Finite Group Databasebeta

← #89856⋊C3 #13051990

Subgroups — #89856⋊C3 #13051990

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, complemented
2C4×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic
3#6383⋊C3C212yesmaximal
4#6382⋊C3C212yesmaximal, complemented
5C4⋊(ΓL(1,𝔽8)⋊C8)C211yesmaximal, characteristic, complemented
6#89856C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)—21no—
8C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
9C2×C4×AGL(1,𝔽7)—21no—
10C4⋊C4×ΓL(1,𝔽8)—21no—
11C4⋊C4×ΓL(1,𝔽8)C2212yes—
12C42×ΓL(1,𝔽8)C2211yescharacteristic
13(C2×AGL(1,𝔽7))⋊C4—21no—
14C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
15C2×ΓL(1,𝔽8)⋊C8C2212yes—
16C7×C4⋊Q8C2×C311yescharacteristic, complemented
17C4×D28C2×C311yescharacteristic
18#6383C2×C312yes—
19#6382C2×C312yescomplemented
20C4⋊(C7⋊C8)C2×C311yescharacteristic, complemented
21C3×#476—71nomaximal, Hall
22C22×AGL(1,𝔽7)—21no—
23Q8×ΓL(1,𝔽8)—42no—
24C2×C4×ΓL(1,𝔽8)—22no—
25C2×C4×ΓL(1,𝔽8)D411yescharacteristic
26C2×C4×ΓL(1,𝔽8)D411yescharacteristic
27C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
28C2×ΓL(1,𝔽8)⋊C4—21no—
29C4⋊AGL(1,𝔽7)—21no—
30C4⋊AGL(1,𝔽7)Q812yescomplemented
31C4×AGL(1,𝔽7)—42no—
32ΓL(1,𝔽8)⋊C8—22no—
33C2×C7×Q8—21no—
34C2×D28C22×C311yescharacteristic
35C2×C4×D7—21no—
36C7×C4⋊C4—21no—
37C7×C4⋊C4C22×C312yes—
38C42×C7C22×C311yescharacteristic
39(C2×D7)⋊C4—21no—
40C4⋊Dic7C22×C311yescharacteristic
41C2×C7⋊C8C22×C312yes—
42C3×C4⋊Q8—71no—
43C3×C4×D4—71no—
44C3×#106—72no—
45C3×C4⋊C8—71no—
46C3×#102—72no—
47C22×ΓL(1,𝔽8)C2×D411yescharacteristic
48C2×AGL(1,𝔽7)—22no—
49C2×AGL(1,𝔽7)—41no—
50C2×AGL(1,𝔽7)—41no—
51C4×ΓL(1,𝔽8)—21no—
52C4×ΓL(1,𝔽8)—42no—
53C4×ΓL(1,𝔽8)SD1612yes—
54C4×ΓL(1,𝔽8)C2×Q811yescharacteristic
55C4×ΓL(1,𝔽8)#4111yescharacteristic
56ΓL(1,𝔽8)⋊C4—41no—
57#476—71noSylow
58C22×D7—21no—
59C7×Q8—42no—
60C2×C4×C7—22no—
61C2×C4×C7C3×D411yescharacteristic
62C2×C4×C7C3×D411yescharacteristic
63C2×C4×C7C23×C311yescharacteristic
64C2×Dic7—21no—
65D28—21no—
66D28C3×Q812yescomplemented
67C4×D7—42no—
68C7⋊C8—22no—
69C2×C3×Q8—141no—
70C2×C3×D4—71no—
71C22×C3×C4—141no—
72C2×C3×C8—72no—
73C3×C4⋊C4—71no—
74C3×C4⋊C4—72no—
75C3×C4⋊C4—141no—
76C3×C22⋊C4—141no—
77C3×C42—71no—
78C2×ΓL(1,𝔽8)C22⋊Q811yescharacteristic
79C2×ΓL(1,𝔽8)C2×SD1611yescharacteristic
80C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
81AGL(1,𝔽7)—42no—
82C4⋊Q8AGL(1,𝔽7)11yescharacteristic, complemented
83C4×D4—71no—
84#106—72no—
85C4⋊C8—71no—
86#102—72no—
87C22×C7C2×C3×D411yescharacteristic
88C2×D7—22no—
89C2×D7—41no—
90C2×D7—41no—
91C4×C7—21no—
92C4×C7—42no—
93C4×C7C3×SD1612yes—
94C4×C7C2×C3×Q811yescharacteristic
95C4×C7C3×#4111yescharacteristic
96Dic7—41no—
97C23×C3—141no—
98C3×Q8—282no—
99C3×D4—72no—
100C3×D4—141no—
101C2×C3×C4—71no—
102C2×C3×C4—71no—
103C2×C3×C4—71no—
104C2×C3×C4—142no—
105C2×C3×C4—141no—
106C2×C3×C4—282no—
107C3×C8—142no—
108ΓL(1,𝔽8)#47611yescharacteristic, complemented, Hall
109C2×Q8—21no—
110C2×D4—71no—
111C22×C4—141no—
112C2×C8—72no—
113C4⋊C4—21no—
114C4⋊C4—71no—
115C4⋊C4C2×AGL(1,𝔽7)12yes—
116C22⋊C4—141no—
117C42C2×AGL(1,𝔽7)11yescharacteristic
118C2×C7C3×C22⋊Q811yescharacteristic
119C2×C7C2×C3×SD1611yescharacteristic
120C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
121D7—42no—
122C22×C3—71no—
123C22×C3—142no—
124C22×C3—281no—
125C22×C3—281no—
126C3×C4—71no—
127C3×C4—71no—
128C3×C4—72no—
129C3×C4—141no—
130C3×C4—282no—
131C3×C4—281no—
132C23—141no—
133Q8—42no—
134D4—72no—
135D4—141no—
136C2×C4—22no—
137C2×C4—141no—
138C2×C4—282no—
139C2×C4(C22⋊D7)⋊C311yescharacteristic
140C2×C4(C22⋊D7)⋊C311yescharacteristic
141C2×C4C22×AGL(1,𝔽7)11yescharacteristic
142C8—142no—
143C7C3×#47611yescharacteristic, complemented, Sylow
144C2×C3—71no—
145C2×C3—71no—
146C2×C3—71no—
147C2×C3—282no—
148C22—142no—
149C22—281no—
150C22—281no—
151C22C2×(C22⋊D7)⋊C311yescharacteristic
152C4—21no—
153C4—42no—
154C4—281no—
155C4#1143⋊C312yes—
156C4Q8×AGL(1,𝔽7)11yescharacteristic
157C4#6665911yescharacteristic
158C3—71noSylow
159C2—282no—
160C2#6495⋊C311yescharacteristic
161C2C2×#1143⋊C311yescharacteristic
162C2#6509⋊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 radicalC2×C4⋊AGL(1,𝔽7)C22
hypercenterC4⋊Q8AGL(1,𝔽7)
hypocenterC7C3×#476
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
1. ΓL(1,𝔽8)×C4⋊Q8C2semidirect
4. #6382⋊C3C2semidirect
5. C4⋊(ΓL(1,𝔽8)⋊C8)C2semidirect
6. #89856C3semidirect
16. C7×C4⋊Q8C2×C3semidirect
19. #6382C2×C3semidirect
20. C4⋊(C7⋊C8)C2×C3semidirect
30. C4⋊AGL(1,𝔽7)Q8semidirect
66. D28C3×Q8semidirect
82. C4⋊Q8AGL(1,𝔽7)semidirect
108. ΓL(1,𝔽8)#476semidirect
143. C7C3×#476semidirect

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
C4×C4⋊AGL(1,𝔽7)2
#6383⋊C32
#6382⋊C32
C4⋊(ΓL(1,𝔽8)⋊C8)2
#898563
C3×#4767