Finite Group Databasebeta

← #763316

Subgroups — #763316

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3#16399C212yesmaximal
4#16264C212yesmaximal
5#15256C211yesmaximal, characteristic, complemented
6#15209C211yesmaximal, characteristic, complemented
7#15207C211yesmaximal, characteristic, complemented
8#15206C212yesmaximal, complemented
9#15173C211yesmaximal, characteristic, complemented
10#15166C212yesmaximal, complemented
11#15114C212yesmaximal
12#14892C212yesmaximal
13#14223C212yesmaximal, complemented
14#14203C214yesmaximal, complemented
15#14202C214yesmaximal
16#14174C212yesmaximal, complemented
17C4×#380C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#398C2212yes—
24C2×#398C2214yes—
25C2×#397C2214yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28#2296—28no—
29C2×#391C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1446C2212yes—
51#1445C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78#1434C2214yes—
79#1434C2214yes—
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—44no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—28no—
96C2×#117—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42—44no—
120C22×C42D418yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#398—216no—
132#398—28no—
133#397—216no—
134#395—28no—
135#391—24no—
136#389—24no—
137#389—24no—
138#389—24no—
139#389—24no—
140#389—24no—
141#387—28no—
142#387—48no—
143#387—48no—
144#385—28no—
145#385—28no—
146#385—28no—
147#380—24no—
148#380—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—44no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—44no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
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—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—42no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C22⋊Q8—232no—
220C4×Q8—416no—
221#117—232no—
222#117—416no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
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—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4#4118yes—
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—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4#4118yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—416no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
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—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95#4118yes—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24C22×D412yes—
339C24C2×#4114yes—
340C24C2511yescharacteristic
341C2×Q8—28no—
342C2×Q8—416no—
343C2×Q8—44no—
344C2×Q8—44no—
345C22×C4—232no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4118yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C42−1+414yes—
440C22×C42−1+414yes—
441C22×C42−1+412yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—216no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C42—216no—
465C42—28no—
466C42—28no—
467C42—44no—
468C42—44no—
469C42—48no—
470C42—48no—
471C42—48no—
472C42—84no—
473C42—84no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23#52514yes—
491C23C23×D411yescharacteristic
492C23C22×#4112yes—
493C23C22×#4112yes—
494C23C22×#4112yes—
495C23C2×H(5,𝔽2)12yes—
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×2−1+412yes—
498C23C2×2−1+412yes—
499C23C2×2−1+411yescharacteristic
500Q8—416no—
501C2×C4—232no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
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—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4#52514yes—
583C2×C4#53214yes—
584C2×C4#53414yes—
585C2×C4#53614yes—
586C2×C4#53614yes—
587C2×C4#53714yes—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—42no—
592C22—48no—
593C22—48no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22C2×#52511yescharacteristic
598C22C2×#52511yescharacteristic
599C22C2×#53211yescharacteristic
600C22C2×#53411yescharacteristic
601C22C2×#53611yescharacteristic
602C22C2×#53611yescharacteristic
603C22C2×#53711yescharacteristic
604C22#348114yes—
605C22#348112yes—
606C22#348212yes—
607C22#348214yes—
608C22#349214yes—
609C22#350912yes—
610C22#350914yes—
611C22#351112yes—
612C22#353612yes—
613C22#353812yes—
614C4—24no—
615C4—24no—
616C4—24no—
617C4—24no—
618C4—24no—
619C4—24no—
620C4—48no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4—82no—
630C2—44no—
631C2#3495711yescharacteristic
632C2#3496411yescharacteristic
633C2#3497111yescharacteristic
634C2#3505711yescharacteristic
635C2#3529711yescharacteristic
636C2#3532211yescharacteristic
637C2#3744211yescharacteristic
638C2#6003014yes—
639C2#6003614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #15256C2semidirect
6. #15209C2semidirect
7. #15207C2semidirect
8. #15206C2semidirect
9. #15173C2semidirect
10. #15166C2semidirect
13. #14223C2semidirect
14. #14203C2semidirect
16. #14174C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#22962
#163992
#162642
#152562
#152092
#152072
#152062
#151732
#151662
#151142
#148922
#142232
#142032
#142022
#141742
C4×#3802