Finite Group Databasebeta

← #89862⋊C3 #13051992

Subgroups — #89862⋊C3 #13051992

Order 1344.

216 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 74 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
2C2×#1144⋊C3C212yesmaximal
3C4×ΓL(1,𝔽8)⋊Q8C211yesmaximal, characteristic
4#6409⋊C3C212yesmaximal
5C4⋊(ΓL(1,𝔽8)⋊C8)C211yesmaximal, characteristic
6#89862C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)C2212yes—
8C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
9C4⋊C4×ΓL(1,𝔽8)—21no—
10C4⋊C4×ΓL(1,𝔽8)—21no—
11C42×ΓL(1,𝔽8)C2211yescharacteristic
12#1144⋊C3—24no—
13C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
14(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
15C4×ΓL(1,𝔽8)⋊C4—21no—
16C2×ΓL(1,𝔽8)⋊C8C2212yes—
17C7×C4⋊Q8C2×C311yescharacteristic
18C2×#1144C2×C312yes—
19C4×Dic14C2×C311yescharacteristic
20#6409C2×C312yes—
21C4⋊(C7⋊C8)C2×C311yescharacteristic
22C3×#462—71nomaximal, Hall
23Q8×ΓL(1,𝔽8)—22no—
24Q8×ΓL(1,𝔽8)—22no—
25C2×C4×ΓL(1,𝔽8)—22no—
26C2×C4×ΓL(1,𝔽8)D411yescharacteristic
27C2×C4×ΓL(1,𝔽8)D411yescharacteristic
28C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
29C2×ΓL(1,𝔽8)⋊C4—21no—
30C2×ΓL(1,𝔽8)⋊C4—21no—
31ΓL(1,𝔽8)⋊Q8—21no—
32ΓL(1,𝔽8)⋊Q8D412yes—
33ΓL(1,𝔽8)⋊C8—22no—
34C2×C7×Q8C22×C312yes—
35C2×Dic14C22×C311yescharacteristic
36C7×C4⋊C4—21no—
37C7×C4⋊C4—21no—
38C42×C7C22×C311yescharacteristic
39#1144—24no—
40C4⋊Dic7C22×C311yescharacteristic
41Dic7⋊C4—21no—
42C4×Dic7—21no—
43C2×C7⋊C8C22×C312yes—
44C2×C3×Q16—72no—
45C3×C4⋊Q8—71no—
46C3×C4×Q8—71no—
47C3×C4⋊C8—71no—
48C3×#103—72no—
49C22×ΓL(1,𝔽8)C2×D411yescharacteristic
50C4×ΓL(1,𝔽8)—21no—
51C4×ΓL(1,𝔽8)—42no—
52C4×ΓL(1,𝔽8)Q1612yescomplemented
53C4×ΓL(1,𝔽8)C2×D411yescharacteristic
54C4×ΓL(1,𝔽8)#4111yescharacteristic
55ΓL(1,𝔽8)⋊C4—22no—
56ΓL(1,𝔽8)⋊C4—41no—
57#462—71noSylow
58C7×Q8—22no—
59C7×Q8—22no—
60C2×C4×C7—22no—
61C2×C4×C7C3×D411yescharacteristic
62C2×C4×C7C3×D411yescharacteristic
63C2×C4×C7C23×C311yescharacteristic
64C2×Dic7—21no—
65C2×Dic7—21no—
66Dic14—21no—
67Dic14C3×D412yes—
68C7⋊C8—22no—
69C2×C3×Q8—71no—
70C2×C3×Q8—72no—
71C3×Q16—144no—
72C2×C3×C8—72no—
73C3×C4⋊C4—71no—
74C3×C4⋊C4—141no—
75C3×C4⋊C4—141no—
76C3×C4⋊C4—141no—
77C3×C42—71no—
78C3×C42—141no—
79C2×ΓL(1,𝔽8)#12111yescharacteristic
80C2×ΓL(1,𝔽8)C2×Q1611yescharacteristic
81C2×ΓL(1,𝔽8)#13711yescharacteristic
82C2×Q16—72no—
83C4⋊Q8AGL(1,𝔽7)11yescharacteristic
84C4×Q8—71no—
85C4⋊C8—71no—
86#103—72no—
87C22×C7C2×C3×D411yescharacteristic
88C4×C7—21no—
89C4×C7—42no—
90C4×C7C3×Q1612yescomplemented
91C4×C7C2×C3×D411yescharacteristic
92C4×C7C3×#4111yescharacteristic
93Dic7—22no—
94Dic7—41no—
95C3×Q8—72no—
96C3×Q8—142no—
97C3×Q8—142no—
98C3×Q8—141no—
99C2×C3×C4—71no—
100C2×C3×C4—71no—
101C2×C3×C4—71no—
102C2×C3×C4—142no—
103C2×C3×C4—141no—
104C2×C3×C4—141no—
105C3×C8—142no—
106ΓL(1,𝔽8)#46211yescharacteristic, complemented, Hall
107C2×Q8—71no—
108C2×Q8C2×AGL(1,𝔽7)12yes—
109Q16—144no—
110C2×C8—72no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—71no—
114C4⋊C4—141no—
115C42—141no—
116C42C2×AGL(1,𝔽7)11yescharacteristic
117C2×C7C3×#12111yescharacteristic
118C2×C7C2×C3×Q1611yescharacteristic
119C2×C7C3×#13711yescharacteristic
120C22×C3—71no—
121C3×C4—71no—
122C3×C4—71no—
123C3×C4—72no—
124C3×C4—141no—
125C3×C4—142no—
126C3×C4—282no—
127C3×C4—281no—
128Q8—22no—
129Q8—22no—
130Q8—72no—
131Q8—141no—
132C2×C4—22no—
133C2×C4—141no—
134C2×C4—141no—
135C2×C4(C22⋊D7)⋊C311yescharacteristic
136C2×C4(C22⋊D7)⋊C311yescharacteristic
137C2×C4C22×AGL(1,𝔽7)11yescharacteristic
138C8—142no—
139C7C3×#46211yescharacteristic, complemented, Sylow
140C2×C3—71no—
141C2×C3—71no—
142C2×C3—71no—
143C22C2×(C22⋊D7)⋊C311yescharacteristic
144C4—21no—
145C4—42no—
146C4—142no—
147C4—281no—
148C4#1144⋊C312yescomplemented
149C4D4×AGL(1,𝔽7)11yescharacteristic
150C4#1159⋊C311yescharacteristic
151C3—71noSylow
152C2#6501⋊C311yescharacteristic
153C2#6505⋊C311yescharacteristic
154C2C2×#1144⋊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 radicalC22×ΓL(1,𝔽8)C2×D4
hypercenterC4⋊Q8AGL(1,𝔽7)
hypocenterC7C3×#462
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
6. #89862C3semidirect
52. C4×ΓL(1,𝔽8)Q16semidirect
90. C4×C7C3×Q16semidirect
106. ΓL(1,𝔽8)#462semidirect
139. C7C3×#462semidirect
148. C4#1144⋊C3semidirect

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
C2×#1144⋊C32
C4×ΓL(1,𝔽8)⋊Q82
#6409⋊C32
C4⋊(ΓL(1,𝔽8)⋊C8)2
#898623
C3×#4627