Finite Group Databasebeta

← #626821

Subgroups — #626821

Order 512.

6106 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic, complemented
2C2×#2848C212yesmaximal, complemented
3C22×#386C211yesmaximal, characteristic
4#15951C218yesmaximal, complemented
5#15323C218yesmaximal, complemented
6#14914C212yesmaximal, complemented
7#14717C212yesmaximal, complemented
8#14338C211yesmaximal, characteristic, complemented
9#14281C212yesmaximal, complemented
10#14127C212yesmaximal, complemented
11#13867C212yesmaximal
12C24×D4C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C2×#543—216no—
15C2×#543—28no—
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2214yescomplemented
19C22×C22⋊Q8C2214yescomplemented
20C22×C22⋊Q8C2214yescomplemented
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2848—216no—
26C2×#393C2214yescomplemented
27C2×#386—28no—
28C2×#386C2218yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#385C2218yescomplemented
33C2×#385C2214yescomplemented
34C2×#380C2212yes—
35C2×#380C2214yescomplemented
36C2×#379C2212yes—
37C2×#379C2214yes—
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1446C2214yescomplemented
41#1446C2214yescomplemented
42#1445C2214yescomplemented
43#1444C2214yescomplemented
44#1441C2218yescomplemented
45#1441C2218yescomplemented
46#1439C2218yescomplemented
47#1439C2218yescomplemented
48#1439C2214yescomplemented
49#1439C2218yescomplemented
50#1438C2218yes—
51C22⋊#95C2212yes—
52C22⋊#95C2214yes—
53C22⋊#95C2214yes—
54C22⋊#95C2214yescomplemented
55C26C2311yescharacteristic
56C26C2311yescharacteristic
57C23×Q8C2312yes—
58C23×Q8C2312yes—
59C23×D4—24no—
60C23×D4—24no—
61C23×D4—44no—
62C23×D4—44no—
63C23×D4Q818yescomplemented
64C23×D4C2314yes—
65C24×C4—28no—
66C24×C4—24no—
67C24×C4—44no—
68C24×C4—44no—
69C24×C4C2311yescharacteristic
70C24×C4C2314yes—
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77#543—2128no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—48no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8D4116yescomplemented
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2318yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—216no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4D418yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116#393—48no—
117#386—232no—
118#386—48no—
119#386—48no—
120#385—232no—
121#385—216no—
122#385—48no—
123#380—28no—
124#380—216no—
125#379—28no—
126#379—216no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95C2318yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2314yes—
138C2×#95C2318yes—
139C25—22no—
140C25—22no—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—42no—
148C25—42no—
149C25—42no—
150C25—42no—
151C25—42no—
152C25—42no—
153C25C2×D414yes—
154C25C2×Q814yes—
155C25C2×Q814yes—
156C25#4114yes—
157C25C2412yes—
158C25C2412yes—
159C25C2411yescharacteristic
160C22×Q8—28no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8C2×D418yescomplemented
169C22×D4—232no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—432no—
177C22×D4—416no—
178C22×D4—416no—
179C22×D4—432no—
180C22×D4C2×Q8116yes—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—416no—
214C23×C4—464no—
215C23×C4—416no—
216C23×C4—416no—
217C23×C4C2×D414yes—
218C23×C4C2×D418yes—
219C23×C4C2×D414yes—
220C23×C4C2×Q814yes—
221C23×C4C2418yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C22⋊Q8—2128no—
230C22⋊Q8—2128no—
231C22⋊Q8—264no—
232C22⋊Q8—264no—
233C22⋊Q8—464no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D418yescomplemented
250C2×C4⋊C4C2×D4116yescomplemented
251C2×C22⋊C4—264no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—232no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—416no—
280C2×C22⋊C4—416no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C2×D4116yes—
284C2×C22⋊C4C2×D4116yes—
285#95—28no—
286#95—28no—
287#95—216no—
288#95—28no—
289#95—44no—
290#95—48no—
291#95—48no—
292#95—48no—
293#95—48no—
294#95—44no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—28no—
300C24—28no—
301C24—216no—
302C24—28no—
303C24—216no—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—28no—
319C24—28no—
320C24—216no—
321C24—216no—
322C24—216no—
323C24—48no—
324C24—44no—
325C24—44no—
326C24—44no—
327C24—48no—
328C24—48no—
329C24—44no—
330C24—48no—
331C24—48no—
332C24—416no—
333C24—416no—
334C24—416no—
335C24—416no—
336C24—44no—
337C24—48no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—48no—
342C24—88no—
343C24—88no—
344C24—88no—
345C24—88no—
346C24C22⋊Q8116yescomplemented
347C24C22×D414yes—
348C24C22×D412yes—
349C24C22×D414yes—
350C24C22×Q814yes—
351C24C22×Q814yes—
352C24C22×Q812yes—
353C24C2×#4114yes—
354C24C2×#4112yes—
355C24H(5,𝔽2)14yes—
356C24H(5,𝔽2)14yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—216no—
360C2×Q8—216no—
361C2×Q8—416no—
362C2×Q8—88no—
363C2×Q8—84no—
364C2×Q8—84no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8—816no—
368C2×D4—232no—
369C2×D4—232no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—4128no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—416no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—4128no—
431C22×C4—44no—
432C22×C4—416no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—816no—
437C22×C4—816no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4#120116yes—
445C22×C4C22×D418yes—
446C22×C4C22×D414yes—
447C22×C4C22×D418yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×Q814yes—
452C22×C4H(5,𝔽2)14yes—
453C4⋊C4—216no—
454C4⋊C4—232no—
455C4⋊C4—216no—
456C4⋊C4—232no—
457C4⋊C4—432no—
458C4⋊C4—416no—
459C22⋊C4—232no—
460C22⋊C4—232no—
461C22⋊C4—232no—
462C22⋊C4—432no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—432no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—88no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—28no—
475C23—22no—
476C23—28no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—232no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—232no—
507C23—416no—
508C23—416no—
509C23—464no—
510C23—416no—
511C23—416no—
512C23—416no—
513C23—464no—
514C23—416no—
515C23—416no—
516C23—42no—
517C23—416no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—48no—
524C23—42no—
525C23—48no—
526C23—44no—
527C23—44no—
528C23—48no—
529C23—44no—
530C23—48no—
531C23—88no—
532C23—832no—
533C23—88no—
534C23—88no—
535C23—832no—
536C23—88no—
537C23C2×#12014yes—
538C23C2×C22⋊Q818yes—
539C23C2×C22⋊Q814yes—
540C23#53614yes—
541C23#54318yescomplemented
542C23#54818yes—
543C23D4×Q818yes—
544C23C23×D411yescharacteristic
545C23C23×D412yes—
546C23C23×Q811yescharacteristic
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553Q8—816no—
554Q8—816no—
555D4—464no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—216no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—216no—
571C2×C4—416no—
572C2×C4—464no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—88no—
600C2×C4—816no—
601C2×C4—816no—
602C2×C4—88no—
603C2×C4C2×#12018yes—
604C22—24no—
605C22—216no—
606C22—24no—
607C22—24no—
608C22—216no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—216no—
613C22—28no—
614C22—28no—
615C22—28no—
616C22—28no—
617C22—464no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—416no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—48no—
628C22—416no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—832no—
634C22#240918yes—
635C22C22×#12011yescharacteristic
636C22C22×C22⋊Q811yescharacteristic
637C22C2×#53611yescharacteristic
638C22C2×#54312yes—
639C22#345912yes—
640C22C2×#54812yes—
641C22C2×D4×Q812yes—
642C22#348414yes—
643C22#348411yescharacteristic
644C22#348411yescharacteristic
645C22#348411yescharacteristic
646C22#348511yescharacteristic
647C22#348814yes—
648C22#349012yes—
649C22#350112yes—
650C22#353514yes—
651C22#353614yes—
652C4—28no—
653C4—44no—
654C4—48no—
655C4—44no—
656C4—44no—
657C4—48no—
658C4—82no—
659C4—88no—
660C2—28no—
661C2—48no—
662C2—48no—
663C2C2×#240911yescharacteristic
664C2#3498211yescharacteristic
665C2#3498511yescharacteristic
666C2#3534611yescharacteristic
667C2#3551211yescharacteristic
668C2#3671811yescharacteristic
669C2#3672011yescharacteristic
670C2#3768414yes—
671C2#3775214yes—
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. C22×#543C2semidirect
2. C2×#2848C2semidirect
4. #15951C2semidirect
5. #15323C2semidirect
6. #14914C2semidirect
7. #14717C2semidirect
8. #14338C2semidirect
9. #14281C2semidirect
10. #14127C2semidirect
18. C22×C22⋊Q8C22semidirect
19. C22×C22⋊Q8C22semidirect
20. C22×C22⋊Q8C22semidirect
26. C2×#393C22semidirect
32. C2×#385C22semidirect
33. C2×#385C22semidirect
35. C2×#380C22semidirect
40. #1446C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
54. C22⋊#95C22semidirect
63. C23×D4Q8semidirect
91. C2×C22⋊Q8D4semidirect
168. C22×Q8C2×D4semidirect
249. C2×C4⋊C4C2×D4semidirect
250. C2×C4⋊C4C2×D4semidirect
346. C24C22⋊Q8semidirect
541. C23#543semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#28482
C22×#3862
#159512
#153232
#149142
#147172
#143382
#142812
#141272
#138672