Finite Group Databasebeta

← #847690

Subgroups — #847690

Order 512.

3128 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic, complemented
3#16653C212yesmaximal, complemented
4#16652C212yesmaximal, complemented
5#15884C212yesmaximal, complemented
6#15833C212yesmaximal, complemented
7#15832C214yesmaximal, complemented
8#14921C212yesmaximal, complemented
9#14913C212yesmaximal
10#14873C212yesmaximal, complemented
11#14696C212yesmaximal, complemented
12#14560C214yesmaximal, complemented
13#14040C212yesmaximal, complemented
14#14020C212yesmaximal, complemented
15#13923C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2591—28no—
21C2×#400C2211yescharacteristic
22C2×#400C2211yescharacteristic
23C2×#394C2211yescharacteristic
24C2×#394C2211yescharacteristic
25C2×#393C2214yes—
26C2×#393C2214yes—
27C2×#391—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#388C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2212yes—
46C22×#95C2212yes—
47#1446C2214yes—
48#1446C2214yes—
49#1445C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2212yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#121—24no—
92C2×#121—42no—
93C2×#121—42no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42—24no—
117C22×C42—44no—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122#400—24no—
123#400—24no—
124#394—24no—
125#394—24no—
126#393—216no—
127#393—48no—
128#391—28no—
129#391—28no—
130#391—28no—
131#391—28no—
132#388—28no—
133#388—28no—
134#388—48no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—44no—
140#386—44no—
141#385—28no—
142#380—42no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—42no—
145C2×#95—28no—
146C2×#95—28no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C22×Q8—232no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—24no—
181C22×Q8—28no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—432no—
189C22×Q8—432no—
190C22×Q8—416no—
191C22×Q8—416no—
192C22×Q8#4118yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—44no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4#4118yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240#121—416no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—416no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42—44no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24C22×D412yes—
346C24C2×#4114yes—
347C24H(5,𝔽2)12yes—
348C24H(5,𝔽2)12yes—
349C242−1+414yes—
350C24C2511yescharacteristic
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—416no—
356C2×Q8—416no—
357C2×Q8—4128no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)14yes—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—88no—
475C4⋊C4—84no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C42—48no—
489C42—48no—
490C42—84no—
491C42—84no—
492C23—22no—
493C23—22no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23#53414yes—
527C23#54218yes—
528C23#54818yes—
529C23#55514yes—
530C23#56314yes—
531C23C23×D411yescharacteristic
532C23C22×#4112yes—
533C23C22×#4112yes—
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23C2×2−1+412yes—
538C23#58514yes—
539Q8—432no—
540Q8—432no—
541D4—88no—
542D4—816no—
543C2×C4—232no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—432no—
567C2×C4—432no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4—88no—
617C2×C4#52514yes—
618C2×C4#53514yes—
619C22—28no—
620C22—28no—
621C22—216no—
622C22—28no—
623C22—24no—
624C22—28no—
625C22—24no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—82no—
637C22#273118yes—
638C22C2×#52511yescharacteristic
639C22C2×#53411yescharacteristic
640C22C2×#53511yescharacteristic
641C22C2×#54212yes—
642C22C2×#54812yes—
643C22C2×#55511yescharacteristic
644C22#348014yes—
645C22#348512yes—
646C22#349012yes—
647C22#349012yes—
648C22#350514yes—
649C22#351614yes—
650C22C2×#56311yescharacteristic
651C22#354114yes—
652C22#354512yes—
653C22#354512yes—
654C4—24no—
655C4—24no—
656C4—48no—
657C4—48no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—82no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4—84no—
667C2—28no—
668C2—162no—
669C2#3505011yescharacteristic
670C2#3658212yes—
671C2#3711911yescharacteristic
672C2#3848014yes—
673C2#3972212yes—
674C2#4634912yes—
675C2C2×#273111yescharacteristic
676C2#5988512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#391C2semidirect
3. #16653C2semidirect
4. #16652C2semidirect
5. #15884C2semidirect
6. #15833C2semidirect
7. #15832C2semidirect
8. #14921C2semidirect
10. #14873C2semidirect
11. #14696C2semidirect
12. #14560C2semidirect
13. #14040C2semidirect
14. #14020C2semidirect
15. #13923C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C22×#3912
#166532
#166522
#158842
#158332
#158322
#149212
#149132
#148732
#146962
#145602
#140402
#140202
#139232