Finite Group Databasebeta

← #5956035

Subgroups — #5956035

Order 512.

2558 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#15103C212yesmaximal, complemented
4#15086C212yesmaximal, complemented
5#15081C212yesmaximal, complemented
6#15070C212yesmaximal
7#15020C212yesmaximal, complemented
8#14933C212yesmaximal, complemented
9#14923C214yesmaximal, complemented
10#14884C211yesmaximal, characteristic, complemented
11#14880C211yesmaximal, characteristic, complemented
12#14872C212yesmaximal, complemented
13#14868C211yesmaximal, characteristic
14#14813C212yesmaximal
15#14807C211yesmaximal, characteristic
16#13789C214yesmaximal, complemented
17C4×#382C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#395C2212yes—
27C2×#394C2212yes—
28C4×#125—28no—
29C4×#124—28no—
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#388C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#385C2214yes—
41C2×#382C2212yes—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#1446C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1438C2212yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78#1434C2211yescharacteristic
79#1434C2211yescharacteristic
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#125—28no—
89C2×#125—24no—
90C2×#124—28no—
91C2×#124—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—44no—
94C2×C4×D4—24no—
95C2×C4×D4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#400—44no—
131#399—44no—
132#398—44no—
133#397—44no—
134#395—44no—
135#394—44no—
136#391—24no—
137#390—24no—
138#389—24no—
139#389—24no—
140#388—216no—
141#388—216no—
142#388—28no—
143#388—28no—
144#387—28no—
145#387—28no—
146#385—216no—
147#382—24no—
148#382—24no—
149#382—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—44no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—28no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C43D418yes—
186C25C2412yes—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—42no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—42no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218#125—232no—
219#124—232no—
220C4×Q8—416no—
221C4×D4—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4#4118yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4#4118yes—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95#4118yes—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24C2×#4114yes—
347C24C2511yescharacteristic
348C2×Q8—28no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×D4—28no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—44no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C2×#4118yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+414yes—
446C22×C42−1+412yes—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—416no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C22⋊C4—232no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C42—28no—
471C42—28no—
472C42—28no—
473C42—28no—
474C42—28no—
475C42—28no—
476C42—44no—
477C42—48no—
478C42—48no—
479C42—48no—
480C42—88no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23C23×D411yescharacteristic
505C23C22×#4112yes—
506C23C22×#4112yes—
507C23C22×#4112yes—
508C23C2×H(5,𝔽2)12yes—
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×2−1+412yes—
511C23C2×2−1+411yescharacteristic
512C23#58512yes—
513Q8—416no—
514D4—416no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
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—44no—
592C2×C4—44no—
593C2×C4—816no—
594C2×C4#52514yes—
595C2×C4#52514yes—
596C2×C4#53214yes—
597C2×C4#53514yes—
598C2×C4#53614yes—
599C2×C4#53714yes—
600C22—44no—
601C22—48no—
602C22—48no—
603C22—44no—
604C22—44no—
605C22—82no—
606C22—82no—
607C22—84no—
608C22—84no—
609C22—82no—
610C22—82no—
611C22C2×#52511yescharacteristic
612C22C2×#52511yescharacteristic
613C22C2×#53211yescharacteristic
614C22#344414yes—
615C22C2×#53511yescharacteristic
616C22C2×#53611yescharacteristic
617C22C2×#53711yescharacteristic
618C22#344911yescharacteristic
619C22#351412yes—
620C22#351414yes—
621C22#351512yes—
622C22#351514yes—
623C22#351912yes—
624C22#352012yes—
625C22#352012yes—
626C22#352214yes—
627C22#352312yes—
628C4—24no—
629C4—24no—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—44no—
635C4—48no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—82no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—82no—
644C2—84no—
645C2#3454711yescharacteristic
646C2#3513011yescharacteristic
647C2#3513111yescharacteristic
648C2#3513811yescharacteristic
649C2#3523611yescharacteristic
650C2#3524111yescharacteristic
651C2#3524911yescharacteristic
652C2#3547814yes—
653C2#3548314yes—
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
1. C2×C4×#125C2semidirect
3. #15103C2semidirect
4. #15086C2semidirect
5. #15081C2semidirect
7. #15020C2semidirect
8. #14933C2semidirect
9. #14923C2semidirect
10. #14884C2semidirect
11. #14880C2semidirect
12. #14872C2semidirect
16. #13789C2semidirect
17. C4×#382C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×C4×#1242
#151032
#150862
#150812
#150702
#150202
#149332
#149232
#148842
#148802
#148722
#148682
#148132
#148072
#137892
C4×#3822