Finite Group Databasebeta

← #13147373

Subgroups — #13147373

Order 1728.

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

normal subgroups only

This group has 283 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#11722451C212yesmaximal, complemented
2#11721808C211yesmaximal, characteristic
3#11720791C212yesmaximal, complemented
4#11720777C211yesmaximal, characteristic, complemented
5C8×#1088C211yesmaximal, characteristic, complemented
6#88748C2212yes—
7C4×#1088C2211yescharacteristic
8#88627—22no—
9#88627C2212yes—
10#88626—22no—
11#88626C2211yescharacteristic
12Q16×H(3,𝔽3)C411yescharacteristic, complemented
13#88549C412yescomplemented
14#88529C411yescharacteristic, complemented
15C8×#272C2211yescharacteristic
16C3×#919—62no—
17C2×#1088—21no—
18Q8×H(3,𝔽3)C2×C412yes—
19C4⋊#272—22no—
20C4⋊#272C2×C412yes—
21C4×#272—22no—
22C4×#272C2311yescharacteristic
23#6188—22no—
24#6188D411yescharacteristic
25#6188D411yescharacteristic
26C8×H(3,𝔽3)C2×C411yescharacteristic
27#6180C2×C411yescharacteristic
28C3×#443—91nomaximal
29C3×#239—64no—
30C32×Q16—22no—
31C3×#215—64no—
32C3×D24—62no—
33C3×D3×C8—62no—
34C2×#272—22no—
35C2×#272#4111yescharacteristic
36#1088—22no—
37C4×H(3,𝔽3)—22no—
38C4×H(3,𝔽3)C22×C411yescharacteristic
39#1084C2×D411yescharacteristic
40C3×#135—91no—
41C3×#131—92no—
42#919—62no—
43C3×#108—91no—
44C3×#104—92no—
45C3×C4×C8—91no—
46C32×Q8—24no—
47C3×D12—64no—
48C3×D12—124no—
49C3×C4×D3—62no—
50C3×C4×D3—124no—
51C32×C8—22no—
52C3×C3⋊C8—62no—
53#443—91noSylow
54C2×H(3,𝔽3)C4×D411yescharacteristic
55#272—21no—
56#272—42no—
57C3×#41—92no—
58#239—64no—
59C3×Q16—62no—
60C3×Q16#15711yescharacteristic
61C3×SD16—92no—
62C3×D8—91no—
63C3×M16—92no—
64C3×M16—182no—
65C2×C3×C8—91no—
66C2×C3×C8—91no—
67C2×C3×C8—182no—
68C3×C42—91no—
69#215—64no—
70D24—62no—
71D3×C8—62no—
72C2×C3×D3—62no—
73C2×C3×D3—124no—
74C32×C4—22no—
75C32×C4—44no—
76C3×Dic3—62no—
77#135—91no—
78#131—92no—
79#108—91no—
80#104—92no—
81C4×C8—91no—
82H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
83C3×Q8—64no—
84C3×Q8C2×#15712yes—
85C3×D4—92no—
86C3×D4—182no—
87C2×C3×C4—91no—
88C2×C3×C4—182no—
89C2×C3×C4—181no—
90D12—64no—
91D12—124no—
92C4×D3—62no—
93C4×D3—124no—
94C3×C8—62no—
95C3×C8—91no—
96C3×C8—91no—
97C3×C8—91no—
98C3×C8—182no—
99C3×C8C2×#15711yescharacteristic
100C3⋊C8—62no—
101C2×C32—22no—
102C3×D3—122no—
103C3×D3—244no—
104#41—92no—
105Q16#108811yescharacteristic, complemented
106SD16—92no—
107D8—91no—
108M16—92no—
109M16—182no—
110C2×C8—91no—
111C2×C8—91no—
112C2×C8—182no—
113C42—91no—
114C22×C3—91no—
115C22×C3—182no—
116C2×D3—62no—
117C2×D3—124no—
118C3×C4—22no—
119C3×C4—62no—
120C3×C4—91no—
121C3×C4—124no—
122C3×C4—182no—
123C3×C4C22×#15711yescharacteristic
124Dic3—62no—
125C32—22no—
126Q8C2×#108812yes—
127D4—92no—
128D4—182no—
129C2×C4—91no—
130C2×C4—182no—
131C2×C4—181no—
132C8—91no—
133C8—91no—
134C8—91no—
135C8—182no—
136C8C2×#108811yescharacteristic
137C2×C3—62no—
138C2×C3—181no—
139C2×C3—362no—
140C2×C3D4×#15711yescharacteristic
141D3—122no—
142D3—244no—
143C22—91no—
144C22—182no—
145C4—22no—
146C4—91no—
147C4—182no—
148C4C22×#108811yescharacteristic
149C3—62no—
150C3#1059378711yescharacteristic
151C2—181no—
152C2—362no—
153C2D4×#108811yescharacteristic
subgroupG/H
center, socleC2×C3D4×#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingQ16×H(3,𝔽3)C4
prime radical#11721808C2
cosocleC4×#272C23
hypercenterC3×Q16#157
hypocenterH(3,𝔽3)#443
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
1. #11722451C2semidirect
3. #11720791C2semidirect
4. #11720777C2semidirect
5. C8×#1088C2semidirect
12. Q16×H(3,𝔽3)C4semidirect
13. #88549C4semidirect
14. #88529C4semidirect
82. H(3,𝔽3)#443semidirect
105. Q16#1088semidirect

Maximal subgroups

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

subgroupindex
#117224512
#117218082
#117207912
#117207772
C8×#10882
C3×#4439