Finite Group Databasebeta

← #72244170

Subgroups — #72244170

Order 2016.

654 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C24×C3×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×#4268⋊C3C213yesmaximal, complemented
3#66636⋊C3C213yesmaximal, complemented
4#10621436C311yesmaximal, characteristic, complemented
5#10620789—31nomaximal
6C23×C3×ΓL(1,𝔽8)—26no—
7C23×C3×ΓL(1,𝔽8)C2213yes—
8C22×C3⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
9#4268⋊C3—26no—
10C2×C3⋊(ΓL(1,𝔽8)⋊C4)C2213yescomplemented
11C24×C3×C7C2×C311yescharacteristic, complemented
12C24×ΓL(1,𝔽8)—21no—
13C24×ΓL(1,𝔽8)D311yescharacteristic, complemented
14C2×#4268C2×C313yescomplemented
15C2×(C22⋊D7)⋊C3—33no—
16#66636C2×C313yescomplemented
17(C22⋊Dic7)⋊C3—33no—
18C3×#953—71nomaximal, Hall
19C22×C3×ΓL(1,𝔽8)—28no—
20C22×C3×ΓL(1,𝔽8)—26no—
21C22×C3×ΓL(1,𝔽8)D416yescomplemented
22C22×C3×ΓL(1,𝔽8)C2311yescharacteristic
23C2×C3⋊AGL(1,𝔽7)—23no—
24C3⋊(ΓL(1,𝔽8)⋊C4)—23no—
25#6516—31noHall
26C23×C3×C7—26no—
27C23×C3×C7C22×C313yes—
28C22×D21C22×C311yescharacteristic, complemented
29C23×ΓL(1,𝔽8)—212no—
30C23×ΓL(1,𝔽8)—26no—
31C23×ΓL(1,𝔽8)—23no—
32C23×ΓL(1,𝔽8)C2×D313yes—
33C22×AGL(1,𝔽7)—31no—
34#4268—26no—
35C2×Dic21C22×C313yescomplemented
36(C22⋊D7)⋊C3—66no—
37C2×ΓL(1,𝔽8)⋊C4—33no—
38C24×C32—71no—
39C2×C3×#67—73no—
40C3×#217—73no—
41C2×C3×ΓL(1,𝔽8)—26no—
42C2×C3×ΓL(1,𝔽8)C2×D413yes—
43C3⋊AGL(1,𝔽7)—41no—
44C24×C7C3×D311yescharacteristic, complemented
45C2×C22⋊D7—33no—
46C22⋊Dic7—33no—
47C3×#120—211no—
48#953—71no—
49C22×C3×C7—28no—
50C22×C3×C7—26no—
51C22×C3×C7C3×D416yescomplemented
52C22×C3×C7C23×C311yescharacteristic
53C2×D21—23no—
54C22×ΓL(1,𝔽8)—216no—
55C22×ΓL(1,𝔽8)—28no—
56C22×ΓL(1,𝔽8)—212no—
57C22×ΓL(1,𝔽8)—26no—
58C22×ΓL(1,𝔽8)—26no—
59C22×ΓL(1,𝔽8)—21no—
60C22×ΓL(1,𝔽8)#6716yescomplemented
61C22×ΓL(1,𝔽8)C22×D311yescharacteristic
62C2×AGL(1,𝔽7)—63no—
63Dic21—23no—
64ΓL(1,𝔽8)⋊C4—63no—
65C23×C32—73no—
66C23×C32—146no—
67C22×C3×D3—71no—
68C3×#67—146no—
69C2×C3×Dic3—73no—
70C3×ΓL(1,𝔽8)#12011yescharacteristic, complemented, Hall
71C23×C7—26no—
72C23×C7C2×C3×D313yes—
73C22×D7—31no—
74C22⋊D7—66no—
75C2×Dic7—33no—
76C24×C3—71no—
77C24×C3—141no—
78C24×C3AGL(1,𝔽7)11yescharacteristic, complemented
79C2×C3×D4—213no—
80C2×#67—73no—
81C3×C22⋊C4—213no—
82#217—73no—
83C2×C3×C7—26no—
84C2×C3×C7C2×C3×D413yes—
85D21—41no—
86C2×ΓL(1,𝔽8)—212no—
87C2×ΓL(1,𝔽8)—26no—
88C2×ΓL(1,𝔽8)—23no—
89C2×ΓL(1,𝔽8)C2×#6713yes—
90AGL(1,𝔽7)—121no—
91C22×C32—76no—
92C22×C32—71no—
93C22×C32—146no—
94C22×C32—148no—
95C2×C3×D3—143no—
96C3×Dic3—143no—
97#120—211noSylow
98C22×C7—28no—
99C22×C7—26no—
100C22×C7C3×#6716yescomplemented
101C22×C7C22×C3×D311yescharacteristic
102C2×D7—63no—
103Dic7—63no—
104C23×C3—26no—
105C23×C3—73no—
106C23×C3—146no—
107C23×C3—1412no—
108C23×C3—143no—
109C23×C3—211no—
110C23×C3C2×AGL(1,𝔽7)13yes—
111C22×D3—71no—
112C3×D4—426no—
113C2×C3×C4—213no—
114#67—146no—
115C2×Dic3—73no—
116C3×C7C3×#12011yescharacteristic, complemented
117ΓL(1,𝔽8)—21no—
118ΓL(1,𝔽8)#95311yescharacteristic, complemented
119C2×C32—73no—
120C2×C32—146no—
121C3×D3—281no—
122C24C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
123C2×D4—213no—
124C22⋊C4—213no—
125C2×C7—26no—
126C2×C7C2×C3×#6713yes—
127D7—121no—
128C22×C3—28no—
129C22×C3—26no—
130C22×C3—71no—
131C22×C3—76no—
132C22×C3—146no—
133C22×C3—148no—
134C22×C3—1416no—
135C22×C3—1412no—
136C22×C3—141no—
137C22×C3—146no—
138C22×C3—423no—
139C22×C3(C22⋊D7)⋊C316yescomplemented
140C22×C3C22×AGL(1,𝔽7)11yescharacteristic
141C2×D3—143no—
142C3×C4—423no—
143Dic3—143no—
144C32—71noSylow
145C23—26no—
146C23—211no—
147C23C2×C3⋊AGL(1,𝔽7)13yes—
148D4—426no—
149C2×C4—213no—
150C7C3×#95311yescharacteristic, complemented, Sylow
151C2×C3—26no—
152C2×C3—73no—
153C2×C3—146no—
154C2×C3—1412no—
155C2×C3—143no—
156C2×C3—841no—
157C2×C3C2×(C22⋊D7)⋊C313yes—
158D3—281no—
159C22—28no—
160C22—26no—
161C22—423no—
162C22#4268⋊C316yescomplemented
163C22C22×C3⋊AGL(1,𝔽7)11yescharacteristic
164C4—423no—
165C3—71no—
166C3—141no—
167C3#1062078911yescharacteristic, complemented
168C2—26no—
169C2—841no—
170C2C2×#4268⋊C313yes—
subgroupG/H
center, FrattiniC22C22×C3⋊AGL(1,𝔽7)
commutator, socle, cosocleC22×C3×C7C23×C3
FittingC24×C3×C7C2×C3
hypercenterC24C3⋊AGL(1,𝔽7)
hypocenterC3×C7C3×#120
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C3×ΓL(1,𝔽8)C2semidirect
2. C2×#4268⋊C3C2semidirect
3. #66636⋊C3C2semidirect
4. #10621436C3semidirect
8. C22×C3⋊AGL(1,𝔽7)C22semidirect
10. C2×C3⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
11. C24×C3×C7C2×C3semidirect
13. C24×ΓL(1,𝔽8)D3semidirect
14. C2×#4268C2×C3semidirect
16. #66636C2×C3semidirect
21. C22×C3×ΓL(1,𝔽8)D4semidirect
28. C22×D21C22×C3semidirect
35. C2×Dic21C22×C3semidirect
44. C24×C7C3×D3semidirect
51. C22×C3×C7C3×D4semidirect
60. C22×ΓL(1,𝔽8)#67semidirect
70. C3×ΓL(1,𝔽8)#120semidirect
78. C24×C3AGL(1,𝔽7)semidirect
100. C22×C7C3×#67semidirect
116. C3×C7C3×#120semidirect
118. ΓL(1,𝔽8)#953semidirect
122. C24C3⋊AGL(1,𝔽7)semidirect
139. C22×C3(C22⋊D7)⋊C3semidirect
150. C7C3×#953semidirect
162. C22#4268⋊C3semidirect
167. C3#10620789semidirect

Maximal subgroups

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

subgroupindex
C24×C3×ΓL(1,𝔽8)2
C2×#4268⋊C32
#66636⋊C32
#106214363
#106207893
C3×#9537