Finite Group Databasebeta

← #4335006

Subgroups — #4335006

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2598C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3#16965C212yesmaximal, complemented
4#16679C212yesmaximal, complemented
5#16473C212yesmaximal, complemented
6#16282C212yesmaximal
7#16057C214yesmaximal, complemented
8#15798C212yesmaximal, complemented
9#15643C212yesmaximal, complemented
10#14888C212yesmaximal, complemented
11#14868C212yesmaximal, complemented
12#14808C212yesmaximal, complemented
13#14776C212yesmaximal, complemented
14#14517C214yesmaximal, complemented
15#13938C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2598—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#394C2211yescharacteristic
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#392C2211yescharacteristic
29#2317—28no—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#384C2212yes—
41C2×#383C2211yescharacteristic
42C2×#383C2211yescharacteristic
43C2×#382C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45#1445C2212yes—
46#1445C2214yes—
47#1445C2212yes—
48#1444C2212yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2212yes—
59#1440C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1437C2214yes—
68#1437C2212yes—
69#1436C2212yes—
70#1436C2214yes—
71#1436C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C2×C4×D4—24no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×#117—24no—
86C2×#117—28no—
87C2×#117—42no—
88C2×#117—42no—
89C2×#117D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#400—24no—
122#399—24no—
123#397—44no—
124#397—44no—
125#394—24no—
126#394—216no—
127#394—48no—
128#392—24no—
129#389—28no—
130#389—28no—
131#388—28no—
132#388—48no—
133#386—28no—
134#386—28no—
135#386—48no—
136#385—28no—
137#385—28no—
138#385—28no—
139#384—28no—
140#383—24no—
141#383—42no—
142#382—28no—
143C4×C4⋊C4—42no—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—22no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—42no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C4×D4—416no—
211#117—232no—
212#117—416no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4#4118yes—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—42no—
287C2×C42—44no—
288C2×C42C2×D414yes—
289C2×C42C2×D414yes—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24C22×D412yes—
333C24C2×#4114yes—
334C24H(5,𝔽2)14yes—
335C24C2511yescharacteristic
336C2×D4—48no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—416no—
343C2×D4—48no—
344C2×D4—84no—
345C2×D4—88no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
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—24no—
411C22×C4—22no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—88no—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4118yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+414yes—
432C4⋊C4—216no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C22⋊C4—216no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—84no—
451C22⋊C4—88no—
452C22⋊C4—84no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C42—216no—
457C42—28no—
458C42—28no—
459C42—84no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—28no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23#52514yes—
499C23C23×D411yescharacteristic
500C23C22×#4112yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)12yes—
504C23C2×2−1+412yes—
505C23#58512yes—
506C23#58514yes—
507D4—416no—
508D4—816no—
509C2×C4—232no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
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—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—816no—
580C2×C4—816no—
581C2×C4#53414yes—
582C2×C4#53614yes—
583C22—28no—
584C22—24no—
585C22—48no—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—44no—
596C22—84no—
597C22—82no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C2×#52511yescharacteristic
603C22C2×#53411yescharacteristic
604C22C2×#53611yescharacteristic
605C22#348812yes—
606C22#349212yes—
607C22#349212yes—
608C22#350012yes—
609C22#351314yes—
610C22#352114yes—
611C22#353711yescharacteristic
612C22#354012yes—
613C22#354011yescharacteristic
614C22#354212yes—
615C22#355112yes—
616C22#355914yes—
617C22#356514yes—
618C4—24no—
619C4—24no—
620C4—42no—
621C4—42no—
622C4—48no—
623C4—42no—
624C4—42no—
625C4—84no—
626C4—82no—
627C4—82no—
628C4—84no—
629C4—84no—
630C4—82no—
631C2—44no—
632C2—84no—
633C2#3505711yescharacteristic
634C2#3597912yes—
635C2#3726911yescharacteristic
636C2#3741911yescharacteristic
637C2#4294412yes—
638C2#4676212yes—
639C2#4715212yes—
640C2#4968414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2317C2semidirect
3. #16965C2semidirect
4. #16679C2semidirect
5. #16473C2semidirect
7. #16057C2semidirect
8. #15798C2semidirect
9. #15643C2semidirect
10. #14888C2semidirect
11. #14868C2semidirect
12. #14808C2semidirect
13. #14776C2semidirect
14. #14517C2semidirect
15. #13938C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25982
C2×#23172
#169652
#166792
#164732
#162822
#160572
#157982
#156432
#148882
#148682
#148082
#147762
#145172
#139382