Finite Group Databasebeta

← #72245538

Subgroups — #72245538

Order 2016.

326 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#66728⋊C3C211yesmaximal, characteristic, complemented
3#66633⋊C3C212yesmaximal, complemented
4#66632⋊C3C212yesmaximal, complemented
5#66627⋊C3C211yesmaximal, characteristic, complemented
6#10621415C311yesmaximal, characteristic, complemented
7#6495⋊C3—31nomaximal
8C23×C3×ΓL(1,𝔽8)—21no—
9C3×D4×ΓL(1,𝔽8)—21no—
10C3×D4×ΓL(1,𝔽8)C2212yes—
11C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
12#4268⋊C3—21no—
13C4⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic, complemented
14C4×C3⋊AGL(1,𝔽7)—21no—
15Dic42⋊C3C2211yescharacteristic, complemented
16C3⋊(ΓL(1,𝔽8)⋊C8)C2212yescomplemented
17C2×C3×C7×D4C2×C311yescharacteristic, complemented
18#66728C2×C311yescharacteristic, complemented
19C2×D4×ΓL(1,𝔽8)—21no—
20C2×D4×ΓL(1,𝔽8)D311yescharacteristic, complemented
21#1157⋊C3—31no—
22#66633C2×C312yescomplemented
23#66632C2×C312yescomplemented
24#66627C2×C311yescharacteristic, complemented
25#1142⋊C3—32no—
26#1141⋊C3—32no—
27#1136⋊C3—31no—
28C3×#932—71nomaximal, Hall
29C22×C3×ΓL(1,𝔽8)—22no—
30C22×C3×ΓL(1,𝔽8)—42no—
31C22×C3×ΓL(1,𝔽8)D411yescharacteristic
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
37C23×C3×C7—21no—
38C23×ΓL(1,𝔽8)—21no—
39C23×ΓL(1,𝔽8)—21no—
40C23×ΓL(1,𝔽8)—21no—
41C3×C7×D4—21no—
42C3×C7×D4C22×C312yes—
43C2×C3×C4×C7C22×C311yescharacteristic
44#4268—21no—
45D84C22×C311yescharacteristic, complemented
46C4×D21—21no—
47Dic42C22×C311yescharacteristic, complemented
48D4×ΓL(1,𝔽8)—21no—
49D4×ΓL(1,𝔽8)—21no—
50D4×ΓL(1,𝔽8)—21no—
51D4×ΓL(1,𝔽8)—22no—
52D4×ΓL(1,𝔽8)C2×D312yes—
53C2×C4×ΓL(1,𝔽8)—21no—
54C2×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
55(C22⋊D7)⋊C3—61no—
56C4⋊AGL(1,𝔽7)—31no—
57C4×AGL(1,𝔽7)—61no—
58ΓL(1,𝔽8)⋊Q8—31no—
59C3⋊(C7⋊C8)C22×C312yescomplemented
60ΓL(1,𝔽8)⋊C8—32no—
61C2×C32×D4—141no—
62C3×#235—71no—
63C3×#214—72no—
64C3×#213—72no—
65C3×#208—71no—
66C2×C3×ΓL(1,𝔽8)—21no—
67C2×C3×ΓL(1,𝔽8)—42no—
68C2×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
69C3⋊AGL(1,𝔽7)—41no—
70C2×C7×D4C3×D311yescharacteristic, complemented
71#1157—31no—
72#1142—32no—
73#1141—32no—
74#1136—31no—
75C3×AGL(1,ℤ/8)—211no—
76#932—71no—
77C22×C3×C7—22no—
78C22×C3×C7—42no—
79C22×C3×C7C3×D411yescharacteristic
80C2×D21—21no—
81C22×ΓL(1,𝔽8)—22no—
82C22×ΓL(1,𝔽8)—22no—
83C22×ΓL(1,𝔽8)—22no—
84C22×ΓL(1,𝔽8)—21no—
85C22×ΓL(1,𝔽8)—42no—
86C22×ΓL(1,𝔽8)—42no—
87C22×ΓL(1,𝔽8)—42no—
88C22×ΓL(1,𝔽8)#6711yescharacteristic
89C2×AGL(1,𝔽7)—61no—
90C3×C4×C7C3×D411yescharacteristic
91C3×C4×C7C23×C311yescharacteristic
92Dic21—21no—
93C4×ΓL(1,𝔽8)—21no—
94C4×ΓL(1,𝔽8)—21no—
95C4×ΓL(1,𝔽8)#6711yescharacteristic
96C4×ΓL(1,𝔽8)C22×D311yescharacteristic
97ΓL(1,𝔽8)⋊C4—61no—
98C23×C32—141no—
99C32×D4—141no—
100C32×D4—142no—
101C2×C32×C4—141no—
102C3×#67—141no—
103C3×D12—71no—
104C3×C4×D3—141no—
105C3×Dic6—71no—
106C3×C3⋊C8—72no—
107C3×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
108C23×C7—21no—
109C7×D4—21no—
110C7×D4C2×C3×D312yes—
111C2×C4×C7C2×C3×D311yescharacteristic
112C22⋊D7—61no—
113D28—31no—
114C4×D7—61no—
115Dic14—31no—
116C7⋊C8—32no—
117C3×#41—211no—
118C2×C3×D4—71no—
119C2×C3×D4—141no—
120C2×C3×D4AGL(1,𝔽7)11yescharacteristic, complemented
121#235—71no—
122C3×SD16—212no—
123C3×D8—212no—
124C3×M16—211no—
125#214—72no—
126#213—72no—
127#208—71no—
128C2×C3×C7—21no—
129C2×C3×C7—42no—
130C2×C3×C7C2×C3×D411yescharacteristic
131D21—41no—
132C2×ΓL(1,𝔽8)—21no—
133C2×ΓL(1,𝔽8)—21no—
134C2×ΓL(1,𝔽8)—21no—
135C2×ΓL(1,𝔽8)—21no—
136C2×ΓL(1,𝔽8)—42no—
137C2×ΓL(1,𝔽8)—42no—
138C2×ΓL(1,𝔽8)—42no—
139C2×ΓL(1,𝔽8)C2×#6711yescharacteristic
140AGL(1,𝔽7)—121no—
141C22×C32—142no—
142C22×C32—141no—
143C22×C32—282no—
144C2×C3×D3—141no—
145C32×C4—141no—
146C32×C4—141no—
147C3×Dic3—141no—
148AGL(1,ℤ/8)—211noSylow
149C22×C7—22no—
150C22×C7—42no—
151C22×C7C3×#6711yescharacteristic
152C2×D7—61no—
153C4×C7C3×#6711yescharacteristic
154C4×C7C22×C3×D311yescharacteristic
155Dic7—61no—
156C23×C3—21no—
157C23×C3—141no—
158C23×C3—141no—
159C23×C3—141no—
160C3×Q8—211no—
161C3×D4—21no—
162C3×D4—72no—
163C3×D4—141no—
164C3×D4—141no—
165C3×D4—141no—
166C3×D4—142no—
167C3×D4—211no—
168C3×D4—421no—
169C3×D4C2×AGL(1,𝔽7)12yes—
170C2×C3×C4—71no—
171C2×C3×C4—141no—
172C2×C3×C4—421no—
173C2×C3×C4C2×AGL(1,𝔽7)11yescharacteristic
174#67—141no—
175D12—71no—
176C4×D3—141no—
177Dic6—71no—
178C3×C8—212no—
179C3⋊C8—72no—
180C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
181ΓL(1,𝔽8)—21no—
182ΓL(1,𝔽8)#93211yescharacteristic, complemented
183C2×C32—141no—
184C2×C32—141no—
185C2×C32—282no—
186C3×D3—281no—
187#41—211no—
188C2×D4C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
189SD16—212no—
190D8—212no—
191M16—211no—
192C2×C7—21no—
193C2×C7—42no—
194C2×C7C2×C3×#6711yescharacteristic
195D7—121no—
196C22×C3—22no—
197C22×C3—42no—
198C22×C3—71no—
199C22×C3—142no—
200C22×C3—142no—
201C22×C3—142no—
202C22×C3—141no—
203C22×C3—282no—
204C22×C3—282no—
205C22×C3—282no—
206C22×C3—421no—
207C22×C3(C22⋊D7)⋊C311yescharacteristic
208C2×D3—141no—
209C3×C4—71no—
210C3×C4—71no—
211C3×C4—141no—
212C3×C4—141no—
213C3×C4—421no—
214C3×C4(C22⋊D7)⋊C311yescharacteristic
215C3×C4C22×AGL(1,𝔽7)11yescharacteristic
216Dic3—141no—
217C32—141noSylow
218C23—21no—
219Q8—211no—
220D4—21no—
221D4—211no—
222D4—421no—
223D4C2×C3⋊AGL(1,𝔽7)12yes—
224C2×C4—421no—
225C2×C4C2×C3⋊AGL(1,𝔽7)11yescharacteristic
226C8—212no—
227C7C3×#93211yescharacteristic, complemented, Sylow
228C2×C3—21no—
229C2×C3—42no—
230C2×C3—71no—
231C2×C3—141no—
232C2×C3—141no—
233C2×C3—141no—
234C2×C3—141no—
235C2×C3—282no—
236C2×C3—282no—
237C2×C3—282no—
238C2×C3—841no—
239C2×C3C2×(C22⋊D7)⋊C311yescharacteristic
240D3—281no—
241C22—22no—
242C22—42no—
243C22—421no—
244C22#4268⋊C311yescharacteristic
245C4—421no—
246C4#4268⋊C311yescharacteristic
247C4C22×C3⋊AGL(1,𝔽7)11yescharacteristic
248C3—71no—
249C3—141no—
250C3#6495⋊C311yescharacteristic, complemented
251C2—21no—
252C2—42no—
253C2—841no—
254C2C2×#4268⋊C311yescharacteristic
subgroupG/H
centerC2C2×#4268⋊C3
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C22×C3⋊AGL(1,𝔽7)
FittingC2×C3×C7×D4C2×C3
socleC2×C3×C7C2×C3×D4
hypercenterC2×D4C3⋊AGL(1,𝔽7)
hypocenterC3×C7C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×ΓL(1,𝔽8)C2semidirect
2. #66728⋊C3C2semidirect
3. #66633⋊C3C2semidirect
4. #66632⋊C3C2semidirect
5. #66627⋊C3C2semidirect
6. #10621415C3semidirect
13. C4⋊(C3⋊AGL(1,𝔽7))C22semidirect
15. Dic42⋊C3C22semidirect
16. C3⋊(ΓL(1,𝔽8)⋊C8)C22semidirect
17. C2×C3×C7×D4C2×C3semidirect
18. #66728C2×C3semidirect
20. C2×D4×ΓL(1,𝔽8)D3semidirect
22. #66633C2×C3semidirect
23. #66632C2×C3semidirect
24. #66627C2×C3semidirect
45. D84C22×C3semidirect
47. Dic42C22×C3semidirect
59. C3⋊(C7⋊C8)C22×C3semidirect
70. C2×C7×D4C3×D3semidirect
107. C3×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
120. C2×C3×D4AGL(1,𝔽7)semidirect
180. C3×C7C3×AGL(1,ℤ/8)semidirect
182. ΓL(1,𝔽8)#932semidirect
188. C2×D4C3⋊AGL(1,𝔽7)semidirect
227. C7C3×#932semidirect
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×C3×D4×ΓL(1,𝔽8)2
#66728⋊C32
#66633⋊C32
#66632⋊C32
#66627⋊C32
#106214153
#6495⋊C33
C3×#9327