Finite Group Databasebeta

← ΓL(1,𝔽8)×#908 #72245512

Subgroups — ΓL(1,𝔽8)×#908 #72245512

Order 2016.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 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)×#235C211yesmaximal, characteristic, complemented
2C2×ΓL(1,𝔽8)×D12C211yesmaximal, characteristic, complemented
3C3×M16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×D24C212yesmaximal, complemented
5ΓL(1,𝔽8)×#204C212yesmaximal, complemented
6C7×#908C311yesmaximal, characteristic, complemented
7ΓL(1,𝔽8)×AGL(1,ℤ/8)—31nomaximal
8C22×D3×ΓL(1,𝔽8)—21no—
9C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
10ΓL(1,𝔽8)×#67—21no—
11ΓL(1,𝔽8)×D12—21no—
12ΓL(1,𝔽8)×D12C2212yes—
13ΓL(1,𝔽8)×D12C2211yescharacteristic, complemented
14C4×D3×ΓL(1,𝔽8)—21no—
15ΓL(1,𝔽8)×Dic6C2211yescharacteristic, complemented
16C3×C8×ΓL(1,𝔽8)C2212yescomplemented
17C7×#235C2×C311yescharacteristic, complemented
18C2×C7×D12C2×C311yescharacteristic, complemented
19#41×ΓL(1,𝔽8)—31no—
20C2×D4×ΓL(1,𝔽8)—31no—
21C3×C7×M16C2×C311yescharacteristic, complemented
22C7×D24C2×C312yescomplemented
23C7×#204C2×C312yescomplemented
24SD16×ΓL(1,𝔽8)—32no—
25D8×ΓL(1,𝔽8)—32no—
26M16×ΓL(1,𝔽8)—21no—
27M16×ΓL(1,𝔽8)D311yescharacteristic, complemented
28C3×#908—71nomaximal, Hall
29C22×C3×ΓL(1,𝔽8)D411yescharacteristic
30C2×D3×ΓL(1,𝔽8)—22no—
31C2×D3×ΓL(1,𝔽8)—21no—
32C2×D3×ΓL(1,𝔽8)—42no—
33C3×C4×ΓL(1,𝔽8)D411yescharacteristic
34C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
35Dic3×ΓL(1,𝔽8)—21no—
36C7×AGL(1,ℤ/8)—31noHall
37C22×D3×C7—21no—
38C23×ΓL(1,𝔽8)—61no—
39C2×C3×C4×C7C22×C311yescharacteristic
40C7×#67—21no—
41C7×D12—21no—
42C7×D12C22×C312yes—
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)—32no—
49D4×ΓL(1,𝔽8)—61no—
50D4×ΓL(1,𝔽8)—61no—
51C2×C4×ΓL(1,𝔽8)—21no—
52C2×C4×ΓL(1,𝔽8)—61no—
53C2×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
54C3×C7×C8C22×C312yescomplemented
55C8×ΓL(1,𝔽8)—22no—
56C8×ΓL(1,𝔽8)C2×D312yescomplemented
57C3×#235—71no—
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
64D3×ΓL(1,𝔽8)—41no—
65D3×ΓL(1,𝔽8)—42no—
66C7×#41—31no—
67C2×C7×D4—31no—
68C7×SD16—32no—
69C7×D8—32no—
70C7×M16C3×D311yescharacteristic, complemented
71C3×AGL(1,ℤ/8)—211no—
72#908ΓL(1,𝔽8)11yescharacteristic, direct product factor
73C22×C3×C7C3×D411yescharacteristic
74C2×D3×C7—21no—
75C2×D3×C7—22no—
76C2×D3×C7—42no—
77C22×ΓL(1,𝔽8)—21no—
78C22×ΓL(1,𝔽8)—61no—
79C22×ΓL(1,𝔽8)—62no—
80C22×ΓL(1,𝔽8)—122no—
81C22×ΓL(1,𝔽8)D1211yescharacteristic
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—71no—
92C3×#67—141no—
93C3×D12—72no—
94C3×D12—71no—
95C3×D12—141no—
96C3×C4×D3—141no—
97C3×Dic6—71no—
98C32×C8—72no—
99C3×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
100C23×C7—61no—
101C7×Q8—31no—
102C7×D4—31no—
103C7×D4—32no—
104C7×D4—61no—
105C7×D4—61no—
106C2×C4×C7—61no—
107C2×C4×C7C2×C3×D311yescharacteristic
108C7×C8C2×C3×D312yescomplemented
109C3×#41—211no—
110C2×C3×D4—211no—
111#235C2×ΓL(1,𝔽8)11yescharacteristic, complemented
112C2×D12C2×ΓL(1,𝔽8)11yescharacteristic, complemented
113C3×SD16—212no—
114C3×D8—212no—
115C3×M16—71no—
116C3×M16—141no—
117C3×M16C2×ΓL(1,𝔽8)11yescharacteristic, complemented
118D24C2×ΓL(1,𝔽8)12yescomplemented
119#204C2×ΓL(1,𝔽8)12yescomplemented
120C2×C3×C7—21no—
121C2×C3×C7C2×C3×D411yescharacteristic
122D3×C7—41no—
123D3×C7—42no—
124C2×ΓL(1,𝔽8)—21no—
125C2×ΓL(1,𝔽8)—21no—
126C2×ΓL(1,𝔽8)—41no—
127C2×ΓL(1,𝔽8)—121no—
128C2×ΓL(1,𝔽8)—122no—
129C2×ΓL(1,𝔽8)C2×D1211yescharacteristic
130C22×C32—71no—
131C2×C3×D3—141no—
132C2×C3×D3—142no—
133C2×C3×D3—282no—
134C32×C4—71no—
135C32×C4—71no—
136C3×Dic3—141no—
137AGL(1,ℤ/8)—31noSylow
138C22×C7—61no—
139C22×C7—62no—
140C22×C7—122no—
141C22×C7C3×D1211yescharacteristic
142C4×C7—61no—
143C4×C7C3×D1211yescharacteristic
144C4×C7C22×C3×D311yescharacteristic
145C23×C3—421no—
146C22×D3—21no—
147C3×Q8—211no—
148C3×D4—211no—
149C3×D4—212no—
150C3×D4—421no—
151C3×D4—421no—
152C2×C3×C4—71no—
153C2×C3×C4—141no—
154C2×C3×C4—421no—
155C2×C3×C4C22×ΓL(1,𝔽8)11yescharacteristic
156#67—21no—
157D12—21no—
158D12C22×ΓL(1,𝔽8)12yes—
159D12C22×ΓL(1,𝔽8)11yescharacteristic, complemented
160C4×D3—21no—
161Dic6C22×ΓL(1,𝔽8)11yescharacteristic, complemented
162C3×C8—72no—
163C3×C8—142no—
164C3×C8C22×ΓL(1,𝔽8)12yescomplemented
165C3×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented
166ΓL(1,𝔽8)—21no—
167ΓL(1,𝔽8)#90811yescharacteristic, direct product factor
168C2×C32—71no—
169C2×C32—141no—
170C3×D3—282no—
171C3×D3—281no—
172#41—31no—
173C2×D4—31no—
174SD16—32no—
175D8—32no—
176M16D3×ΓL(1,𝔽8)11yescharacteristic, complemented
177C2×C7—21no—
178C2×C7—122no—
179C2×C7—121no—
180C2×C7C2×C3×D1211yescharacteristic
181C22×C3—71no—
182C22×C3—141no—
183C22×C3—421no—
184C22×C3—422no—
185C22×C3—842no—
186C22×C3D4×ΓL(1,𝔽8)11yescharacteristic
187C2×D3—21no—
188C2×D3—22no—
189C2×D3—42no—
190C3×C4—71no—
191C3×C4—71no—
192C3×C4—141no—
193C3×C4—141no—
194C3×C4—421no—
195C3×C4D4×ΓL(1,𝔽8)11yescharacteristic
196C3×C4C23×ΓL(1,𝔽8)11yescharacteristic
197Dic3—21no—
198C32—71noSylow
199C23—61no—
200Q8—31no—
201D4—31no—
202D4—32no—
203D4—61no—
204D4—61no—
205C2×C4—61no—
206C2×C4C2×D3×ΓL(1,𝔽8)11yescharacteristic
207C8C2×D3×ΓL(1,𝔽8)12yescomplemented
208C7C3×#90811yescharacteristic, complemented, Sylow
209C2×C3—21no—
210C2×C3—71no—
211C2×C3—141no—
212C2×C3—141no—
213C2×C3—281no—
214C2×C3—842no—
215C2×C3—841no—
216C2×C3C2×D4×ΓL(1,𝔽8)11yescharacteristic
217D3—42no—
218D3—41no—
219C22—61no—
220C22—62no—
221C22—122no—
222C22ΓL(1,𝔽8)×D1211yescharacteristic
223C4—61no—
224C4ΓL(1,𝔽8)×D1211yescharacteristic
225C4C22×D3×ΓL(1,𝔽8)11yescharacteristic
226C3—71no—
227C3—141no—
228C3ΓL(1,𝔽8)×AGL(1,ℤ/8)11yescharacteristic, complemented
229C2—21no—
230C2—122no—
231C2—121no—
232C2C2×ΓL(1,𝔽8)×D1211yescharacteristic
subgroupG/H
centerC2C2×ΓL(1,𝔽8)×D12
commutator, cosocleC3×C4×C7C23×C3
FrattiniC4C22×D3×ΓL(1,𝔽8)
FittingC3×C7×M16C2×C3
socleC2×C3×C7C2×C3×D4
hypercenterM16D3×ΓL(1,𝔽8)
hypocenterC3×C7C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#235C2semidirect
2. C2×ΓL(1,𝔽8)×D12C2semidirect
3. C3×M16×ΓL(1,𝔽8)C2semidirect
4. ΓL(1,𝔽8)×D24C2semidirect
5. ΓL(1,𝔽8)×#204C2semidirect
6. C7×#908C3semidirect
13. ΓL(1,𝔽8)×D12C22semidirect
15. ΓL(1,𝔽8)×Dic6C22semidirect
16. C3×C8×ΓL(1,𝔽8)C22semidirect
17. C7×#235C2×C3semidirect
18. C2×C7×D12C2×C3semidirect
21. C3×C7×M16C2×C3semidirect
22. C7×D24C2×C3semidirect
23. C7×#204C2×C3semidirect
27. M16×ΓL(1,𝔽8)D3semidirect
43. C7×D12C22×C3semidirect
45. C7×Dic6C22×C3semidirect
54. C3×C7×C8C22×C3semidirect
56. C8×ΓL(1,𝔽8)C2×D3semidirect
70. C7×M16C3×D3semidirect
72. #908ΓL(1,𝔽8)direct (N × H)
99. C3×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
108. C7×C8C2×C3×D3semidirect
111. #235C2×ΓL(1,𝔽8)semidirect
112. C2×D12C2×ΓL(1,𝔽8)semidirect
117. C3×M16C2×ΓL(1,𝔽8)semidirect
118. D24C2×ΓL(1,𝔽8)semidirect
119. #204C2×ΓL(1,𝔽8)semidirect
159. D12C22×ΓL(1,𝔽8)semidirect
161. Dic6C22×ΓL(1,𝔽8)semidirect
164. C3×C8C22×ΓL(1,𝔽8)semidirect
165. C3×C7C3×AGL(1,ℤ/8)semidirect
167. ΓL(1,𝔽8)#908direct (N × H)
176. M16D3×ΓL(1,𝔽8)semidirect
207. C8C2×D3×ΓL(1,𝔽8)semidirect
208. C7C3×#908semidirect
228. C3ΓL(1,𝔽8)×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#2352
C2×ΓL(1,𝔽8)×D122
C3×M16×ΓL(1,𝔽8)2
ΓL(1,𝔽8)×D242
ΓL(1,𝔽8)×#2042
C7×#9083
ΓL(1,𝔽8)×AGL(1,ℤ/8)3
C3×#9087