Finite Group Databasebeta

← #13147366

Subgroups — #13147366

Order 1728.

280 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 47 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#11722449C212yesmaximal, complemented
2#11721794C211yesmaximal, characteristic
3#11720789C212yesmaximal, complemented
4#11720776C211yesmaximal, characteristic, complemented
5#11720773C211yesmaximal, characteristic, complemented
6#88744C2212yes—
7#88631—22no—
8C2×#6188—22no—
9C4×#1088C2211yescharacteristic
10#88627C2212yes—
11#88626C2211yescharacteristic
12D8×H(3,𝔽3)C411yescharacteristic, complemented
13#88546C412yescomplemented
14#88530C411yescharacteristic, complemented
15C8×#272C2211yescharacteristic
16C3×#912—62no—
17C2×#1088—21no—
18D4×H(3,𝔽3)C2×C412yes—
19#6235—22no—
20C2×#1084—22no—
21C4×#272C2311yescharacteristic
22#6229C2×C412yes—
23#6188—22no—
24#6188D412yes—
25C8×H(3,𝔽3)C2×C411yescharacteristic
26#6180C2×C411yescharacteristic
27C3×#444—91nomaximal
28C3×#237—64no—
29C32×D8—22no—
30C3×#214—64no—
31C3×Dic12—62no—
32C3×D3×C8—62no—
33C22×H(3,𝔽3)—22no—
34C2×#272#4111yescharacteristic
35#1088—41no—
36C4×H(3,𝔽3)C22×C411yescharacteristic
37#1084—22no—
38#1084C2×D411yescharacteristic
39C3×#135—91no—
40C3×#131—92no—
41#912—62no—
42C3×#108—91no—
43C3×#104—92no—
44C3×#97—91no—
45C32×D4—24no—
46C3×#67—124no—
47C2×C3×Dic3—124no—
48C3×C4×D3—62no—
49C3×Dic6—64no—
50C32×C8—22no—
51C3×C3⋊C8—62no—
52#444—91noSylow
53C2×H(3,𝔽3)—42no—
54C2×H(3,𝔽3)C4×D411yescharacteristic
55#272—21no—
56C3×#41—92no—
57#237—64no—
58C3×Q16—91no—
59C3×SD16—92no—
60C3×D8—62no—
61C3×D8#15711yescharacteristic
62C3×M16—92no—
63C3×M16—182no—
64C2×C3×C8—91no—
65C2×C3×C8—91no—
66C2×C3×C8—182no—
67C3×C42—91no—
68#214—64no—
69Dic12—62no—
70D3×C8—62no—
71C22×C32—44no—
72C2×C3×D3—62no—
73C32×C4—22no—
74C3×Dic3—62no—
75C3×Dic3—124no—
76#135—91no—
77#131—92no—
78#108—91no—
79#104—92no—
80#97—91no—
81H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
82C3×Q8—92no—
83C3×D4—64no—
84C3×D4—182no—
85C3×D4C2×#15712yes—
86C2×C3×C4—91no—
87C2×C3×C4—182no—
88C2×C3×C4—181no—
89#67—124no—
90C2×Dic3—124no—
91C4×D3—62no—
92Dic6—64no—
93C3×C8—62no—
94C3×C8—92no—
95C3×C8—91no—
96C3×C8—182no—
97C3×C8C2×#15711yescharacteristic
98C3⋊C8—62no—
99C2×C32—22no—
100C2×C32—84no—
101C3×D3—122no—
102#41—92no—
103Q16—91no—
104SD16—92no—
105D8#108811yescharacteristic, complemented
106M16—92no—
107M16—182no—
108C2×C8—91no—
109C2×C8—91no—
110C2×C8—182no—
111C42—91no—
112C22×C3—22no—
113C22×C3—91no—
114C22×C3—124no—
115C2×D3—62no—
116C3×C4—62no—
117C3×C4—91no—
118C3×C4—182no—
119C3×C4—361no—
120C3×C4C22×#15711yescharacteristic
121Dic3—62no—
122Dic3—124no—
123C32—22no—
124Q8—92no—
125D4—182no—
126D4C2×#108812yes—
127C2×C4—91no—
128C2×C4—182no—
129C2×C4—181no—
130C8—92no—
131C8—91no—
132C8—182no—
133C8C2×#108811yescharacteristic
134C2×C3—42no—
135C2×C3—62no—
136C2×C3—181no—
137C2×C3—244no—
138C2×C3D4×#15711yescharacteristic
139D3—122no—
140C22—22no—
141C22—91no—
142C4—91no—
143C4—182no—
144C4—361no—
145C4C22×#108811yescharacteristic
146C3—62no—
147C3#1059378011yescharacteristic
148C2—42no—
149C2—181no—
150C2D4×#108811yescharacteristic
subgroupG/H
center, socleC2×C3D4×#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingD8×H(3,𝔽3)C4
prime radical#11721794C2
cosocleC4×#272C23
hypercenterC3×D8#157
hypocenterH(3,𝔽3)#444
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8

Complements

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

NH ≅ G/Ntype
1. #11722449C2semidirect
3. #11720789C2semidirect
4. #11720776C2semidirect
5. #11720773C2semidirect
12. D8×H(3,𝔽3)C4semidirect
13. #88546C4semidirect
14. #88530C4semidirect
81. H(3,𝔽3)#444semidirect
105. D8#1088semidirect

Maximal subgroups

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

subgroupindex
#117224492
#117217942
#117207892
#117207762
#117207732
C3×#4449