Finite Group Databasebeta

← #4335009

Subgroups — #4335009

Order 512.

2622 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2598C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3#16962C212yesmaximal, complemented
4#16676C212yesmaximal, complemented
5#16321C212yesmaximal, complemented
6#16288C212yesmaximal
7#15946C212yesmaximal, complemented
8#15941C212yesmaximal, complemented
9#15392C214yesmaximal, complemented
10#14888C212yesmaximal, complemented
11#14870C212yesmaximal, complemented
12#14807C212yesmaximal, complemented
13#14777C212yesmaximal, complemented
14#14535C214yesmaximal, complemented
15#13938C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2598—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#394C2214yes—
28C2×#394C2211yescharacteristic
29C2×#392C2211yescharacteristic
30#2317—28no—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#385C2214yes—
43C2×#384C2212yes—
44C2×#383C2211yescharacteristic
45C2×#383C2211yescharacteristic
46C2×#382C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1438C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1437C2212yes—
72#1436C2214yes—
73#1436C2212yes—
74C4⋊#95C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
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⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#400—24no—
124#399—24no—
125#399—44no—
126#399—44no—
127#398—48no—
128#396—48no—
129#394—24no—
130#394—216no—
131#392—24no—
132#389—28no—
133#389—28no—
134#388—28no—
135#386—28no—
136#386—28no—
137#386—48no—
138#385—28no—
139#385—28no—
140#385—28no—
141#385—48no—
142#385—48no—
143#385—48no—
144#384—28no—
145#383—24no—
146#383—42no—
147#382—28no—
148C4×C4⋊C4—42no—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—22no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—44no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C2×D414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4×D4—416no—
214#117—232no—
215#117—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
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⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×D414yes—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4#4118yes—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323C24—24no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24C22×D412yes—
340C24C2×#4114yes—
341C242−1+414yes—
342C24C2511yescharacteristic
343C2×D4—48no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—416no—
350C2×D4—48no—
351C2×D4—84no—
352C2×D4—88no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
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—24no—
400C22×C4—24no—
401C22×C4—24no—
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—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4118yes—
435C22×C4H(5,𝔽2)18yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+414yes—
439C4⋊C4—216no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C22⋊C4—216no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—84no—
462C22⋊C4—88no—
463C22⋊C4—84no—
464C22⋊C4—88no—
465C42—216no—
466C42—28no—
467C42—28no—
468C42—84no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—24no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23#52514yes—
508C23C23×D411yescharacteristic
509C23C22×#4112yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)14yes—
513C23C2×2−1+412yes—
514C23C2×2−1+412yes—
515C23#58512yes—
516D4—416no—
517D4—816no—
518C2×C4—232no—
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—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—41no—
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—42no—
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—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
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—816no—
589C2×C4—816no—
590C2×C4#53414yes—
591C2×C4#53614yes—
592C22—28no—
593C22—24no—
594C22—48no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—48no—
601C22—44no—
602C22—44no—
603C22—48no—
604C22—44no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—84no—
609C22—82no—
610C22—82no—
611C22C2×#52511yescharacteristic
612C22C2×#53411yescharacteristic
613C22C2×#53611yescharacteristic
614C22#348912yes—
615C22#349212yes—
616C22#349212yes—
617C22#349812yes—
618C22#351314yes—
619C22#352014yes—
620C22#353712yes—
621C22#353811yescharacteristic
622C22#354011yescharacteristic
623C22#354112yes—
624C22#354912yes—
625C22#355214yes—
626C22#355914yes—
627C4—24no—
628C4—24no—
629C4—48no—
630C4—48no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—84no—
640C2—44no—
641C2—84no—
642C2#3505711yescharacteristic
643C2#3599012yes—
644C2#3729711yescharacteristic
645C2#3737411yescharacteristic
646C2#4291412yes—
647C2#4675812yes—
648C2#4676212yes—
649C2#4968514yes—
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. #16962C2semidirect
4. #16676C2semidirect
5. #16321C2semidirect
7. #15946C2semidirect
8. #15941C2semidirect
9. #15392C2semidirect
10. #14888C2semidirect
11. #14870C2semidirect
12. #14807C2semidirect
13. #14777C2semidirect
14. #14535C2semidirect
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
#169622
#166762
#163212
#162882
#159462
#159412
#153922
#148882
#148702
#148072
#147772
#145352
#139382