Finite Group Databasebeta

← #10589409⋊C3 #13135622

Subgroups — #10589409⋊C3 #13135622

Order 1728.

266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic
2#63630⋊C3C212yesmaximal
3#10589409C311yesmaximal, characteristic, complemented
4C3×#4543—31nomaximal
5C4⋊C4×H(3,𝔽3)—22no—
6C4⋊C4×H(3,𝔽3)C412yes—
7C42×H(3,𝔽3)C2211yescharacteristic
8C2×(C3⋊(C3⋊C8))⋊C3—22no—
9C32×#125—21no—
10C32×#125D311yescharacteristic
11C32×#125C2×C311yescharacteristic
12#63630C2×C312yes—
13C3×#803—32no—
14C2×C4×H(3,𝔽3)—22no—
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16C2×C4×H(3,𝔽3)D412yes—
17(C3⋊(C3⋊C8))⋊C3—24no—
18C3×#330—91no—
19#4543—31no—
20#4543—91no—
21C32×C4⋊C4—24no—
22C32×C4⋊C4—22no—
23C32×C4⋊C4—22no—
24C32×C4⋊C4—22no—
25C32×C4⋊C4Dic312yes—
26C32×C4⋊C4C3×C412yes—
27C32×C42—21no—
28C32×C42C2×D311yescharacteristic
29C32×C42C22×C311yescharacteristic
30C2×C3⋊(C3⋊C8)—22no—
31C2×C3×C3⋊C8—62no—
32C22×H(3,𝔽3)C22⋊C411yescharacteristic
33C4×H(3,𝔽3)—21no—
34C4×H(3,𝔽3)—22no—
35C4×H(3,𝔽3)—42no—
36C3×#125—31no—
37C3×#125—31no—
38C3×#125—61no—
39C3×#125C3×D311yescharacteristic
40C3×#97—92no—
41#803—32no—
42#803—92no—
43C2×C32×C4—24no—
44C2×C32×C4—22no—
45C2×C32×C4—22no—
46C2×C32×C4—21no—
47C2×C32×C4—22no—
48C2×C32×C4C2×Dic311yescharacteristic
49C2×C32×C4#6712yes—
50C2×C32×C4C2×C3×C411yescharacteristic
51C2×C32×C4C3×D412yes—
52C3⋊(C3⋊C8)—24no—
53C3×C3⋊C8—64no—
54#330—91noSylow
55C2×H(3,𝔽3)#10111yescharacteristic
56C2×H(3,𝔽3)#10412yes—
57C2×C3×C8—182no—
58C3×C4⋊C4—22no—
59C3×C4⋊C4—32no—
60C3×C4⋊C4—32no—
61C3×C4⋊C4—62no—
62C3×C4⋊C4—62no—
63C3×C4⋊C4—64no—
64C3×C4⋊C4—62no—
65C3×C4⋊C4C3×Dic312yes—
66C3×C42—31no—
67C3×C42—31no—
68C3×C42—61no—
69C3×C42C2×C3×D311yescharacteristic
70C2×C3⋊C8—62no—
71C2×C3⋊C8—182no—
72C22×C32—21no—
73C22×C32#21711yescharacteristic
74C22×C32C3×C22⋊C411yescharacteristic
75C32×C4—22no—
76C32×C4—21no—
77C32×C4—22no—
78C32×C4—21no—
79C32×C4—22no—
80C32×C4—42no—
81C32×C4—44no—
82C32×C4—42no—
83C32×C4—42no—
84#125(C3⋊D3)⋊C311yescharacteristic
85#97—92no—
86H(3,𝔽3)#33011yescharacteristic, complemented, Sylow
87C2×C3×C4—22no—
88C2×C3×C4—31no—
89C2×C3×C4—31no—
90C2×C3×C4—32no—
91C2×C3×C4—32no—
92C2×C3×C4—62no—
93C2×C3×C4—62no—
94C2×C3×C4—64no—
95C2×C3×C4—61no—
96C2×C3×C4—62no—
97C2×C3×C4C2×C3×Dic311yescharacteristic
98C2×C3×C4C3×#6712yes—
99C3×C8—184no—
100C3⋊C8—64no—
101C3⋊C8—184no—
102C2×C32—21no—
103C2×C32—22no—
104C2×C32#83611yescharacteristic
105C2×C32#83712yes—
106C2×C32C3×#10111yescharacteristic
107C2×C32C3×#10412yes—
108C2×C8—182no—
109C4⋊C4—22no—
110C4⋊C4(C3⋊Dic3)⋊C312yes—
111C42C2×(C3⋊D3)⋊C311yescharacteristic
112C22×C3—31no—
113C22×C3—31no—
114C22×C3—61no—
115C22×C3C3×#21711yescharacteristic
116C3×C4—21no—
117C3×C4—22no—
118C3×C4—42no—
119C3×C4—62no—
120C3×C4—62no—
121C3×C4—61no—
122C3×C4—61no—
123C3×C4—62no—
124C3×C4—122no—
125C3×C4—122no—
126C3×C4—122no—
127C3×C4—124no—
128C32—21no—
129C32#454311yescharacteristic, complemented
130C32C3×#33011yescharacteristic, complemented
131C2×C4—22no—
132C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
133C2×C4#638⋊C312yes—
134C8—184no—
135C2×C3—31no—
136C2×C3—31no—
137C2×C3—32no—
138C2×C3—32no—
139C2×C3—61no—
140C2×C3—62no—
141C2×C3C3×#83611yescharacteristic
142C2×C3C3×#83712yes—
143C22#3757⋊C311yescharacteristic
144C4—21no—
145C4—22no—
146C4—42no—
147C3—31no—
148C3—31no—
149C3—61no—
150C3C3×#454311yescharacteristic
151C2#63663⋊C311yescharacteristic
152C2#63664⋊C312yes—
subgroupG/H
centerC22#3757⋊C3
commutatorC2×C32×C4C2×C3×C4
FrattiniC3×C42C2×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C3×#217
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC32×C42C22×C3
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#330
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
3. #10589409C3semidirect
86. H(3,𝔽3)#330semidirect
129. C32#4543semidirect
130. C32C3×#330semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#63630⋊C32
#105894093
C3×#45433