Finite Group Databasebeta

← #379⋊ΓL(1,𝔽8) #13055062

Subgroups — #379⋊ΓL(1,𝔽8) #13055062

Order 1344.

212 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 19 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×C22⋊(C22⋊ΓL(1,𝔽8))C211yesmaximal, characteristic
2C7×#379C311yesmaximal, characteristic, complemented, Hall
3C22⋊(C22⋊ΓL(1,𝔽8))C411yescharacteristic, complemented
4C4×C22⋊ΓL(1,𝔽8)—41nomaximal
5C25×C7C2×C311yescharacteristic
6C2×C7×C22⋊C4—31no—
7C2×C7×C22⋊C4—31no—
8#379⋊C3—71nomaximal, Hall
9C2×C22⋊ΓL(1,𝔽8)—41no—
10C2×C22⋊ΓL(1,𝔽8)—82no—
11C24×C7—31no—
12C24×C7—31no—
13C24×C7—62no—
14C24×C7—62no—
15C24×C7C3×C411yescharacteristic, complemented
16C22×C4×C7—31no—
17C22×C4×C7A411yescharacteristic, complemented
18C7×C22⋊C4—62no—
19C7×C22⋊C4—62no—
20C2×C22⋊A4—71no—
21C22⋊ΓL(1,𝔽8)—41no—
22C22⋊ΓL(1,𝔽8)—82no—
23C4×ΓL(1,𝔽8)—161no—
24#379ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
25C23×C7—22no—
26C23×C7—32no—
27C23×C7—32no—
28C23×C7—31no—
29C23×C7—31no—
30C23×C7—62no—
31C23×C7—62no—
32C23×C7—62no—
33C23×C7—62no—
34C23×C7—62no—
35C23×C7—64no—
36C23×C7—62no—
37C23×C7—62no—
38C23×C7—64no—
39C23×C7C2×A411yescharacteristic
40C2×C4×C7—61no—
41C2×C4×C7—62no—
42C2×C4×C7—61no—
43C22⋊A4—71no—
44C4×A4—281no—
45C2×ΓL(1,𝔽8)—161no—
46C25C2×ΓL(1,𝔽8)11yescharacteristic
47C2×C22⋊C4—31no—
48C2×C22⋊C4—31no—
49C22×C7—22no—
50C22×C7—31no—
51C22×C7—31no—
52C22×C7—32no—
53C22×C7—32no—
54C22×C7—64no—
55C22×C7—62no—
56C22×C7—62no—
57C22×C7—62no—
58C22×C7—62no—
59C22×C7—62no—
60C22×C7—62no—
61C22×C7—64no—
62C22×C7—62no—
63C22×C7C4×A411yescharacteristic
64C4×C7—41no—
65C4×C7—121no—
66C2×A4—281no—
67C2×A4—562no—
68ΓL(1,𝔽8)—161noHall
69C24—31no—
70C24—31no—
71C24—62no—
72C24—62no—
73C24C4×ΓL(1,𝔽8)11yescharacteristic, complemented
74C22×C4—31no—
75C22×C4C22⋊ΓL(1,𝔽8)11yescharacteristic, complemented
76C22⋊C4—62no—
77C22⋊C4—62no—
78C2×C7—31no—
79C2×C7—31no—
80C2×C7—62no—
81C2×C7—62no—
82C2×C7#120⋊C311yescharacteristic
83A4—281no—
84A4—562no—
85C3×C4—1121no—
86C23—22no—
87C23—32no—
88C23—32no—
89C23—31no—
90C23—31no—
91C23—62no—
92C23—62no—
93C23—62no—
94C23—62no—
95C23—62no—
96C23—64no—
97C23—62no—
98C23—62no—
99C23—64no—
100C23C2×C22⋊ΓL(1,𝔽8)11yescharacteristic
101C2×C4—61no—
102C2×C4—62no—
103C2×C4—61no—
104C7#379⋊C311yescharacteristic, complemented, Sylow
105C2×C3—1121no—
106C22—22no—
107C22—31no—
108C22—31no—
109C22—32no—
110C22—32no—
111C22—64no—
112C22—62no—
113C22—62no—
114C22—62no—
115C22—62no—
116C22—62no—
117C22—62no—
118C22—64no—
119C22—62no—
120C22C4×C22⋊ΓL(1,𝔽8)11yescharacteristic
121C4—41no—
122C4—121no—
123C3—1121noSylow
124C2—31no—
125C2—31no—
126C2—62no—
127C2—62no—
128C2#120⋊ΓL(1,𝔽8)11yescharacteristic
subgroupG/H
center, hypercenterC2#120⋊ΓL(1,𝔽8)
commutator, hypocenterC24×C7C3×C4
FrattiniC23C2×C22⋊ΓL(1,𝔽8)
FittingC7×#379C3
socleC23×C7C2×A4
prime radicalC2×C22⋊(C22⋊ΓL(1,𝔽8))C2
cosocleC25×C7C2×C3
LCSG ▷ C24×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C7×#379C3semidirect
3. C22⋊(C22⋊ΓL(1,𝔽8))C4semidirect
15. C24×C7C3×C4semidirect
17. C22×C4×C7A4semidirect
24. #379ΓL(1,𝔽8)semidirect
73. C24C4×ΓL(1,𝔽8)semidirect
75. C22×C4C22⋊ΓL(1,𝔽8)semidirect
104. C7#379⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C22⋊ΓL(1,𝔽8))2
C7×#3793
C4×C22⋊ΓL(1,𝔽8)4
#379⋊C37