Finite Group Databasebeta

← #13147372

Subgroups — #13147372

Order 1728.

282 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 49 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
1Q8×#1088C212yesmaximal
2Q16×#272C211yesmaximal, characteristic
3#11720790C212yesmaximal
4#11720774C211yesmaximal, characteristic
5#11720773C211yesmaximal, characteristic
6Q8×#272C2212yes—
7C4⋊#1088—22no—
8C4⋊#1088C2212yes—
9C4×#1088—22no—
10C4×#1088C2211yescharacteristic
11#88626C2211yescharacteristic
12Q16×H(3,𝔽3)C411yescharacteristic, complemented
13#88550C412yescomplemented
14#88530C411yescharacteristic, complemented
15C8×#272C2211yescharacteristic
16C3×D3×Q16—62no—
17C2×#1088—22no—
18C2×#1088D412yes—
19Q8×H(3,𝔽3)C2×C412yes—
20C4×#272—22no—
21C4×#272C2311yescharacteristic
22#6229—22no—
23#6229C2×C412yes—
24#6188—21no—
25C8×H(3,𝔽3)C2×C411yescharacteristic
26#6180C2×C411yescharacteristic
27C3×#441—91nomaximal
28C3×D3×Q8—64no—
29C32×Q16—22no—
30C3×#216—64no—
31C3×Dic12—62no—
32C3×D3×C8—62no—
33C2×#272C2×D411yescharacteristic
34#1088—22no—
35#1088—42no—
36C4×H(3,𝔽3)—22no—
37C4×H(3,𝔽3)C22×C411yescharacteristic
38#1084—22no—
39#1084#4111yescharacteristic
40C2×C3×Q16—91no—
41C3×C4×Q8—92no—
42D3×Q16—62no—
43C3×#106—91no—
44C3×#103—92no—
45C3×#97—91no—
46C32×Q8—24no—
47C3×C4×D3—62no—
48C3×C4×D3—124no—
49C3×Dic6—64no—
50C3×Dic6—124no—
51C32×C8—22no—
52C3×C3⋊C8—62no—
53#441—91noSylow
54C2×H(3,𝔽3)C4×D411yescharacteristic
55#272#13712yes—
56C2×C3×Q8—92no—
57D3×Q8—64no—
58C3×Q16—62no—
59C3×Q16—92no—
60C3×Q16—91no—
61C3×Q16#15711yescharacteristic
62C2×C3×C8—91no—
63C2×C3×C8—91no—
64C3×C4⋊C4—92no—
65C3×C4⋊C4—182no—
66C3×C42—91no—
67C3×C42—182no—
68#216—64no—
69Dic12—62no—
70D3×C8—62no—
71C2×C3×D3—62no—
72C32×C4—22no—
73C32×C4—44no—
74C3×Dic3—62no—
75C3×Dic3—124no—
76C2×Q16—91no—
77C4×Q8—92no—
78#106—91no—
79#103—92no—
80#97—91no—
81H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
82C3×Q8—64no—
83C3×Q8—92no—
84C3×Q8—182no—
85C3×Q8C2×#15712yes—
86C2×C3×C4—91no—
87C2×C3×C4—92no—
88C2×C3×C4—182no—
89C2×C3×C4—182no—
90C4×D3—62no—
91C4×D3—124no—
92Dic6—64no—
93Dic6—124no—
94C3×C8—62no—
95C3×C8—91no—
96C3×C8—181no—
97C3×C8C2×#15711yescharacteristic
98C3⋊C8—62no—
99C2×C32—22no—
100C3×D3—64no—
101C2×Q8—92no—
102Q16—92no—
103Q16—91no—
104Q16#108811yescharacteristic, complemented
105C2×C8—91no—
106C2×C8—91no—
107C4⋊C4—92no—
108C4⋊C4—182no—
109C42—91no—
110C42—182no—
111C22×C3—91no—
112C2×D3—62no—
113C3×C4—22no—
114C3×C4—62no—
115C3×C4—91no—
116C3×C4—124no—
117C3×C4—182no—
118C3×C4—182no—
119C3×C4—362no—
120C3×C4C22×#15711yescharacteristic
121Dic3—62no—
122Dic3—124no—
123C32—22no—
124Q8—92no—
125Q8—182no—
126Q8C2×#108812yes—
127C2×C4—91no—
128C2×C4—92no—
129C2×C4—182no—
130C2×C4—182no—
131C8—91no—
132C8—181no—
133C8C2×#108811yescharacteristic
134C2×C3—62no—
135C2×C3—92no—
136C2×C3D4×#15711yescharacteristic
137D3—64no—
138C22—91no—
139C4—22no—
140C4—91no—
141C4—182no—
142C4—182no—
143C4—362no—
144C4C22×#108811yescharacteristic
145C3—62no—
146C3#1059378611yescharacteristic
147C2—92no—
148C2D4×#108811yescharacteristic
subgroupG/H
center, socleC2×C3D4×#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingQ16×H(3,𝔽3)C4
prime radicalC2×#272C2×D4
cosocleC4×#272C23
hypercenterC3×Q16#157
hypocenterH(3,𝔽3)#441
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q16

Complements

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

NH ≅ G/Ntype
12. Q16×H(3,𝔽3)C4semidirect
13. #88550C4semidirect
14. #88530C4semidirect
81. H(3,𝔽3)#441semidirect
104. Q16#1088semidirect

Maximal subgroups

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

subgroupindex
Q8×#10882
Q16×#2722
#117207902
#117207742
#117207732
C3×#4419