Finite Group Databasebeta

← #760860

Subgroups — #760860

Order 512.

4090 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2437C212yesmaximal
2#16825C212yesmaximal, complemented
3#16824C211yesmaximal, characteristic, complemented
4#16823C211yesmaximal, characteristic, complemented
5#15165C212yesmaximal, complemented
6#15161C212yesmaximal, complemented
7#14920C214yesmaximal, complemented
8#14914C214yesmaximal, complemented
9#14127C212yesmaximal
10#14052C214yesmaximal, complemented
11#14004C214yesmaximal, complemented
12C4×#379C211yesmaximal, characteristic
13C2×C22⋊#95C212yesmaximal
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8C2214yes—
16C23×C22⋊C4C2214yes—
17#2437—216no—
18C2×#399C2212yes—
19C2×#396C2214yes—
20C2×#393C2212yes—
21C2×#391C2212yescomplemented
22C2×#389C2212yescomplemented
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#380C2214yes—
28C2×#379C2211yescharacteristic
29C2×#379C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2214yes—
33C22×#95C2214yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36#1446C2212yescomplemented
37#1446C2214yescomplemented
38#1445C2212yescomplemented
39#1445C2214yescomplemented
40#1444C2212yescomplemented
41#1444C2214yescomplemented
42#1442C2212yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1441C2214yescomplemented
46#1441C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1439C2214yes—
51#1439C2212yes—
52#1439C2212yes—
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55C4⋊#95C2212yescomplemented
56C4⋊#95C2212yescomplemented
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2212yes—
61C22⋊#95C2214yes—
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2211yescharacteristic
68C26C2311yescharacteristic
69C23×Q8C2312yes—
70C24×C4—42no—
71C24×C4—42no—
72C24×C4—48no—
73C24×C4Q818yescomplemented
74C24×C4C2314yes—
75C24×C4C2314yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—48no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—216no—
92C22×C22⋊C4—216no—
93C22×C22⋊C4—48no—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106#399—28no—
107#396—216no—
108#393—28no—
109#391—44no—
110#389—44no—
111#385—216no—
112#385—216no—
113#385—216no—
114#385—48no—
115#380—216no—
116#379D418yes—
117#379Q818yescomplemented
118C4×C22⋊C4—42no—
119C4×C22⋊C4—42no—
120C4×C22⋊C4—48no—
121C2×#95—44no—
122C2×#95—44no—
123C2×#95—48no—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25—28no—
150C25—28no—
151C25—41no—
152C25—41no—
153C25—41no—
154C25—41no—
155C25C2×D414yes—
156C25C2×Q814yes—
157C25C2×Q814yes—
158C25C2411yescharacteristic
159C25C2412yes—
160C22×Q8—24no—
161C22×Q8—28no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—42no—
165C23×C4—28no—
166C23×C4—28no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—416no—
180C23×C4—432no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—432no—
196C23×C4—432no—
197C23×C4C2×Q818yes—
198C23×C4C2×Q818yes—
199C23×C4C2×Q814yes—
200C23×C4C2×Q814yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C22⋊Q8—464no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—464no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4C2×D4116yes—
260C2×C22⋊C4C2×D414yes—
261C2×C22⋊C4C2×D414yes—
262C2×C22⋊C4C2×Q8116yes—
263C2×C22⋊C4C2×Q814yes—
264C2×C22⋊C4C2×Q814yes—
265C2×C22⋊C4#41116yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—44no—
269C2×C42—44no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—42no—
285#95—44no—
286#95—42no—
287#95—42no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—24no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24—216no—
302C24—216no—
303C24—232no—
304C24—232no—
305C24—24no—
306C24—21no—
307C24—21no—
308C24—24no—
309C24—21no—
310C24—21no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—416no—
322C24—416no—
323C24C4⋊Q8116yes—
324C24C22×D412yes—
325C24C22×D414yes—
326C24C22×Q812yes—
327C24C22×Q812yes—
328C24C22×Q814yes—
329C24C22×Q814yes—
330C24C2×#4118yes—
331C24H(5,𝔽2)18yes—
332C24C2511yescharacteristic
333C2×Q8—416no—
334C2×Q8—48no—
335C2×Q8—84no—
336C2×Q8—88no—
337C2×Q8—84no—
338C22×C4—24no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—24no—
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—24no—
391C22×C4—24no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—4128no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×Q814yes—
424C22×C4C22×Q814yes—
425C22×C4C22×Q814yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C2×#4118yes—
429C22×C4C2×#4118yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C4⋊C4—48no—
433C4⋊C4—416no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—88no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C4⋊C4—84no—
440C22⋊C4—232no—
441C22⋊C4—232no—
442C22⋊C4—232no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C42—84no—
451C42—84no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—28no—
475C23—28no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—264no—
481C23—416no—
482C23—416no—
483C23—432no—
484C23—432no—
485C23—416no—
486C23—432no—
487C23—432no—
488C23—432no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23C2×C4⋊Q814yes—
496C23C2×C4⋊Q818yes—
497C23#53318yes—
498C23#54318yes—
499C23#546116yes—
500C23#554116yes—
501C23C23×D411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C23×Q811yescharacteristic
504C23C22×#4114yes—
505C23C2×H(5,𝔽2)14yes—
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23#58511yescharacteristic
509C23#58511yescharacteristic
510Q8—816no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—432no—
534C2×C4—416no—
535C2×C4—432no—
536C2×C4—416no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
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—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
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—44no—
566C2×C4—44no—
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—832no—
576C2×C4—832no—
577C2×C4#54314yes—
578C22—28no—
579C22—216no—
580C22—216no—
581C22—28no—
582C22—28no—
583C22—216no—
584C22—216no—
585C22—28no—
586C22—28no—
587C22—28no—
588C22—28no—
589C22—24no—
590C22—464no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22#2549116yes—
599C22C22×C4⋊Q811yescharacteristic
600C22C2×#53312yes—
601C22C2×#54312yes—
602C22C2×#54311yescharacteristic
603C22#345811yescharacteristic
604C22C2×#54614yes—
605C22C2×#55414yes—
606C22#348112yes—
607C22#348112yes—
608C22#349611yescharacteristic
609C22#349611yescharacteristic
610C22#350111yescharacteristic
611C22#350111yescharacteristic
612C22#351314yes—
613C22#351314yes—
614C22#353212yes—
615C22#353212yes—
616C4—24no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—48no—
622C4—44no—
623C4—44no—
624C4—48no—
625C4—84no—
626C4—84no—
627C2—216no—
628C2—44no—
629C2C2×#254912yes—
630C2#3495811yescharacteristic
631C2#3500211yescharacteristic
632C2#3506512yes—
633C2#3747511yescharacteristic
634C2#5980212yes—
635C2#5980312yes—
636C2#5991412yes—
637C2#5991512yes—
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
2. #16825C2semidirect
3. #16824C2semidirect
4. #16823C2semidirect
5. #15165C2semidirect
6. #15161C2semidirect
7. #14920C2semidirect
8. #14914C2semidirect
10. #14052C2semidirect
11. #14004C2semidirect
21. C2×#391C22semidirect
22. C2×#389C22semidirect
36. #1446C22semidirect
37. #1446C22semidirect
38. #1445C22semidirect
39. #1445C22semidirect
40. #1444C22semidirect
41. #1444C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. C4⋊#95C22semidirect
56. C4⋊#95C22semidirect
73. C24×C4Q8semidirect
117. #379Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24372
#168252
#168242
#168232
#151652
#151612
#149202
#149142
#141272
#140522
#140042
C4×#3792
C2×C22⋊#952