Finite Group Databasebeta

← #72277208

Subgroups — #72277208

Order 2688.

1254 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 36 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×C22⋊(C22⋊ΓL(1,𝔽8))C211yesmaximal, characteristic, complemented
2C2×#120⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3#379⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4C7×#2846C311yesmaximal, characteristic, complemented, Hall
5C2×C22⋊(C22⋊ΓL(1,𝔽8))—21no—
6C2×C22⋊(C22⋊ΓL(1,𝔽8))C2211yescharacteristic
7D4×C22⋊ΓL(1,𝔽8)—41nomaximal
8#120⋊ΓL(1,𝔽8)—21no—
9C26×C7C2×C311yescharacteristic, complemented
10C2×C7×#120—31no—
11C2×C7×#120—31no—
12C2×C7×#120C2×C311yescharacteristic, complemented
13C7×#379—31no—
14C7×#379—31no—
15C7×#379C2×C311yescharacteristic, complemented
16AΣL(1,𝔽64)—71nomaximal, Hall
17C22⋊(C22⋊ΓL(1,𝔽8))D411yescharacteristic, complemented
18C22×C22⋊ΓL(1,𝔽8)—41no—
19C22×C22⋊ΓL(1,𝔽8)—41no—
20C22×C22⋊ΓL(1,𝔽8)—82no—
21C4×C22⋊ΓL(1,𝔽8)—41no—
22C25×C7—21no—
23C25×C7—31no—
24C25×C7—31no—
25C25×C7—62no—
26C25×C7—62no—
27C25×C7—64no—
28C25×C7—61no—
29C25×C7C22×C311yescharacteristic
30C22×C7×D4—31no—
31C22×C7×D4—31no—
32C22×C7×D4A411yescharacteristic, complemented
33C7×#120—21no—
34C7×#120—62no—
35C7×#120—64no—
36C7×#120—62no—
37C7×#120—61no—
38C2×C7×C22⋊C4—32no—
39C2×C7×C22⋊C4—31no—
40C2×C7×C22⋊C4—31no—
41C2×C7×C22⋊C4—31no—
42C2×C7×C22⋊C4—31no—
43C2×C7×C22⋊C4—31no—
44C22×C22⋊A4—71no—
45C2×#120⋊C3—71no—
46#379⋊C3—71no—
47C2×C22⋊ΓL(1,𝔽8)—41no—
48C2×C22⋊ΓL(1,𝔽8)—82no—
49C2×C22⋊ΓL(1,𝔽8)—81no—
50C2×C22⋊ΓL(1,𝔽8)—81no—
51C2×C22⋊ΓL(1,𝔽8)—84no—
52D4×ΓL(1,𝔽8)—161no—
53#2846ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
54C24×C7—22no—
55C24×C7—31no—
56C24×C7—31no—
57C24×C7—34no—
58C24×C7—32no—
59C24×C7—32no—
60C24×C7—31no—
61C24×C7—61no—
62C24×C7—61no—
63C24×C7—62no—
64C24×C7—64no—
65C24×C7—64no—
66C24×C7—62no—
67C24×C7—64no—
68C24×C7—62no—
69C24×C7—62no—
70C24×C7—64no—
71C24×C7—64no—
72C24×C7—62no—
73C24×C7—62no—
74C24×C7—61no—
75C24×C7—61no—
76C24×C7—62no—
77C24×C7—64no—
78C24×C7—64no—
79C24×C7—64no—
80C24×C7—64no—
81C24×C7—68no—
82C24×C7—68no—
83C24×C7—68no—
84C24×C7—64no—
85C24×C7—68no—
86C24×C7—64no—
87C24×C7—62no—
88C24×C7—64no—
89C24×C7—62no—
90C24×C7C3×D411yescharacteristic, complemented
91C24×C7C2×A411yescharacteristic
92C24×C7C2×A411yescharacteristic, complemented
93C2×C7×D4—61no—
94C2×C7×D4—61no—
95C2×C7×D4—61no—
96C2×C7×D4—62no—
97C2×C7×D4—62no—
98C2×C7×D4—64no—
99C2×C7×D4—62no—
100C2×C7×D4—61no—
101C22×C4×C7—31no—
102C22×C4×C7—31no—
103C22×C4×C7C2×A411yescharacteristic, complemented
104C7×C22⋊C4—62no—
105C7×C22⋊C4—62no—
106C7×C22⋊C4—64no—
107C7×C22⋊C4—62no—
108C7×C22⋊C4—61no—
109C7×C22⋊C4—61no—
110C7×C22⋊C4—61no—
111C7×C22⋊C4—61no—
112C2×C22⋊A4—71no—
113C2×C22⋊A4—141no—
114D4×A4—281no—
115#120⋊C3—141no—
116C22⋊ΓL(1,𝔽8)—41no—
117C22⋊ΓL(1,𝔽8)—82no—
118C22×ΓL(1,𝔽8)—161no—
119C22×ΓL(1,𝔽8)—161no—
120C4×ΓL(1,𝔽8)—161no—
121C26C2×ΓL(1,𝔽8)11yescharacteristic, complemented
122C2×#120—31no—
123C2×#120—31no—
124C2×#120C2×ΓL(1,𝔽8)11yescharacteristic, complemented
125#379—31no—
126#379—31no—
127#379C2×ΓL(1,𝔽8)11yescharacteristic, complemented
128C23×C7—24no—
129C23×C7—22no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—34no—
133C23×C7—32no—
134C23×C7—32no—
135C23×C7—32no—
136C23×C7—31no—
137C23×C7—31no—
138C23×C7—31no—
139C23×C7—31no—
140C23×C7—62no—
141C23×C7—62no—
142C23×C7—64no—
143C23×C7—61no—
144C23×C7—61no—
145C23×C7—61no—
146C23×C7—62no—
147C23×C7—62no—
148C23×C7—62no—
149C23×C7—62no—
150C23×C7—62no—
151C23×C7—64no—
152C23×C7—62no—
153C23×C7—62no—
154C23×C7—64no—
155C23×C7—68no—
156C23×C7—62no—
157C23×C7—64no—
158C23×C7—64no—
159C23×C7—64no—
160C23×C7—68no—
161C23×C7—62no—
162C23×C7—64no—
163C23×C7—61no—
164C23×C7—64no—
165C23×C7—62no—
166C23×C7—62no—
167C23×C7—64no—
168C23×C7—62no—
169C23×C7—62no—
170C23×C7—64no—
171C23×C7—64no—
172C23×C7—68no—
173C23×C7—64no—
174C23×C7—64no—
175C23×C7—68no—
176C23×C7—64no—
177C23×C7—64no—
178C23×C7—68no—
179C23×C7—64no—
180C23×C7—68no—
181C23×C7—64no—
182C23×C7—64no—
183C23×C7—68no—
184C23×C7—64no—
185C23×C7—68no—
186C23×C7—68no—
187C23×C7—64no—
188C23×C7—68no—
189C23×C7—68no—
190C23×C7—68no—
191C23×C7—68no—
192C23×C7—64no—
193C23×C7—62no—
194C23×C7—61no—
195C23×C7C22×A411yescharacteristic
196C7×D4—41no—
197C7×D4—121no—
198C7×D4—122no—
199C7×D4—122no—
200C7×D4—124no—
201C2×C4×C7—61no—
202C2×C4×C7—62no—
203C2×C4×C7—61no—
204C2×C4×C7—61no—
205C2×C4×C7—61no—
206C2×C4×C7—61no—
207C22⋊A4—71no—
208C22×A4—281no—
209C22×A4—281no—
210C22×A4—562no—
211C4×A4—281no—
212C2×ΓL(1,𝔽8)—161no—
213C2×ΓL(1,𝔽8)—321no—
214C2×ΓL(1,𝔽8)—321no—
215C25—21no—
216C25—31no—
217C25—31no—
218C25—62no—
219C25—62no—
220C25—64no—
221C25—61no—
222C25C22×ΓL(1,𝔽8)11yescharacteristic
223C22×D4—31no—
224C22×D4—31no—
225C22×D4C22⋊ΓL(1,𝔽8)11yescharacteristic, complemented
226#120—21no—
227#120—62no—
228#120—64no—
229#120—62no—
230#120—61no—
231C2×C22⋊C4—32no—
232C2×C22⋊C4—31no—
233C2×C22⋊C4—31no—
234C2×C22⋊C4—31no—
235C2×C22⋊C4—31no—
236C2×C22⋊C4—31no—
237C22×C7—22no—
238C22×C7—31no—
239C22×C7—31no—
240C22×C7—34no—
241C22×C7—32no—
242C22×C7—32no—
243C22×C7—31no—
244C22×C7—41no—
245C22×C7—61no—
246C22×C7—61no—
247C22×C7—64no—
248C22×C7—64no—
249C22×C7—61no—
250C22×C7—61no—
251C22×C7—64no—
252C22×C7—62no—
253C22×C7—64no—
254C22×C7—62no—
255C22×C7—64no—
256C22×C7—62no—
257C22×C7—62no—
258C22×C7—64no—
259C22×C7—62no—
260C22×C7—62no—
261C22×C7—62no—
262C22×C7—64no—
263C22×C7—64no—
264C22×C7—64no—
265C22×C7—64no—
266C22×C7—68no—
267C22×C7—68no—
268C22×C7—64no—
269C22×C7—68no—
270C22×C7—68no—
271C22×C7—64no—
272C22×C7—62no—
273C22×C7—62no—
274C22×C7—121no—
275C22×C7—121no—
276C22×C7#120⋊C311yescharacteristic, complemented
277C22×C7D4×A411yescharacteristic
278C4×C7—41no—
279C4×C7—121no—
280C4×C7—121no—
281C2×A4—281no—
282C2×A4—562no—
283C2×A4—561no—
284C2×A4—561no—
285C2×A4—564no—
286C3×D4—1121no—
287ΓL(1,𝔽8)—161noHall
288C24—22no—
289C24—31no—
290C24—31no—
291C24—34no—
292C24—32no—
293C24—32no—
294C24—31no—
295C24—62no—
296C24—61no—
297C24—61no—
298C24—64no—
299C24—64no—
300C24—62no—
301C24—62no—
302C24—61no—
303C24—61no—
304C24—62no—
305C24—64no—
306C24—62no—
307C24—64no—
308C24—64no—
309C24—62no—
310C24—62no—
311C24—64no—
312C24—62no—
313C24—64no—
314C24—62no—
315C24—64no—
316C24—64no—
317C24—64no—
318C24—64no—
319C24—68no—
320C24—68no—
321C24—68no—
322C24—64no—
323C24—68no—
324C24D4×ΓL(1,𝔽8)11yescharacteristic, complemented
325C24C2×C22⋊ΓL(1,𝔽8)11yescharacteristic
326C24C2×C22⋊ΓL(1,𝔽8)11yescharacteristic, complemented
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—64no—
334C2×D4—61no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4C2×C22⋊ΓL(1,𝔽8)11yescharacteristic, complemented
338C22⋊C4—62no—
339C22⋊C4—62no—
340C22⋊C4—64no—
341C22⋊C4—62no—
342C22⋊C4—61no—
343C22⋊C4—61no—
344C22⋊C4—61no—
345C22⋊C4—61no—
346C2×C7—21no—
347C2×C7—31no—
348C2×C7—31no—
349C2×C7—62no—
350C2×C7—64no—
351C2×C7—62no—
352C2×C7—61no—
353C2×C7—81no—
354C2×C7C2×#120⋊C311yescharacteristic
355C22×C3—1121no—
356C22×C3—1121no—
357A4—281no—
358A4—562no—
359C3×C4—1121no—
360C23—24no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—32no—
365C23—34no—
366C23—32no—
367C23—32no—
368C23—31no—
369C23—31no—
370C23—31no—
371C23—31no—
372C23—62no—
373C23—62no—
374C23—64no—
375C23—61no—
376C23—61no—
377C23—62no—
378C23—61no—
379C23—62no—
380C23—62no—
381C23—62no—
382C23—62no—
383C23—64no—
384C23—62no—
385C23—62no—
386C23—64no—
387C23—68no—
388C23—62no—
389C23—64no—
390C23—64no—
391C23—64no—
392C23—68no—
393C23—62no—
394C23—64no—
395C23—61no—
396C23—64no—
397C23—62no—
398C23—62no—
399C23—64no—
400C23—62no—
401C23—62no—
402C23—64no—
403C23—64no—
404C23—68no—
405C23—64no—
406C23—64no—
407C23—68no—
408C23—64no—
409C23—64no—
410C23—68no—
411C23—64no—
412C23—68no—
413C23—64no—
414C23—68no—
415C23—64no—
416C23—64no—
417C23—68no—
418C23—68no—
419C23—64no—
420C23—68no—
421C23—68no—
422C23—68no—
423C23—68no—
424C23—64no—
425C23—61no—
426C23—62no—
427C23C22×C22⋊ΓL(1,𝔽8)11yescharacteristic
428D4—41no—
429D4—121no—
430D4—122no—
431D4—122no—
432D4—124no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—62no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C7AΣL(1,𝔽64)11yescharacteristic, complemented, Sylow
440C2×C3—1121no—
441C2×C3—2241no—
442C2×C3—2241no—
443C22—22no—
444C22—31no—
445C22—31no—
446C22—32no—
447C22—32no—
448C22—34no—
449C22—31no—
450C22—41no—
451C22—61no—
452C22—61no—
453C22—64no—
454C22—64no—
455C22—61no—
456C22—61no—
457C22—62no—
458C22—64no—
459C22—62no—
460C22—62no—
461C22—64no—
462C22—62no—
463C22—64no—
464C22—64no—
465C22—62no—
466C22—62no—
467C22—62no—
468C22—64no—
469C22—64no—
470C22—64no—
471C22—64no—
472C22—68no—
473C22—68no—
474C22—64no—
475C22—68no—
476C22—68no—
477C22—64no—
478C22—62no—
479C22—62no—
480C22—121no—
481C22—121no—
482C22#120⋊ΓL(1,𝔽8)11yescharacteristic, complemented
483C22D4×C22⋊ΓL(1,𝔽8)11yescharacteristic
484C4—41no—
485C4—121no—
486C4—121no—
487C3—1121noSylow
488C2—21no—
489C2—31no—
490C2—31no—
491C2—62no—
492C2—64no—
493C2—62no—
494C2—61no—
495C2—81no—
496C2C2×#120⋊ΓL(1,𝔽8)11yescharacteristic
subgroupG/H
centerC2C2×#120⋊ΓL(1,𝔽8)
commutator, cosocleC25×C7C22×C3
FrattiniC23C22×C22⋊ΓL(1,𝔽8)
FittingC7×#2846C3
socleC23×C7C22×A4
hypercenterC22#120⋊ΓL(1,𝔽8)
hypocenterC24×C7C3×D4
LCSG ▷ C25×C7 ▷ C24×C7
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C22⋊ΓL(1,𝔽8))C2semidirect
2. C2×#120⋊ΓL(1,𝔽8)C2semidirect
3. #379⋊ΓL(1,𝔽8)C2semidirect
4. C7×#2846C3semidirect
9. C26×C7C2×C3semidirect
12. C2×C7×#120C2×C3semidirect
15. C7×#379C2×C3semidirect
17. C22⋊(C22⋊ΓL(1,𝔽8))D4semidirect
32. C22×C7×D4A4semidirect
53. #2846ΓL(1,𝔽8)semidirect
90. C24×C7C3×D4semidirect
92. C24×C7C2×A4semidirect
103. C22×C4×C7C2×A4semidirect
121. C26C2×ΓL(1,𝔽8)semidirect
124. C2×#120C2×ΓL(1,𝔽8)semidirect
127. #379C2×ΓL(1,𝔽8)semidirect
225. C22×D4C22⋊ΓL(1,𝔽8)semidirect
276. C22×C7#120⋊C3semidirect
324. C24D4×ΓL(1,𝔽8)semidirect
326. C24C2×C22⋊ΓL(1,𝔽8)semidirect
337. C22×C4C2×C22⋊ΓL(1,𝔽8)semidirect
439. C7AΣL(1,𝔽64)semidirect
482. C22#120⋊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×C22⋊(C22⋊ΓL(1,𝔽8))2
C2×#120⋊ΓL(1,𝔽8)2
#379⋊ΓL(1,𝔽8)2
C7×#28463
D4×C22⋊ΓL(1,𝔽8)4
AΣL(1,𝔽64)7