Finite Group Databasebeta

← #11711539

Subgroups — #11711539

Order 768.

164 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 37 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×#504⋊C3C211yesmaximal, characteristic
2#73120C212yesmaximal
3#12841—31nomaximal, Sylow
4C2×#131⋊C3C2211yescharacteristic
5#5423—22no—
6#504⋊C3C411yescharacteristic, complemented
7#504⋊C3C411yescharacteristic, complemented
8#4506—41nomaximal
9C2×#504D311yescharacteristic
10C2×#368—31no—
11#2158—31no—
12#1814—32no—
13#1389—32no—
14C2×#41⋊C3D411yescharacteristic
15#131⋊C3C2×C411yescharacteristic
16#131⋊C3Q811yescharacteristic
17GU(2,𝔽3)—42no—
18C2×C3×C16—41no—
19#819—42no—
20C2×#131C2×D311yescharacteristic
21#504—31no—
22#504—31no—
23#504Dic311yescharacteristic
24#504Dic311yescharacteristic
25C2×M32—31no—
26C22×C16—31no—
27C2×#108—32no—
28#427—32no—
29#421—62no—
30#368—61no—
31#368—61no—
32#367—61no—
33#366—61no—
34#344—62no—
35#41⋊C3C4⋊C411yescharacteristic
36#41⋊C3D811yescharacteristic
37C2×SL(2,𝔽3)Q1611yescharacteristic
38C2×C3×C8—41no—
39#208—82no—
40C3×C16—41no—
41C3×C16—41no—
42C2×#41D1211yescharacteristic
43#131—31no—
44#131—31no—
45#131Dic611yescharacteristic
46#131C2×Dic311yescharacteristic
47C2×M16—31no—
48C2×M16—62no—
49C22×C8—31no—
50#117—62no—
51M32—31no—
52M32—31no—
53M32—61no—
54C2×C16—31no—
55C2×C16—31no—
56C2×C16—31no—
57C2×C16—31no—
58C2×C16—31no—
59C2×C16S411yescharacteristic
60#108—62no—
61#108—62no—
62#107—62no—
63#106—62no—
64#104—122no—
65#104—122no—
66C2×C3×C4—41no—
67SL(2,𝔽3)#10711yescharacteristic
68C3×C8—41no—
69C3×C8—41no—
70C3⋊C8—162no—
71#41—31no—
72#41—31no—
73#41D2411yescharacteristic
74#41C4⋊Dic311yescharacteristic
75C2×Q8Dic1211yescharacteristic
76C2×D4—31no—
77C22×C4—31no—
78M16—31no—
79M16—31no—
80M16—61no—
81M16—122no—
82M16—122no—
83C2×C8—31no—
84C2×C8—31no—
85C2×C8—31no—
86C2×C8—31no—
87C2×C8—31no—
88C2×C8—122no—
89C2×C8C2×S411yescharacteristic
90C4⋊C4—122no—
91C22⋊C4—122no—
92C42—122no—
93C16—31no—
94C16—31no—
95C16SL(2,ℤ/4)11yescharacteristic
96C16SL(2,ℤ/4)11yescharacteristic
97C22×C3—41no—
98C3×C4—41no—
99C3×C4—41no—
100C23—31no—
101Q8—31no—
102Q8#81811yescharacteristic
103D4—31no—
104D4—31no—
105D4—61no—
106C2×C4—31no—
107C2×C4—31no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—31no—
111C2×C4—242no—
112C2×C4C4⋊S411yescharacteristic
113C8—31no—
114C8—31no—
115C8—242no—
116C8A4⋊Q811yescharacteristic
117C8C2×SL(2,ℤ/4)11yescharacteristic
118C2×C3—41no—
119C2×C3—81no—
120C22—31no—
121C22—31no—
122C22—121no—
123C22#539811yescharacteristic
124C4—31no—
125C4—31no—
126C4—482no—
127C4#540211yescharacteristic
128C4C4⋊SL(2,ℤ/4)11yescharacteristic
129C3—41noSylow
130C2—21no—
131C2—61no—
132C2—61no—
133C2#7311111yescharacteristic
subgroupG/H
centerC4#5402
commutator#131⋊C3C2×C4
FrattiniC2×C8C2×S4
FittingC2×#504D3
socleC2#73111
prime radicalC2×#41⋊C3D4
cosocleC2×#131⋊C3C22
hypercenterC2×C16S4
hypocenterSL(2,𝔽3)#107
LCSG ▷ #131⋊C3 ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16

Complements

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

NH ≅ G/Ntype
6. #504⋊C3C4semidirect
7. #504⋊C3C4semidirect

Maximal subgroups

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

subgroupindex
C2×#504⋊C32
#731202
#128413
#45064