Finite Group Databasebeta

← #89620⋊C3 #13051749

Subgroups — #89620⋊C3 #13051749

Order 1344.

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

normal subgroups only

This group has 256 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×#1132⋊C3C212yesmaximal, complemented
2#10620710C211yesmaximal, characteristic
3C4×ΓL(1,𝔽8)⋊Q8C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×C4⋊C8C211yesmaximal, characteristic, complemented
5#6395⋊C3C212yesmaximal, complemented
6#89620C311yesmaximal, characteristic, complemented, Hall
7C2×C4⋊AGL(1,𝔽7)—21no—
8C2×C4⋊AGL(1,𝔽7)—21no—
9C2×C4⋊AGL(1,𝔽7)C2212yes—
10C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
11C2×C8×ΓL(1,𝔽8)C2212yes—
12C42×ΓL(1,𝔽8)C2211yescharacteristic
13C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
14(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
15C4×ΓL(1,𝔽8)⋊C4—21no—
16#1132⋊C3—24no—
17C2×#1132C2×C312yescomplemented
18#6437C2×C311yescharacteristic
19C4×Dic14C2×C311yescharacteristic, complemented
20C7×C4⋊C8C2×C311yescharacteristic, complemented
21#6395C2×C312yescomplemented
22C3×#460—71nomaximal, Hall
23C22×AGL(1,𝔽7)—22no—
24C2×C4×ΓL(1,𝔽8)D411yescharacteristic
25C2×C4×ΓL(1,𝔽8)D411yescharacteristic
26C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
27C2×ΓL(1,𝔽8)⋊C4—21no—
28C2×ΓL(1,𝔽8)⋊C4—21no—
29C4⋊AGL(1,𝔽7)—22no—
30C4⋊AGL(1,𝔽7)—22no—
31C4⋊AGL(1,𝔽7)—42no—
32C4⋊AGL(1,𝔽7)—42no—
33ΓL(1,𝔽8)⋊Q8—21no—
34ΓL(1,𝔽8)⋊Q8D412yescomplemented
35C8×ΓL(1,𝔽8)—22no—
36C2×D28—21no—
37C2×D28—21no—
38C2×D28C22×C312yes—
39C2×Dic14C22×C311yescharacteristic
40C2×C7×C8C22×C312yes—
41C42×C7C22×C311yescharacteristic
42C4⋊Dic7C22×C311yescharacteristic
43Dic7⋊C4—21no—
44C4×Dic7—21no—
45#1132—24no—
46C2×C3×SD16—72no—
47C3×#127—71no—
48C3×C4×Q8—71no—
49C3×C4⋊C8—71no—
50C3×#102—72no—
51C22×ΓL(1,𝔽8)C2×D411yescharacteristic
52C2×AGL(1,𝔽7)—42no—
53C2×AGL(1,𝔽7)—42no—
54C2×AGL(1,𝔽7)—42no—
55C4×ΓL(1,𝔽8)—21no—
56C4×ΓL(1,𝔽8)SD1612yescomplemented
57C4×ΓL(1,𝔽8)C2×D411yescharacteristic
58C4×ΓL(1,𝔽8)#4111yescharacteristic
59ΓL(1,𝔽8)⋊C4—22no—
60ΓL(1,𝔽8)⋊C4—41no—
61#460—71noSylow
62C22×D7—22no—
63C2×C4×C7C3×D411yescharacteristic
64C2×C4×C7C3×D411yescharacteristic
65C2×C4×C7C23×C311yescharacteristic
66C2×Dic7—21no—
67C2×Dic7—21no—
68D28—22no—
69D28—22no—
70D28—42no—
71D28—42no—
72Dic14—21no—
73Dic14C3×D412yescomplemented
74C7×C8—22no—
75C2×C3×Q8—71no—
76C2×C3×D4—72no—
77C2×C3×D4—141no—
78C2×C3×D4—141no—
79C3×SD16—144no—
80C2×C3×C8—72no—
81C3×C4⋊C4—71no—
82C3×C4⋊C4—141no—
83C3×C42—71no—
84C3×C42—141no—
85C2×ΓL(1,𝔽8)#12111yescharacteristic
86C2×ΓL(1,𝔽8)C2×SD1611yescharacteristic
87C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
88AGL(1,𝔽7)—82no—
89C2×SD16—72no—
90#127—71no—
91C4×Q8—71no—
92C4⋊C8AGL(1,𝔽7)11yescharacteristic, complemented
93#102—72no—
94C22×C7C2×C3×D411yescharacteristic
95C2×D7—42no—
96C2×D7—42no—
97C2×D7—42no—
98C4×C7—21no—
99C4×C7C3×SD1612yescomplemented
100C4×C7C2×C3×D411yescharacteristic
101C4×C7C3×#4111yescharacteristic
102Dic7—22no—
103Dic7—41no—
104C23×C3—142no—
105C3×Q8—72no—
106C3×Q8—141no—
107C3×D4—142no—
108C3×D4—142no—
109C3×D4—282no—
110C3×D4—282no—
111C2×C3×C4—71no—
112C2×C3×C4—71no—
113C2×C3×C4—71no—
114C2×C3×C4—141no—
115C2×C3×C4—141no—
116C3×C8—142no—
117ΓL(1,𝔽8)#46011yescharacteristic, complemented, Hall
118C2×Q8—71no—
119C2×D4—72no—
120C2×D4—141no—
121C2×D4—141no—
122SD16—144no—
123C2×C8C2×AGL(1,𝔽7)12yes—
124C4⋊C4—71no—
125C4⋊C4—141no—
126C42—141no—
127C42C2×AGL(1,𝔽7)11yescharacteristic
128C2×C7C3×#12111yescharacteristic
129C2×C7C2×C3×SD1611yescharacteristic
130C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
131D7—82no—
132C22×C3—71no—
133C22×C3—282no—
134C22×C3—282no—
135C22×C3—282no—
136C3×C4—71no—
137C3×C4—71no—
138C3×C4—72no—
139C3×C4—141no—
140C3×C4—142no—
141C3×C4—281no—
142C23—142no—
143Q8—72no—
144Q8—141no—
145D4—142no—
146D4—142no—
147D4—282no—
148D4—282no—
149C2×C4—141no—
150C2×C4—141no—
151C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
152C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
153C2×C4C22×AGL(1,𝔽7)11yescharacteristic
154C8—22no—
155C7C3×#46011yescharacteristic, complemented, Sylow
156C2×C3—71no—
157C2×C3—71no—
158C2×C3—71no—
159C2×C3—562no—
160C22—282no—
161C22—282no—
162C22—282no—
163C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
164C4—21no—
165C4—142no—
166C4—281no—
167C4#1132⋊C312yescomplemented
168C4D4×AGL(1,𝔽7)11yescharacteristic
169C4#6665911yescharacteristic
170C3—71noSylow
171C2—562no—
172C2#6458⋊C311yescharacteristic
173C2C2×#1132⋊C311yescharacteristic
174C2#6471⋊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 radical#10620710C2
hypercenterC4⋊C8AGL(1,𝔽7)
hypocenterC7C3×#460
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
1. C2×#1132⋊C3C2semidirect
3. C4×ΓL(1,𝔽8)⋊Q8C2semidirect
4. ΓL(1,𝔽8)×C4⋊C8C2semidirect
5. #6395⋊C3C2semidirect
6. #89620C3semidirect
17. C2×#1132C2×C3semidirect
19. C4×Dic14C2×C3semidirect
20. C7×C4⋊C8C2×C3semidirect
21. #6395C2×C3semidirect
34. ΓL(1,𝔽8)⋊Q8D4semidirect
56. C4×ΓL(1,𝔽8)SD16semidirect
73. Dic14C3×D4semidirect
92. C4⋊C8AGL(1,𝔽7)semidirect
99. C4×C7C3×SD16semidirect
117. ΓL(1,𝔽8)#460semidirect
155. C7C3×#460semidirect
167. C4#1132⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1132⋊C32
#106207102
C4×ΓL(1,𝔽8)⋊Q82
ΓL(1,𝔽8)×C4⋊C82
#6395⋊C32
#896203
C3×#4607