Finite Group Databasebeta

← #13147374

Subgroups — #13147374

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#11722451C212yesmaximal, complemented
2#11721808C211yesmaximal, characteristic
3#11720791C212yesmaximal, complemented
4#11720776C211yesmaximal, characteristic, complemented
5#11720773C211yesmaximal, 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#6188D412yes—
25C8×H(3,𝔽3)C2×C411yescharacteristic
26#6180C2×C411yescharacteristic
27C3×#444—91nomaximal
28C3×#239—64no—
29C32×Q16—22no—
30C3×#215—64no—
31C3×D24—62no—
32C3×D3×C8—62no—
33C2×#272—22no—
34C2×#272#4111yescharacteristic
35#1088—41no—
36C4×H(3,𝔽3)—22no—
37C4×H(3,𝔽3)C22×C411yescharacteristic
38#1084C2×D411yescharacteristic
39C3×#135—91no—
40C3×#131—92no—
41#919—62no—
42C3×#108—91no—
43C3×#104—92no—
44C3×#97—91no—
45C32×Q8—24no—
46C3×D12—64no—
47C3×D12—124no—
48C3×C4×D3—62no—
49C3×C4×D3—124no—
50C32×C8—22no—
51C3×C3⋊C8—62no—
52#444—91noSylow
53C2×H(3,𝔽3)C4×D411yescharacteristic
54#272—21no—
55#272—42no—
56C3×#41—92no—
57#239—64no—
58C3×Q16—62no—
59C3×Q16#15711yescharacteristic
60C3×SD16—92no—
61C3×D8—91no—
62C3×M16—92no—
63C3×M16—182no—
64C2×C3×C8—91no—
65C2×C3×C8—91no—
66C2×C3×C8—182no—
67C3×C42—91no—
68#215—64no—
69D24—62no—
70D3×C8—62no—
71C2×C3×D3—62no—
72C2×C3×D3—124no—
73C32×C4—22no—
74C32×C4—44no—
75C3×Dic3—62no—
76#135—91no—
77#131—92no—
78#108—91no—
79#104—92no—
80#97—91no—
81H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
82C3×Q8—64no—
83C3×Q8C2×#15712yes—
84C3×D4—92no—
85C3×D4—182no—
86C2×C3×C4—91no—
87C2×C3×C4—182no—
88C2×C3×C4—181no—
89D12—64no—
90D12—124no—
91C4×D3—62no—
92C4×D3—124no—
93C3×C8—62no—
94C3×C8—92no—
95C3×C8—91no—
96C3×C8—182no—
97C3×C8C2×#15711yescharacteristic
98C3⋊C8—62no—
99C2×C32—22no—
100C3×D3—122no—
101C3×D3—244no—
102#41—92no—
103Q16#108811yescharacteristic, complemented
104SD16—92no—
105D8—91no—
106M16—92no—
107M16—182no—
108C2×C8—91no—
109C2×C8—91no—
110C2×C8—182no—
111C42—91no—
112C22×C3—91no—
113C22×C3—182no—
114C2×D3—62no—
115C2×D3—124no—
116C3×C4—22no—
117C3×C4—62no—
118C3×C4—91no—
119C3×C4—124no—
120C3×C4—361no—
121C3×C4C22×#15711yescharacteristic
122Dic3—62no—
123C32—22no—
124Q8C2×#108812yes—
125D4—92no—
126D4—182no—
127C2×C4—91no—
128C2×C4—182no—
129C2×C4—181no—
130C8—92no—
131C8—91no—
132C8—182no—
133C8C2×#108811yescharacteristic
134C2×C3—62no—
135C2×C3—181no—
136C2×C3—362no—
137C2×C3D4×#15711yescharacteristic
138D3—122no—
139D3—244no—
140C22—91no—
141C22—182no—
142C4—22no—
143C4—91no—
144C4—361no—
145C4C22×#108811yescharacteristic
146C3—62no—
147C3#1059378811yescharacteristic
148C2—181no—
149C2—362no—
150C2D4×#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)#444
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. #11720776C2semidirect
5. #11720773C2semidirect
12. Q16×H(3,𝔽3)C4semidirect
13. #88549C4semidirect
14. #88529C4semidirect
81. H(3,𝔽3)#444semidirect
103. 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
#117207762
#117207732
C3×#4449