Finite Group Databasebeta

← ΓL(1,𝔽8)×#447 #13054978

Subgroups — ΓL(1,𝔽8)×#447 #13054978

Order 1344.

396 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×D8×ΓL(1,𝔽8)C212yesmaximal, complemented
3ΓL(1,𝔽8)×#121C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#102C212yesmaximal, complemented
5ΓL(1,𝔽8)×C22⋊C8C211yesmaximal, characteristic, complemented
6C7×#447C311yesmaximal, characteristic, complemented, Hall
7C24×ΓL(1,𝔽8)—21no—
8C2×D4×ΓL(1,𝔽8)—24no—
9C2×D4×ΓL(1,𝔽8)—21no—
10C2×D4×ΓL(1,𝔽8)—21no—
11C2×D4×ΓL(1,𝔽8)C2212yes—
12C2×D4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
13C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
14D8×ΓL(1,𝔽8)—24no—
15C2×C8×ΓL(1,𝔽8)C2212yescomplemented
16C4⋊C4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
17C22⋊C4×ΓL(1,𝔽8)—21no—
18C22×C7×D4C2×C311yescharacteristic, complemented
19C2×C7×D8C2×C312yescomplemented
20C7×#121C2×C311yescharacteristic, complemented
21C7×#102C2×C312yescomplemented
22C7×C22⋊C8C2×C311yescharacteristic, complemented
23C3×#447—71nomaximal, Hall
24C23×ΓL(1,𝔽8)—24no—
25C23×ΓL(1,𝔽8)—22no—
26C23×ΓL(1,𝔽8)—21no—
27C23×ΓL(1,𝔽8)—42no—
28C23×ΓL(1,𝔽8)—42no—
29C23×ΓL(1,𝔽8)D411yescharacteristic
30D4×ΓL(1,𝔽8)—24no—
31D4×ΓL(1,𝔽8)—22no—
32D4×ΓL(1,𝔽8)—22no—
33D4×ΓL(1,𝔽8)—42no—
34D4×ΓL(1,𝔽8)D414yescomplemented
35C2×C4×ΓL(1,𝔽8)—22no—
36C2×C4×ΓL(1,𝔽8)—21no—
37C2×C4×ΓL(1,𝔽8)D411yescharacteristic
38C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
39C8×ΓL(1,𝔽8)—22no—
40C24×C7—21no—
41C2×C7×D4—24no—
42C2×C7×D4—21no—
43C2×C7×D4—21no—
44C2×C7×D4C22×C312yes—
45C2×C7×D4C22×C311yescharacteristic, complemented
46C22×C4×C7C22×C311yescharacteristic
47C7×D8—24no—
48C2×C7×C8C22×C312yescomplemented
49C7×C4⋊C4C22×C311yescharacteristic, complemented
50C7×C22⋊C4—21no—
51C22×C3×D4—71no—
52C2×C3×D8—72no—
53C3×#121—71no—
54C3×#102—72no—
55C3×C22⋊C8—71no—
56C22×ΓL(1,𝔽8)—21no—
57C22×ΓL(1,𝔽8)—21no—
58C22×ΓL(1,𝔽8)—24no—
59C22×ΓL(1,𝔽8)—48no—
60C22×ΓL(1,𝔽8)—42no—
61C22×ΓL(1,𝔽8)—42no—
62C22×ΓL(1,𝔽8)—41no—
63C22×ΓL(1,𝔽8)—41no—
64C22×ΓL(1,𝔽8)—41no—
65C22×ΓL(1,𝔽8)D812yescomplemented
66C22×ΓL(1,𝔽8)C2×D411yescharacteristic
67C4×ΓL(1,𝔽8)—21no—
68C4×ΓL(1,𝔽8)—41no—
69C4×ΓL(1,𝔽8)C2×D412yes—
70#447ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
71C23×C7—24no—
72C23×C7—22no—
73C23×C7—21no—
74C23×C7—42no—
75C23×C7—42no—
76C23×C7C3×D411yescharacteristic
77C7×D4—24no—
78C7×D4—22no—
79C7×D4—22no—
80C7×D4—42no—
81C7×D4C3×D414yescomplemented
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7C3×D411yescharacteristic
85C2×C4×C7C23×C311yescharacteristic
86C7×C8—22no—
87C24×C3—141no—
88C2×C3×D4—72no—
89C2×C3×D4—71no—
90C2×C3×D4—141no—
91C2×C3×D4—144no—
92C2×C3×D4—141no—
93C22×C3×C4—71no—
94C3×D8—144no—
95C2×C3×C8—72no—
96C3×C4⋊C4—71no—
97C3×C22⋊C4—141no—
98C2×ΓL(1,𝔽8)—22no—
99C2×ΓL(1,𝔽8)—44no—
100C2×ΓL(1,𝔽8)—81no—
101C2×ΓL(1,𝔽8)#12011yescharacteristic
102C2×ΓL(1,𝔽8)C2×D811yescharacteristic
103C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
104C22×D4C2×ΓL(1,𝔽8)11yescharacteristic, complemented
105C2×D8C2×ΓL(1,𝔽8)12yescomplemented
106#121C2×ΓL(1,𝔽8)11yescharacteristic, complemented
107#102C2×ΓL(1,𝔽8)12yescomplemented
108C22⋊C8C2×ΓL(1,𝔽8)11yescharacteristic, complemented
109C22×C7—21no—
110C22×C7—21no—
111C22×C7—24no—
112C22×C7—48no—
113C22×C7—42no—
114C22×C7—42no—
115C22×C7—41no—
116C22×C7—41no—
117C22×C7—41no—
118C22×C7C3×D812yescomplemented
119C22×C7C2×C3×D411yescharacteristic
120C4×C7—21no—
121C4×C7—41no—
122C4×C7C2×C3×D412yes—
123C23×C3—71no—
124C23×C3—142no—
125C23×C3—141no—
126C23×C3—144no—
127C23×C3—282no—
128C23×C3—282no—
129C3×D4—74no—
130C3×D4—142no—
131C3×D4—142no—
132C3×D4—144no—
133C3×D4—282no—
134C2×C3×C4—71no—
135C2×C3×C4—71no—
136C2×C3×C4—142no—
137C2×C3×C4—141no—
138C3×C8—142no—
139ΓL(1,𝔽8)#44711yescharacteristic, direct product factor, Hall
140C24—21no—
141C2×D4—24no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C22×ΓL(1,𝔽8)12yes—
145C2×D4C22×ΓL(1,𝔽8)11yescharacteristic, complemented
146C22×C4C22×ΓL(1,𝔽8)11yescharacteristic
147D8—24no—
148C2×C8C22×ΓL(1,𝔽8)12yescomplemented
149C4⋊C4C22×ΓL(1,𝔽8)11yescharacteristic, complemented
150C22⋊C4—21no—
151C2×C7—22no—
152C2×C7—44no—
153C2×C7—81no—
154C2×C7C3×#12011yescharacteristic
155C2×C7C2×C3×D811yescharacteristic
156C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
157C22×C3—72no—
158C22×C3—71no—
159C22×C3—141no—
160C22×C3—141no—
161C22×C3—144no—
162C22×C3—282no—
163C22×C3—282no—
164C22×C3—281no—
165C22×C3—281no—
166C22×C3—281no—
167C22×C3—288no—
168C3×C4—72no—
169C3×C4—141no—
170C3×C4—281no—
171C23—24no—
172C23—22no—
173C23—21no—
174C23—42no—
175C23—42no—
176C23D4×ΓL(1,𝔽8)11yescharacteristic
177D4—24no—
178D4—22no—
179D4—22no—
180D4—42no—
181D4D4×ΓL(1,𝔽8)14yescomplemented
182C2×C4—22no—
183C2×C4—21no—
184C2×C4D4×ΓL(1,𝔽8)11yescharacteristic
185C2×C4C23×ΓL(1,𝔽8)11yescharacteristic
186C8—22no—
187C7C3×#44711yescharacteristic, complemented, Sylow
188C2×C3—71no—
189C2×C3—71no—
190C2×C3—71no—
191C2×C3—142no—
192C2×C3—284no—
193C2×C3—561no—
194C22—21no—
195C22—21no—
196C22—24no—
197C22—48no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22D8×ΓL(1,𝔽8)12yescomplemented
204C22C2×D4×ΓL(1,𝔽8)11yescharacteristic
205C4—21no—
206C4—41no—
207C4C2×D4×ΓL(1,𝔽8)12yes—
208C3—71noSylow
209C2—22no—
210C2—44no—
211C2—81no—
212C2ΓL(1,𝔽8)×#12011yescharacteristic
213C2C2×D8×ΓL(1,𝔽8)11yescharacteristic
214C2ΓL(1,𝔽8)×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×ΓL(1,𝔽8)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C23×ΓL(1,𝔽8)
FittingC7×#447C3
socleC22×C7C2×C3×D4
hypercenter#447ΓL(1,𝔽8)
hypocenterC7C3×#447
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×D4×ΓL(1,𝔽8)C2semidirect
2. C2×D8×ΓL(1,𝔽8)C2semidirect
3. ΓL(1,𝔽8)×#121C2semidirect
4. ΓL(1,𝔽8)×#102C2semidirect
5. ΓL(1,𝔽8)×C22⋊C8C2semidirect
6. C7×#447C3semidirect
12. C2×D4×ΓL(1,𝔽8)C22semidirect
15. C2×C8×ΓL(1,𝔽8)C22semidirect
16. C4⋊C4×ΓL(1,𝔽8)C22semidirect
18. C22×C7×D4C2×C3semidirect
19. C2×C7×D8C2×C3semidirect
20. C7×#121C2×C3semidirect
21. C7×#102C2×C3semidirect
22. C7×C22⋊C8C2×C3semidirect
34. D4×ΓL(1,𝔽8)D4semidirect
45. C2×C7×D4C22×C3semidirect
48. C2×C7×C8C22×C3semidirect
49. C7×C4⋊C4C22×C3semidirect
65. C22×ΓL(1,𝔽8)D8semidirect
70. #447ΓL(1,𝔽8)direct (N × H)
81. C7×D4C3×D4semidirect
104. C22×D4C2×ΓL(1,𝔽8)semidirect
105. C2×D8C2×ΓL(1,𝔽8)semidirect
106. #121C2×ΓL(1,𝔽8)semidirect
107. #102C2×ΓL(1,𝔽8)semidirect
108. C22⋊C8C2×ΓL(1,𝔽8)semidirect
118. C22×C7C3×D8semidirect
139. ΓL(1,𝔽8)#447direct (N × H)
145. C2×D4C22×ΓL(1,𝔽8)semidirect
148. C2×C8C22×ΓL(1,𝔽8)semidirect
149. C4⋊C4C22×ΓL(1,𝔽8)semidirect
181. D4D4×ΓL(1,𝔽8)semidirect
187. C7C3×#447semidirect
203. C22D8×ΓL(1,𝔽8)semidirect

Maximal subgroups

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

subgroupindex
C22×D4×ΓL(1,𝔽8)2
C2×D8×ΓL(1,𝔽8)2
ΓL(1,𝔽8)×#1212
ΓL(1,𝔽8)×#1022
ΓL(1,𝔽8)×C22⋊C82
C7×#4473
C3×#4477