Finite Group Databasebeta

← #625935

Subgroups — #625935

Order 512.

5598 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C212yesmaximal, complemented
2C2×#2539C212yesmaximal, complemented
3C2×#2656C211yesmaximal, characteristic, complemented
4C2×#2618C212yesmaximal, complemented
5C22×#385C211yesmaximal, characteristic, complemented
6#15505C218yesmaximal, complemented
7#15315C218yesmaximal, complemented
8#14926C212yesmaximal, complemented
9#14716C212yesmaximal, complemented
10#14137C212yesmaximal, complemented
11#13778C211yesmaximal, characteristic, complemented
12C25×C4C2211yescharacteristic
13C22×#123C2214yes—
14C22×#121C2212yes—
15C22×#121C2212yes—
16C22×#121C2214yes—
17C22×#121C2212yes—
18C22×#120C2214yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2847—216no—
24#2656—28no—
25#2618—216no—
26#2539—216no—
27C2×#399C2214yes—
28C2×#399C2214yescomplemented
29C2×#398C2214yescomplemented
30C2×#396C2218yescomplemented
31C2×#394C2214yescomplemented
32C2×#392C2214yes—
33C2×#392C2214yes—
34C2×#392C2214yescomplemented
35C2×#390C2218yes—
36C2×#390C2214yescomplemented
37C2×#386C2214yes—
38C2×#386C2218yes—
39C2×#386C2214yes—
40C2×#385—28no—
41C2×#385C2218yes—
42C2×#380C2214yes—
43C2×#380C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1444C2214yescomplemented
48#1439C2214yes—
49#1439C2218yescomplemented
50#1439C2218yescomplemented
51#1438C2218yes—
52#1438C2214yescomplemented
53#1436C2218yescomplemented
54C4⋊#95C2214yescomplemented
55C22⋊#95C2212yes—
56C26C2311yescharacteristic
57C23×D4C2314yes—
58C23×D4C2318yes—
59C23×D4C2312yes—
60C23×D4C2312yes—
61C23×D4C2312yes—
62C24×C4—28no—
63C24×C4—24no—
64C24×C4—44no—
65C24×C4—44no—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2314yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C2×#123—216no—
74C2×#123—216no—
75C2×#123—48no—
76C2×#121—28no—
77C2×#121—28no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—216no—
84C2×#121—48no—
85C2×#121—48no—
86C2×#121—48no—
87C2×#120—216no—
88C2×#120—216no—
89C2×#120—48no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4D418yescomplemented
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2314yes—
117#399—216no—
118#399—216no—
119#398—216no—
120#396—232no—
121#394—216no—
122#392—216no—
123#392—216no—
124#392—48no—
125#390—232no—
126#390—48no—
127#386—216no—
128#386—216no—
129#386—232no—
130#385—232no—
131#380—216no—
132#380D4116yescomplemented
133C4×C22⋊C4—28no—
134C2×#95—24no—
135C2×#95D418yescomplemented
136C2×#95C2314yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C25—22no—
145C25—22no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—42no—
151C25—42no—
152C25C2×D414yes—
153C25C2414yes—
154C25C2412yes—
155C25C2411yescharacteristic
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—216no—
164C22×D4—216no—
165C22×D4—216no—
166C22×D4—216no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—48no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—48no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—48no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—416no—
211C23×C4—464no—
212C23×C4—416no—
213C23×C4—416no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D418yes—
220C23×C4#4114yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2418yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229#123—464no—
230#121—464no—
231#121—432no—
232#121—432no—
233#121—432no—
234#120—464no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4C2×D4116yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—232no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—216no—
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—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4C2×D4116yes—
288C2×C22⋊C4C2×D4116yes—
289C2×C42—28no—
290C2×C42—44no—
291C2×C42—44no—
292#95—28no—
293#95—28no—
294#95—216no—
295#95—216no—
296#95—44no—
297#95—44no—
298#95C2×D4116yes—
299#95C2×D4116yes—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—28no—
305C24—28no—
306C24—216no—
307C24—28no—
308C24—216no—
309C24—22no—
310C24—24no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—22no—
316C24—24no—
317C24—216no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—28no—
324C24—24no—
325C24—28no—
326C24—24no—
327C24—28no—
328C24—28no—
329C24—48no—
330C24—416no—
331C24—416no—
332C24—416no—
333C24—416no—
334C24—44no—
335C24—48no—
336C24—44no—
337C24—44no—
338C24—44no—
339C24—48no—
340C24C22×D414yes—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C22×D414yes—
344C24C2×#4112yes—
345C24H(5,𝔽2)14yes—
346C24H(5,𝔽2)14yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349C2×D4—432no—
350C2×D4—432no—
351C2×D4—432no—
352C2×D4—416no—
353C2×D4—48no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—432no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—84no—
366C2×D4—88no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—88no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—816no—
374C2×D4—816no—
375C2×D4—816no—
376C2×D4—816no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—232no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—416no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—44no—
430C22×C4—4128no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4#121116yescomplemented
442C22×C4#127116yes—
443C22×C4C22×D414yes—
444C22×C4C22×D4116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)14yes—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—432no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C22⋊C4—232no—
461C22⋊C4—232no—
462C22⋊C4—432no—
463C22⋊C4—432no—
464C22⋊C4—416no—
465C22⋊C4—432no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—816no—
470C42—88no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—232no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—464no—
497C23—416no—
498C23—416no—
499C23—416no—
500C23—416no—
501C23—464no—
502C23—416no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—416no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—48no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—48no—
524C23—48no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23C2×#12114yes—
538C23C2×#12714yes—
539C23#53514yes—
540C23#53718yes—
541C23D4218yes—
542C23#55018yes—
543C23C23×D411yescharacteristic
544C23C23×D411yescharacteristic
545C23C23×D412yes—
546C23C22×#4111yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552D4—816no—
553D4—816no—
554D4—864no—
555D4—816no—
556D4—832no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—216no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—216no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—216no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—464no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—416no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4C2×#12118yes—
605C2×C4C2×#12718yes—
606C22—216no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—216no—
612C22—24no—
613C22—24no—
614C22—216no—
615C22—28no—
616C22—464no—
617C22—48no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—48no—
624C22—48no—
625C22—44no—
626C22—88no—
627C22—88no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—88no—
634C22—88no—
635C22—84no—
636C22—84no—
637C22#254118yes—
638C22C22×#12111yescharacteristic
639C22C22×#12711yescharacteristic
640C22C2×#53511yescharacteristic
641C22C2×#53712yes—
642C22#345912yes—
643C22C2×D4212yes—
644C22C2×#55012yes—
645C22#348414yes—
646C22#348411yescharacteristic
647C22#348411yescharacteristic
648C22#348511yescharacteristic
649C22#348611yescharacteristic
650C22#348914yes—
651C22#349212yes—
652C22#349612yes—
653C22#353414yes—
654C22#353614yes—
655C4—28no—
656C4—28no—
657C4—416no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—84no—
663C2—28no—
664C2—48no—
665C2—164no—
666C2C2×#254111yescharacteristic
667C2#3498411yescharacteristic
668C2#3522311yescharacteristic
669C2#3539011yescharacteristic
670C2#3563411yescharacteristic
671C2#3670711yescharacteristic
672C2#3672111yescharacteristic
673C2#3767314yes—
674C2#3776814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2847C2semidirect
2. C2×#2539C2semidirect
3. C2×#2656C2semidirect
4. C2×#2618C2semidirect
5. C22×#385C2semidirect
6. #15505C2semidirect
7. #15315C2semidirect
8. #14926C2semidirect
9. #14716C2semidirect
10. #14137C2semidirect
11. #13778C2semidirect
28. C2×#399C22semidirect
29. C2×#398C22semidirect
30. C2×#396C22semidirect
31. C2×#394C22semidirect
34. C2×#392C22semidirect
36. C2×#390C22semidirect
47. #1444C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
52. #1438C22semidirect
53. #1436C22semidirect
54. C4⋊#95C22semidirect
105. C22×C22⋊C4D4semidirect
132. #380D4semidirect
135. C2×#95D4semidirect
441. C22×C4#121semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C2×#25392
C2×#26562
C2×#26182
C22×#3852
#155052
#153152
#149262
#147162
#141372
#137782