Finite Group Databasebeta

← ΓL(1,𝔽8)×#2021 #72276589

Subgroups — ΓL(1,𝔽8)×#2021 #72276589

Order 2688.

1276 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 138 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)×#534C211yesmaximal, characteristic, complemented
2C2×ΓL(1,𝔽8)×#120C212yesmaximal, complemented
3C2×ΓL(1,𝔽8)×#99C212yesmaximal, complemented
4ΓL(1,𝔽8)×#379C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×#342C211yesmaximal, characteristic, complemented
6C7×#2021C311yesmaximal, characteristic, complemented, Hall
7C25×ΓL(1,𝔽8)C2211yescharacteristic
8C22×D4×ΓL(1,𝔽8)—22no—
9C22×D4×ΓL(1,𝔽8)C2212yes—
10ΓL(1,𝔽8)×#123—22no—
11ΓL(1,𝔽8)×#121—22no—
12ΓL(1,𝔽8)×#120—24no—
13ΓL(1,𝔽8)×#120—22no—
14ΓL(1,𝔽8)×#120—24no—
15C2×C22⋊C4×ΓL(1,𝔽8)—22no—
16C2×C22⋊C4×ΓL(1,𝔽8)—22no—
17C2×C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
18C2×C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
19C2×C22⋊C4×ΓL(1,𝔽8)C2212yes—
20ΓL(1,𝔽8)×#99—24no—
21C7×#534C2×C311yescharacteristic, complemented
22C2×C7×#120C2×C312yescomplemented
23C2×C7×#99C2×C312yescomplemented
24C7×#379C2×C311yescharacteristic, complemented
25C7×#342C2×C311yescharacteristic, complemented
26C3×#2021—71nomaximal, Hall
27C24×ΓL(1,𝔽8)—22no—
28C24×ΓL(1,𝔽8)—21no—
29C24×ΓL(1,𝔽8)—22no—
30C24×ΓL(1,𝔽8)—21no—
31C24×ΓL(1,𝔽8)—44no—
32C24×ΓL(1,𝔽8)D412yescomplemented
33C24×ΓL(1,𝔽8)D412yescomplemented
34C24×ΓL(1,𝔽8)C2311yescharacteristic
35C2×D4×ΓL(1,𝔽8)—28no—
36C2×D4×ΓL(1,𝔽8)—22no—
37C2×D4×ΓL(1,𝔽8)—22no—
38C2×D4×ΓL(1,𝔽8)—22no—
39C2×D4×ΓL(1,𝔽8)—22no—
40C2×D4×ΓL(1,𝔽8)—22no—
41C2×D4×ΓL(1,𝔽8)—22no—
42C2×D4×ΓL(1,𝔽8)—22no—
43C2×D4×ΓL(1,𝔽8)—48no—
44C22×C4×ΓL(1,𝔽8)—22no—
45C22×C4×ΓL(1,𝔽8)—21no—
46C22×C4×ΓL(1,𝔽8)D412yescomplemented
47C22×C4×ΓL(1,𝔽8)D412yescomplemented
48C4⋊C4×ΓL(1,𝔽8)—22no—
49C22⋊C4×ΓL(1,𝔽8)—28no—
50C22⋊C4×ΓL(1,𝔽8)—24no—
51C22⋊C4×ΓL(1,𝔽8)—22no—
52C22⋊C4×ΓL(1,𝔽8)—22no—
53C22⋊C4×ΓL(1,𝔽8)—22no—
54C22⋊C4×ΓL(1,𝔽8)—44no—
55C22⋊C4×ΓL(1,𝔽8)D414yescomplemented
56C25×C7C22×C311yescharacteristic
57C22×C7×D4—22no—
58C22×C7×D4C22×C312yes—
59C7×#123—22no—
60C7×#121—22no—
61C7×#120—24no—
62C7×#120—22no—
63C7×#120—24no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4C22×C311yescharacteristic
67C2×C7×C22⋊C4C22×C311yescharacteristic
68C2×C7×C22⋊C4C22×C312yes—
69C7×#99—24no—
70C3×#534—71no—
71C2×C3×#120—72no—
72C2×C3×#99—72no—
73C3×#379—71no—
74C3×#342—71no—
75C23×ΓL(1,𝔽8)—24no—
76C23×ΓL(1,𝔽8)—28no—
77C23×ΓL(1,𝔽8)—24no—
78C23×ΓL(1,𝔽8)—22no—
79C23×ΓL(1,𝔽8)—22no—
80C23×ΓL(1,𝔽8)—22no—
81C23×ΓL(1,𝔽8)—22no—
82C23×ΓL(1,𝔽8)—44no—
83C23×ΓL(1,𝔽8)—42no—
84C23×ΓL(1,𝔽8)—42no—
85C23×ΓL(1,𝔽8)—44no—
86C23×ΓL(1,𝔽8)—48no—
87C23×ΓL(1,𝔽8)—44no—
88C23×ΓL(1,𝔽8)—44no—
89C23×ΓL(1,𝔽8)—41no—
90C23×ΓL(1,𝔽8)—42no—
91C23×ΓL(1,𝔽8)—41no—
92C23×ΓL(1,𝔽8)C2×D412yes—
93C23×ΓL(1,𝔽8)C2×D411yescharacteristic
94C23×ΓL(1,𝔽8)C2×D411yescharacteristic
95C23×ΓL(1,𝔽8)C2×D412yes—
96C23×ΓL(1,𝔽8)#4111yescharacteristic
97D4×ΓL(1,𝔽8)—416no—
98D4×ΓL(1,𝔽8)—42no—
99D4×ΓL(1,𝔽8)—42no—
100D4×ΓL(1,𝔽8)—42no—
101D4×ΓL(1,𝔽8)—42no—
102D4×ΓL(1,𝔽8)—82no—
103C2×C4×ΓL(1,𝔽8)—22no—
104C2×C4×ΓL(1,𝔽8)—22no—
105C2×C4×ΓL(1,𝔽8)—24no—
106C2×C4×ΓL(1,𝔽8)—22no—
107C2×C4×ΓL(1,𝔽8)—22no—
108C2×C4×ΓL(1,𝔽8)—44no—
109C2×C4×ΓL(1,𝔽8)—42no—
110C2×C4×ΓL(1,𝔽8)—42no—
111C2×C4×ΓL(1,𝔽8)—41no—
112#2021ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
113C24×C7—21no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—21no—
117C24×C7—44no—
118C24×C7C3×D412yescomplemented
119C24×C7C3×D412yescomplemented
120C24×C7C23×C311yescharacteristic
121C2×C7×D4—28no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—48no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7C3×D412yescomplemented
133C22×C4×C7C3×D412yescomplemented
134C7×C4⋊C4—22no—
135C7×C22⋊C4—28no—
136C7×C22⋊C4—24no—
137C7×C22⋊C4—22no—
138C7×C22⋊C4—22no—
139C7×C22⋊C4—22no—
140C7×C22⋊C4—44no—
141C7×C22⋊C4C3×D414yescomplemented
142C25×C3—71no—
143C22×C3×D4—72no—
144C22×C3×D4—142no—
145C3×#123—142no—
146C3×#121—142no—
147C3×#120—144no—
148C3×#120—144no—
149C3×#120—142no—
150C2×C3×C22⋊C4—72no—
151C2×C3×C22⋊C4—71no—
152C2×C3×C22⋊C4—71no—
153C2×C3×C22⋊C4—142no—
154C2×C3×C22⋊C4—142no—
155C3×#99—144no—
156C22×ΓL(1,𝔽8)—22no—
157C22×ΓL(1,𝔽8)—24no—
158C22×ΓL(1,𝔽8)—28no—
159C22×ΓL(1,𝔽8)—24no—
160C22×ΓL(1,𝔽8)—41no—
161C22×ΓL(1,𝔽8)—42no—
162C22×ΓL(1,𝔽8)—41no—
163C22×ΓL(1,𝔽8)—44no—
164C22×ΓL(1,𝔽8)—44no—
165C22×ΓL(1,𝔽8)—42no—
166C22×ΓL(1,𝔽8)—48no—
167C22×ΓL(1,𝔽8)—44no—
168C22×ΓL(1,𝔽8)—42no—
169C22×ΓL(1,𝔽8)—44no—
170C22×ΓL(1,𝔽8)—42no—
171C22×ΓL(1,𝔽8)—44no—
172C22×ΓL(1,𝔽8)—82no—
173C22×ΓL(1,𝔽8)#12012yes—
174C22×ΓL(1,𝔽8)#12011yescharacteristic
175C22×ΓL(1,𝔽8)#12112yes—
176C22×ΓL(1,𝔽8)#12111yescharacteristic
177C22×ΓL(1,𝔽8)#12711yescharacteristic
178C4×ΓL(1,𝔽8)—44no—
179C4×ΓL(1,𝔽8)—42no—
180C4×ΓL(1,𝔽8)—42no—
181C4×ΓL(1,𝔽8)—81no—
182#534C2×ΓL(1,𝔽8)11yescharacteristic, complemented
183C2×#120C2×ΓL(1,𝔽8)12yescomplemented
184C2×#99C2×ΓL(1,𝔽8)12yescomplemented
185#379C2×ΓL(1,𝔽8)11yescharacteristic, complemented
186#342C2×ΓL(1,𝔽8)11yescharacteristic, complemented
187C23×C7—24no—
188C23×C7—28no—
189C23×C7—24no—
190C23×C7—22no—
191C23×C7—22no—
192C23×C7—22no—
193C23×C7—22no—
194C23×C7—44no—
195C23×C7—42no—
196C23×C7—42no—
197C23×C7—44no—
198C23×C7—48no—
199C23×C7—44no—
200C23×C7—44no—
201C23×C7—41no—
202C23×C7—42no—
203C23×C7—41no—
204C23×C7C2×C3×D412yes—
205C23×C7C2×C3×D411yescharacteristic
206C23×C7C2×C3×D411yescharacteristic
207C23×C7C2×C3×D412yes—
208C23×C7C3×#4111yescharacteristic
209C7×D4—416no—
210C7×D4—42no—
211C7×D4—42no—
212C7×D4—42no—
213C7×D4—42no—
214C7×D4—82no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—24no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—44no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—41no—
224C24×C3—72no—
225C24×C3—72no—
226C24×C3—71no—
227C24×C3—141no—
228C24×C3—142no—
229C24×C3—141no—
230C24×C3—142no—
231C24×C3—284no—
232C2×C3×D4—142no—
233C2×C3×D4—142no—
234C2×C3×D4—142no—
235C2×C3×D4—142no—
236C2×C3×D4—142no—
237C2×C3×D4—142no—
238C2×C3×D4—148no—
239C2×C3×D4—142no—
240C2×C3×D4—288no—
241C22×C3×C4—72no—
242C22×C3×C4—72no—
243C22×C3×C4—142no—
244C22×C3×C4—141no—
245C3×C4⋊C4—142no—
246C3×C22⋊C4—74no—
247C3×C22⋊C4—142no—
248C3×C22⋊C4—142no—
249C3×C22⋊C4—148no—
250C3×C22⋊C4—144no—
251C3×C22⋊C4—142no—
252C3×C22⋊C4—284no—
253C2×ΓL(1,𝔽8)—22no—
254C2×ΓL(1,𝔽8)—21no—
255C2×ΓL(1,𝔽8)—21no—
256C2×ΓL(1,𝔽8)—22no—
257C2×ΓL(1,𝔽8)—44no—
258C2×ΓL(1,𝔽8)—82no—
259C2×ΓL(1,𝔽8)#39211yescharacteristic
260C2×ΓL(1,𝔽8)B(4,𝔽2)12yes—
261C25C22×ΓL(1,𝔽8)11yescharacteristic
262C22×D4—22no—
263C22×D4C22×ΓL(1,𝔽8)12yes—
264#123—22no—
265#121—22no—
266#120—24no—
267#120—22no—
268#120—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4C22×ΓL(1,𝔽8)11yescharacteristic
272C2×C22⋊C4C22×ΓL(1,𝔽8)11yescharacteristic
273C2×C22⋊C4C22×ΓL(1,𝔽8)12yes—
274#99—24no—
275C22×C7—22no—
276C22×C7—24no—
277C22×C7—28no—
278C22×C7—24no—
279C22×C7—41no—
280C22×C7—42no—
281C22×C7—41no—
282C22×C7—44no—
283C22×C7—44no—
284C22×C7—42no—
285C22×C7—48no—
286C22×C7—44no—
287C22×C7—42no—
288C22×C7—44no—
289C22×C7—42no—
290C22×C7—44no—
291C22×C7—82no—
292C22×C7C3×#12012yes—
293C22×C7C3×#12011yescharacteristic
294C22×C7C3×#12112yes—
295C22×C7C3×#12111yescharacteristic
296C22×C7C3×#12711yescharacteristic
297C4×C7—44no—
298C4×C7—42no—
299C4×C7—42no—
300C4×C7—81no—
301C23×C3—72no—
302C23×C3—71no—
303C23×C3—72no—
304C23×C3—71no—
305C23×C3—71no—
306C23×C3—144no—
307C23×C3—142no—
308C23×C3—142no—
309C23×C3—142no—
310C23×C3—142no—
311C23×C3—144no—
312C23×C3—148no—
313C23×C3—281no—
314C23×C3—282no—
315C23×C3—281no—
316C23×C3—284no—
317C23×C3—282no—
318C23×C3—282no—
319C23×C3—284no—
320C23×C3—288no—
321C23×C3—284no—
322C23×C3—284no—
323C3×D4—282no—
324C3×D4—282no—
325C3×D4—282no—
326C3×D4—282no—
327C3×D4—2816no—
328C3×D4—562no—
329C2×C3×C4—144no—
330C2×C3×C4—142no—
331C2×C3×C4—142no—
332C2×C3×C4—142no—
333C2×C3×C4—142no—
334C2×C3×C4—284no—
335C2×C3×C4—282no—
336C2×C3×C4—281no—
337C2×C3×C4—282no—
338ΓL(1,𝔽8)#202111yescharacteristic, direct product factor, Hall
339C24—21no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—44no—
344C24D4×ΓL(1,𝔽8)12yescomplemented
345C24D4×ΓL(1,𝔽8)12yescomplemented
346C24C23×ΓL(1,𝔽8)11yescharacteristic
347C2×D4—28no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—48no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4D4×ΓL(1,𝔽8)12yescomplemented
359C22×C4D4×ΓL(1,𝔽8)12yescomplemented
360C4⋊C4—22no—
361C22⋊C4—28no—
362C22⋊C4—24no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—44no—
367C22⋊C4D4×ΓL(1,𝔽8)14yescomplemented
368C2×C7—22no—
369C2×C7—21no—
370C2×C7—21no—
371C2×C7—22no—
372C2×C7—44no—
373C2×C7—82no—
374C2×C7C3×#39211yescharacteristic
375C2×C7C3×B(4,𝔽2)12yes—
376C22×C3—72no—
377C22×C3—71no—
378C22×C3—72no—
379C22×C3—71no—
380C22×C3—71no—
381C22×C3—142no—
382C22×C3—144no—
383C22×C3—148no—
384C22×C3—144no—
385C22×C3—284no—
386C22×C3—282no—
387C22×C3—282no—
388C22×C3—284no—
389C22×C3—281no—
390C22×C3—282no—
391C22×C3—281no—
392C22×C3—284no—
393C22×C3—284no—
394C22×C3—282no—
395C22×C3—288no—
396C22×C3—284no—
397C22×C3—562no—
398C3×C4—282no—
399C3×C4—282no—
400C3×C4—284no—
401C3×C4—561no—
402C23—24no—
403C23—28no—
404C23—24no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—44no—
410C23—42no—
411C23—42no—
412C23—44no—
413C23—48no—
414C23—44no—
415C23—44no—
416C23—41no—
417C23—42no—
418C23—41no—
419C23C2×D4×ΓL(1,𝔽8)12yes—
420C23C2×D4×ΓL(1,𝔽8)11yescharacteristic
421C23C2×D4×ΓL(1,𝔽8)11yescharacteristic
422C23C2×D4×ΓL(1,𝔽8)12yes—
423C23#41×ΓL(1,𝔽8)11yescharacteristic
424D4—416no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—82no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C7C3×#202111yescharacteristic, complemented, Sylow
440C2×C3—72no—
441C2×C3—71no—
442C2×C3—142no—
443C2×C3—141no—
444C2×C3—141no—
445C2×C3—142no—
446C2×C3—284no—
447C2×C3—562no—
448C22—22no—
449C22—24no—
450C22—28no—
451C22—24no—
452C22—41no—
453C22—42no—
454C22—41no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22—48no—
459C22—44no—
460C22—44no—
461C22—42no—
462C22—44no—
463C22—42no—
464C22—82no—
465C22ΓL(1,𝔽8)×#12012yes—
466C22ΓL(1,𝔽8)×#12011yescharacteristic
467C22ΓL(1,𝔽8)×#12112yes—
468C22ΓL(1,𝔽8)×#12111yescharacteristic
469C22ΓL(1,𝔽8)×#12711yescharacteristic
470C4—44no—
471C4—42no—
472C4—42no—
473C4—81no—
474C3—71noSylow
475C2—22no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—44no—
480C2—82no—
481C2ΓL(1,𝔽8)×#39211yescharacteristic
482C2ΓL(1,𝔽8)×B(4,𝔽2)12yes—
subgroupG/H
centerC22ΓL(1,𝔽8)×#120
commutator, cosocleC24×C7C23×C3
FrattiniC24C23×ΓL(1,𝔽8)
FittingC7×#2021C3
socleC22×C7C3×#120
hypercenter#2021ΓL(1,𝔽8)
hypocenterC7C3×#2021
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ #2021

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#534C2semidirect
2. C2×ΓL(1,𝔽8)×#120C2semidirect
3. C2×ΓL(1,𝔽8)×#99C2semidirect
4. ΓL(1,𝔽8)×#379C2semidirect
5. ΓL(1,𝔽8)×#342C2semidirect
6. C7×#2021C3semidirect
21. C7×#534C2×C3semidirect
22. C2×C7×#120C2×C3semidirect
23. C2×C7×#99C2×C3semidirect
24. C7×#379C2×C3semidirect
25. C7×#342C2×C3semidirect
32. C24×ΓL(1,𝔽8)D4semidirect
33. C24×ΓL(1,𝔽8)D4semidirect
46. C22×C4×ΓL(1,𝔽8)D4semidirect
47. C22×C4×ΓL(1,𝔽8)D4semidirect
55. C22⋊C4×ΓL(1,𝔽8)D4semidirect
112. #2021ΓL(1,𝔽8)direct (N × H)
118. C24×C7C3×D4semidirect
119. C24×C7C3×D4semidirect
132. C22×C4×C7C3×D4semidirect
133. C22×C4×C7C3×D4semidirect
141. C7×C22⋊C4C3×D4semidirect
182. #534C2×ΓL(1,𝔽8)semidirect
183. C2×#120C2×ΓL(1,𝔽8)semidirect
184. C2×#99C2×ΓL(1,𝔽8)semidirect
185. #379C2×ΓL(1,𝔽8)semidirect
186. #342C2×ΓL(1,𝔽8)semidirect
338. ΓL(1,𝔽8)#2021direct (N × H)
344. C24D4×ΓL(1,𝔽8)semidirect
345. C24D4×ΓL(1,𝔽8)semidirect
358. C22×C4D4×ΓL(1,𝔽8)semidirect
359. C22×C4D4×ΓL(1,𝔽8)semidirect
367. C22⋊C4D4×ΓL(1,𝔽8)semidirect
439. C7C3×#2021semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#5342
C2×ΓL(1,𝔽8)×#1202
C2×ΓL(1,𝔽8)×#992
ΓL(1,𝔽8)×#3792
ΓL(1,𝔽8)×#3422
C7×#20213
C3×#20217