Finite Group Databasebeta

← #89867⋊C3 #13051995

Subgroups — #89867⋊C3 #13051995

Order 1344.

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#10621516C211yesmaximal, characteristic
2ΓL(1,𝔽8)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#6505⋊C3C212yesmaximal, complemented
4#6410⋊C3C211yesmaximal, characteristic, complemented
5#6379⋊C3C212yesmaximal, complemented
6#89867C311yesmaximal, characteristic, complemented, Hall
7C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic
8C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic
9#66659—22no—
10Q8×AGL(1,𝔽7)—22no—
11#1157⋊C3—22no—
12#1157⋊C3C2212yes—
13C4⋊C4×ΓL(1,𝔽8)—22no—
14C42×ΓL(1,𝔽8)C2211yescharacteristic
15#1144⋊C3—22no—
16#1143⋊C3—22no—
17#1136⋊C3C2212yes—
18#6551C2×C311yescharacteristic
19C7×C4⋊Q8C2×C311yescharacteristic, complemented
20#6505C2×C312yescomplemented
21#6410C2×C311yescharacteristic, complemented
22#6379C2×C312yescomplemented
23C3×#456—71nomaximal, Hall
24Q8×ΓL(1,𝔽8)—22no—
25Q8×ΓL(1,𝔽8)—22no—
26C2×C4×ΓL(1,𝔽8)—21no—
27C2×C4×ΓL(1,𝔽8)—21no—
28C2×C4×ΓL(1,𝔽8)D412yes—
29C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
30(C22⋊D7)⋊C3—22no—
31C4⋊AGL(1,𝔽7)—22no—
32C4⋊AGL(1,𝔽7)D412yes—
33C4×AGL(1,𝔽7)—24no—
34C4×AGL(1,𝔽7)—22no—
35ΓL(1,𝔽8)⋊Q8—22no—
36ΓL(1,𝔽8)⋊Q8D412yes—
37ΓL(1,𝔽8)⋊C8—22no—
38C2×C7×Q8C22×C311yescharacteristic
39C2×C7×Q8C22×C311yescharacteristic
40#1161—22no—
41Q8×D7—22no—
42#1157—22no—
43#1157C22×C312yes—
44C7×C4⋊C4—22no—
45C42×C7C22×C311yescharacteristic
46#1144—22no—
47#1143—22no—
48#1136C22×C312yes—
49C3×2−1+4—71no—
50C3×#137—72no—
51C3×C4⋊Q8—71no—
52C3×#104—72no—
53C3×#101—71no—
54C22×ΓL(1,𝔽8)C2×D411yescharacteristic
55C2×AGL(1,𝔽7)—22no—
56C4×ΓL(1,𝔽8)—22no—
57C4×ΓL(1,𝔽8)—22no—
58C4×ΓL(1,𝔽8)—41no—
59C4×ΓL(1,𝔽8)C2×D412yes—
60ΓL(1,𝔽8)⋊C4—22no—
61#456—71noSylow
62C7×Q8—22no—
63C7×Q8—22no—
64C2×C4×C7—21no—
65C2×C4×C7—21no—
66C2×C4×C7C3×D412yes—
67C2×C4×C7C23×C311yescharacteristic
68C22⋊D7—22no—
69D28—22no—
70D28C3×D412yes—
71C4×D7—24no—
72C4×D7—22no—
73Dic14—22no—
74Dic14C3×D412yes—
75C7⋊C8—22no—
76C3×#41—72no—
77C3×#41—142no—
78C3×#41—142no—
79C2×C3×Q8—71no—
80C2×C3×Q8—71no—
81C2×C3×Q8—142no—
82C3×Q16—142no—
83C3×SD16—142no—
84C3×M16—72no—
85C3×C4⋊C4—142no—
86C3×C42—71no—
87C2×ΓL(1,𝔽8)—21no—
88C2×ΓL(1,𝔽8)#12011yescharacteristic
89AGL(1,𝔽7)—42no—
902−1+4—71no—
91#137—72no—
92C4⋊Q8AGL(1,𝔽7)11yescharacteristic, complemented
93#104—72no—
94#101—71no—
95C22×C7C2×C3×D411yescharacteristic
96C2×D7—22no—
97C4×C7—22no—
98C4×C7—22no—
99C4×C7—41no—
100C4×C7C2×C3×D412yes—
101Dic7—22no—
102C3×Q8—72no—
103C3×Q8—142no—
104C3×Q8—142no—
105C3×Q8—142no—
106C3×D4—72no—
107C3×D4—142no—
108C3×D4—142no—
109C2×C3×C4—72no—
110C2×C3×C4—71no—
111C2×C3×C4—141no—
112C2×C3×C4—141no—
113C2×C3×C4—142no—
114C2×C3×C4—144no—
115C3×C8—142no—
116ΓL(1,𝔽8)#45611yescharacteristic, complemented, Hall
117#41—72no—
118#41—142no—
119#41—142no—
120C2×Q8—142no—
121C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
122C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
123Q16—142no—
124SD16—142no—
125M16—72no—
126C4⋊C4—22no—
127C42C2×AGL(1,𝔽7)11yescharacteristic
128C2×C7—21no—
129C2×C7C3×#12011yescharacteristic
130D7—42no—
131C22×C3—71no—
132C22×C3—142no—
133C3×C4—72no—
134C3×C4—142no—
135C3×C4—142no—
136C3×C4—142no—
137C3×C4—281no—
138Q8—22no—
139Q8—22no—
140Q8—72no—
141Q8—142no—
142D4—72no—
143D4—142no—
144D4—142no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—144no—
148C2×C4—142no—
149C2×C4(C22⋊D7)⋊C312yes—
150C2×C4C22×AGL(1,𝔽7)11yescharacteristic
151C8—142no—
152C7C3×#45611yescharacteristic, complemented, Sylow
153C2×C3—71no—
154C2×C3—141no—
155C2×C3—282no—
156C22—142no—
157C22C2×(C22⋊D7)⋊C311yescharacteristic
158C4—22no—
159C4—22no—
160C4—41no—
161C4—142no—
162C4D4×AGL(1,𝔽7)12yes—
163C3—71noSylow
164C2—21no—
165C2—282no—
166C2#6500⋊C311yescharacteristic
subgroupG/H
centerC2#6500⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C4⋊Q8C2×C3
socleC2×C7C3×#120
prime radical#10621516C2
hypercenterC4⋊Q8AGL(1,𝔽7)
hypocenterC7C3×#456
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×C4⋊Q8C2semidirect
3. #6505⋊C3C2semidirect
4. #6410⋊C3C2semidirect
5. #6379⋊C3C2semidirect
6. #89867C3semidirect
19. C7×C4⋊Q8C2×C3semidirect
20. #6505C2×C3semidirect
21. #6410C2×C3semidirect
22. #6379C2×C3semidirect
92. C4⋊Q8AGL(1,𝔽7)semidirect
116. ΓL(1,𝔽8)#456semidirect
152. C7C3×#456semidirect

Maximal subgroups

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

subgroupindex
#106215162
ΓL(1,𝔽8)×C4⋊Q82
#6505⋊C32
#6410⋊C32
#6379⋊C32
#898673
C3×#4567