Finite Group Databasebeta

← #622394

Subgroups — #622394

Order 512.

3353 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2614C211yesmaximal, characteristic
2C2×#2623C211yesmaximal, characteristic
3C2×#2310C211yesmaximal, characteristic, complemented
4C22×#391C211yesmaximal, characteristic, complemented
5#15504C214yesmaximal, complemented
6#15503C214yesmaximal, complemented
7#15500C214yesmaximal, complemented
8#15499C214yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14912C212yesmaximal
11#14706C211yesmaximal, characteristic
12#14705C211yesmaximal, characteristic, complemented
13#14342C211yesmaximal, characteristic, complemented
14#14135C212yesmaximal
15#13962C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#123C2211yescharacteristic
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2623—28no—
24#2614—28no—
25C2×#400C2212yes—
26C2×#399C2212yes—
27C2×#398C2212yes—
28C2×#397C2212yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33#2310—28no—
34C2×#391—28no—
35C2×#391C2214yes—
36C2×#391C2214yes—
37C2×#391C2214yes—
38C2×#391C2214yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2211yescharacteristic
49C22×#95C2212yes—
50C22×#95C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1437C2214yes—
66#1437C2212yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2212yes—
72#1436C2212yes—
73C4⋊#95C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—28no—
82C23×Q8—44no—
83C23×Q8D418yes—
84C23×Q8C2312yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—24no—
97C2×#123—28no—
98C2×#123—44no—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117—44no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42—24no—
126C22×C42—44no—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130#400—28no—
131#399—28no—
132#398—28no—
133#397—28no—
134#395—28no—
135#395—28no—
136#393—28no—
137#393—28no—
138#391—216no—
139#391—216no—
140#391—216no—
141#391—216no—
142#386—28no—
143#386—44no—
144#385—28no—
145#385—44no—
146#381—24no—
147#381—24no—
148#381—24no—
149#381—24no—
150#381—28no—
151#381D418yes—
152C2×#95—28no—
153C2×#95—28no—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—232no—
185C22×Q8—232no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—416no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8—432no—
200C22×Q8—432no—
201C22×Q8C2×D418yes—
202C22×Q8C2×D418yes—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4#4114yes—
232C23×C4#4118yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246#123—416no—
247#123—416no—
248#123—416no—
249#117—432no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×D418yes—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—416no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C2×D418yes—
344#95C2×D418yes—
345C24—24no—
346C24—24no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C2×#4112yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)14yes—
359C24C2511yescharacteristic
360C2×Q8—216no—
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—4128no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—48no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4#121116yes—
467C22×C4#123116yes—
468C22×C4C22×D414yes—
469C22×C4C22×D418yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4114yes—
479C22×C42−1+418yes—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—416no—
501C42—48no—
502C42—48no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23C2×#12114yes—
529C23C2×#12314yes—
530C23#53414yes—
531C23#54618yes—
532C23#55118yes—
533C23C23×D411yescharacteristic
534C23C23×D411yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4112yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×2−1+414yes—
539C23#58514yes—
540Q8—432no—
541Q8—432no—
542D4—88no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—232no—
547C2×C4—232no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—432no—
569C2×C4—416no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—816no—
620C2×C4C2×#12114yes—
621C2×C4C2×#12114yes—
622C2×C4C2×#12318yes—
623C22—216no—
624C22—28no—
625C22—24no—
626C22—24no—
627C22—28no—
628C22—216no—
629C22—84no—
630C22—82no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—82no—
635C22—84no—
636C22#241218yes—
637C22C22×#12111yescharacteristic
638C22C22×#12311yescharacteristic
639C22C2×#53411yescharacteristic
640C22#346114yes—
641C22C2×#54612yes—
642C22C2×#55112yes—
643C22#348612yes—
644C22#348712yes—
645C22#349912yes—
646C22#350012yes—
647C22#350714yes—
648C22#350912yes—
649C22#351012yes—
650C22#353514yes—
651C22#353514yes—
652C4—24no—
653C4—24no—
654C4—28no—
655C4—48no—
656C4—48no—
657C4—44no—
658C4—84no—
659C4—82no—
660C4—81no—
661C4—81no—
662C4—84no—
663C4—81no—
664C4—82no—
665C2—28no—
666C2—161no—
667C2C2×#241211yescharacteristic
668C2#3518212yes—
669C2#3566711yescharacteristic
670C2#3567711yescharacteristic
671C2#3718612yes—
672C2#3943114yes—
673C2#3955914yes—
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
3. C2×#2310C2semidirect
4. C22×#391C2semidirect
5. #15504C2semidirect
6. #15503C2semidirect
7. #15500C2semidirect
8. #15499C2semidirect
9. #14919C2semidirect
12. #14705C2semidirect
13. #14342C2semidirect
15. #13962C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26142
C2×#26232
C2×#23102
C22×#3912
#155042
#155032
#155002
#154992
#149192
#149122
#147062
#147052
#143422
#141352
#139622