Finite Group Databasebeta

← #72245554

Subgroups — #72245554

Order 2016.

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

normal subgroups only

This group has 312 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×ΓL(1,𝔽8)×D12C211yesmaximal, characteristic, complemented
2C3×#1157⋊C3C211yesmaximal, characteristic, complemented
3#66627⋊C3C211yesmaximal, characteristic, complemented
4#66564⋊C3C212yesmaximal, complemented
5#66560⋊C3C212yesmaximal, complemented
6#10620861C311yesmaximal, characteristic, complemented
7#6495⋊C3—31nomaximal
8C22×D3×ΓL(1,𝔽8)—21no—
9C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
10ΓL(1,𝔽8)×D12—21no—
11ΓL(1,𝔽8)×D12C2212yes—
12C3×(C22⋊D7)⋊C3—21no—
13C3×C4⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
14C3×C4×AGL(1,𝔽7)—21no—
15C3×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic, complemented
16C3⋊(ΓL(1,𝔽8)⋊C8)C2212yescomplemented
17C2×C7×D12C2×C311yescharacteristic, complemented
18C3×#1157C2×C311yescharacteristic, complemented
19C2×D4×ΓL(1,𝔽8)—31no—
20#1157⋊C3—21no—
21#1157⋊C3D311yescharacteristic, complemented
22#66627C2×C311yescharacteristic, complemented
23#66564C2×C312yescomplemented
24#66560C2×C312yescomplemented
25#1142⋊C3—32no—
26#1141⋊C3—32no—
27#1136⋊C3—31no—
28C3×#949—71nomaximal, Hall
29C22×C3×ΓL(1,𝔽8)D411yescharacteristic
30C2×D3×ΓL(1,𝔽8)—22no—
31C2×D3×ΓL(1,𝔽8)—42no—
32C2×C3×AGL(1,𝔽7)—21no—
33C3×C4×ΓL(1,𝔽8)D411yescharacteristic
34C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
35C3×ΓL(1,𝔽8)⋊C4—21no—
36#6495—31noHall
37C22×D3×C7—21no—
38C23×ΓL(1,𝔽8)—61no—
39C2×C3×C4×C7C22×C311yescharacteristic
40C7×D12—21no—
41C7×D12C22×C312yes—
42C3×C22⋊D7—21no—
43C3×D28C22×C311yescharacteristic, complemented
44C3×C4×D7—21no—
45C3×Dic14C22×C311yescharacteristic, complemented
46D4×ΓL(1,𝔽8)—32no—
47D4×ΓL(1,𝔽8)—61no—
48C2×C4×ΓL(1,𝔽8)—21no—
49C2×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
50(C22⋊D7)⋊C3—21no—
51(C22⋊D7)⋊C3—21no—
52(C22⋊D7)⋊C3—21no—
53C4⋊AGL(1,𝔽7)—21no—
54C4⋊AGL(1,𝔽7)C2×D311yescharacteristic, complemented
55C4×AGL(1,𝔽7)—21no—
56C4×AGL(1,𝔽7)—21no—
57C4×AGL(1,𝔽7)—21no—
58ΓL(1,𝔽8)⋊Q8—21no—
59ΓL(1,𝔽8)⋊Q8C2×D311yescharacteristic, complemented
60C3⋊(C7⋊C8)C22×C312yescomplemented
61ΓL(1,𝔽8)⋊C8—32no—
62C32×#41—71no—
63C2×C3×D12—141no—
64C3×#215—72no—
65C3×#213—72no—
66C3×#208—71no—
67C2×C3×ΓL(1,𝔽8)—21no—
68C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
69D3×ΓL(1,𝔽8)—42no—
70C3×AGL(1,𝔽7)—41no—
71C2×C7×D4—31no—
72#1157C3×D311yescharacteristic, complemented
73#1142—32no—
74#1141—32no—
75#1136—31no—
76C3×AGL(1,ℤ/8)—211no—
77#949—71no—
78C22×C3×C7C3×D411yescharacteristic
79C2×D3×C7—22no—
80C2×D3×C7—42no—
81C2×C3×D7—21no—
82C22×ΓL(1,𝔽8)—21no—
83C22×ΓL(1,𝔽8)—62no—
84C22×ΓL(1,𝔽8)—122no—
85C22×ΓL(1,𝔽8)#6711yescharacteristic
86C2×AGL(1,𝔽7)—21no—
87C2×AGL(1,𝔽7)—21no—
88C2×AGL(1,𝔽7)—21no—
89C3×C4×C7C3×D411yescharacteristic
90C3×C4×C7C23×C311yescharacteristic
91C3×Dic7—21no—
92C4×ΓL(1,𝔽8)—21no—
93C4×ΓL(1,𝔽8)—21no—
94C4×ΓL(1,𝔽8)#6711yescharacteristic
95C4×ΓL(1,𝔽8)C22×D311yescharacteristic
96ΓL(1,𝔽8)⋊C4—21no—
97ΓL(1,𝔽8)⋊C4—21no—
98ΓL(1,𝔽8)⋊C4—21no—
99C22×C3×D3—141no—
100C32×Q8—71no—
101C32×D4—71no—
102C32×D4—141no—
103C2×C32×C4—141no—
104C2×C32×C4—141no—
105C3×D12—141no—
106C3×D12—142no—
107C3×C3⋊C8—72no—
108C3×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
109C23×C7—61no—
110C7×D4—32no—
111C7×D4—61no—
112C2×C4×C7C2×C3×D311yescharacteristic
113C22⋊D7—21no—
114D28C2×C3×D311yescharacteristic, complemented
115C4×D7—21no—
116Dic14C2×C3×D311yescharacteristic, complemented
117C7⋊C8—32no—
118C3×#41—71no—
119C3×#41—71no—
120C3×#41—141no—
121C2×C3×D4—211no—
122C2×D12AGL(1,𝔽7)11yescharacteristic, complemented
123C3×SD16—212no—
124C3×D8—212no—
125C3×M16—211no—
126#215—72no—
127#213—72no—
128#208—71no—
129C2×C3×C7—21no—
130C2×C3×C7C2×C3×D411yescharacteristic
131C3×D7—41no—
132D3×C7—42no—
133C2×ΓL(1,𝔽8)—21no—
134C2×ΓL(1,𝔽8)—21no—
135C2×ΓL(1,𝔽8)—41no—
136C2×ΓL(1,𝔽8)—122no—
137C2×ΓL(1,𝔽8)C2×#6711yescharacteristic
138AGL(1,𝔽7)—41no—
139AGL(1,𝔽7)—41no—
140AGL(1,𝔽7)—41no—
141C22×C32—141no—
142C22×C32—141no—
143C2×C3×D3—142no—
144C2×C3×D3—282no—
145C32×C4—141no—
146C32×C4—141no—
147C32×C4—141no—
148AGL(1,ℤ/8)—211noSylow
149C22×C7—62no—
150C22×C7—122no—
151C22×C7C3×#6711yescharacteristic
152C2×D7—21no—
153C4×C7C3×#6711yescharacteristic
154C4×C7C22×C3×D311yescharacteristic
155Dic7—21no—
156C23×C3—421no—
157C22×D3—21no—
158C3×Q8—71no—
159C3×Q8—71no—
160C3×Q8—141no—
161C3×D4—71no—
162C3×D4—71no—
163C3×D4—141no—
164C3×D4—141no—
165C3×D4—141no—
166C3×D4—141no—
167C3×D4—141no—
168C3×D4—212no—
169C3×D4—421no—
170C2×C3×C4—71no—
171C2×C3×C4—141no—
172C2×C3×C4—141no—
173C2×C3×C4—141no—
174C2×C3×C4—141no—
175C2×C3×C4—281no—
176C2×C3×C4C2×AGL(1,𝔽7)11yescharacteristic
177D12—21no—
178D12C2×AGL(1,𝔽7)12yes—
179C3×C8—212no—
180C3⋊C8—72no—
181C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
182ΓL(1,𝔽8)—21no—
183ΓL(1,𝔽8)#94911yescharacteristic, complemented
184C2×C32—141no—
185C2×C32—141no—
186C2×C32—281no—
187C3×D3—282no—
188#41—71no—
189C2×D4—31no—
190SD16—212no—
191D8—212no—
192M16—211no—
193C2×C7—21no—
194C2×C7—122no—
195C2×C7C2×C3×#6711yescharacteristic
196D7—41no—
197C22×C3—71no—
198C22×C3—141no—
199C22×C3—141no—
200C22×C3—141no—
201C22×C3—141no—
202C22×C3—281no—
203C22×C3—422no—
204C22×C3—842no—
205C22×C3(C22⋊D7)⋊C311yescharacteristic
206C2×D3—22no—
207C2×D3—42no—
208C3×C4—71no—
209C3×C4—71no—
210C3×C4—141no—
211C3×C4—141no—
212C3×C4—141no—
213C3×C4—141no—
214C3×C4—281no—
215C3×C4—281no—
216C3×C4(C22⋊D7)⋊C311yescharacteristic
217C3×C4C22×AGL(1,𝔽7)11yescharacteristic
218C32—141noSylow
219C23—61no—
220Q8—71no—
221D4—32no—
222D4—61no—
223D4—71no—
224D4—141no—
225C2×C4—141no—
226C2×C4D3×AGL(1,𝔽7)11yescharacteristic
227C8—212no—
228C7C3×#94911yescharacteristic, complemented, Sylow
229C2×C3—21no—
230C2×C3—71no—
231C2×C3—141no—
232C2×C3—281no—
233C2×C3—281no—
234C2×C3—281no—
235C2×C3—281no—
236C2×C3—281no—
237C2×C3—281no—
238C2×C3—842no—
239C2×C3C2×(C22⋊D7)⋊C311yescharacteristic
240D3—42no—
241C22—62no—
242C22—122no—
243C22—141no—
244C22#4245⋊C311yescharacteristic
245C4—141no—
246C4#4245⋊C311yescharacteristic
247C4C2×D3×AGL(1,𝔽7)11yescharacteristic
248C3—71no—
249C3—281no—
250C3#6495⋊C311yescharacteristic, complemented
251C2—21no—
252C2—122no—
253C2—281no—
254C2C2×#4245⋊C311yescharacteristic
subgroupG/H
centerC2C2×#4245⋊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×ΓL(1,𝔽8)×D12C2semidirect
2. C3×#1157⋊C3C2semidirect
3. #66627⋊C3C2semidirect
4. #66564⋊C3C2semidirect
5. #66560⋊C3C2semidirect
6. #10620861C3semidirect
13. C3×C4⋊AGL(1,𝔽7)C22semidirect
15. C3×ΓL(1,𝔽8)⋊Q8C22semidirect
16. C3⋊(ΓL(1,𝔽8)⋊C8)C22semidirect
17. C2×C7×D12C2×C3semidirect
18. C3×#1157C2×C3semidirect
21. #1157⋊C3D3semidirect
22. #66627C2×C3semidirect
23. #66564C2×C3semidirect
24. #66560C2×C3semidirect
43. C3×D28C22×C3semidirect
45. C3×Dic14C22×C3semidirect
54. C4⋊AGL(1,𝔽7)C2×D3semidirect
59. ΓL(1,𝔽8)⋊Q8C2×D3semidirect
60. C3⋊(C7⋊C8)C22×C3semidirect
72. #1157C3×D3semidirect
108. C3×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
114. D28C2×C3×D3semidirect
116. Dic14C2×C3×D3semidirect
122. C2×D12AGL(1,𝔽7)semidirect
181. C3×C7C3×AGL(1,ℤ/8)semidirect
183. ΓL(1,𝔽8)#949semidirect
228. C7C3×#949semidirect
250. C3#6495⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×D122
C3×#1157⋊C32
#66627⋊C32
#66564⋊C32
#66560⋊C32
#106208613
#6495⋊C33
C3×#9497