Finite Group Databasebeta

← #4389611

Subgroups — #4389611

Order 512.

2962 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2640C211yesmaximal, characteristic, complemented
3C2×#2631C211yesmaximal, characteristic
4C2×#2337C211yesmaximal, characteristic, complemented
5#17060C212yesmaximal, complemented
6#16799C212yesmaximal, complemented
7#15613C214yesmaximal, complemented
8#15499C214yesmaximal, complemented
9#15493C214yesmaximal, complemented
10#15323C214yesmaximal, complemented
11#15160C212yesmaximal, complemented
12#15150C212yesmaximal, complemented
13#14663C211yesmaximal, characteristic, complemented
14#13964C211yesmaximal, characteristic, complemented
15#13959C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2214yes—
19C22×C4×D4C2211yescharacteristic
20#2640—28no—
21#2631—28no—
22#2542—28no—
23C2×#399C2214yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2214yes—
28C2×#394C2214yes—
29C2×#393C2212yes—
30C2×#393C2214yes—
31C2×#392C2212yes—
32C2×#392C2212yes—
33#2337—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#386C2214yes—
42C2×#386C2212yes—
43C2×#386C2214yes—
44C2×#385C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#383C2211yescharacteristic
48C2×#383C2211yescharacteristic
49C2×#380C2214yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1446C2212yes—
54#1446C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2214yes—
58#1445C2212yes—
59#1445C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2214yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2312yes—
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2314yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×#124—24no—
90C2×#124—24no—
91C2×#124D418yes—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—48no—
94C2×C22⋊Q8—48no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#399—216no—
127#397—28no—
128#397—28no—
129#397—28no—
130#397—48no—
131#394—48no—
132#393—28no—
133#393—216no—
134#392—28no—
135#392—28no—
136#391—28no—
137#391—28no—
138#390—28no—
139#390—24no—
140#390—24no—
141#388—28no—
142#388—28no—
143#386—216no—
144#386—28no—
145#386—48no—
146#385—28no—
147#384—24no—
148#384—24no—
149#383—24no—
150#383—24no—
151#380—48no—
152C4×C4⋊C4D418yescomplemented
153C4×C22⋊C4—28no—
154#376—24no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2412yes—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2×D414yes—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4C2×D414yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C4⋊Q8—416no—
230#124—232no—
231C22⋊Q8—464no—
232C4×D4—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D418yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
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⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D418yes—
298C2×C22⋊C4C2×D418yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4114yes—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—44no—
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—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24C22×D414yes—
355C242−1+414yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—48no—
359C2×Q8—416no—
360C2×Q8—48no—
361C2×Q8—88no—
362C2×Q8—84no—
363C2×Q8—84no—
364C2×Q8—84no—
365C2×Q8—84no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—44no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
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—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—84no—
441C22×C4—88no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—88no—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)14yes—
460C4⋊C4—216no—
461C4⋊C4—416no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—84no—
476C22⋊C4—88no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—44no—
484C42—44no—
485C42—48no—
486C42—84no—
487C42#121116yes—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+412yes—
540C23#58512yes—
541C23#58514yes—
542Q8—88no—
543Q8—816no—
544D4—416no—
545D4—88no—
546D4—832no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—816no—
612C2×C4—816no—
613C2×C4C2×#12114yes—
614C2×C4C2×#12114yes—
615C2×C4C2×#12114yes—
616C2×C4#53518yes—
617C22—44no—
618C22—44no—
619C22—48no—
620C22—44no—
621C22—44no—
622C22—48no—
623C22—48no—
624C22—48no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—84no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—84no—
633C22—82no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22C22×#12111yescharacteristic
640C22C2×#53512yes—
641C22#348512yes—
642C22#348511yescharacteristic
643C22#348611yescharacteristic
644C22#348912yes—
645C22#349512yes—
646C22#349812yes—
647C22#350012yes—
648C22#350811yescharacteristic
649C22#350811yescharacteristic
650C22#351012yes—
651C22#353314yes—
652C22#354114yes—
653C22#356014yes—
654C22#356114yes—
655C4—24no—
656C4—24no—
657C4—28no—
658C4—24no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—84no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C2—84no—
669C2—84no—
670C2#3527211yescharacteristic
671C2#3561711yescharacteristic
672C2#3563311yescharacteristic
673C2#4137112yes—
674C2#4278812yes—
675C2#4769014yes—
676C2#5091612yes—
677C2#5122412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2631C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2542C2semidirect
2. C2×#2640C2semidirect
4. C2×#2337C2semidirect
5. #17060C2semidirect
6. #16799C2semidirect
7. #15613C2semidirect
8. #15499C2semidirect
9. #15493C2semidirect
10. #15323C2semidirect
11. #15160C2semidirect
12. #15150C2semidirect
13. #14663C2semidirect
14. #13964C2semidirect
15. #13959C2semidirect
152. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26402
C2×#26312
C2×#23372
#170602
#167992
#156132
#154992
#154932
#153232
#151602
#151502
#146632
#139642
#139592