Finite Group Databasebeta

← #4341757

Subgroups — #4341757

Order 512.

2566 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×#2731C212yesmaximal
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2326C211yesmaximal, characteristic
4#16596C212yesmaximal
5#16563C212yesmaximal, complemented
6#16518C212yesmaximal, complemented
7#16506C212yesmaximal, complemented
8#15824C214yesmaximal, complemented
9#15590C214yesmaximal, complemented
10#15122C212yesmaximal, complemented
11#15078C212yesmaximal, complemented
12#14868C212yesmaximal
13#14808C212yesmaximal, complemented
14#14450C212yesmaximal, complemented
15#13942C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2731—216no—
22#2546—28no—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#394C2214yes—
29C2×#393C2214yes—
30#2326—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49#1446C2214yes—
50#1445C2212yes—
51#1445C2212yes—
52#1444C2212yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73#1434C2212yes—
74#1434C2214yes—
75#1434C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2312yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—42no—
88C2×#124—24no—
89C2×#124—24no—
90C2×#124D418yes—
91C2×C4×Q8—42no—
92C2×C4×Q8—42no—
93C2×C4×Q8D418yes—
94C2×C4×D4—24no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#398—28no—
128#398—28no—
129#397—44no—
130#397—44no—
131#396—216no—
132#394—216no—
133#393—216no—
134#391—24no—
135#391—28no—
136#391—28no—
137#390—24no—
138#389—28no—
139#389—28no—
140#388—216no—
141#388—216no—
142#388—28no—
143#388—28no—
144#387—216no—
145#385—28no—
146#385—28no—
147#384—24no—
148#382—42no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—42no—
151#376—24no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8C2×D414yes—
190C22×Q8C2×D414yes—
191C22×Q8#4114yes—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—41no—
198C22×D4C2×D414yes—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—416no—
219C4⋊Q8—432no—
220#124—232no—
221C4×Q8—416no—
222C4×D4—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—41no—
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—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×D418yes—
285C2×C22⋊C4C2×D418yes—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—41no—
300C2×C42—41no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24C22×D414yes—
343C24C2511yescharacteristic
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×Q8#121116yes—
358C2×D4—28no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—48no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4C22×D414yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+418yes—
450C4⋊C4—216no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—416no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—28no—
474C42—28no—
475C42—28no—
476C42—28no—
477C42—44no—
478C42—48no—
479C42—84no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×2−1+414yes—
509C23#58511yescharacteristic
510C23#58511yescharacteristic
511C23#58514yes—
512Q8—416no—
513Q8—816no—
514D4—416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—816no—
592C2×C4C2×#12114yes—
593C2×C4C2×#12114yes—
594C2×C4C2×#12114yes—
595C2×C4#53414yes—
596C22—48no—
597C22—44no—
598C22—44no—
599C22—48no—
600C22—44no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—84no—
607C22C22×#12111yescharacteristic
608C22C2×#53411yescharacteristic
609C22#344911yescharacteristic
610C22#348612yes—
611C22#348914yes—
612C22#349611yescharacteristic
613C22#349611yescharacteristic
614C22#350012yes—
615C22#350511yescharacteristic
616C22#350511yescharacteristic
617C22#351614yes—
618C22#354012yes—
619C22#354412yes—
620C22#354412yes—
621C22#355112yes—
622C22#355714yes—
623C22#356514yes—
624C4—24no—
625C4—24no—
626C4—24no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—48no—
633C4—82no—
634C4—84no—
635C4—82no—
636C4—84no—
637C4—84no—
638C2—84no—
639C2#3500011yescharacteristic
640C2#3573712yes—
641C2#3725811yescharacteristic
642C2#3741911yescharacteristic
643C2#4215412yes—
644C2#4753812yes—
645C2#4778812yes—
646C2#5158612yes—
647C2#5199912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
5. #16563C2semidirect
6. #16518C2semidirect
7. #16506C2semidirect
8. #15824C2semidirect
9. #15590C2semidirect
10. #15122C2semidirect
11. #15078C2semidirect
13. #14808C2semidirect
14. #14450C2semidirect
15. #13942C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27312
C2×#25462
C2×#23262
#165962
#165632
#165182
#165062
#158242
#155902
#151222
#150782
#148682
#148082
#144502
#139422