Finite Group Databasebeta

← #4397055

Subgroups — #4397055

Order 512.

2784 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic
4C2×#2332C211yesmaximal, characteristic, complemented
5#17061C212yesmaximal, complemented
6#16800C212yesmaximal, complemented
7#15765C214yesmaximal, complemented
8#15612C214yesmaximal, complemented
9#15499C214yesmaximal, complemented
10#15397C214yesmaximal, complemented
11#15163C212yesmaximal, complemented
12#15151C212yesmaximal, complemented
13#14342C211yesmaximal, characteristic, complemented
14#14290C211yesmaximal, characteristic
15#13953C211yesmaximal, characteristic, complemented
16C22×#123C2214yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2816—28no—
21#2502—28no—
22#2501—28no—
23C2×#399C2212yes—
24C2×#399C2214yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#396C2214yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#393C2212yes—
33#2332—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#389C2214yes—
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#388C2214yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#385C2214yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#381C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1445C2212yes—
54#1445C2212yes—
55#1445C2214yes—
56#1445C2214yes—
57#1445C2212yes—
58#1444C2214yes—
59#1444C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2212yes—
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×#123—216no—
85C2×#123—48no—
86C2×#123—48no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8D418yes—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#399—28no—
129#399—216no—
130#398—28no—
131#398—28no—
132#398—28no—
133#398—48no—
134#396—48no—
135#394—28no—
136#394—216no—
137#393—28no—
138#391—28no—
139#391—28no—
140#390—24no—
141#390—24no—
142#389—48no—
143#388—28no—
144#388—28no—
145#388—48no—
146#387—28no—
147#387—28no—
148#385—216no—
149#384—28no—
150#384—24no—
151#384—24no—
152#381—24no—
153#381—24no—
154C4×C4⋊C4—28no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4D418yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8C2×D414yes—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—42no—
195C22×D4—42no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4#4114yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219#123—464no—
220C22⋊Q8—232no—
221#121—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D418yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C2×D418yes—
299C2×C22⋊C4C2×D418yes—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24C22×D412yes—
349C24C22×D412yes—
350C24C2×#4112yes—
351C24C2511yescharacteristic
352C2×Q8—28no—
353C2×Q8—48no—
354C2×Q8—82no—
355C2×Q8—82no—
356C2×Q8—84no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—44no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—88no—
365C2×D4—84no—
366C2×D4—84no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
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—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—88no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4#121116yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+418yes—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—82no—
473C22⋊C4—84no—
474C22⋊C4—82no—
475C42—44no—
476C42—44no—
477C42—48no—
478C42—48no—
479C42—84no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23C2×#12114yes—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C22×#4111yescharacteristic
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×2−1+414yes—
523C23#58512yes—
524C23#58512yes—
525Q8—88no—
526D4—88no—
527D4—816no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—816no—
598C2×C4C2×#12114yes—
599C2×C4C2×#12114yes—
600C2×C4#53518yes—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—84no—
610C22—82no—
611C22—82no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22C22×#12111yescharacteristic
619C22C2×#53512yes—
620C22#348511yescharacteristic
621C22#348511yescharacteristic
622C22#348512yes—
623C22#348612yes—
624C22#349112yes—
625C22#349812yes—
626C22#349912yes—
627C22#350811yescharacteristic
628C22#350811yescharacteristic
629C22#350912yes—
630C22#353514yes—
631C22#354114yes—
632C22#355414yes—
633C22#355914yes—
634C4—24no—
635C4—24no—
636C4—28no—
637C4—44no—
638C4—44no—
639C4—48no—
640C4—44no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4—82no—
647C2—44no—
648C2—162no—
649C2#3527211yescharacteristic
650C2#3544711yescharacteristic
651C2#3566711yescharacteristic
652C2#4137512yes—
653C2#4264812yes—
654C2#4685412yes—
655C2#4722412yes—
656C2#4882314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2502C2semidirect
4. C2×#2332C2semidirect
5. #17061C2semidirect
6. #16800C2semidirect
7. #15765C2semidirect
8. #15612C2semidirect
9. #15499C2semidirect
10. #15397C2semidirect
11. #15163C2semidirect
12. #15151C2semidirect
13. #14342C2semidirect
15. #13953C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#25022
C2×#25012
C2×#23322
#170612
#168002
#157652
#156122
#154992
#153972
#151632
#151512
#143422
#142902
#139532