Finite Group Databasebeta

← #89494⋊C3 #13051623

Subgroups — #89494⋊C3 #13051623

Order 1344.

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

normal subgroups only

This group has 208 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
1C2×C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic
2#6487⋊C3C211yesmaximal, characteristic
3ΓL(1,𝔽8)×C22⋊C8C211yesmaximal, characteristic
4#6392⋊C3C212yesmaximal, complemented
5#6390⋊C3C212yesmaximal, complemented
6#89494C311yesmaximal, characteristic, complemented, Hall
7C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
8C22×ΓL(1,𝔽8)⋊C4—21no—
9C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
10C2×C8×ΓL(1,𝔽8)C2212yes—
11(C22⋊Dic7)⋊C3—21no—
12C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
13C4⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
14C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
15(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
16C2×C4⋊Dic7C2×C311yescharacteristic
17#6487C2×C311yescharacteristic
18C7×C22⋊C8C2×C311yescharacteristic
19#6392C2×C312yescomplemented
20#6390C2×C312yescomplemented
21C3×#484—71nomaximal, Hall
22C23×ΓL(1,𝔽8)D411yescharacteristic
23C2×C4×ΓL(1,𝔽8)—22no—
24C2×C4×ΓL(1,𝔽8)D411yescharacteristic
25C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
26C2×ΓL(1,𝔽8)⋊C4—21no—
27C2×ΓL(1,𝔽8)⋊C4—22no—
28C2×ΓL(1,𝔽8)⋊C4—21no—
29C2×ΓL(1,𝔽8)⋊C4—42no—
30ΓL(1,𝔽8)⋊Q8—22no—
31C8×ΓL(1,𝔽8)—22no—
32C22×C4×C7C22×C311yescharacteristic
33C22×Dic7—21no—
34C2×Dic14C22×C311yescharacteristic
35C2×C7×C8C22×C312yes—
36C22⋊Dic7—21no—
37C4⋊Dic7—21no—
38C4⋊Dic7C22×C311yescharacteristic
39C4⋊Dic7C22×C312yes—
40Dic7⋊C4—21no—
41C3×C22⋊Q8—71no—
42C2×C3×C4⋊C4—71no—
43C3×#107—72no—
44C3×#103—72no—
45C3×C22⋊C8—71no—
46C22×ΓL(1,𝔽8)—21no—
47C22×ΓL(1,𝔽8)—21no—
48C22×ΓL(1,𝔽8)Q1612yes—
49C22×ΓL(1,𝔽8)C2×D411yescharacteristic
50C4×ΓL(1,𝔽8)—21no—
51C4×ΓL(1,𝔽8)#4112yes—
52ΓL(1,𝔽8)⋊C4—41no—
53ΓL(1,𝔽8)⋊C4—41no—
54ΓL(1,𝔽8)⋊C4—42no—
55#484—71noSylow
56C23×C7C3×D411yescharacteristic
57C2×C4×C7—22no—
58C2×C4×C7C3×D411yescharacteristic
59C2×C4×C7C23×C311yescharacteristic
60C2×Dic7—21no—
61C2×Dic7—21no—
62C2×Dic7—22no—
63C2×Dic7—42no—
64Dic14—22no—
65C7×C8—22no—
66C2×C3×Q8—71no—
67C22×C3×C4—71no—
68C22×C3×C4—141no—
69C2×C3×C8—72no—
70C3×C4⋊C4—72no—
71C3×C4⋊C4—71no—
72C3×C4⋊C4—141no—
73C3×C4⋊C4—141no—
74C3×C22⋊C4—141no—
75C2×ΓL(1,𝔽8)—22no—
76C2×ΓL(1,𝔽8)#12311yescharacteristic
77C2×ΓL(1,𝔽8)C2×Q1611yescharacteristic
78C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
79C22⋊Q8—71no—
80C2×C4⋊C4—71no—
81#107—72no—
82#103—72no—
83C22⋊C8AGL(1,𝔽7)11yescharacteristic
84C22×C7—21no—
85C22×C7—21no—
86C22×C7C3×Q1612yes—
87C22×C7C2×C3×D411yescharacteristic
88C4×C7—21no—
89C4×C7C3×#4112yes—
90Dic7—41no—
91Dic7—42no—
92Dic7—41no—
93C23×C3—71no—
94C3×Q8—142no—
95C2×C3×C4—71no—
96C2×C3×C4—71no—
97C2×C3×C4—142no—
98C2×C3×C4—141no—
99C2×C3×C4—141no—
100C2×C3×C4—142no—
101C2×C3×C4—282no—
102C3×C8—142no—
103ΓL(1,𝔽8)#48411yescharacteristic, complemented, Hall
104C2×Q8—71no—
105C22×C4—141no—
106C22×C4C2×AGL(1,𝔽7)11yescharacteristic
107C2×C8C2×AGL(1,𝔽7)12yes—
108C4⋊C4—72no—
109C4⋊C4—71no—
110C4⋊C4—141no—
111C4⋊C4—141no—
112C22⋊C4—141no—
113C2×C7—22no—
114C2×C7C3×#12311yescharacteristic
115C2×C7C2×C3×Q1611yescharacteristic
116C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
117C22×C3—71no—
118C22×C3—72no—
119C22×C3—141no—
120C22×C3—141no—
121C3×C4—72no—
122C3×C4—141no—
123C3×C4—282no—
124C3×C4—281no—
125C3×C4—281no—
126C23C4⋊AGL(1,𝔽7)11yescharacteristic
127Q8—142no—
128C2×C4—22no—
129C2×C4—141no—
130C2×C4—141no—
131C2×C4—142no—
132C2×C4—282no—
133C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
134C2×C4C22×AGL(1,𝔽7)11yescharacteristic
135C8—22no—
136C7C3×#48411yescharacteristic, complemented, Sylow
137C2×C3—71no—
138C2×C3—71no—
139C2×C3—71no—
140C2×C3—142no—
141C22—21no—
142C22—21no—
143C22Dic28⋊C312yes—
144C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
145C4—21no—
146C4—281no—
147C4—282no—
148C4—281no—
149C4#1159⋊C312yes—
150C3—71noSylow
151C2—22no—
152C2#6449⋊C311yescharacteristic
153C2C2×Dic28⋊C311yescharacteristic
154C2#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 radicalC23×ΓL(1,𝔽8)D4
hypercenterC22⋊C8AGL(1,𝔽7)
hypocenterC7C3×#484
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
4. #6392⋊C3C2semidirect
5. #6390⋊C3C2semidirect
6. #89494C3semidirect
19. #6392C2×C3semidirect
20. #6390C2×C3semidirect
103. ΓL(1,𝔽8)#484semidirect
136. C7C3×#484semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(ΓL(1,𝔽8)⋊C4)2
#6487⋊C32
ΓL(1,𝔽8)×C22⋊C82
#6392⋊C32
#6390⋊C32
#894943
C3×#4847