Finite Group Databasebeta

← #89507⋊C3 #13051635

Subgroups — #89507⋊C3 #13051635

Order 1344.

228 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#6493⋊C3C211yesmaximal, characteristic
2C2×C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×C22⋊C8C211yesmaximal, characteristic, complemented
4#6395⋊C3C212yesmaximal, complemented
5#6391⋊C3C212yesmaximal, complemented
6#89507C311yesmaximal, characteristic, complemented, Hall
7C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
8C2×(C22⋊D7)⋊C3—21no—
9C22×ΓL(1,𝔽8)⋊C4—21no—
10C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
11C2×C8×ΓL(1,𝔽8)C2212yes—
12(C2×AGL(1,𝔽7))⋊C4—21no—
13C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
14C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
15C4⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
16#6493C2×C311yescharacteristic
17C2×C4⋊Dic7C2×C311yescharacteristic, complemented
18C7×C22⋊C8C2×C311yescharacteristic, complemented
19#6395C2×C312yescomplemented
20#6391C2×C312yescomplemented
21C3×#481—71nomaximal, Hall
22C23×ΓL(1,𝔽8)D411yescharacteristic
23C22×AGL(1,𝔽7)—21no—
24C2×C4×ΓL(1,𝔽8)—22no—
25C2×C4×ΓL(1,𝔽8)D411yescharacteristic
26C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
27(C22⋊D7)⋊C3—42no—
28C2×ΓL(1,𝔽8)⋊C4—22no—
29C2×ΓL(1,𝔽8)⋊C4—21no—
30C2×ΓL(1,𝔽8)⋊C4—42no—
31C4⋊AGL(1,𝔽7)—22no—
32C8×ΓL(1,𝔽8)—22no—
33C22×C4×C7C22×C311yescharacteristic
34C2×C22⋊D7—21no—
35C22×Dic7—21no—
36C2×D28C22×C311yescharacteristic
37C2×C7×C8C22×C312yes—
38(C2×D7)⋊C4—21no—
39C4⋊Dic7—21no—
40C4⋊Dic7C22×C312yes—
41C4⋊Dic7C22×C311yescharacteristic
42C3×#121—71no—
43C2×C3×C4⋊C4—71no—
44C3×#106—72no—
45C3×#102—72no—
46C3×C22⋊C8—71no—
47C22×ΓL(1,𝔽8)—21no—
48C22×ΓL(1,𝔽8)—21no—
49C22×ΓL(1,𝔽8)SD1612yes—
50C22×ΓL(1,𝔽8)C2×D411yescharacteristic
51C2×AGL(1,𝔽7)—41no—
52C2×AGL(1,𝔽7)—41no—
53C2×AGL(1,𝔽7)—41no—
54C4×ΓL(1,𝔽8)—21no—
55C4×ΓL(1,𝔽8)#4112yes—
56ΓL(1,𝔽8)⋊C4—41no—
57ΓL(1,𝔽8)⋊C4—42no—
58#481—71noSylow
59C23×C7C3×D411yescharacteristic
60C22×D7—21no—
61C2×C4×C7—22no—
62C2×C4×C7C3×D411yescharacteristic
63C2×C4×C7C23×C311yescharacteristic
64C22⋊D7—42no—
65C2×Dic7—22no—
66C2×Dic7—21no—
67C2×Dic7—42no—
68D28—22no—
69C7×C8—22no—
70C2×C3×D4—71no—
71C2×C3×D4—141no—
72C22×C3×C4—71no—
73C22×C3×C4—141no—
74C2×C3×C8—72no—
75C3×C4⋊C4—72no—
76C3×C4⋊C4—71no—
77C3×C4⋊C4—141no—
78C3×C22⋊C4—141no—
79C2×ΓL(1,𝔽8)—22no—
80C2×ΓL(1,𝔽8)#12311yescharacteristic
81C2×ΓL(1,𝔽8)C2×SD1611yescharacteristic
82C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
83AGL(1,𝔽7)—81no—
84#121—71no—
85C2×C4⋊C4—71no—
86#106—72no—
87#102—72no—
88C22⋊C8AGL(1,𝔽7)11yescharacteristic, complemented
89C22×C7—21no—
90C22×C7—21no—
91C22×C7C3×SD1612yes—
92C22×C7C2×C3×D411yescharacteristic
93C2×D7—41no—
94C2×D7—41no—
95C2×D7—41no—
96C4×C7—21no—
97C4×C7C3×#4112yes—
98Dic7—41no—
99Dic7—42no—
100C23×C3—71no—
101C23×C3—141no—
102C3×D4—142no—
103C3×D4—282no—
104C2×C3×C4—71no—
105C2×C3×C4—71no—
106C2×C3×C4—142no—
107C2×C3×C4—141no—
108C2×C3×C4—142no—
109C2×C3×C4—282no—
110C3×C8—142no—
111ΓL(1,𝔽8)#48111yescharacteristic, complemented, Hall
112C2×D4—71no—
113C2×D4—141no—
114C22×C4—141no—
115C22×C4C2×AGL(1,𝔽7)11yescharacteristic
116C2×C8C2×AGL(1,𝔽7)12yes—
117C4⋊C4—72no—
118C4⋊C4—71no—
119C4⋊C4—141no—
120C22⋊C4—141no—
121C2×C7—22no—
122C2×C7C3×#12311yescharacteristic
123C2×C7C2×C3×SD1611yescharacteristic
124C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
125D7—81no—
126C22×C3—71no—
127C22×C3—72no—
128C22×C3—141no—
129C22×C3—141no—
130C22×C3—281no—
131C22×C3—281no—
132C22×C3—281no—
133C3×C4—72no—
134C3×C4—141no—
135C3×C4—282no—
136C3×C4—281no—
137C23—141no—
138C23C4⋊AGL(1,𝔽7)11yescharacteristic
139D4—142no—
140D4—282no—
141C2×C4—22no—
142C2×C4—141no—
143C2×C4—142no—
144C2×C4—282no—
145C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
146C2×C4C22×AGL(1,𝔽7)11yescharacteristic
147C8—22no—
148C7C3×#48111yescharacteristic, complemented, Sylow
149C2×C3—71no—
150C2×C3—71no—
151C2×C3—71no—
152C2×C3—142no—
153C2×C3—561no—
154C22—21no—
155C22—21no—
156C22—281no—
157C22—281no—
158C22—281no—
159C22#1132⋊C312yes—
160C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
161C4—21no—
162C4—281no—
163C4—282no—
164C4#1159⋊C312yes—
165C3—71noSylow
166C2—22no—
167C2—561no—
168C2#6449⋊C311yescharacteristic
169C2C2×#1132⋊C311yescharacteristic
170C2#6471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C22⋊C8C2×C3
socleC22×C7C2×C3×D4
prime radical#6493⋊C3C2
hypercenterC22⋊C8AGL(1,𝔽7)
hypocenterC7C3×#481
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
3. ΓL(1,𝔽8)×C22⋊C8C2semidirect
4. #6395⋊C3C2semidirect
5. #6391⋊C3C2semidirect
6. #89507C3semidirect
17. C2×C4⋊Dic7C2×C3semidirect
18. C7×C22⋊C8C2×C3semidirect
19. #6395C2×C3semidirect
20. #6391C2×C3semidirect
88. C22⋊C8AGL(1,𝔽7)semidirect
111. ΓL(1,𝔽8)#481semidirect
148. C7C3×#481semidirect

Maximal subgroups

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

subgroupindex
#6493⋊C32
C2×C4⋊(ΓL(1,𝔽8)⋊C4)2
ΓL(1,𝔽8)×C22⋊C82
#6395⋊C32
#6391⋊C32
#895073
C3×#4817