—
37C2×C32×C4—62no—
38C2×C32×C4—61no—
39C3×SL(2,𝔽3)—86no—
40#395H(3,𝔽3)11yescharacteristic, complemented, Sylow
41C2×H(3,𝔽3)—161no—
42C2×C3×Q8—62no—
43C2×C3×Q8—66no—
44C22×C3×C4—33no—
45C22×C3×C4—33no—
46C22×C3×C4—31no—
47C22×C3×C4—31no—
48C22×C3×C4—31no—
49C22×C3×C4C3×A411yescharacteristic
50C4×A4—123no—
51C3×C4⋊C4—62no—
52C3×C4⋊C4—66no—
53C22×C32—31no—
54C22×C32—31no—
55C22×C32#41⋊C311yescharacteristic
56C3×A4—43no—
57C32×C4—41no—
58C32×C4—62no—
59C32×C4—62no—
60C32×C4—163no—
61C22×Q8C2×H(3,𝔽3)11yescharacteristic
62C2×C4⋊C4—31no—
63#95—31no—
64H(3,𝔽3)—161noSylow
65C23×C3—31no—
66C23×C3C2×C3×A411yescharacteristic
67C2×A4—123no—
68C3×Q8—22no—
69C3×Q8—62no—
70C3×Q8—62no—
71C3×Q8—66no—
72C2×C3×C4—36no—
73C2×C3×C4—32no—
74C2×C3×C4—32no—
75C2×C3×C4—36no—
76C2×C3×C4—61no—
77C2×C3×C4—63no—
78C2×C3×C4—61no—
79C2×C3×C4—63no—
80C2×C3×C4—66no—
81C2×C3×C4—62no—
82C2×C3×C4—61no—
83C2×C3×C4—63no—
84SL(2,𝔽3)—246no—
85C2×C32—31no—
86C2×C32—31no—
87C2×C32—163no—
88C2×C32#120⋊C311yescharacteristic
89C2×Q8—62no—
90C22×C4—31no—
91C22×C4—31no—
92C22×C4C22⋊H(3,𝔽3)11yescharacteristic
93C4⋊C4—62no—
94C22×C3—31no—
95C22×C3—33no—
96C22×C3—33no—
97C22×C3—31no—
98C22×C3—31no—
99C22×C3C3×#41⋊C311yescharacteristic
100A4—123no—
101C3×C4—41no—
102C3×C4—66no—
103C3×C4—66no—
104C3×C4—62no—
105C3×C4—62no—
106C3×C4—121no—
107C3×C4—483no—
108C32—163no—
109C32#395⋊C311yescharacteristic, complemented
110C23C2×C22⋊H(3,𝔽3)11yescharacteristic
111Q8—22no—
112Q8—62no—
113C2×C4—32no—
114C2×C4—32no—
115C2×C4—61no—
116C2×C4—61no—
117C2×C4—62no—
118C2×C4—61no—
119C2×C3—33no—
120C2×C3—33no—
121C2×C3—31no—
122C2×C3—31no—
123C2×C3—31no—
124C2×C3—483no—
125C2×C3C3×#120⋊C311yescharacteristic
126C22—31no—
127C22—31no—
128C22#41⋊H(3,𝔽3)11yescharacteristic
129C4—41no—
130C4—62no—
131C4—62no—
132C3—31no—
133C3—483no—
134C3C3×#395⋊C311yescharacteristic
135C2—31no—
136C2—31no—
137C2#120⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C3×#120⋊C3
commutator, cosocleC22×C3×Q8C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#395C3
prime radical#11722587C2
hypercenterC2×C32#120⋊C3
hypocenterC22×Q8C2×H(3,𝔽3)
LCSG ▷ C22×C3×Q8 ▷ C22×Q8
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C32×#395C3semidirect
3. C3×#395⋊C3C3semidirect
40. #395H(3,𝔽3)semidirect
109. C32#395⋊C3semidirect

Maximal subgroups

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

subgroupindex
#117225872
C32×#3953
C3×#395⋊C33
C4×C22⋊H(3,𝔽3)4