Finite Group Databasebeta

← #626957

Subgroups — #626957

Order 512.

5222 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C212yesmaximal, complemented
2C2×#2659C211yesmaximal, characteristic, complemented
3C2×#2626C212yesmaximal, complemented
4C22×#386C211yesmaximal, characteristic
5#15505C218yesmaximal, complemented
6#15325C218yesmaximal, complemented
7#14734C212yesmaximal, complemented
8#14734C212yesmaximal, complemented
9#13965C212yesmaximal, complemented
10#13902C212yesmaximal
11#13778C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2214yescomplemented
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2848—216no—
20#2659—28no—
21#2626—216no—
22C2×#399C2214yescomplemented
23C2×#399C2214yescomplemented
24C2×#399C2214yescomplemented
25C2×#398C2214yescomplemented
26C2×#396C2218yescomplemented
27C2×#395C2214yescomplemented
28C2×#393C2214yescomplemented
29C2×#390C2214yes—
30C2×#390C2214yes—
31C2×#386—28no—
32C2×#386C2218yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2218yescomplemented
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2214yescomplemented
40C2×#379C2214yes—
41C2×#379C2214yes—
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1444C2214yescomplemented
46#1444C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2218yescomplemented
49#1439C2218yescomplemented
50#1439C2214yescomplemented
51#1438C2218yes—
52#1436C2218yescomplemented
53C4⋊#95C2218yescomplemented
54C22⋊#95C2212yes—
55C26C2311yescharacteristic
56C26C2311yescharacteristic
57C23×Q8C2312yes—
58C23×D4—24no—
59C23×D4—24no—
60C23×D4—44no—
61C23×D4—44no—
62C23×D4Q818yescomplemented
63C23×D4C2314yes—
64C24×C4—28no—
65C24×C4—24no—
66C24×C4—44no—
67C24×C4—44no—
68C24×C4C2311yescharacteristic
69C24×C4C2314yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—48no—
79C2×C22⋊Q8—48no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2318yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2318yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2318yes—
104C22×C42C2314yes—
105#399—216no—
106#399—48no—
107#399—48no—
108#398—216no—
109#396—232no—
110#395—216no—
111#393—216no—
112#390—48no—
113#390—48no—
114#386—232no—
115#385—216no—
116#385—216no—
117#385—232no—
118#380—28no—
119#380—28no—
120#380—216no—
121#380D4116yescomplemented
122#379—216no—
123#379—216no—
124C4×C22⋊C4—28no—
125C2×#95—24no—
126C2×#95D418yes—
127C2×#95C2318yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2318yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C25—22no—
138C25—22no—
139C25—24no—
140C25—24no—
141C25—22no—
142C25—24no—
143C25—22no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—42no—
148C25—42no—
149C25—42no—
150C25—42no—
151C25—42no—
152C25—42no—
153C25C2×Q814yes—
154C25C2×Q814yes—
155C25C2412yes—
156C25C2412yes—
157C25C2411yescharacteristic
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—232no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—432no—
171C22×D4—416no—
172C22×D4—416no—
173C22×D4—432no—
174C22×D4C2×Q8116yes—
175C23×C4—22no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—416no—
198C23×C4—464no—
199C23×C4—416no—
200C23×C4—416no—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D418yes—
205C23×C4C2×Q814yes—
206C23×C4#4114yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2418yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C22⋊Q8—464no—
216C22⋊Q8—432no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—216no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D4116yescomplemented
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—216no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—416no—
263C2×C22⋊C4—416no—
264C2×C22⋊C4—416no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4C2×D4116yes—
270C2×C42—28no—
271C2×C42—44no—
272C2×C42—44no—
273#95—28no—
274#95—216no—
275#95—28no—
276#95—28no—
277#95—216no—
278#95—28no—
279#95—44no—
280#95—44no—
281#95C2×D4116yes—
282#95C2×D4116yescomplemented
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—28no—
288C24—28no—
289C24—216no—
290C24—28no—
291C24—216no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—24no—
300C24—28no—
301C24—216no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—216no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—416no—
315C24—416no—
316C24—416no—
317C24—416no—
318C24—44no—
319C24—48no—
320C24—44no—
321C24—44no—
322C24—44no—
323C24—44no—
324C24—48no—
325C24—48no—
326C24—44no—
327C24—44no—
328C24—44no—
329C24—48no—
330C24—44no—
331C24—44no—
332C24—48no—
333C24—88no—
334C24—88no—
335C24—88no—
336C24—88no—
337C24C22×D412yes—
338C24C22×D414yes—
339C24C22×Q814yes—
340C24C22×Q814yes—
341C24C22×Q812yes—
342C24C2×#4112yes—
343C24H(5,𝔽2)14yes—
344C24H(5,𝔽2)14yes—
345C242−1+414yes—
346C24C2511yescharacteristic
347C2×Q8—416no—
348C2×Q8—48no—
349C2×Q8—88no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×D4—232no—
353C2×D4—232no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—4128no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—232no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—416no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—44no—
410C22×C4—4128no—
411C22×C4—44no—
412C22×C4—416no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4C22⋊Q8116yescomplemented
419C22×C4#127116yes—
420C22×C4C22×D414yes—
421C22×C4C22×D4116yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×Q814yes—
427C22×C4C2×#4114yes—
428C22×C4H(5,𝔽2)14yes—
429C4⋊C4—416no—
430C4⋊C4—432no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—816no—
438C22⋊C4—232no—
439C22⋊C4—416no—
440C22⋊C4—432no—
441C22⋊C4—416no—
442C22⋊C4—432no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—816no—
446C22⋊C4—816no—
447C42—88no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—28no—
454C23—22no—
455C23—28no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—232no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—464no—
478C23—416no—
479C23—416no—
480C23—416no—
481C23—416no—
482C23—464no—
483C23—416no—
484C23—416no—
485C23—416no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—416no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—48no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—416no—
511C23—832no—
512C23—88no—
513C23—88no—
514C23—88no—
515C23—832no—
516C23—88no—
517C23C2×C22⋊Q814yes—
518C23C2×#12714yes—
519C23#53614yes—
520C23#54318yes—
521C23#54718yes—
522C23D4×Q818yes—
523C23C23×D411yescharacteristic
524C23C23×D412yes—
525C23C23×Q811yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23#58512yes—
532Q8—816no—
533D4—464no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—216no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—216no—
548C2×C4—28no—
549C2×C4—216no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—464no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4C2×C22⋊Q818yes—
579C2×C4C2×#12718yes—
580C22—216no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—216no—
586C22—24no—
587C22—24no—
588C22—216no—
589C22—28no—
590C22—464no—
591C22—48no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—48no—
598C22—48no—
599C22—44no—
600C22—416no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—832no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—88no—
609C22—82no—
610C22#254218yes—
611C22C22×C22⋊Q811yescharacteristic
612C22C22×#12711yescharacteristic
613C22C2×#53611yescharacteristic
614C22C2×#54312yes—
615C22#346012yes—
616C22C2×#54712yes—
617C22C2×D4×Q812yes—
618C22#348414yes—
619C22#348411yescharacteristic
620C22#348411yescharacteristic
621C22#348411yescharacteristic
622C22#348711yescharacteristic
623C22#348814yes—
624C22#349012yes—
625C22#350112yes—
626C22#353514yes—
627C22#353614yes—
628C4—28no—
629C4—28no—
630C4—44no—
631C4—44no—
632C4—416no—
633C4—84no—
634C4—84no—
635C4—82no—
636C2—28no—
637C2—48no—
638C2—84no—
639C2C2×#254211yescharacteristic
640C2#3498211yescharacteristic
641C2#3517411yescharacteristic
642C2#3539011yescharacteristic
643C2#3556511yescharacteristic
644C2#3671111yescharacteristic
645C2#3671811yescharacteristic
646C2#3768414yes—
647C2#3776114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2848C2semidirect
2. C2×#2659C2semidirect
3. C2×#2626C2semidirect
5. #15505C2semidirect
6. #15325C2semidirect
7. #14734C2semidirect
8. #14734C2semidirect
9. #13965C2semidirect
11. #13778C2semidirect
15. C22×C22⋊Q8C22semidirect
22. C2×#399C22semidirect
23. C2×#399C22semidirect
24. C2×#399C22semidirect
25. C2×#398C22semidirect
26. C2×#396C22semidirect
27. C2×#395C22semidirect
28. C2×#393C22semidirect
35. C2×#385C22semidirect
39. C2×#380C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
52. #1436C22semidirect
53. C4⋊#95C22semidirect
62. C23×D4Q8semidirect
121. #380D4semidirect
239. C2×C4⋊C4C2×D4semidirect
282. #95C2×D4semidirect
418. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#26592
C2×#26262
C22×#3862
#155052
#153252
#147342
#139652
#139022
#137782