Finite Group Databasebeta

← ΓL(1,𝔽8)×#2846 #72276514

Subgroups — ΓL(1,𝔽8)×#2846 #72276514

Order 2688.

6844 conjugacy classes of subgroups; only the 540 normal subgroups are shown.

normal subgroups only

This group has 6844 conjugacy classes of subgroups — too many to draw. Only its 540 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
1C26×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×ΓL(1,𝔽8)×#120C217yesmaximal, complemented
3ΓL(1,𝔽8)×#379C217yesmaximal, complemented
4C7×#2846C311yesmaximal, characteristic, complemented, Hall
5C25×ΓL(1,𝔽8)—228no—
6C25×ΓL(1,𝔽8)C2217yes—
7C22×D4×ΓL(1,𝔽8)C2217yescomplemented
8ΓL(1,𝔽8)×#120—228no—
9C2×C22⋊C4×ΓL(1,𝔽8)C22121yescomplemented
10C26×C7C2×C311yescharacteristic, complemented
11C2×C7×#120C2×C317yescomplemented
12C7×#379C2×C317yescomplemented
13C3×#2846—71nomaximal, Hall
14C24×ΓL(1,𝔽8)—2224no—
15C24×ΓL(1,𝔽8)—284no—
16C24×ΓL(1,𝔽8)D4128yescomplemented
17C24×ΓL(1,𝔽8)C2317yes—
18C24×ΓL(1,𝔽8)C2311yescharacteristic, complemented
19C2×D4×ΓL(1,𝔽8)—242no—
20C22×C4×ΓL(1,𝔽8)C2317yescomplemented
21C22⋊C4×ΓL(1,𝔽8)—242no—
22C25×C7—228no—
23C25×C7C22×C317yes—
24C22×C7×D4C22×C317yescomplemented
25C7×#120—228no—
26C2×C7×C22⋊C4C22×C3121yescomplemented
27C26×C3—71no—
28C2×C3×#120—77no—
29C3×#379—77no—
30C23×ΓL(1,𝔽8)—2256no—
31C23×ΓL(1,𝔽8)—2224no—
32C23×ΓL(1,𝔽8)—2168no—
33C23×ΓL(1,𝔽8)—228no—
34C23×ΓL(1,𝔽8)—27no—
35C23×ΓL(1,𝔽8)C2×D4142yes—
36C23×ΓL(1,𝔽8)C2411yescharacteristic
37D4×ΓL(1,𝔽8)—428no—
38C2×C4×ΓL(1,𝔽8)—221no—
39#2846ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
40C24×C7—2224no—
41C24×C7—284no—
42C24×C7C3×D4128yescomplemented
43C24×C7C23×C317yes—
44C24×C7C23×C311yescharacteristic, complemented
45C2×C7×D4—242no—
46C22×C4×C7C23×C317yescomplemented
47C7×C22⋊C4—242no—
48C25×C3—77no—
49C25×C3—1428no—
50C22×C3×D4—77no—
51C3×#120—1428no—
52C2×C3×C22⋊C4—721no—
53C22×ΓL(1,𝔽8)—2224no—
54C22×ΓL(1,𝔽8)—284no—
55C22×ΓL(1,𝔽8)—47no—
56C22×ΓL(1,𝔽8)#120128yescomplemented
57C22×ΓL(1,𝔽8)C22×D417yes—
58C4×ΓL(1,𝔽8)—47no—
59C26C2×ΓL(1,𝔽8)11yescharacteristic, complemented
60C2×#120C2×ΓL(1,𝔽8)17yescomplemented
61#379C2×ΓL(1,𝔽8)17yescomplemented
62C23×C7—2256no—
63C23×C7—2224no—
64C23×C7—2168no—
65C23×C7—228no—
66C23×C7—27no—
67C23×C7C2×C3×D4142yes—
68C23×C7C24×C311yescharacteristic
69C7×D4—428no—
70C2×C4×C7—221no—
71C24×C3—728no—
72C24×C3—77no—
73C24×C3—71no—
74C24×C3—1484no—
75C24×C3—14224no—
76C2×C3×D4—1442no—
77C22×C3×C4—77no—
78C3×C22⋊C4—1442no—
79C2×ΓL(1,𝔽8)—228no—
80C2×ΓL(1,𝔽8)—81no—
81C2×ΓL(1,𝔽8)C2×#12017yes—
82C25—228no—
83C25C22×ΓL(1,𝔽8)17yes—
84C22×D4C22×ΓL(1,𝔽8)17yescomplemented
85#120—228no—
86C2×C22⋊C4C22×ΓL(1,𝔽8)121yescomplemented
87C22×C7—2224no—
88C22×C7—284no—
89C22×C7—47no—
90C22×C7C3×#120128yescomplemented
91C22×C7C22×C3×D417yes—
92C4×C7—47no—
93C23×C3—742no—
94C23×C3—71no—
95C23×C3—1428no—
96C23×C3—14168no—
97C23×C3—147no—
98C23×C3—14224no—
99C23×C3—14256no—
100C3×D4—2828no—
101C2×C3×C4—1421no—
102ΓL(1,𝔽8)#284611yescharacteristic, direct product factor, Hall
103C24—2224no—
104C24—284no—
105C24D4×ΓL(1,𝔽8)128yescomplemented
106C24C23×ΓL(1,𝔽8)17yes—
107C24C23×ΓL(1,𝔽8)11yescharacteristic, complemented
108C2×D4—242no—
109C22×C4C23×ΓL(1,𝔽8)17yescomplemented
110C22⋊C4—242no—
111C2×C7—228no—
112C2×C7—81no—
113C2×C7C2×C3×#12017yes—
114C22×C3—77no—
115C22×C3—728no—
116C22×C3—1484no—
117C22×C3—14224no—
118C22×C3—287no—
119C3×C4—287no—
120C23—2256no—
121C23—2224no—
122C23—228no—
123C23—2168no—
124C23—27no—
125C23C2×D4×ΓL(1,𝔽8)142yes—
126C23C24×ΓL(1,𝔽8)11yescharacteristic
127D4—428no—
128C2×C4—221no—
129C7C3×#284611yescharacteristic, complemented, Sylow
130C2×C3—77no—
131C2×C3—1428no—
132C2×C3—561no—
133C22—2224no—
134C22—284no—
135C22—47no—
136C22ΓL(1,𝔽8)×#120128yescomplemented
137C22C22×D4×ΓL(1,𝔽8)17yes—
138C4—47no—
139C3—71noSylow
140C2—228no—
141C2—81no—
142C2C2×ΓL(1,𝔽8)×#12017yes—
subgroupG/H
center, FrattiniC23C24×ΓL(1,𝔽8)
commutator, socle, cosocleC23×C7C24×C3
FittingC7×#2846C3
hypercenter#2846ΓL(1,𝔽8)
hypocenterC7C3×#2846
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2846

Complements

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

NH ≅ G/Ntype
1. C26×ΓL(1,𝔽8)C2semidirect
2. C2×ΓL(1,𝔽8)×#120C2semidirect
3. ΓL(1,𝔽8)×#379C2semidirect
4. C7×#2846C3semidirect
7. C22×D4×ΓL(1,𝔽8)C22semidirect
9. C2×C22⋊C4×ΓL(1,𝔽8)C22semidirect
10. C26×C7C2×C3semidirect
11. C2×C7×#120C2×C3semidirect
12. C7×#379C2×C3semidirect
16. C24×ΓL(1,𝔽8)D4semidirect
18. C24×ΓL(1,𝔽8)C23semidirect
20. C22×C4×ΓL(1,𝔽8)C23semidirect
24. C22×C7×D4C22×C3semidirect
26. C2×C7×C22⋊C4C22×C3semidirect
39. #2846ΓL(1,𝔽8)direct (N × H)
42. C24×C7C3×D4semidirect
44. C24×C7C23×C3semidirect
46. C22×C4×C7C23×C3semidirect
56. C22×ΓL(1,𝔽8)#120semidirect
59. C26C2×ΓL(1,𝔽8)semidirect
60. C2×#120C2×ΓL(1,𝔽8)semidirect
61. #379C2×ΓL(1,𝔽8)semidirect
84. C22×D4C22×ΓL(1,𝔽8)semidirect
86. C2×C22⋊C4C22×ΓL(1,𝔽8)semidirect
90. C22×C7C3×#120semidirect
102. ΓL(1,𝔽8)#2846direct (N × H)
105. C24D4×ΓL(1,𝔽8)semidirect
107. C24C23×ΓL(1,𝔽8)semidirect
109. C22×C4C23×ΓL(1,𝔽8)semidirect
129. C7C3×#2846semidirect
136. C22ΓL(1,𝔽8)×#120semidirect

Maximal subgroups

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

subgroupindex
C26×ΓL(1,𝔽8)2
C2×ΓL(1,𝔽8)×#1202
ΓL(1,𝔽8)×#3792
C7×#28463
C3×#28467