Finite Group Databasebeta

← #89621⋊C3 #13051751

Subgroups — #89621⋊C3 #13051751

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
1C2×Dic28⋊C3C212yesmaximal
2#6432⋊C3C211yesmaximal, characteristic
3C4×ΓL(1,𝔽8)⋊Q8C211yesmaximal, characteristic
4ΓL(1,𝔽8)×C4⋊C8C211yesmaximal, characteristic
5#6390⋊C3C212yesmaximal
6#89621C311yesmaximal, characteristic, complemented, Hall
7C2×ΓL(1,𝔽8)⋊Q8C2212yes—
8C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
9C2×C8×ΓL(1,𝔽8)C2212yes—
10C42×ΓL(1,𝔽8)C2211yescharacteristic
11C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
12C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
13C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
14(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
15C4×ΓL(1,𝔽8)⋊C4—21no—
16Dic28⋊C3—24no—
17C2×Dic28C2×C312yes—
18#6432C2×C311yescharacteristic
19C4×Dic14C2×C311yescharacteristic
20C7×C4⋊C8C2×C311yescharacteristic
21#6390C2×C312yes—
22C3×#462—71nomaximal, Hall
23C2×C4×ΓL(1,𝔽8)D411yescharacteristic
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—
29ΓL(1,𝔽8)⋊Q8—21no—
30ΓL(1,𝔽8)⋊Q8—22no—
31ΓL(1,𝔽8)⋊Q8—22no—
32ΓL(1,𝔽8)⋊Q8D412yes—
33C8×ΓL(1,𝔽8)—22no—
34C2×Dic14C22×C312yes—
35C2×Dic14C22×C311yescharacteristic
36C2×C7×C8C22×C312yes—
37C42×C7C22×C311yescharacteristic
38C4⋊Dic7—21no—
39C4⋊Dic7—21no—
40C4⋊Dic7C22×C311yescharacteristic
41Dic7⋊C4—21no—
42C4×Dic7—21no—
43Dic28—24no—
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)Q1612yescomplemented
52C4×ΓL(1,𝔽8)C2×D411yescharacteristic
53C4×ΓL(1,𝔽8)#4111yescharacteristic
54ΓL(1,𝔽8)⋊C4—22no—
55ΓL(1,𝔽8)⋊C4—42no—
56ΓL(1,𝔽8)⋊C4—41no—
57#462—71noSylow
58C2×C4×C7C3×D411yescharacteristic
59C2×C4×C7C3×D411yescharacteristic
60C2×C4×C7C23×C311yescharacteristic
61C2×Dic7—21no—
62C2×Dic7—21no—
63C2×Dic7—22no—
64Dic14—21no—
65Dic14—22no—
66Dic14—22no—
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⋊Q8—71no—
84C4×Q8—71no—
85C4⋊C8AGL(1,𝔽7)11yescharacteristic
86#103—72no—
87C22×C7C2×C3×D411yescharacteristic
88C4×C7—21no—
89C4×C7C3×Q1612yescomplemented
90C4×C7C2×C3×D411yescharacteristic
91C4×C7C3×#4111yescharacteristic
92Dic7—22no—
93Dic7—42no—
94Dic7—41no—
95C3×Q8—72no—
96C3×Q8—141no—
97C3×Q8—142no—
98C3×Q8—142no—
99C2×C3×C4—71no—
100C2×C3×C4—71no—
101C2×C3×C4—71no—
102C2×C3×C4—141no—
103C2×C3×C4—141no—
104C2×C3×C4—142no—
105C3×C8—142no—
106ΓL(1,𝔽8)#46211yescharacteristic, complemented, Hall
107C2×Q8—71no—
108C2×Q8—72no—
109Q16—144no—
110C2×C8C2×AGL(1,𝔽7)12yes—
111C4⋊C4—71no—
112C4⋊C4—141no—
113C4⋊C4—141no—
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—72no—
129Q8—141no—
130Q8—142no—
131Q8—142no—
132C2×C4—141no—
133C2×C4—141no—
134C2×C4—142no—
135C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
136C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
137C2×C4C22×AGL(1,𝔽7)11yescharacteristic
138C8—22no—
139C7C3×#46211yescharacteristic, complemented, Sylow
140C2×C3—71no—
141C2×C3—71no—
142C2×C3—71no—
143C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
144C4—21no—
145C4—142no—
146C4—282no—
147C4—281no—
148C4Dic28⋊C312yescomplemented
149C4D4×AGL(1,𝔽7)11yescharacteristic
150C4#6665911yescharacteristic
151C3—71noSylow
152C2#6458⋊C311yescharacteristic
153C2C2×Dic28⋊C311yescharacteristic
154C2#6472⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C4⋊C8C2×C3
socleC22×C7C2×C3×D4
prime radicalC22×ΓL(1,𝔽8)C2×D4
hypercenterC4⋊C8AGL(1,𝔽7)
hypocenterC7C3×#462
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
6. #89621C3semidirect
51. C4×ΓL(1,𝔽8)Q16semidirect
89. C4×C7C3×Q16semidirect
106. ΓL(1,𝔽8)#462semidirect
139. C7C3×#462semidirect
148. C4Dic28⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×Dic28⋊C32
#6432⋊C32
C4×ΓL(1,𝔽8)⋊Q82
ΓL(1,𝔽8)×C4⋊C82
#6390⋊C32
#896213
C3×#4627