Finite Group Databasebeta

← #763171

Subgroups — #763171

Order 512.

3590 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16902C212yesmaximal, complemented
2#16824C211yesmaximal, characteristic, complemented
3#16823C211yesmaximal, characteristic, complemented
4#15246C214yesmaximal, complemented
5#15161C214yesmaximal, complemented
6#14734C212yesmaximal
7#14717C212yesmaximal
8#14215C212yesmaximal, complemented
9#14213C212yesmaximal, complemented
10#14052C214yesmaximal, complemented
11#13980C214yesmaximal, complemented
12C4×#379C212yesmaximal
13C2×C22⋊#95C211yesmaximal, characteristic
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C2×#399C2212yes—
19C2×#399C2212yes—
20C2×#393C2212yes—
21C2×#393C2212yes—
22C2×#391C2214yescomplemented
23C2×#387C2214yescomplemented
24C2×#385C2214yes—
25C2×#380C2212yes—
26C2×#379C2214yes—
27C2×C4×C22⋊C4C2214yes—
28C2×C4×C22⋊C4C2214yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C22×#95C2214yes—
32#1446C2214yescomplemented
33#1445C2214yescomplemented
34#1445C2214yescomplemented
35#1443C2214yescomplemented
36#1442C2214yescomplemented
37#1441C2214yescomplemented
38#1441C2214yescomplemented
39#1441C2214yescomplemented
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2212yes—
45#1439C2212yes—
46#1439C2212yes—
47#1439C2212yes—
48#1436C2214yescomplemented
49#1436C2214yescomplemented
50#1436C2214yescomplemented
51#1436C2214yescomplemented
52C4⋊#95C2214yescomplemented
53#1434C2214yescomplemented
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2212yes—
57C22⋊#95C2212yes—
58C22⋊#95C2214yes—
59C22⋊#95C2212yes—
60C22⋊#95C2214yes—
61C22⋊#95C2211yescharacteristic
62C22⋊#95C2211yescharacteristic
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2211yescharacteristic
67C26C2311yescharacteristic
68C23×Q8C2312yes—
69C24×C4—48no—
70C24×C4—42no—
71C24×C4—42no—
72C24×C4Q818yescomplemented
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—44no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—42no—
90C22×C22⋊C4—42no—
91C22×C22⋊C4—42no—
92C22×C22⋊C4—42no—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C42C2314yes—
106C22×C42C2314yes—
107#399—28no—
108#399—44no—
109#393—28no—
110#393—44no—
111#391—48no—
112#387—48no—
113#385—216no—
114#380—44no—
115#379—216no—
116C4×C22⋊C4—48no—
117C4×C22⋊C4—48no—
118C4×C22⋊C4—44no—
119C4×C22⋊C4—44no—
120C2×#95—48no—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C25—22no—
147C25—22no—
148C25—22no—
149C25—22no—
150C25—22no—
151C25—22no—
152C25—28no—
153C25—44no—
154C25C2×Q814yes—
155C25C2411yescharacteristic
156C25C2411yescharacteristic
157C25C2411yescharacteristic
158C22×Q8—28no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—42no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—28no—
166C23×C4—28no—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—432no—
175C23×C4—416no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—44no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—432no—
190C23×C4—432no—
191C23×C4C2×Q818yes—
192C23×C4C2×Q814yes—
193C23×C4C2×Q818yes—
194C23×C4C2×Q814yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C22⋊Q8—432no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—416no—
256C2×C22⋊C4—416no—
257C2×C22⋊C4#41116yes—
258C2×C22⋊C4#4114yes—
259C2×C22⋊C4#4114yes—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—42no—
291C24—21no—
292C24—21no—
293C24—24no—
294C24—21no—
295C24—21no—
296C24—28no—
297C24—28no—
298C24—28no—
299C24—28no—
300C24—28no—
301C24—28no—
302C24—28no—
303C24—24no—
304C24—21no—
305C24—21no—
306C24—24no—
307C24—21no—
308C24—24no—
309C24—24no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—216no—
316C24—48no—
317C24—416no—
318C24—416no—
319C24—416no—
320C24—416no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24C22×Q812yes—
326C24C22×Q812yes—
327C24C22×Q812yes—
328C24C2×#4112yes—
329C24C2×#4112yes—
330C24C2×#4118yes—
331C24H(5,𝔽2)12yes—
332C24H(5,𝔽2)12yes—
333C24H(5,𝔽2)12yes—
334C242−1+412yes—
335C24C2511yescharacteristic
336C2×Q8—48no—
337C2×Q8—416no—
338C2×Q8—84no—
339C2×Q8—88no—
340C2×Q8—84no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—216no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—4128no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4C22×Q814yes—
422C22×C4C22×Q814yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4118yes—
425C22×C4C2×#4118yes—
426C22×C4C2×#4114yes—
427C22×C4H(5,𝔽2)18yes—
428C4⋊C4—416no—
429C4⋊C4—48no—
430C4⋊C4—416no—
431C4⋊C4—84no—
432C4⋊C4—88no—
433C4⋊C4—84no—
434C22⋊C4—232no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—416no—
446C42—88no—
447C42—88no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—28no—
452C23—28no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—28no—
458C23—28no—
459C23—24no—
460C23—24no—
461C23—232no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—28no—
471C23—28no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—432no—
483C23—432no—
484C23—416no—
485C23—432no—
486C23—432no—
487C23—432no—
488C23—432no—
489C23—416no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23#53314yes—
503C23#53318yes—
504C23#54314yes—
505C23#54414yes—
506C23#55118yes—
507C23#55918yes—
508C23C23×Q811yescharacteristic
509C23C22×#4114yes—
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)14yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×2−1+411yescharacteristic
517Q8—816no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—416no—
536C2×C4—432no—
537C2×C4—432no—
538C2×C4—416no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—816no—
580C2×C4—816no—
581C2×C4#54318yes—
582C22—28no—
583C22—24no—
584C22—28no—
585C22—28no—
586C22—28no—
587C22—28no—
588C22—28no—
589C22—24no—
590C22—28no—
591C22—28no—
592C22—24no—
593C22—24no—
594C22—464no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22#260718yes—
610C22C2×#53312yes—
611C22C2×#53311yescharacteristic
612C22C2×#54312yes—
613C22C2×#54311yescharacteristic
614C22C2×#54411yescharacteristic
615C22C2×#55112yes—
616C22C2×#55912yes—
617C22#348114yes—
618C22#348114yes—
619C22#349011yescharacteristic
620C22#349011yescharacteristic
621C22#349211yescharacteristic
622C22#349211yescharacteristic
623C22#351914yes—
624C22#352014yes—
625C22#353212yes—
626C22#353212yes—
627C4—28no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—48no—
632C4—48no—
633C4—84no—
634C4—82no—
635C4—84no—
636C4—82no—
637C2—28no—
638C2—44no—
639C2—44no—
640C2C2×#260711yescharacteristic
641C2#3495812yes—
642C2#3505211yescharacteristic
643C2#3507912yes—
644C2#3747511yescharacteristic
645C2#5937012yes—
646C2#5937112yes—
647C2#5962312yes—
648C2#5962412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16902C2semidirect
2. #16824C2semidirect
3. #16823C2semidirect
4. #15246C2semidirect
5. #15161C2semidirect
8. #14215C2semidirect
9. #14213C2semidirect
10. #14052C2semidirect
11. #13980C2semidirect
22. C2×#391C22semidirect
23. C2×#387C22semidirect
32. #1446C22semidirect
33. #1445C22semidirect
34. #1445C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1441C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
48. #1436C22semidirect
49. #1436C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
52. C4⋊#95C22semidirect
53. #1434C22semidirect
72. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
#169022
#168242
#168232
#152462
#151612
#147342
#147172
#142152
#142132
#140522
#139802
C4×#3792
C2×C22⋊#952