Finite Group Databasebeta

← #89816⋊C3 #13051944

Subgroups — #89816⋊C3 #13051944

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
1ΓL(1,𝔽8)×C22⋊Q8C211yesmaximal, characteristic
2C2×C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic
3#6409⋊C3C212yesmaximal, complemented
4(C22⋊(C7⋊C8))⋊C3C211yesmaximal, characteristic
5#6381⋊C3C212yesmaximal, complemented
6#89816C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic
8C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
9C22×ΓL(1,𝔽8)⋊C4—21no—
10C4⋊C4×ΓL(1,𝔽8)—21no—
11C4⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
12C22⋊C4×ΓL(1,𝔽8)—21no—
13C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
14C4⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
15C2×ΓL(1,𝔽8)⋊C8C2212yes—
16C7×C22⋊Q8C2×C311yescharacteristic
17C2×C4⋊Dic7C2×C311yescharacteristic
18#6409C2×C312yescomplemented
19C22⋊(C7⋊C8)C2×C311yescharacteristic
20#6381C2×C312yescomplemented
21C3×#484—71nomaximal, Hall
22C23×ΓL(1,𝔽8)D411yescharacteristic
23Q8×ΓL(1,𝔽8)—22no—
24C2×C4×ΓL(1,𝔽8)—22no—
25C2×C4×ΓL(1,𝔽8)—21no—
26C2×C4×ΓL(1,𝔽8)—21no—
27C2×C4×ΓL(1,𝔽8)D411yescharacteristic
28C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
29C2×ΓL(1,𝔽8)⋊C4—22no—
30C2×ΓL(1,𝔽8)⋊C4—42no—
31ΓL(1,𝔽8)⋊C8—22no—
32C2×C7×Q8C22×C311yescharacteristic
33C22×C4×C7C22×C311yescharacteristic
34C22×Dic7—21no—
35C7×C4⋊C4—21no—
36C7×C4⋊C4C22×C311yescharacteristic
37C7×C22⋊C4—21no—
38C4⋊Dic7—21no—
39C4⋊Dic7C22×C312yes—
40C2×C7⋊C8C22×C312yes—
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)—41no—
52C4×ΓL(1,𝔽8)—41no—
53C4×ΓL(1,𝔽8)#4112yes—
54ΓL(1,𝔽8)⋊C4—42no—
55#484—71noSylow
56C23×C7C3×D411yescharacteristic
57C7×Q8—22no—
58C2×C4×C7—22no—
59C2×C4×C7—21no—
60C2×C4×C7—21no—
61C2×C4×C7C3×D411yescharacteristic
62C2×C4×C7C23×C311yescharacteristic
63C2×Dic7—22no—
64C2×Dic7—42no—
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⋊Q8AGL(1,𝔽7)11yescharacteristic
80C2×C4⋊C4—71no—
81#107—72no—
82#103—72no—
83C22⋊C8—71no—
84C22×C7—21no—
85C22×C7—21no—
86C22×C7C3×Q1612yes—
87C22×C7C2×C3×D411yescharacteristic
88C4×C7—21no—
89C4×C7—41no—
90C4×C7—41no—
91C4×C7C3×#4112yes—
92Dic7—42no—
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×Q8C2×AGL(1,𝔽7)11yescharacteristic
105C22×C4—141no—
106C22×C4C2×AGL(1,𝔽7)11yescharacteristic
107C2×C8—72no—
108C4⋊C4—21no—
109C4⋊C4—72no—
110C4⋊C4—141no—
111C4⋊C4C2×AGL(1,𝔽7)11yescharacteristic
112C22⋊C4—21no—
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—281no—
124C3×C4—281no—
125C3×C4—282no—
126C23(C22⋊D7)⋊C311yescharacteristic
127Q8—22no—
128C2×C4—22no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—142no—
132C2×C4—282no—
133C2×C4(C22⋊D7)⋊C311yescharacteristic
134C2×C4C22×AGL(1,𝔽7)11yescharacteristic
135C8—142no—
136C7C3×#48411yescharacteristic, complemented, Sylow
137C2×C3—71no—
138C2×C3—71no—
139C2×C3—71no—
140C2×C3—142no—
141C22—21no—
142C22—21no—
143C22#1144⋊C312yes—
144C22C2×(C22⋊D7)⋊C311yescharacteristic
145C4—21no—
146C4—41no—
147C4—41no—
148C4—282no—
149C4#1159⋊C312yes—
150C3—71noSylow
151C2—22no—
152C2#6498⋊C311yescharacteristic
153C2C2×#1144⋊C311yescharacteristic
154C2#6512⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C22⋊D7)⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C22⋊Q8C2×C3
socleC22×C7C2×C3×D4
prime radicalC23×ΓL(1,𝔽8)D4
hypercenterC22⋊Q8AGL(1,𝔽7)
hypocenterC7C3×#484
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #6409⋊C3C2semidirect
5. #6381⋊C3C2semidirect
6. #89816C3semidirect
18. #6409C2×C3semidirect
20. #6381C2×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
ΓL(1,𝔽8)×C22⋊Q82
C2×C4⋊(ΓL(1,𝔽8)⋊C4)2
#6409⋊C32
(C22⋊(C7⋊C8))⋊C32
#6381⋊C32
#898163
C3×#4847