Finite Group Databasebeta

← ΓL(1,𝔽8)×#560⋊C3 #72277350

Subgroups — ΓL(1,𝔽8)×#560⋊C3 #72277350

Order 4032.

328 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 58 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)×#126⋊C3C212yesmaximal, complemented
2ΓL(1,𝔽8)×#127⋊C3C211yesmaximal, characteristic, complemented
3#560⋊ΓL(1,𝔽8)C312yesmaximal, complemented
4C7×#560⋊C3C311yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×#560C311yesmaximal, characteristic, complemented
6C22×A4×ΓL(1,𝔽8)—41nomaximal
7ΓL(1,𝔽8)×C42⋊C3C2211yescharacteristic, complemented
8#127⋊ΓL(1,𝔽8)C2×C312yescomplemented
9#10621175C2×C312yescomplemented
10#10621174C2×C312yescomplemented
11C7×#127⋊C3C2×C311yescharacteristic, complemented
12C7×#126⋊C3C2×C312yescomplemented
13ΓL(1,𝔽8)×#127C2×C311yescharacteristic, complemented
14ΓL(1,𝔽8)×#126C2×C312yescomplemented
15ΓL(1,𝔽8)×#123—31no—
16ΓL(1,𝔽8)×#121—32no—
17ΓL(1,𝔽8)×#120—31no—
18C3×#560⋊C3—71nomaximal, Hall
19C2×A4×ΓL(1,𝔽8)—41no—
20C2×A4×ΓL(1,𝔽8)—42no—
21C7×#560C3211yescharacteristic, complemented, Hall
22C22×C22⋊ΓL(1,𝔽8)—42no—
23C22×C7×A4—41no—
24C24×ΓL(1,𝔽8)A411yescharacteristic
25C2×D4×ΓL(1,𝔽8)—32no—
26C2×D4×ΓL(1,𝔽8)—31no—
27C2×D4×ΓL(1,𝔽8)—31no—
28C22×C4×ΓL(1,𝔽8)—31no—
29C7⋊(C42⋊C3)C22×C312yescomplemented
30C7×C42⋊C3C22×C311yescharacteristic, complemented
31C4⋊C4×ΓL(1,𝔽8)—32no—
32C22⋊C4×ΓL(1,𝔽8)—31no—
33C22⋊C4×ΓL(1,𝔽8)—32no—
34C22⋊C4×ΓL(1,𝔽8)—31no—
35C42×ΓL(1,𝔽8)C22×C311yescharacteristic, complemented
36C3×#127⋊C3—71no—
37C3×#126⋊C3—72no—
38C22×C3×ΓL(1,𝔽8)—161no—
39A4×ΓL(1,𝔽8)—41no—
40C7×#127C2×C3211yescharacteristic, complemented
41C7×#126C2×C3212yescomplemented
42C7×#123—31no—
43C7×#121—32no—
44C7×#120—31no—
45C3×#560—71no—
46#560⋊C3—72no—
47#560⋊C3ΓL(1,𝔽8)11yescharacteristic, direct product factor
48C2×C22⋊ΓL(1,𝔽8)—42no—
49C2×C22⋊ΓL(1,𝔽8)—42no—
50C2×C22⋊ΓL(1,𝔽8)—42no—
51C2×C7×A4—42no—
52C2×C7×A4—41no—
53C23×ΓL(1,𝔽8)—31no—
54C23×ΓL(1,𝔽8)—121no—
55C23×ΓL(1,𝔽8)C2×A412yes—
56C23×ΓL(1,𝔽8)C2×A411yescharacteristic
57D4×ΓL(1,𝔽8)—122no—
58D4×ΓL(1,𝔽8)—121no—
59D4×ΓL(1,𝔽8)—121no—
60C2×C4×ΓL(1,𝔽8)—31no—
61C2×C4×ΓL(1,𝔽8)—32no—
62C2×C4×ΓL(1,𝔽8)—121no—
63C22×C3×A4—281no—
64C3×C42⋊C3—71no—
65C2×C3×ΓL(1,𝔽8)—161no—
66C2×C3×ΓL(1,𝔽8)—162no—
67C24×C7C3×A411yescharacteristic
68C2×C7×D4—32no—
69C2×C7×D4—31no—
70C2×C7×D4—31no—
71C22×C4×C7—31no—
72C7×C4⋊C4—32no—
73C7×C22⋊C4—31no—
74C7×C22⋊C4—32no—
75C7×C22⋊C4—31no—
76C42×C7C22×C3211yescharacteristic, complemented
77C3×#127—71no—
78C3×#126—72no—
79C3×#123—211no—
80C3×#121—212no—
81C3×#120—211no—
82#127⋊C3—72no—
83#127⋊C3C2×ΓL(1,𝔽8)11yescharacteristic, complemented
84#126⋊C3—72no—
85#126⋊C3—72no—
86#126⋊C3C2×ΓL(1,𝔽8)12yescomplemented
87C22×C3×C7—161no—
88C22⋊ΓL(1,𝔽8)—42no—
89C7×A4—41no—
90C22×ΓL(1,𝔽8)—41no—
91C22×ΓL(1,𝔽8)—62no—
92C22×ΓL(1,𝔽8)—61no—
93C22×ΓL(1,𝔽8)—61no—
94C22×ΓL(1,𝔽8)—62no—
95C22×ΓL(1,𝔽8)—121no—
96C22×ΓL(1,𝔽8)—162no—
97C22×ΓL(1,𝔽8)C22×A411yescharacteristic
98C4×ΓL(1,𝔽8)—62no—
99C4×ΓL(1,𝔽8)—61no—
100C2×C3×A4—282no—
101C2×C3×A4—281no—
102#560C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
103C3×ΓL(1,𝔽8)—161noHall
104C23×C7—31no—
105C23×C7—121no—
106C23×C7C2×C3×A412yes—
107C23×C7C2×C3×A411yescharacteristic
108C7×D4—122no—
109C7×D4—121no—
110C7×D4—121no—
111C2×C4×C7—31no—
112C2×C4×C7—32no—
113C2×C4×C7—121no—
114C24×C3—71no—
115C22×A4—41no—
116C22×A4—282no—
117C2×C3×D4—212no—
118C2×C3×D4—211no—
119C2×C3×D4—211no—
120C22×C3×C4—211no—
121C3×C4⋊C4—212no—
122C3×C22⋊C4—212no—
123C3×C22⋊C4—211no—
124C3×C22⋊C4—211no—
125C3×C42—71no—
126C42⋊C3—72no—
127C42⋊C3C22×ΓL(1,𝔽8)11yescharacteristic, complemented
128C2×C3×C7—162no—
129C2×C3×C7—161no—
130C2×ΓL(1,𝔽8)—31no—
131C2×ΓL(1,𝔽8)—41no—
132C2×ΓL(1,𝔽8)—42no—
133C2×ΓL(1,𝔽8)—121no—
134C2×ΓL(1,𝔽8)—162no—
135C2×ΓL(1,𝔽8)—162no—
136C2×ΓL(1,𝔽8)—162no—
137C22×C32—1121no—
138C3×A4—281no—
139#127C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
140#126C2×C3×ΓL(1,𝔽8)12yescomplemented
141#123—31no—
142#121—32no—
143#120—31no—
144C22×C7—41no—
145C22×C7—62no—
146C22×C7—61no—
147C22×C7—61no—
148C22×C7—62no—
149C22×C7—121no—
150C22×C7C22×C3×A411yescharacteristic
151C4×C7—62no—
152C4×C7—61no—
153C23×C3—72no—
154C23×C3—71no—
155C23×C3—211no—
156C23×C3—841no—
157C2×A4—42no—
158C2×A4—41no—
159C2×A4—282no—
160C2×A4—282no—
161C2×A4—282no—
162C3×D4—842no—
163C3×D4—841no—
164C3×D4—841no—
165C2×C3×C4—211no—
166C2×C3×C4—212no—
167C2×C3×C4—841no—
168C3×C7—161no—
169ΓL(1,𝔽8)—162no—
170ΓL(1,𝔽8)#560⋊C311yescharacteristic, direct product factor
171C2×C32—1122no—
172C2×C32—1121no—
173C24A4×ΓL(1,𝔽8)11yescharacteristic
174C2×D4—31no—
175C2×D4—32no—
176C2×D4—31no—
177C22×C4—31no—
178C4⋊C4—32no—
179C22⋊C4—31no—
180C22⋊C4—32no—
181C22⋊C4—31no—
182C42C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
183C2×C7—31no—
184C2×C7—42no—
185C2×C7—41no—
186C2×C7—121no—
187C22×C3—71no—
188C22×C3—161no—
189C22×C3—281no—
190C22×C3—422no—
191C22×C3—421no—
192C22×C3—421no—
193C22×C3—422no—
194C22×C3—841no—
195C22×C3—1122no—
196A4—41no—
197A4—282no—
198C3×C4—422no—
199C3×C4—421no—
200C32—1121noSylow
201C23—31no—
202C23—121no—
203C23C2×A4×ΓL(1,𝔽8)12yes—
204C23C2×A4×ΓL(1,𝔽8)11yescharacteristic
205D4—122no—
206D4—121no—
207D4—121no—
208C2×C4—31no—
209C2×C4—32no—
210C2×C4—121no—
211C7C3×#560⋊C311yescharacteristic, complemented, Sylow
212C2×C3—162no—
213C2×C3—161no—
214C2×C3—211no—
215C2×C3—282no—
216C2×C3—281no—
217C2×C3—841no—
218C2×C3—1122no—
219C2×C3—1122no—
220C2×C3—1122no—
221C22—41no—
222C22—62no—
223C22—61no—
224C22—61no—
225C22—62no—
226C22—121no—
227C22C22×A4×ΓL(1,𝔽8)11yescharacteristic
228C4—62no—
229C4—61no—
230C3—71no—
231C3—161no—
232C3—1122no—
233C2—31no—
234C2—42no—
235C2—41no—
236C2—121no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC42×C7C22×C32
FrattiniC22C22×A4×ΓL(1,𝔽8)
FittingC7×#560C32
socleC22×C7C22×C3×A4
LCSG ▷ C42×C7
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#126⋊C3C2semidirect
2. ΓL(1,𝔽8)×#127⋊C3C2semidirect
3. #560⋊ΓL(1,𝔽8)C3semidirect
4. C7×#560⋊C3C3semidirect
5. ΓL(1,𝔽8)×#560C3semidirect
7. ΓL(1,𝔽8)×C42⋊C3C22semidirect
8. #127⋊ΓL(1,𝔽8)C2×C3semidirect
9. #10621175C2×C3semidirect
10. #10621174C2×C3semidirect
11. C7×#127⋊C3C2×C3semidirect
12. C7×#126⋊C3C2×C3semidirect
13. ΓL(1,𝔽8)×#127C2×C3semidirect
14. ΓL(1,𝔽8)×#126C2×C3semidirect
21. C7×#560C32semidirect
29. C7⋊(C42⋊C3)C22×C3semidirect
30. C7×C42⋊C3C22×C3semidirect
35. C42×ΓL(1,𝔽8)C22×C3semidirect
40. C7×#127C2×C32semidirect
41. C7×#126C2×C32semidirect
47. #560⋊C3ΓL(1,𝔽8)direct (N × H)
76. C42×C7C22×C32semidirect
83. #127⋊C3C2×ΓL(1,𝔽8)semidirect
86. #126⋊C3C2×ΓL(1,𝔽8)semidirect
102. #560C3×ΓL(1,𝔽8)semidirect
127. C42⋊C3C22×ΓL(1,𝔽8)semidirect
139. #127C2×C3×ΓL(1,𝔽8)semidirect
140. #126C2×C3×ΓL(1,𝔽8)semidirect
170. ΓL(1,𝔽8)#560⋊C3direct (N × H)
182. C42C22×C3×ΓL(1,𝔽8)semidirect
211. C7C3×#560⋊C3semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#126⋊C32
ΓL(1,𝔽8)×#127⋊C32
#560⋊ΓL(1,𝔽8)3
C7×#560⋊C33
ΓL(1,𝔽8)×#5603
C22×A4×ΓL(1,𝔽8)4
C3×#560⋊C37