Finite Group Databasebeta

← #13136955

Subgroups — #13136955

Order 1728.

208 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 35 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#11721869C211yesmaximal, characteristic
2#11720779C211yesmaximal, characteristic, complemented
3C2×#88410C211yesmaximal, characteristic, complemented
4#41×H(3,𝔽3)C411yescharacteristic
5#88538—21no—
6#88538C411yescharacteristic
7C2×#6180—21no—
8C2×#6180C2211yescharacteristic
9#88410—21no—
10#88410—21no—
11C3×#948—62no—
12Q8×H(3,𝔽3)C811yescharacteristic
13D4×H(3,𝔽3)—21no—
14D4×H(3,𝔽3)C811yescharacteristic
15C2×C4×H(3,𝔽3)—21no—
16C2×C4×H(3,𝔽3)C2×C411yescharacteristic
17#6180—21no—
18#6180D411yescharacteristic
19#6180D411yescharacteristic
20C3×#350—91nomaximal
21C32×#41—22no—
22C3×#208—62no—
23C3×#208—64no—
24C2×C3×C3⋊C8—62no—
25C2×C3×C3⋊C8—64no—
26C22×H(3,𝔽3)—21no—
27C22×H(3,𝔽3)M1611yescharacteristic
28C4×H(3,𝔽3)—21no—
29C4×H(3,𝔽3)C22⋊C411yescharacteristic
30C4×H(3,𝔽3)C2×C811yescharacteristic
31C3×#131—91no—
32#948—62no—
33C3×M32—91no—
34C2×C3×C16—91no—
35C32×Q8—22no—
36C32×D4—24no—
37C32×D4—22no—
38C2×C32×C4—24no—
39C2×C32×C4—22no—
40C3×C3⋊C8—62no—
41C3×C3⋊C8—62no—
42C3×C3⋊C8—64no—
43#350—91noSylow
44C2×H(3,𝔽3)—21no—
45C2×H(3,𝔽3)—41no—
46C2×H(3,𝔽3)C22⋊C811yescharacteristic
47C3×#41—62no—
48C3×#41#15711yescharacteristic
49C3×M16—91no—
50C3×M16—181no—
51C2×C3×C8—91no—
52C2×C3×C8—181no—
53#208—62no—
54#208—64no—
55C2×C3⋊C8—62no—
56C2×C3⋊C8—64no—
57C3×C16—181no—
58C3×C16—181no—
59C22×C32—24no—
60C22×C32—22no—
61C32×C4—24no—
62C32×C4—22no—
63C32×C4—22no—
64#131—91no—
65M32—91no—
66C2×C16—91no—
67H(3,𝔽3)#35011yescharacteristic, complemented, Sylow
68C3×Q8—62no—
69C3×Q8#62211yescharacteristic
70C3×D4—21no—
71C3×D4—64no—
72C3×D4—62no—
73C3×D4#62211yescharacteristic
74C2×C3×C4—21no—
75C2×C3×C4—64no—
76C2×C3×C4—62no—
77C2×C3×C4C2×#15711yescharacteristic
78C3×C8—91no—
79C3×C8—91no—
80C3×C8—181no—
81C3⋊C8—62no—
82C3⋊C8—62no—
83C3⋊C8—64no—
84C2×C32—22no—
85C2×C32—42no—
86C2×C32—44no—
87#41#108811yescharacteristic
88M16—91no—
89M16—181no—
90C2×C8—91no—
91C2×C8—181no—
92C16—181no—
93C16—181no—
94C22×C3—21no—
95C22×C3—64no—
96C22×C3—62no—
97C22×C3#379211yescharacteristic
98C3×C4—21no—
99C3×C4—64no—
100C3×C4—62no—
101C3×C4—62no—
102C3×C4C2×#62211yescharacteristic
103C3×C4C22⋊#15711yescharacteristic
104C32—22no—
105Q8#618811yescharacteristic
106D4—21no—
107D4#618811yescharacteristic
108C2×C4—21no—
109C2×C4C2×#108811yescharacteristic
110C8—91no—
111C8—91no—
112C8—181no—
113C2×C3—21no—
114C2×C3—41no—
115C2×C3—62no—
116C2×C3—122no—
117C2×C3—124no—
118C2×C3C22⋊#62211yescharacteristic
119C22—21no—
120C22#8863111yescharacteristic
121C4—21no—
122C4C2×#618811yescharacteristic
123C4C22⋊#108811yescharacteristic
124C3—62no—
125C3#1059022111yescharacteristic
126C2—21no—
127C2—41no—
128C2C22⋊#618811yescharacteristic
subgroupG/H
centerC3×C4C22⋊#157
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC2×C3×C4C2×#157
Fitting, prime radical#41×H(3,𝔽3)C4
socleC2×C3C22⋊#622
cosocleC2×#6180C22
hypercenterC3×#41#157
hypocenterH(3,𝔽3)#350
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
2. #11720779C2semidirect
3. C2×#88410C2semidirect
67. H(3,𝔽3)#350semidirect

Maximal subgroups

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

subgroupindex
#117218692
#117207792
C2×#884102
C3×#3509