Finite Group Databasebeta

← #4416351

Subgroups — #4416351

Order 512.

4164 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#17057C212yesmaximal, complemented
4#16895C214yesmaximal, complemented
5#16395C214yesmaximal, complemented
6#16362C212yesmaximal, complemented
7#15069C212yesmaximal, complemented
8#15019C212yesmaximal, complemented
9#14926C212yesmaximal, complemented
10#14911C214yesmaximal, complemented
11#14776C212yesmaximal, complemented
12#14681C211yesmaximal, characteristic, complemented
13#14674C212yesmaximal, complemented
14#14659C212yesmaximal, complemented
15C25×C4C2211yescharacteristic
16C22×#123C2214yes—
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#120C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24#2847—28no—
25C2×#397C2214yescomplemented
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#381C2212yes—
31C2×#381C2212yes—
32C2×#381C2214yes—
33C2×#380—28no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#379C2212yes—
42C22×#95C2212yes—
43#1445C2214yescomplemented
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1444C2214yescomplemented
47#1444C2214yescomplemented
48#1443C2218yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1439C2218yes—
52#1439C2218yes—
53#1439C2218yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1438C2218yes—
58#1438C2218yes—
59#1438C2214yescomplemented
60C22⋊#95C2214yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C26C2311yescharacteristic
66C23×D4C2312yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4—28no—
71C24×C4—44no—
72C24×C4—44no—
73C24×C4D418yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#123—48no—
83C2×#123—48no—
84C2×#123—48no—
85C2×#121—24no—
86C2×#121—24no—
87C2×#121—24no—
88C2×#121—28no—
89C2×#121—44no—
90C2×#121—44no—
91C2×#120—28no—
92C2×#120—48no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2318yes—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—48no—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2318yes—
120#397—48no—
121#386—28no—
122#386—28no—
123#386—28no—
124#386—48no—
125#381—28no—
126#381—44no—
127#381—48no—
128#380—24no—
129#380—24no—
130#380—24no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—28no—
135#379—44no—
136C2×#95—28no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25—24no—
156C25—42no—
157C25—42no—
158C25C2×D414yes—
159C25C2412yes—
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—44no—
179C22×D4—44no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—232no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—416no—
199C23×C4—416no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—464no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#4118yes—
217C23×C4C2418yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#123—464no—
228#121—416no—
229#121—416no—
230#121—416no—
231#120—432no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—416no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—432no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—48no—
292#95—44no—
293#95—48no—
294#95—48no—
295#95—48no—
296#95—48no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—232no—
302C24—28no—
303C24—28no—
304C24—28no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—28no—
318C24—28no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—44no—
329C24—44no—
330C24—416no—
331C24—416no—
332C24—416no—
333C24—416no—
334C24—416no—
335C24—416no—
336C24C22×D412yes—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C2×#4114yes—
340C24H(5,𝔽2)14yes—
341C24H(5,𝔽2)14yes—
342C24H(5,𝔽2)14yes—
343C242−1+414yes—
344C24C2511yescharacteristic
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—44no—
359C2×D4—88no—
360C2×D4—816no—
361C2×D4—816no—
362C22×C4—232no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—4128no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—816no—
434C22×C4#123116yescomplemented
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)14yes—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—432no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—432no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—816no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—232no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—416no—
493C23—416no—
494C23—416no—
495C23—416no—
496C23—416no—
497C23—4128no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—416no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—48no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—44no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—44no—
523C23—44no—
524C23C2×#12314yes—
525C23#53414yes—
526C23#53514yes—
527C23#53514yes—
528C23#53614yes—
529C23#560116yes—
530C23C23×D411yescharacteristic
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)12yes—
535C23C2×H(5,𝔽2)12yes—
536C23C2×2−1+412yes—
537C23#58512yes—
538D4—88no—
539D4—816no—
540D4—816no—
541D4—816no—
542C2×C4—232no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—416no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—416no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—816no—
592C2×C4—816no—
593C2×C4C2×#12314yes—
594C2×C4C2×#12314yes—
595C2×C4#53414yes—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—232no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—416no—
617C22—416no—
618C22—42no—
619C22—42no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—88no—
625C22—88no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22#281518yes—
630C22C22×#12311yescharacteristic
631C22C2×#53411yescharacteristic
632C22C2×#53411yescharacteristic
633C22C2×#53511yescharacteristic
634C22C2×#53511yescharacteristic
635C22C2×#53611yescharacteristic
636C22#344911yescharacteristic
637C22#349014yes—
638C22#349214yes—
639C22#349414yes—
640C22C2×#56014yes—
641C22#353514yes—
642C22#353614yes—
643C22#354214yes—
644C4—24no—
645C4—24no—
646C4—24no—
647C4—42no—
648C4—84no—
649C4—82no—
650C4—84no—
651C4—82no—
652C4—88no—
653C4—82no—
654C2—28no—
655C2—44no—
656C2—44no—
657C2—162no—
658C2#3499211yescharacteristic
659C2#3522711yescharacteristic
660C2#3526211yescharacteristic
661C2C2×#281511yescharacteristic
662C2#3743711yescharacteristic
663C2#3744711yescharacteristic
664C2#3745111yescharacteristic
665C2#4546518yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#380C2semidirect
3. #17057C2semidirect
4. #16895C2semidirect
5. #16395C2semidirect
6. #16362C2semidirect
7. #15069C2semidirect
8. #15019C2semidirect
9. #14926C2semidirect
10. #14911C2semidirect
11. #14776C2semidirect
12. #14681C2semidirect
13. #14674C2semidirect
14. #14659C2semidirect
25. C2×#397C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
59. #1438C22semidirect
434. C22×C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C22×#3802
#170572
#168952
#163952
#163622
#150692
#150192
#149262
#149112
#147762
#146812
#146742
#146592