Finite Group Databasebeta

← #13164676

Subgroups — #13164676

Order 1728.

715 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×#88748C211yesmaximal, characteristic
2#11722451C218yesmaximal, complemented
3#11722445C216yesmaximal, complemented
4C2×Q8×H(3,𝔽3)C411yescharacteristic
5#88748C414yes—
6#88748C2214yes—
7C2×C4⋊#272C413yes—
8C2×C4×#272C2213yes—
9#88631C2214yes—
10#88627C22112yescomplemented
11#88626C22112yescomplemented
12C2×C3×#239—62no—
13C22×#272C2×C413yes—
14Q8×H(3,𝔽3)C2×C414yes—
15C2×C4×H(3,𝔽3)C2×C413yes—
16C2×#1084C2311yescharacteristic
17C4⋊#272C2×C4112yes—
18C4×#272C2×C416yes—
19C4×#272C2316yes—
20#6188C2318yes—
21C3×#568—91nomaximal
22C2×C32×Q8—22no—
23C3×#239—68no—
24C3×#239—68no—
25C2×C3×D12—66no—
26C2×C3×C4×D3—66no—
27C22×H(3,𝔽3)C22×C411yescharacteristic
28C2×#272—43no—
29C2×#272C22×C416yes—
30C4×H(3,𝔽3)C22×C416yes—
31#1084C22×C411yescharacteristic
32#1084C2411yescharacteristic
33C2×C3×#41—91no—
34C2×#239—62no—
35C3×#131—98no—
36C2×C3×M16—96no—
37C22×C3×D3—66no—
38C32×Q8—28no—
39C2×C32×C4—26no—
40C2×C3×Dic3—62no—
41C3×D12—624no—
42C3×C4×D3—612no—
43C3×C4×D3—612no—
44#568—91noSylow
45C2×H(3,𝔽3)—21no—
46C2×H(3,𝔽3)C23×C411yescharacteristic
47#272—26no—
48C3×#41—94no—
49C3×#41—94no—
50C2×C3×Q8—62no—
51C2×C3×Q8#15711yescharacteristic
52C2×C3×D4—93no—
53C22×C3×C4—93no—
54#239—68no—
55#239—68no—
56C2×D12—66no—
57C2×C4×D3—66no—
58C3×M16—912no—
59C3×M16—94no—
60C2×C3×C8—912no—
61C22×C32—22no—
62C2×C3×D3—612no—
63C2×C3×D3—1212no—
64C32×C4—212no—
65C3×Dic3—62no—
66C3×Dic3—62no—
67C2×#41—91no—
68#131—98no—
69C2×M16—96no—
70H(3,𝔽3)#56811yescharacteristic, complemented, Sylow
71C23×C3—93no—
72C22×D3—66no—
73C3×Q8—68no—
74C3×Q8C2×#15714yes—
75C3×D4—912no—
76C2×C3×C4—66no—
77C2×C3×C4—91no—
78C2×C3×C4—96no—
79C2×C3×C4—96no—
80C2×C3×C4C2×#15713yes—
81C2×Dic3—62no—
82D12—624no—
83C4×D3—612no—
84C4×D3—612no—
85C3×C8—98no—
86C2×C32—24no—
87C2×C32—22no—
88C3×D3—1212no—
89#41—94no—
90#41—94no—
91C2×Q8#108811yescharacteristic
92C2×D4—93no—
93C22×C4—93no—
94M16—94no—
95M16—912no—
96C2×C8—912no—
97C22×C3—62no—
98C22×C3—96no—
99C22×C3—363no—
100C22×C3C22×#15711yescharacteristic
101C2×D3—612no—
102C2×D3—1212no—
103C3×C4—612no—
104C3×C4—91no—
105C3×C4—91no—
106C3×C4C22×#15716yes—
107Dic3—62no—
108Dic3—62no—
109C32—22no—
110C23—93no—
111Q8C2×#108814yes—
112D4—912no—
113C2×C4—91no—
114C2×C4—96no—
115C2×C4—96no—
116C2×C4C2×#108813yes—
117C8—98no—
118C2×C3—21no—
119C2×C3—64no—
120C2×C3—62no—
121C2×C3—186no—
122C2×C3C23×#15711yescharacteristic
123D3—1212no—
124C22—96no—
125C22—363no—
126C22C22×#108811yescharacteristic
127C4—91no—
128C4—91no—
129C4C22×#108816yes—
130C3—62no—
131C3#1059664411yescharacteristic
132C2—21no—
133C2—186no—
134C2C23×#108811yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C23×#157
commutatorC2×H(3,𝔽3)C23×C4
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC2×#88748C2
cosocle#1084C24
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#568
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. #11722451C2semidirect
3. #11722445C2semidirect
10. #88627C22semidirect
11. #88626C22semidirect
70. H(3,𝔽3)#568semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
#117224512
#117224452
C3×#5689