Finite Group Databasebeta

← #89861⋊C3 #13051991

Subgroups — #89861⋊C3 #13051991

Order 1344.

200 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 64 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
1ΓL(1,𝔽8)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#6438⋊C3C211yesmaximal, characteristic
3#6384⋊C3C212yesmaximal, complemented
4#6383⋊C3C212yesmaximal
5#6377⋊C3C211yesmaximal, characteristic, complemented
6#89861C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)—21no—
8C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
9C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
10C4⋊C4×ΓL(1,𝔽8)—21no—
11C4⋊C4×ΓL(1,𝔽8)C2212yes—
12C42×ΓL(1,𝔽8)C2211yescharacteristic
13(C2×AGL(1,𝔽7))⋊C4—21no—
14(C2×AGL(1,𝔽7))⋊C4—21no—
15C2×ΓL(1,𝔽8)⋊C8C2212yes—
16C7×C4⋊Q8C2×C311yescharacteristic, complemented
17#6438C2×C311yescharacteristic
18#6384C2×C312yescomplemented
19#6383C2×C312yes—
20#6377C2×C311yescharacteristic, complemented
21C3×#489—71nomaximal, Hall
22C22×AGL(1,𝔽7)—21no—
23Q8×ΓL(1,𝔽8)—42no—
24C2×C4×ΓL(1,𝔽8)—22no—
25C2×C4×ΓL(1,𝔽8)D411yescharacteristic
26C2×C4×ΓL(1,𝔽8)D411yescharacteristic
27C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
28C2×ΓL(1,𝔽8)⋊C4—21no—
29C4⋊AGL(1,𝔽7)—22no—
30ΓL(1,𝔽8)⋊Q8—22no—
31ΓL(1,𝔽8)⋊C8—22no—
32C2×C7×Q8—21no—
33C2×D28C22×C311yescharacteristic
34C2×Dic14C22×C311yescharacteristic
35C7×C4⋊C4—21no—
36C7×C4⋊C4C22×C312yes—
37C42×C7C22×C311yescharacteristic
38(C2×D7)⋊C4—21no—
39(C2×D7)⋊C4—21no—
40C2×C7⋊C8C22×C312yes—
41C3×C4⋊Q8—71no—
42C3×#124—71no—
43C3×#103—72no—
44C3×#102—72no—
45C3×#97—71no—
46C22×ΓL(1,𝔽8)C2×D411yescharacteristic
47C2×AGL(1,𝔽7)—41no—
48C2×AGL(1,𝔽7)—41no—
49C2×AGL(1,𝔽7)—41no—
50C4×ΓL(1,𝔽8)—21no—
51C4×ΓL(1,𝔽8)—21no—
52C4×ΓL(1,𝔽8)—42no—
53C4×ΓL(1,𝔽8)#4112yes—
54ΓL(1,𝔽8)⋊C4—41no—
55#489—71noSylow
56C22×D7—21no—
57C7×Q8—42no—
58C2×C4×C7—22no—
59C2×C4×C7C3×D411yescharacteristic
60C2×C4×C7C3×D411yescharacteristic
61C2×C4×C7C23×C311yescharacteristic
62C2×Dic7—21no—
63D28—22no—
64Dic14—22no—
65C7⋊C8—22no—
66C2×C3×Q8—71no—
67C2×C3×Q8—141no—
68C2×C3×D4—71no—
69C2×C3×C8—72no—
70C3×C4⋊C4—72no—
71C3×C4⋊C4—141no—
72C3×C22⋊C4—141no—
73C3×C22⋊C4—141no—
74C3×C42—71no—
75C2×ΓL(1,𝔽8)#12411yescharacteristic
76C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
77C2×ΓL(1,𝔽8)#13711yescharacteristic
78AGL(1,𝔽7)—81no—
79C4⋊Q8AGL(1,𝔽7)11yescharacteristic, complemented
80#124—71no—
81#103—72no—
82#102—72no—
83#97—71no—
84C22×C7C2×C3×D411yescharacteristic
85C2×D7—41no—
86C2×D7—41no—
87C2×D7—41no—
88C4×C7—21no—
89C4×C7—21no—
90C4×C7—42no—
91C4×C7C3×#4112yes—
92Dic7—41no—
93C23×C3—141no—
94C3×Q8—142no—
95C3×Q8—282no—
96C3×D4—142no—
97C2×C3×C4—71no—
98C2×C3×C4—71no—
99C2×C3×C4—71no—
100C2×C3×C4—142no—
101C2×C3×C4—141no—
102C3×C8—142no—
103ΓL(1,𝔽8)#48911yescharacteristic, complemented, Hall
104C2×Q8—21no—
105C2×Q8—71no—
106C2×D4—71no—
107C2×C8—72no—
108C4⋊C4—21no—
109C4⋊C4C2×AGL(1,𝔽7)12yes—
110C22⋊C4—141no—
111C22⋊C4—141no—
112C42C2×AGL(1,𝔽7)11yescharacteristic
113C2×C7C3×#12411yescharacteristic
114C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
115C2×C7C3×#13711yescharacteristic
116D7—81no—
117C22×C3—71no—
118C22×C3—281no—
119C22×C3—281no—
120C22×C3—281no—
121C3×C4—72no—
122C3×C4—141no—
123C3×C4—141no—
124C3×C4—282no—
125C3×C4—281no—
126C23—141no—
127Q8—42no—
128Q8—142no—
129D4—142no—
130C2×C4—22no—
131C2×C4—141no—
132C2×C4(C22⋊D7)⋊C311yescharacteristic
133C2×C4(C22⋊D7)⋊C311yescharacteristic
134C2×C4C22×AGL(1,𝔽7)11yescharacteristic
135C8—142no—
136C7C3×#48911yescharacteristic, complemented, Sylow
137C2×C3—71no—
138C2×C3—71no—
139C2×C3—71no—
140C2×C3—561no—
141C22—281no—
142C22—281no—
143C22—281no—
144C22C2×(C22⋊D7)⋊C311yescharacteristic
145C4—21no—
146C4—21no—
147C4—42no—
148C4—281no—
149C4#6665912yes—
150C3—71noSylow
151C2—561no—
152C2#6495⋊C311yescharacteristic
153C2#6505⋊C311yescharacteristic
154C2#6510⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C22⋊D7)⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C4⋊Q8C2×C3
socleC22×C7C2×C3×D4
prime radicalC2×C4⋊AGL(1,𝔽7)C22
hypercenterC4⋊Q8AGL(1,𝔽7)
hypocenterC7C3×#489
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×C4⋊Q8C2semidirect
3. #6384⋊C3C2semidirect
5. #6377⋊C3C2semidirect
6. #89861C3semidirect
16. C7×C4⋊Q8C2×C3semidirect
18. #6384C2×C3semidirect
20. #6377C2×C3semidirect
79. C4⋊Q8AGL(1,𝔽7)semidirect
103. ΓL(1,𝔽8)#489semidirect
136. C7C3×#489semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×C4⋊Q82
#6438⋊C32
#6384⋊C32
#6383⋊C32
#6377⋊C32
#898613
C3×#4897