Finite Group Databasebeta

← #72245518

Subgroups — #72245518

Order 2016.

292 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 73 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×C4⋊(C3⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)×#235C211yesmaximal, characteristic, complemented
3C3×#1136⋊C3C211yesmaximal, characteristic, complemented
4#66568⋊C3C212yesmaximal, complemented
5#66562⋊C3C212yesmaximal, complemented
6#10620867C311yesmaximal, characteristic, complemented
7#6512⋊C3—31nomaximal
8C22×C3⋊AGL(1,𝔽7)—21no—
9C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C4⋊(C3⋊AGL(1,𝔽7))—21no—
11C4⋊(C3⋊AGL(1,𝔽7))C2212yes—
12ΓL(1,𝔽8)×#67—21no—
13ΓL(1,𝔽8)×D12C2211yescharacteristic, complemented
14C4×D3×ΓL(1,𝔽8)—21no—
15ΓL(1,𝔽8)×Dic6C2211yescharacteristic, complemented
16C3×ΓL(1,𝔽8)⋊C8C2212yescomplemented
17C2×D84C2×C311yescharacteristic, complemented
18C7×#235C2×C311yescharacteristic, complemented
19#41×ΓL(1,𝔽8)—31no—
20C2×C4⋊AGL(1,𝔽7)—31no—
21C3×#1136C2×C311yescharacteristic, complemented
22#66568C2×C312yescomplemented
23#66562C2×C312yescomplemented
24#1143⋊C3—32no—
25#1141⋊C3—32no—
26#1136⋊C3—21no—
27#1136⋊C3D311yescharacteristic, complemented
28C3×#908—71nomaximal, Hall
29C22×C3×ΓL(1,𝔽8)D411yescharacteristic
30C2×C3⋊AGL(1,𝔽7)—22no—
31C2×C3⋊AGL(1,𝔽7)—42no—
32C2×D3×ΓL(1,𝔽8)—21no—
33C3×C4×ΓL(1,𝔽8)D411yescharacteristic
34C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
35Dic3×ΓL(1,𝔽8)—21no—
36#6512—31noHall
37C22×D21—21no—
38C22×AGL(1,𝔽7)—61no—
39C2×C3×C4×C7C22×C311yescharacteristic
40D84—21no—
41D84C22×C312yes—
42C7×#67—21no—
43C7×D12C22×C311yescharacteristic, complemented
44C4×D3×C7—21no—
45C7×Dic6C22×C311yescharacteristic, complemented
46Q8×ΓL(1,𝔽8)—31no—
47D4×ΓL(1,𝔽8)—31no—
48D4×ΓL(1,𝔽8)—61no—
49C2×C4×ΓL(1,𝔽8)—21no—
50C2×C4×ΓL(1,𝔽8)—61no—
51C2×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
52C4⋊AGL(1,𝔽7)—32no—
53C4⋊AGL(1,𝔽7)—61no—
54C3×C7⋊C8C22×C312yescomplemented
55ΓL(1,𝔽8)⋊C8—22no—
56ΓL(1,𝔽8)⋊C8C2×D312yescomplemented
57C3×#235—141no—
58C2×C3×D12—71no—
59C32×M16—71no—
60C3×D24—72no—
61C3×#204—72no—
62C2×C3×ΓL(1,𝔽8)—21no—
63C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
64C3⋊AGL(1,𝔽7)—42no—
65D3×ΓL(1,𝔽8)—41no—
66C7×#41—31no—
67C2×D28—31no—
68#1143—32no—
69#1141—32no—
70#1136C3×D311yescharacteristic, complemented
71C3×AGL(1,ℤ/8)—211no—
72#908—71no—
73C22×C3×C7C3×D411yescharacteristic
74C2×D21—22no—
75C2×D21—42no—
76C2×D3×C7—21no—
77C22×ΓL(1,𝔽8)—21no—
78C22×ΓL(1,𝔽8)—61no—
79C22×ΓL(1,𝔽8)D1211yescharacteristic
80C2×AGL(1,𝔽7)—62no—
81C2×AGL(1,𝔽7)—122no—
82C3×C4×C7C3×D411yescharacteristic
83C3×C4×C7C23×C311yescharacteristic
84C7×Dic3—21no—
85C4×ΓL(1,𝔽8)—21no—
86C4×ΓL(1,𝔽8)—21no—
87C4×ΓL(1,𝔽8)—61no—
88C4×ΓL(1,𝔽8)D1211yescharacteristic
89C4×ΓL(1,𝔽8)C22×D311yescharacteristic
90C22×C3×D3—141no—
91C2×C32×C4—141no—
92C3×#67—141no—
93C3×D12—72no—
94C3×D12—141no—
95C3×D12—141no—
96C3×C4×D3—141no—
97C3×Dic6—141no—
98C32×C8—72no—
99C3×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
100C22×D7—61no—
101C7×Q8—31no—
102C7×D4—31no—
103C7×D4—61no—
104C2×C4×C7—61no—
105C2×C4×C7C2×C3×D311yescharacteristic
106D28—32no—
107D28—61no—
108C7⋊C8C2×C3×D312yescomplemented
109C3×#41—421no—
110C2×C3×D4—211no—
111#235AGL(1,𝔽7)11yescharacteristic, complemented
112C2×D12—71no—
113C3×SD16—212no—
114C3×D8—212no—
115C3×M16—71no—
116C3×M16—71no—
117C3×M16—141no—
118D24—72no—
119#204—72no—
120C2×C3×C7—21no—
121C2×C3×C7C2×C3×D411yescharacteristic
122D21—42no—
123D3×C7—41no—
124C2×ΓL(1,𝔽8)—21no—
125C2×ΓL(1,𝔽8)—21no—
126C2×ΓL(1,𝔽8)—41no—
127C2×ΓL(1,𝔽8)—121no—
128C2×ΓL(1,𝔽8)C2×D1211yescharacteristic
129AGL(1,𝔽7)—122no—
130C22×C32—141no—
131C2×C3×D3—141no—
132C2×C3×D3—142no—
133C2×C3×D3—282no—
134C32×C4—141no—
135C32×C4—141no—
136C3×Dic3—141no—
137AGL(1,ℤ/8)—211noSylow
138C22×C7—61no—
139C22×C7C3×D1211yescharacteristic
140C2×D7—62no—
141C2×D7—122no—
142C4×C7—61no—
143C4×C7C3×D1211yescharacteristic
144C4×C7C22×C3×D311yescharacteristic
145C23×C3—421no—
146C22×D3—141no—
147C3×Q8—421no—
148C3×D4—212no—
149C3×D4—421no—
150C3×D4—421no—
151C3×D4—421no—
152C2×C3×C4—141no—
153C2×C3×C4—281no—
154C2×C3×C4—421no—
155C2×C3×C4C2×AGL(1,𝔽7)11yescharacteristic
156#67—21no—
157D12—72no—
158D12—141no—
159D12C2×AGL(1,𝔽7)11yescharacteristic, complemented
160C4×D3—21no—
161Dic6C2×AGL(1,𝔽7)11yescharacteristic, complemented
162C3×C8—72no—
163C3×C8—72no—
164C3×C8—142no—
165C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
166ΓL(1,𝔽8)—21no—
167ΓL(1,𝔽8)#90811yescharacteristic, complemented
168C2×C32—141no—
169C2×C32—281no—
170C3×D3—281no—
171C3×D3—282no—
172#41—31no—
173C2×D4—211no—
174SD16—212no—
175D8—212no—
176M16—71no—
177C2×C7—21no—
178C2×C7—121no—
179C2×C7C2×C3×D1211yescharacteristic
180D7—122no—
181C22×C3—141no—
182C22×C3—281no—
183C22×C3—421no—
184C22×C3—422no—
185C22×C3—842no—
186C22×C3(C22⋊D7)⋊C311yescharacteristic
187C2×D3—21no—
188C2×D3—142no—
189C2×D3—282no—
190C3×C4—141no—
191C3×C4—141no—
192C3×C4—281no—
193C3×C4—281no—
194C3×C4—421no—
195C3×C4(C22⋊D7)⋊C311yescharacteristic
196C3×C4C22×AGL(1,𝔽7)11yescharacteristic
197Dic3—21no—
198C32—141noSylow
199C23—421no—
200Q8—31no—
201D4—31no—
202D4—61no—
203D4—212no—
204D4—421no—
205C2×C4—61no—
206C2×C4D3×AGL(1,𝔽7)11yescharacteristic
207C8—72no—
208C7C3×#90811yescharacteristic, complemented, Sylow
209C2×C3—21no—
210C2×C3—141no—
211C2×C3—281no—
212C2×C3—281no—
213C2×C3—281no—
214C2×C3—841no—
215C2×C3—842no—
216C2×C3C2×(C22⋊D7)⋊C311yescharacteristic
217D3—41no—
218D3—282no—
219C22—61no—
220C22—422no—
221C22—842no—
222C22#4247⋊C311yescharacteristic
223C4—61no—
224C4#4247⋊C311yescharacteristic
225C4C2×D3×AGL(1,𝔽7)11yescharacteristic
226C3—141no—
227C3—281no—
228C3#6512⋊C311yescharacteristic, complemented
229C2—21no—
230C2—121no—
231C2—842no—
232C2C2×#4247⋊C311yescharacteristic
subgroupG/H
centerC2C2×#4247⋊C3
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C2×D3×AGL(1,𝔽7)
FittingC2×C3×C4×C7C22×C3
socleC2×C3×C7C2×C3×D4
hypercenterC2×C4D3×AGL(1,𝔽7)
hypocenterC3×C7C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C3⋊AGL(1,𝔽7))C2semidirect
2. ΓL(1,𝔽8)×#235C2semidirect
3. C3×#1136⋊C3C2semidirect
4. #66568⋊C3C2semidirect
5. #66562⋊C3C2semidirect
6. #10620867C3semidirect
13. ΓL(1,𝔽8)×D12C22semidirect
15. ΓL(1,𝔽8)×Dic6C22semidirect
16. C3×ΓL(1,𝔽8)⋊C8C22semidirect
17. C2×D84C2×C3semidirect
18. C7×#235C2×C3semidirect
21. C3×#1136C2×C3semidirect
22. #66568C2×C3semidirect
23. #66562C2×C3semidirect
27. #1136⋊C3D3semidirect
43. C7×D12C22×C3semidirect
45. C7×Dic6C22×C3semidirect
54. C3×C7⋊C8C22×C3semidirect
56. ΓL(1,𝔽8)⋊C8C2×D3semidirect
70. #1136C3×D3semidirect
99. C3×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
108. C7⋊C8C2×C3×D3semidirect
111. #235AGL(1,𝔽7)semidirect
159. D12C2×AGL(1,𝔽7)semidirect
161. Dic6C2×AGL(1,𝔽7)semidirect
165. C3×C7C3×AGL(1,ℤ/8)semidirect
167. ΓL(1,𝔽8)#908semidirect
208. C7C3×#908semidirect
228. C3#6512⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊AGL(1,𝔽7))2
ΓL(1,𝔽8)×#2352
C3×#1136⋊C32
#66568⋊C32
#66562⋊C32
#106208673
#6512⋊C33
C3×#9087