Finite Group Databasebeta

← #11714085

Subgroups — #11714085

Order 768.

374 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C42×SL(2,𝔽3)C211yesmaximal, characteristic
2#73062C213yesmaximal
3#73061C213yesmaximal
4#33083—31nomaximal, Sylow
5C22×#226C2211yescharacteristic
6C2×C4×SL(2,𝔽3)C2213yes—
7C2×#859C2213yes—
8#4937—41nomaximal
9#2863—31no—
10C42×Q8D311yescharacteristic
11#1928—33no—
12#1921—33no—
13#1878—33no—
14#1863—33no—
15#1839—31no—
16C22×SL(2,𝔽3)C2311yescharacteristic
17C2×#226—23no—
18C2×C3×C42—41no—
19C4×SL(2,𝔽3)—23no—
20#859—23no—
21C3⋊#95—43no—
22C3⋊#95—43no—
23C22×Q16—31no—
24C2×C4×Q8C2×D313yes—
25C2×#107—33no—
26C2×#103—33no—
27C2×#103—33no—
28#391—33no—
29#389—33no—
30#387—63no—
31#383—61no—
32C4×C4⋊C4—33no—
33C43—31no—
34#340—33no—
35#340—33no—
36#336—33no—
37#336—33no—
38C22×C3×C4—43no—
39C22×Dic3—41no—
40C22×Dic3—43no—
41C2×SL(2,𝔽3)#4113yes—
42#226—41no—
43C3×C42—44no—
44C22×Q8—31no—
45C22×Q8C22×D311yescharacteristic
46C2×Q16—63no—
47C2×Q16—63no—
48C22×C8—31no—
49C22×C8—33no—
50C4×Q8—23no—
51C4×Q8—63no—
52C2×C4⋊C4—33no—
53C2×C4⋊C4—33no—
54C2×C4⋊C4—63no—
55C2×C42—33no—
56C2×C42—33no—
57C2×C42S411yescharacteristic
58#107—66no—
59#103—66no—
60#103—63no—
61#103—63no—
62#95—61no—
63#95—63no—
64#95—63no—
65#95—63no—
66C23×C3—41no—
67C2×C3×C4—46no—
68C2×C3×C4—83no—
69C2×C3×C4—83no—
70C2×Dic3—83no—
71C2×Dic3—83no—
72C2×Dic3—86no—
73SL(2,𝔽3)#12611yescharacteristic
74C2×Q8—33no—
75C2×Q8—63no—
76C2×Q8—63no—
77C2×Q8#23513yes—
78C22×C4—31no—
79C22×C4—33no—
80C22×C4—61no—
81C22×C4—63no—
82C22×C4C2×S413yes—
83Q16—123no—
84Q16—121no—
85C2×C8—63no—
86C2×C8—63no—
87C2×C8—63no—
88C2×C8—63no—
89C4⋊C4—36no—
90C4⋊C4—63no—
91C4⋊C4—63no—
92C4⋊C4—63no—
93C4⋊C4—126no—
94C42—63no—
95C42—63no—
96C42—123no—
97C42#22614yescomplemented
98C22×C3—43no—
99C22×C3—41no—
100C22×C3—43no—
101C3×C4—86no—
102Dic3—163no—
103Dic3—161no—
104C23C22×S411yescharacteristic
105Q8—33no—
106Q8—61no—
107Q8—121no—
108Q8—123no—
109Q8#87611yescharacteristic
110C2×C4—23no—
111C2×C4—23no—
112C2×C4—33no—
113C2×C4—33no—
114C2×C4—63no—
115C2×C4—63no—
116C2×C4—63no—
117C2×C4—123no—
118C2×C4—123no—
119C2×C4—126no—
120C2×C4C2×#22616yes—
121C8—123no—
122C8—121no—
123C2×C3—41no—
124C2×C3—43no—
125C2×C3—43no—
126C22#591213yes—
127C22C22×#22611yescharacteristic
128C22#592313yes—
129C4—26no—
130C4—31no—
131C4—33no—
132C4—123no—
133C4—241no—
134C4—243no—
135C3—41noSylow
136C2#8529111yescharacteristic
137C2#8532213yes—
138C2#8534913yes—
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
97. C42#226semidirect

Maximal subgroups

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

subgroupindex
C42×SL(2,𝔽3)2
#730622
#730612
#330833
#49374