Finite Group Databasebeta

← #72228933

Subgroups — #72228933

Order 2688.

1656 conjugacy classes of subgroups; only the 381 normal subgroups are shown.

normal subgroups only

This group has 1656 conjugacy classes of subgroups — too many to draw. Only its 381 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×C42×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C22×(C2×AGL(1,𝔽7))⋊C4C213yesmaximal
3C2×ΓL(1,𝔽8)⋊#95C213yesmaximal, complemented
4#11729291C311yesmaximal, characteristic, complemented, Hall
5C24×AGL(1,𝔽7)C2211yescharacteristic
6C23×C4×ΓL(1,𝔽8)C2213yes—
7C23×ΓL(1,𝔽8)⋊C4C2213yes—
8C2×C42×ΓL(1,𝔽8)—26no—
9C2×(C2×AGL(1,𝔽7))⋊C4—26no—
10C2×(C2×AGL(1,𝔽7))⋊C4—26no—
11C2×(C2×AGL(1,𝔽7))⋊C4C4112yescomplemented
12ΓL(1,𝔽8)⋊#95—26no—
13C22×C42×C7C2×C311yescharacteristic, complemented
14C22×(C2×D7)⋊C4C2×C313yes—
15C2×C7⋊#95C2×C313yescomplemented
16C3×#1437—71nomaximal, Hall
17C24×ΓL(1,𝔽8)C2311yescharacteristic
18C23×AGL(1,𝔽7)—23no—
19C23×AGL(1,𝔽7)—23no—
20C23×AGL(1,𝔽7)—26no—
21C23×AGL(1,𝔽7)C2×C416yes—
22C22×C4×ΓL(1,𝔽8)—26no—
23C22×C4×ΓL(1,𝔽8)—26no—
24C22×C4×ΓL(1,𝔽8)C2×C416yes—
25C22×C4×ΓL(1,𝔽8)D416yes—
26C22×C4×ΓL(1,𝔽8)D416yes—
27C22×ΓL(1,𝔽8)⋊C4—26no—
28C22×ΓL(1,𝔽8)⋊C4—23no—
29C22×ΓL(1,𝔽8)⋊C4—23no—
30C22×ΓL(1,𝔽8)⋊C4—26no—
31C22×ΓL(1,𝔽8)⋊C4C2×C416yes—
32C42×ΓL(1,𝔽8)—28no—
33(C2×AGL(1,𝔽7))⋊C4—224no—
34C24×D7C22×C311yescharacteristic
35C23×C4×C7C22×C313yes—
36C23×Dic7C22×C313yes—
37C2×C42×C7—26no—
38C2×(C2×D7)⋊C4—26no—
39C2×(C2×D7)⋊C4—26no—
40C2×(C2×D7)⋊C4C3×C4112yescomplemented
41C7⋊#95—26no—
42C22×C3×C22⋊C4—73no—
43C22×C3×C42—71no—
44C2×C3×#95—73no—
45C23×ΓL(1,𝔽8)C22×C413yes—
46C23×ΓL(1,𝔽8)C2×D413yes—
47C23×ΓL(1,𝔽8)C2×D413yes—
48C23×ΓL(1,𝔽8)#4116yes—
49C22×AGL(1,𝔽7)—26no—
50C22×AGL(1,𝔽7)—26no—
51C22×AGL(1,𝔽7)—212no—
52C22×AGL(1,𝔽7)—212no—
53C22×AGL(1,𝔽7)—41no—
54C22×AGL(1,𝔽7)—46no—
55C22×AGL(1,𝔽7)—43no—
56C22×AGL(1,𝔽7)—43no—
57C22×AGL(1,𝔽7)—42no—
58C22×AGL(1,𝔽7)—41no—
59C22×AGL(1,𝔽7)C4214yescomplemented
60C2×C4×ΓL(1,𝔽8)—212no—
61C2×C4×ΓL(1,𝔽8)—26no—
62C2×C4×ΓL(1,𝔽8)—212no—
63C2×C4×ΓL(1,𝔽8)—26no—
64C2×C4×ΓL(1,𝔽8)C22⋊C4112yescomplemented
65C2×ΓL(1,𝔽8)⋊C4—26no—
66C2×ΓL(1,𝔽8)⋊C4—212no—
67C2×ΓL(1,𝔽8)⋊C4—46no—
68C2×ΓL(1,𝔽8)⋊C4—43no—
69C2×ΓL(1,𝔽8)⋊C4—43no—
70#1437—71noSylow
71C24×C7C23×C311yescharacteristic
72C23×D7—23no—
73C23×D7—23no—
74C23×D7—26no—
75C23×D7C2×C3×C416yes—
76C22×C4×C7—26no—
77C22×C4×C7—26no—
78C22×C4×C7C2×C3×C416yes—
79C22×C4×C7C3×D416yes—
80C22×C4×C7C3×D416yes—
81C22×Dic7—26no—
82C22×Dic7—23no—
83C22×Dic7—23no—
84C22×Dic7—26no—
85C22×Dic7C2×C3×C416yes—
86C42×C7—28no—
87(C2×D7)⋊C4—224no—
88C25×C3—71no—
89C23×C3×C4—73no—
90C23×C3×C4—73no—
91C2×C3×C22⋊C4—712no—
92C2×C3×C22⋊C4—146no—
93C2×C3×C22⋊C4—146no—
94C2×C3×C42—146no—
95C3×#95—146no—
96C22×ΓL(1,𝔽8)C2×C4211yescharacteristic
97C22×ΓL(1,𝔽8)C2×C22⋊C413yes—
98C22×ΓL(1,𝔽8)#11713yes—
99C22×ΓL(1,𝔽8)C4×D416yes—
100C22×ΓL(1,𝔽8)C4×D416yes—
101C22×ΓL(1,𝔽8)#12011yescharacteristic
102C22×ΓL(1,𝔽8)#12116yes—
103C22×ΓL(1,𝔽8)#12313yes—
104C22×ΓL(1,𝔽8)#12413yes—
105C22×ΓL(1,𝔽8)#12612yes—
106C22×ΓL(1,𝔽8)#12711yescharacteristic
107C2×AGL(1,𝔽7)—212no—
108C2×AGL(1,𝔽7)—46no—
109C2×AGL(1,𝔽7)—412no—
110C2×AGL(1,𝔽7)—46no—
111C4×ΓL(1,𝔽8)—212no—
112ΓL(1,𝔽8)⋊C4—46no—
113C22×C22⋊C4—73no—
114C22×C42AGL(1,𝔽7)11yescharacteristic, complemented
115C2×#95—73no—
116C23×C7C22×C3×C413yes—
117C23×C7C2×C3×D413yes—
118C23×C7C2×C3×D413yes—
119C23×C7C3×#4116yes—
120C22×D7—26no—
121C22×D7—26no—
122C22×D7—212no—
123C22×D7—212no—
124C22×D7—41no—
125C22×D7—46no—
126C22×D7—43no—
127C22×D7—43no—
128C22×D7—42no—
129C22×D7—41no—
130C22×D7C3×C4214yescomplemented
131C2×C4×C7—212no—
132C2×C4×C7—26no—
133C2×C4×C7—212no—
134C2×C4×C7—26no—
135C2×C4×C7C3×C22⋊C4112yescomplemented
136C2×Dic7—26no—
137C2×Dic7—212no—
138C2×Dic7—46no—
139C2×Dic7—43no—
140C2×Dic7—43no—
141C24×C3—76no—
142C24×C3—71no—
143C24×C3—143no—
144C24×C3—143no—
145C24×C3—146no—
146C22×C3×C4—76no—
147C22×C3×C4—76no—
148C22×C3×C4—76no—
149C22×C3×C4—76no—
150C22×C3×C4—146no—
151C22×C3×C4—146no—
152C22×C3×C4—146no—
153C22×C3×C4—143no—
154C22×C3×C4—143no—
155C22×C3×C4—146no—
156C3×C22⋊C4—1424no—
157C3×C42—148no—
158C2×ΓL(1,𝔽8)C4×C22⋊C413yes—
159C2×ΓL(1,𝔽8)#38613yes—
160C2×ΓL(1,𝔽8)#38816yes—
161C2×ΓL(1,𝔽8)#39013yes—
162AGL(1,𝔽7)—44no—
163C25—71no—
164C23×C4—73no—
165C23×C4C2×AGL(1,𝔽7)13yes—
166C2×C22⋊C4—712no—
167C2×C22⋊C4—146no—
168C2×C22⋊C4—146no—
169C2×C42—26no—
170#95—146no—
171C22×C7C2×C3×C4211yescharacteristic
172C22×C7C2×C3×C22⋊C413yes—
173C22×C7C3×#11713yes—
174C22×C7C3×C4×D416yes—
175C22×C7C3×C4×D416yes—
176C22×C7C3×#12011yescharacteristic
177C22×C7C3×#12116yes—
178C22×C7C3×#12313yes—
179C22×C7C3×#12413yes—
180C22×C7C3×#12612yes—
181C22×C7C3×#12711yescharacteristic
182C2×D7—212no—
183C2×D7—46no—
184C2×D7—412no—
185C2×D7—46no—
186C4×C7—212no—
187Dic7—46no—
188C23×C3—74no—
189C23×C3—73no—
190C23×C3—73no—
191C23×C3—76no—
192C23×C3—73no—
193C23×C3—146no—
194C23×C3—146no—
195C23×C3—1412no—
196C23×C3—1412no—
197C23×C3—281no—
198C23×C3—283no—
199C23×C3—282no—
200C23×C3—283no—
201C23×C3—286no—
202C23×C3—281no—
203C2×C3×C4—712no—
204C2×C3×C4—146no—
205C2×C3×C4—1412no—
206C2×C3×C4—146no—
207C2×C3×C4—1412no—
208C2×C3×C4—146no—
209C2×C3×C4—1412no—
210C2×C3×C4—283no—
211C2×C3×C4—283no—
212C2×C3×C4—286no—
213ΓL(1,𝔽8)#143711yescharacteristic, complemented, Hall
214C24—76no—
215C24—143no—
216C24—146no—
217C24—143no—
218C24C22×AGL(1,𝔽7)11yescharacteristic
219C22×C4—26no—
220C22×C4—26no—
221C22×C4—76no—
222C22×C4—146no—
223C22×C4—143no—
224C22×C4—146no—
225C22×C4—143no—
226C22×C4C4×AGL(1,𝔽7)16yes—
227C22×C4C4⋊AGL(1,𝔽7)16yes—
228C22×C4(C22⋊D7)⋊C316yes—
229C22⋊C4—1424no—
230C42—28no—
231C2×C7C3×C4×C22⋊C413yes—
232C2×C7C3×#38613yes—
233C2×C7C3×#38816yes—
234C2×C7C3×#39013yes—
235D7—44no—
236C22×C3—71no—
237C22×C3—73no—
238C22×C3—76no—
239C22×C3—71no—
240C22×C3—73no—
241C22×C3—72no—
242C22×C3—73no—
243C22×C3—76no—
244C22×C3—73no—
245C22×C3—76no—
246C22×C3—71no—
247C22×C3—1412no—
248C22×C3—286no—
249C22×C3—2812no—
250C22×C3—286no—
251C3×C4—1412no—
252C3×C4—286no—
253C23—74no—
254C23—146no—
255C23—1412no—
256C23—146no—
257C23—1412no—
258C23—281no—
259C23—283no—
260C23—282no—
261C23—283no—
262C23—286no—
263C23—281no—
264C23C2×C4×AGL(1,𝔽7)13yes—
265C23C2×C4⋊AGL(1,𝔽7)13yes—
266C23#1157⋊C316yes—
267C23C2×(C22⋊D7)⋊C313yes—
268C2×C4—212no—
269C2×C4—26no—
270C2×C4—212no—
271C2×C4—26no—
272C2×C4—1412no—
273C2×C4—146no—
274C2×C4—283no—
275C2×C4—283no—
276C2×C4—286no—
277C2×C4(C2×AGL(1,𝔽7))⋊C4112yescomplemented
278C7C3×#143711yescharacteristic, complemented, Sylow
279C2×C3—73no—
280C2×C3—73no—
281C2×C3—76no—
282C2×C3—73no—
283C2×C3—284no—
284C22—1412no—
285C22—286no—
286C22—2812no—
287C22—286no—
288C22C42×AGL(1,𝔽7)11yescharacteristic
289C22#6435⋊C313yes—
290C22C4×C4⋊AGL(1,𝔽7)16yes—
291C22#1062071011yescharacteristic
292C22#6438⋊C313yes—
293C22#1062071212yes—
294C22C2×(C2×AGL(1,𝔽7))⋊C413yes—
295C22C4×(C22⋊D7)⋊C316yes—
296C22#6492⋊C313yes—
297C22#6493⋊C316yes—
298C22#1062078911yescharacteristic
299C4—212no—
300C4—286no—
301C3—71noSylow
302C2—284no—
303C2C4×(C2×AGL(1,𝔽7))⋊C413yes—
304C2#89711⋊C313yes—
305C2#89712⋊C316yes—
306C2#89985⋊C313yes—
subgroupG/H
center, FrattiniC24C22×AGL(1,𝔽7)
commutatorC22×C7C2×C3×C42
FittingC22×C42×C7C2×C3
socle, cosocleC24×C7C23×C3
prime radicalC24×AGL(1,𝔽7)C22
hypercenterC22×C42AGL(1,𝔽7)
hypocenterC7C3×#1437
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. C22×C42×ΓL(1,𝔽8)C2semidirect
3. C2×ΓL(1,𝔽8)⋊#95C2semidirect
4. #11729291C3semidirect
11. C2×(C2×AGL(1,𝔽7))⋊C4C4semidirect
13. C22×C42×C7C2×C3semidirect
15. C2×C7⋊#95C2×C3semidirect
40. C2×(C2×D7)⋊C4C3×C4semidirect
59. C22×AGL(1,𝔽7)C42semidirect
64. C2×C4×ΓL(1,𝔽8)C22⋊C4semidirect
114. C22×C42AGL(1,𝔽7)semidirect
130. C22×D7C3×C42semidirect
135. C2×C4×C7C3×C22⋊C4semidirect
213. ΓL(1,𝔽8)#1437semidirect
277. C2×C4(C2×AGL(1,𝔽7))⋊C4semidirect
278. C7C3×#1437semidirect

Maximal subgroups

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

subgroupindex
C22×C42×ΓL(1,𝔽8)2
C22×(C2×AGL(1,𝔽7))⋊C42
C2×ΓL(1,𝔽8)⋊#952
#117292913
C3×#14377