Finite Group Databasebeta

← ΓL(1,𝔽8)×#379⋊C3 #72276777

Subgroups — ΓL(1,𝔽8)×#379⋊C3 #72276777

Order 4032.

476 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 39 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)×C22⋊A4C211yesmaximal, characteristic
2#379⋊ΓL(1,𝔽8)C312yesmaximal, complemented
3C7×#379⋊C3C311yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#379C311yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×C22⋊A4C411yescharacteristic, complemented
6C4×A4×ΓL(1,𝔽8)—41nomaximal
7C2×C22⋊(C22⋊ΓL(1,𝔽8))C2×C312yes—
8C2×C7×C22⋊A4C2×C311yescharacteristic
9C25×ΓL(1,𝔽8)C2×C311yescharacteristic
10C2×C22⋊C4×ΓL(1,𝔽8)—31no—
11C2×C22⋊C4×ΓL(1,𝔽8)—31no—
12C3×#379⋊C3—71nomaximal, Hall
13C2×A4×ΓL(1,𝔽8)—41no—
14C2×A4×ΓL(1,𝔽8)—82no—
15C7×#379C3211yescharacteristic, complemented, Hall
16C22⋊(C22⋊ΓL(1,𝔽8))C3×C412yescomplemented
17C7×C22⋊A4C3×C411yescharacteristic, complemented
18C24×ΓL(1,𝔽8)—31no—
19C24×ΓL(1,𝔽8)—31no—
20C24×ΓL(1,𝔽8)—62no—
21C24×ΓL(1,𝔽8)—62no—
22C24×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
23C4×C22⋊ΓL(1,𝔽8)—42no—
24C4×C7×A4—41no—
25C22×C4×ΓL(1,𝔽8)—31no—
26C22×C4×ΓL(1,𝔽8)A411yescharacteristic, complemented
27C22⋊C4×ΓL(1,𝔽8)—62no—
28C22⋊C4×ΓL(1,𝔽8)—62no—
29C2×C3×C22⋊A4—71no—
30A4×ΓL(1,𝔽8)—41no—
31A4×ΓL(1,𝔽8)—82no—
32C3×C4×ΓL(1,𝔽8)—161no—
33C25×C7C2×C3211yescharacteristic
34C2×C7×C22⋊C4—31no—
35C2×C7×C22⋊C4—31no—
36C3×#379—71no—
37#379⋊C3—72no—
38#379⋊C3ΓL(1,𝔽8)11yescharacteristic, direct product factor
39C2×C22⋊ΓL(1,𝔽8)—42no—
40C2×C22⋊ΓL(1,𝔽8)—84no—
41C2×C7×A4—41no—
42C2×C7×A4—82no—
43C23×ΓL(1,𝔽8)—22no—
44C23×ΓL(1,𝔽8)—32no—
45C23×ΓL(1,𝔽8)—32no—
46C23×ΓL(1,𝔽8)—31no—
47C23×ΓL(1,𝔽8)—31no—
48C23×ΓL(1,𝔽8)—62no—
49C23×ΓL(1,𝔽8)—62no—
50C23×ΓL(1,𝔽8)—62no—
51C23×ΓL(1,𝔽8)—62no—
52C23×ΓL(1,𝔽8)—62no—
53C23×ΓL(1,𝔽8)—64no—
54C23×ΓL(1,𝔽8)—62no—
55C23×ΓL(1,𝔽8)—62no—
56C23×ΓL(1,𝔽8)—64no—
57C23×ΓL(1,𝔽8)C2×A411yescharacteristic
58C2×C4×ΓL(1,𝔽8)—61no—
59C2×C4×ΓL(1,𝔽8)—62no—
60C2×C4×ΓL(1,𝔽8)—61no—
61C3×C22⋊A4—71no—
62C3×C4×A4—281no—
63C2×C3×ΓL(1,𝔽8)—161no—
64C24×C7—31no—
65C24×C7—31no—
66C24×C7—62no—
67C24×C7—62no—
68C24×C7C32×C411yescharacteristic, complemented
69C22×C4×C7—31no—
70C22×C4×C7C3×A411yescharacteristic, complemented
71C7×C22⋊C4—62no—
72C7×C22⋊C4—62no—
73C25×C3—71no—
74C2×C22⋊A4—72no—
75C2×C22⋊A4C2×ΓL(1,𝔽8)11yescharacteristic
76C2×C3×C22⋊C4—211no—
77C2×C3×C22⋊C4—211no—
78C22⋊ΓL(1,𝔽8)—42no—
79C22⋊ΓL(1,𝔽8)—84no—
80C7×A4—41no—
81C7×A4—82no—
82C22×ΓL(1,𝔽8)—22no—
83C22×ΓL(1,𝔽8)—31no—
84C22×ΓL(1,𝔽8)—31no—
85C22×ΓL(1,𝔽8)—32no—
86C22×ΓL(1,𝔽8)—32no—
87C22×ΓL(1,𝔽8)—64no—
88C22×ΓL(1,𝔽8)—62no—
89C22×ΓL(1,𝔽8)—62no—
90C22×ΓL(1,𝔽8)—62no—
91C22×ΓL(1,𝔽8)—62no—
92C22×ΓL(1,𝔽8)—62no—
93C22×ΓL(1,𝔽8)—62no—
94C22×ΓL(1,𝔽8)—64no—
95C22×ΓL(1,𝔽8)—62no—
96C22×ΓL(1,𝔽8)C4×A411yescharacteristic
97C3×C4×C7—161no—
98C4×ΓL(1,𝔽8)—41no—
99C4×ΓL(1,𝔽8)—121no—
100C4×ΓL(1,𝔽8)—162no—
101C2×C3×A4—281no—
102C2×C3×A4—562no—
103#379C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
104C3×ΓL(1,𝔽8)—161noHall
105C23×C7—22no—
106C23×C7—32no—
107C23×C7—32no—
108C23×C7—31no—
109C23×C7—31no—
110C23×C7—62no—
111C23×C7—62no—
112C23×C7—62no—
113C23×C7—62no—
114C23×C7—62no—
115C23×C7—64no—
116C23×C7—62no—
117C23×C7—62no—
118C23×C7—64no—
119C23×C7C2×C3×A411yescharacteristic
120C2×C4×C7—61no—
121C2×C4×C7—62no—
122C2×C4×C7—61no—
123C24×C3—71no—
124C24×C3—211no—
125C24×C3—211no—
126C24×C3—422no—
127C24×C3—422no—
128C22⋊A4—72no—
129C22⋊A4C4×ΓL(1,𝔽8)11yescharacteristic, complemented
130C22×C3×C4—71no—
131C22×C3×C4—211no—
132C4×A4—41no—
133C4×A4—282no—
134C3×C22⋊C4—422no—
135C3×C22⋊C4—422no—
136C2×C3×C7—161no—
137C2×ΓL(1,𝔽8)—31no—
138C2×ΓL(1,𝔽8)—31no—
139C2×ΓL(1,𝔽8)—62no—
140C2×ΓL(1,𝔽8)—62no—
141C2×ΓL(1,𝔽8)—162no—
142C2×ΓL(1,𝔽8)#120⋊C311yescharacteristic
143C3×A4—281no—
144C3×A4—562no—
145C32×C4—1121no—
146C25C2×C3×ΓL(1,𝔽8)11yescharacteristic
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C22×C7—22no—
150C22×C7—31no—
151C22×C7—31no—
152C22×C7—32no—
153C22×C7—32no—
154C22×C7—64no—
155C22×C7—62no—
156C22×C7—62no—
157C22×C7—62no—
158C22×C7—62no—
159C22×C7—62no—
160C22×C7—62no—
161C22×C7—64no—
162C22×C7—62no—
163C22×C7C3×C4×A411yescharacteristic
164C4×C7—41no—
165C4×C7—121no—
166C23×C3—71no—
167C23×C3—142no—
168C23×C3—211no—
169C23×C3—211no—
170C23×C3—212no—
171C23×C3—212no—
172C23×C3—422no—
173C23×C3—422no—
174C23×C3—422no—
175C23×C3—422no—
176C23×C3—422no—
177C23×C3—424no—
178C23×C3—422no—
179C23×C3—422no—
180C23×C3—424no—
181C2×A4—41no—
182C2×A4—82no—
183C2×A4—282no—
184C2×A4—564no—
185C2×C3×C4—421no—
186C2×C3×C4—421no—
187C2×C3×C4—422no—
188C3×C7—161no—
189ΓL(1,𝔽8)—162no—
190ΓL(1,𝔽8)#379⋊C311yescharacteristic, direct product factor
191C2×C32—1121no—
192C24—31no—
193C24—31no—
194C24—62no—
195C24—62no—
196C24C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
197C22×C4—31no—
198C22×C4A4×ΓL(1,𝔽8)11yescharacteristic, complemented
199C22⋊C4—62no—
200C22⋊C4—62no—
201C2×C7—31no—
202C2×C7—31no—
203C2×C7—62no—
204C2×C7—62no—
205C2×C7C3×#120⋊C311yescharacteristic
206C22×C3—71no—
207C22×C3—142no—
208C22×C3—211no—
209C22×C3—211no—
210C22×C3—212no—
211C22×C3—212no—
212C22×C3—422no—
213C22×C3—422no—
214C22×C3—424no—
215C22×C3—422no—
216C22×C3—422no—
217C22×C3—422no—
218C22×C3—422no—
219C22×C3—422no—
220C22×C3—424no—
221A4—41no—
222A4—82no—
223A4—282no—
224A4—564no—
225C3×C4—161no—
226C3×C4—281no—
227C3×C4—841no—
228C3×C4—1122no—
229C32—1121noSylow
230C23—22no—
231C23—32no—
232C23—32no—
233C23—31no—
234C23—31no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—62no—
240C23—64no—
241C23—62no—
242C23—62no—
243C23—64no—
244C23C2×A4×ΓL(1,𝔽8)11yescharacteristic
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—61no—
248C7C3×#379⋊C311yescharacteristic, complemented, Sylow
249C2×C3—71no—
250C2×C3—161no—
251C2×C3—211no—
252C2×C3—211no—
253C2×C3—422no—
254C2×C3—422no—
255C2×C3—1122no—
256C22—22no—
257C22—31no—
258C22—31no—
259C22—32no—
260C22—32no—
261C22—64no—
262C22—62no—
263C22—62no—
264C22—62no—
265C22—62no—
266C22—62no—
267C22—62no—
268C22—64no—
269C22—62no—
270C22C4×A4×ΓL(1,𝔽8)11yescharacteristic
271C4—41no—
272C4—121no—
273C3—71no—
274C3—161no—
275C3—1122no—
276C2—31no—
277C2—31no—
278C2—62no—
279C2—62no—
280C2ΓL(1,𝔽8)×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×#120⋊C3
commutator, hypocenterC24×C7C32×C4
FrattiniC23C2×A4×ΓL(1,𝔽8)
FittingC7×#379C32
socleC23×C7C2×C3×A4
prime radicalC2×ΓL(1,𝔽8)×C22⋊A4C2
cosocleC25×C7C2×C32
LCSG ▷ C24×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #379⋊ΓL(1,𝔽8)C3semidirect
3. C7×#379⋊C3C3semidirect
4. ΓL(1,𝔽8)×#379C3semidirect
5. ΓL(1,𝔽8)×C22⋊A4C4semidirect
15. C7×#379C32semidirect
16. C22⋊(C22⋊ΓL(1,𝔽8))C3×C4semidirect
17. C7×C22⋊A4C3×C4semidirect
22. C24×ΓL(1,𝔽8)C3×C4semidirect
26. C22×C4×ΓL(1,𝔽8)A4semidirect
38. #379⋊C3ΓL(1,𝔽8)direct (N × H)
68. C24×C7C32×C4semidirect
70. C22×C4×C7C3×A4semidirect
103. #379C3×ΓL(1,𝔽8)semidirect
129. C22⋊A4C4×ΓL(1,𝔽8)semidirect
190. ΓL(1,𝔽8)#379⋊C3direct (N × H)
196. C24C3×C4×ΓL(1,𝔽8)semidirect
198. C22×C4A4×ΓL(1,𝔽8)semidirect
248. C7C3×#379⋊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)×C22⋊A42
#379⋊ΓL(1,𝔽8)3
C7×#379⋊C33
ΓL(1,𝔽8)×#3793
C4×A4×ΓL(1,𝔽8)4
C3×#379⋊C37