Finite Group Databasebeta

← ΓL(1,𝔽8)×#3025 #72277507

Subgroups — ΓL(1,𝔽8)×#3025 #72277507

Order 2688.

1532 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 1532 conjugacy classes of subgroups — too many to draw. Only its 318 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)×#585C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)×#525C213yesmaximal, complemented
3ΓL(1,𝔽8)×#458C214yesmaximal, complemented
4ΓL(1,𝔽8)×B(4,𝔽2)C214yesmaximal, complemented
5ΓL(1,𝔽8)×#410C213yesmaximal, complemented
6C7×#3025C311yesmaximal, characteristic, complemented, Hall
7ΓL(1,𝔽8)×2−1+4—23no—
8ΓL(1,𝔽8)×H(5,𝔽2)—23no—
9ΓL(1,𝔽8)×H(5,𝔽2)C2214yes—
10C2×#41×ΓL(1,𝔽8)—26no—
11C2×#41×ΓL(1,𝔽8)C2213yes—
12C23×C4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
13ΓL(1,𝔽8)×#123C2216yescomplemented
14ΓL(1,𝔽8)×C22⋊Q8—23no—
15ΓL(1,𝔽8)×#121—23no—
16ΓL(1,𝔽8)×#120C2216yescomplemented
17C4×D4×ΓL(1,𝔽8)—26no—
18ΓL(1,𝔽8)×#117C2213yescomplemented
19ΓL(1,𝔽8)×#99C22112yescomplemented
20C7×#585C2×C311yescharacteristic, complemented
21C7×#525C2×C313yescomplemented
22C7×#458C2×C314yescomplemented
23C7×B(4,𝔽2)C2×C314yescomplemented
24C7×#410C2×C313yescomplemented
25C3×#3025—71nomaximal, Hall
26C24×ΓL(1,𝔽8)C2311yescharacteristic
27#41×ΓL(1,𝔽8)—24no—
28#41×ΓL(1,𝔽8)—212no—
29#41×ΓL(1,𝔽8)—26no—
30#41×ΓL(1,𝔽8)—212no—
31#41×ΓL(1,𝔽8)—26no—
32C2×Q8×ΓL(1,𝔽8)—26no—
33C2×Q8×ΓL(1,𝔽8)D413yescomplemented
34C2×D4×ΓL(1,𝔽8)—26no—
35C2×D4×ΓL(1,𝔽8)—212no—
36C2×D4×ΓL(1,𝔽8)—26no—
37C2×D4×ΓL(1,𝔽8)D413yescomplemented
38C2×D4×ΓL(1,𝔽8)C2316yes—
39C22×C4×ΓL(1,𝔽8)—24no—
40C22×C4×ΓL(1,𝔽8)—23no—
41C22×C4×ΓL(1,𝔽8)—23no—
42C22×C4×ΓL(1,𝔽8)D416yescomplemented
43C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
44C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
45C4⋊C4×ΓL(1,𝔽8)—23no—
46C4⋊C4×ΓL(1,𝔽8)—26no—
47C22⋊C4×ΓL(1,𝔽8)—26no—
48C22⋊C4×ΓL(1,𝔽8)—26no—
49C22⋊C4×ΓL(1,𝔽8)C2316yes—
50C42×ΓL(1,𝔽8)—23no—
51C7×2−1+4—23no—
52C7×H(5,𝔽2)—23no—
53C7×H(5,𝔽2)C22×C314yes—
54C2×C7×#41—26no—
55C2×C7×#41C22×C313yes—
56C23×C4×C7C22×C311yescharacteristic, complemented
57C7×#123C22×C316yescomplemented
58C7×C22⋊Q8—23no—
59C7×#121—23no—
60C7×#120C22×C316yescomplemented
61C4×C7×D4—26no—
62C7×#117C22×C313yescomplemented
63C7×#99C22×C3112yescomplemented
64C3×#585—71no—
65C3×#525—73no—
66C3×#458—74no—
67C3×B(4,𝔽2)—74no—
68C3×#410—73no—
69C23×ΓL(1,𝔽8)—24no—
70C23×ΓL(1,𝔽8)—23no—
71C23×ΓL(1,𝔽8)—42no—
72C23×ΓL(1,𝔽8)C2×D416yes—
73C23×ΓL(1,𝔽8)C2411yescharacteristic
74Q8×ΓL(1,𝔽8)—24no—
75Q8×ΓL(1,𝔽8)—26no—
76D4×ΓL(1,𝔽8)—212no—
77D4×ΓL(1,𝔽8)—212no—
78D4×ΓL(1,𝔽8)—26no—
79D4×ΓL(1,𝔽8)—412no—
80C2×C4×ΓL(1,𝔽8)—212no—
81C2×C4×ΓL(1,𝔽8)—26no—
82C2×C4×ΓL(1,𝔽8)—26no—
83C2×C4×ΓL(1,𝔽8)—26no—
84C2×C4×ΓL(1,𝔽8)—23no—
85C2×C4×ΓL(1,𝔽8)—41no—
86C2×C4×ΓL(1,𝔽8)—43no—
87C2×C4×ΓL(1,𝔽8)C2×D416yes—
88C2×C4×ΓL(1,𝔽8)C2×D413yes—
89C2×C4×ΓL(1,𝔽8)C2×D413yes—
90#3025ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
91C24×C7C23×C311yescharacteristic
92C7×#41—24no—
93C7×#41—212no—
94C7×#41—26no—
95C7×#41—212no—
96C7×#41—26no—
97C2×C7×Q8—26no—
98C2×C7×Q8C3×D413yescomplemented
99C2×C7×D4—26no—
100C2×C7×D4—212no—
101C2×C7×D4—26no—
102C2×C7×D4C3×D413yescomplemented
103C2×C7×D4C23×C316yes—
104C22×C4×C7—24no—
105C22×C4×C7—23no—
106C22×C4×C7—23no—
107C22×C4×C7C3×D416yescomplemented
108C22×C4×C7C23×C311yescharacteristic
109C22×C4×C7C23×C311yescharacteristic
110C7×C4⋊C4—23no—
111C7×C4⋊C4—26no—
112C7×C22⋊C4—26no—
113C7×C22⋊C4—26no—
114C7×C22⋊C4C23×C316yes—
115C42×C7—23no—
116C3×2−1+4—143no—
117C3×H(5,𝔽2)—74no—
118C3×H(5,𝔽2)—143no—
119C2×C3×#41—73no—
120C2×C3×#41—146no—
121C23×C3×C4—71no—
122C3×#123—76no—
123C3×C22⋊Q8—143no—
124C3×#121—143no—
125C3×#120—76no—
126C3×C4×D4—146no—
127C3×#117—73no—
128C3×#99—712no—
129C22×ΓL(1,𝔽8)—26no—
130C22×ΓL(1,𝔽8)—26no—
131C22×ΓL(1,𝔽8)—41no—
132C22×ΓL(1,𝔽8)—46no—
133C22×ΓL(1,𝔽8)—43no—
134C22×ΓL(1,𝔽8)—41no—
135C22×ΓL(1,𝔽8)—84no—
136C22×ΓL(1,𝔽8)C22×D413yes—
137C4×ΓL(1,𝔽8)—26no—
138C4×ΓL(1,𝔽8)—46no—
139C4×ΓL(1,𝔽8)—41no—
140C4×ΓL(1,𝔽8)#12011yescharacteristic
141C4×ΓL(1,𝔽8)#12013yes—
142#585C2×ΓL(1,𝔽8)11yescharacteristic, complemented
143#525C2×ΓL(1,𝔽8)13yescomplemented
144#458C2×ΓL(1,𝔽8)14yescomplemented
145B(4,𝔽2)C2×ΓL(1,𝔽8)14yescomplemented
146#410C2×ΓL(1,𝔽8)13yescomplemented
147C23×C7—24no—
148C23×C7—23no—
149C23×C7—42no—
150C23×C7C2×C3×D416yes—
151C23×C7C24×C311yescharacteristic
152C7×Q8—24no—
153C7×Q8—26no—
154C7×D4—212no—
155C7×D4—212no—
156C7×D4—26no—
157C7×D4—412no—
158C2×C4×C7—212no—
159C2×C4×C7—26no—
160C2×C4×C7—26no—
161C2×C4×C7—26no—
162C2×C4×C7—23no—
163C2×C4×C7—43no—
164C2×C4×C7—41no—
165C2×C4×C7C2×C3×D416yes—
166C2×C4×C7C2×C3×D413yes—
167C2×C4×C7C2×C3×D413yes—
168C24×C3—71no—
169C3×#41—146no—
170C3×#41—146no—
171C3×#41—1412no—
172C3×#41—144no—
173C3×#41—1412no—
174C2×C3×Q8—73no—
175C2×C3×Q8—146no—
176C2×C3×D4—73no—
177C2×C3×D4—76no—
178C2×C3×D4—1412no—
179C2×C3×D4—146no—
180C2×C3×D4—146no—
181C22×C3×C4—76no—
182C22×C3×C4—71no—
183C22×C3×C4—71no—
184C22×C3×C4—143no—
185C22×C3×C4—143no—
186C22×C3×C4—144no—
187C3×C4⋊C4—143no—
188C3×C4⋊C4—146no—
189C3×C22⋊C4—76no—
190C3×C22⋊C4—146no—
191C3×C22⋊C4—146no—
192C3×C42—143no—
193C2×ΓL(1,𝔽8)—23no—
194C2×ΓL(1,𝔽8)—42no—
195C2×ΓL(1,𝔽8)—46no—
196C2×ΓL(1,𝔽8)C2×#12011yescharacteristic
1972−1+4—23no—
198H(5,𝔽2)—23no—
199H(5,𝔽2)C22×ΓL(1,𝔽8)14yes—
200C2×#41—26no—
201C2×#41C22×ΓL(1,𝔽8)13yes—
202C23×C4C22×ΓL(1,𝔽8)11yescharacteristic, complemented
203#123C22×ΓL(1,𝔽8)16yescomplemented
204C22⋊Q8—23no—
205#121—23no—
206#120C22×ΓL(1,𝔽8)16yescomplemented
207C4×D4—26no—
208#117C22×ΓL(1,𝔽8)13yescomplemented
209#99C22×ΓL(1,𝔽8)112yescomplemented
210C22×C7—26no—
211C22×C7—26no—
212C22×C7—41no—
213C22×C7—46no—
214C22×C7—43no—
215C22×C7—41no—
216C22×C7—84no—
217C22×C7C22×C3×D413yes—
218C4×C7—26no—
219C4×C7—46no—
220C4×C7—41no—
221C4×C7C3×#12013yes—
222C4×C7C3×#12011yescharacteristic
223C23×C3—76no—
224C23×C3—71no—
225C23×C3—143no—
226C23×C3—144no—
227C23×C3—282no—
228C3×Q8—146no—
229C3×Q8—144no—
230C3×D4—146no—
231C3×D4—1412no—
232C3×D4—1412no—
233C3×D4—2812no—
234C2×C3×C4—73no—
235C2×C3×C4—73no—
236C2×C3×C4—76no—
237C2×C3×C4—146no—
238C2×C3×C4—146no—
239C2×C3×C4—1412no—
240C2×C3×C4—146no—
241C2×C3×C4—143no—
242C2×C3×C4—283no—
243C2×C3×C4—281no—
244ΓL(1,𝔽8)#302511yescharacteristic, direct product factor, Hall
245C24C23×ΓL(1,𝔽8)11yescharacteristic
246#41—24no—
247#41—212no—
248#41—26no—
249#41—212no—
250#41—26no—
251C2×Q8—26no—
252C2×Q8D4×ΓL(1,𝔽8)13yescomplemented
253C2×D4—26no—
254C2×D4—212no—
255C2×D4—26no—
256C2×D4D4×ΓL(1,𝔽8)13yescomplemented
257C2×D4C23×ΓL(1,𝔽8)16yes—
258C22×C4—24no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4D4×ΓL(1,𝔽8)16yescomplemented
262C22×C4C23×ΓL(1,𝔽8)11yescharacteristic
263C22×C4C23×ΓL(1,𝔽8)11yescharacteristic
264C4⋊C4—23no—
265C4⋊C4—26no—
266C22⋊C4—26no—
267C22⋊C4—26no—
268C22⋊C4C23×ΓL(1,𝔽8)16yes—
269C42—23no—
270C2×C7—23no—
271C2×C7—42no—
272C2×C7—46no—
273C2×C7C2×C3×#12011yescharacteristic
274C22×C3—73no—
275C22×C3—146no—
276C22×C3—146no—
277C22×C3—281no—
278C22×C3—281no—
279C22×C3—286no—
280C22×C3—283no—
281C22×C3—564no—
282C3×C4—73no—
283C3×C4—71no—
284C3×C4—146no—
285C3×C4—281no—
286C3×C4—286no—
287C23—24no—
288C23—23no—
289C23—42no—
290C23C2×D4×ΓL(1,𝔽8)16yes—
291C23C24×ΓL(1,𝔽8)11yescharacteristic
292Q8—24no—
293Q8—26no—
294D4—212no—
295D4—212no—
296D4—26no—
297D4—412no—
298C2×C4—212no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—23no—
303C2×C4—43no—
304C2×C4—41no—
305C2×C4C2×D4×ΓL(1,𝔽8)16yes—
306C2×C4C2×D4×ΓL(1,𝔽8)13yes—
307C2×C4C2×D4×ΓL(1,𝔽8)13yes—
308C7C3×#302511yescharacteristic, complemented, Sylow
309C2×C3—71no—
310C2×C3—143no—
311C2×C3—282no—
312C2×C3—286no—
313C22—26no—
314C22—26no—
315C22—41no—
316C22—46no—
317C22—43no—
318C22—41no—
319C22—84no—
320C22C22×D4×ΓL(1,𝔽8)13yes—
321C4—26no—
322C4—46no—
323C4—41no—
324C4ΓL(1,𝔽8)×#12013yes—
325C4ΓL(1,𝔽8)×#12011yescharacteristic
326C3—71noSylow
327C2—23no—
328C2—42no—
329C2—46no—
330C2C2×ΓL(1,𝔽8)×#12011yescharacteristic
subgroupG/H
centerC4ΓL(1,𝔽8)×#120
commutator, cosocleC23×C7C24×C3
FrattiniC23C24×ΓL(1,𝔽8)
FittingC7×#3025C3
socleC2×C7C2×C3×#120
hypercenter#3025ΓL(1,𝔽8)
hypocenterC7C3×#3025
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3025

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#585C2semidirect
2. ΓL(1,𝔽8)×#525C2semidirect
3. ΓL(1,𝔽8)×#458C2semidirect
4. ΓL(1,𝔽8)×B(4,𝔽2)C2semidirect
5. ΓL(1,𝔽8)×#410C2semidirect
6. C7×#3025C3semidirect
12. C23×C4×ΓL(1,𝔽8)C22semidirect
13. ΓL(1,𝔽8)×#123C22semidirect
16. ΓL(1,𝔽8)×#120C22semidirect
18. ΓL(1,𝔽8)×#117C22semidirect
19. ΓL(1,𝔽8)×#99C22semidirect
20. C7×#585C2×C3semidirect
21. C7×#525C2×C3semidirect
22. C7×#458C2×C3semidirect
23. C7×B(4,𝔽2)C2×C3semidirect
24. C7×#410C2×C3semidirect
33. C2×Q8×ΓL(1,𝔽8)D4semidirect
37. C2×D4×ΓL(1,𝔽8)D4semidirect
42. C22×C4×ΓL(1,𝔽8)D4semidirect
56. C23×C4×C7C22×C3semidirect
57. C7×#123C22×C3semidirect
60. C7×#120C22×C3semidirect
62. C7×#117C22×C3semidirect
63. C7×#99C22×C3semidirect
90. #3025ΓL(1,𝔽8)direct (N × H)
98. C2×C7×Q8C3×D4semidirect
102. C2×C7×D4C3×D4semidirect
107. C22×C4×C7C3×D4semidirect
142. #585C2×ΓL(1,𝔽8)semidirect
143. #525C2×ΓL(1,𝔽8)semidirect
144. #458C2×ΓL(1,𝔽8)semidirect
145. B(4,𝔽2)C2×ΓL(1,𝔽8)semidirect
146. #410C2×ΓL(1,𝔽8)semidirect
202. C23×C4C22×ΓL(1,𝔽8)semidirect
203. #123C22×ΓL(1,𝔽8)semidirect
206. #120C22×ΓL(1,𝔽8)semidirect
208. #117C22×ΓL(1,𝔽8)semidirect
209. #99C22×ΓL(1,𝔽8)semidirect
244. ΓL(1,𝔽8)#3025direct (N × H)
252. C2×Q8D4×ΓL(1,𝔽8)semidirect
256. C2×D4D4×ΓL(1,𝔽8)semidirect
261. C22×C4D4×ΓL(1,𝔽8)semidirect
308. C7C3×#3025semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#5852
ΓL(1,𝔽8)×#5252
ΓL(1,𝔽8)×#4582
ΓL(1,𝔽8)×B(4,𝔽2)2
ΓL(1,𝔽8)×#4102
C7×#30253
C3×#30257