Finite Group Databasebeta

← #4929586

Subgroups — #4929586

Order 512.

3328 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2540C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic, complemented
3#15001C212yesmaximal, complemented
4#14996C212yesmaximal, complemented
5#14926C212yesmaximal, complemented
6#14914C212yesmaximal, complemented
7#14793C214yesmaximal, complemented
8#14721C214yesmaximal, complemented
9#14288C211yesmaximal, characteristic, complemented
10#14195C214yesmaximal, complemented
11#14169C211yesmaximal, characteristic, complemented
12#14164C211yesmaximal, characteristic, complemented
13#14095C214yesmaximal, complemented
14#13959C211yesmaximal, characteristic, complemented
15#13856C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×#123C2214yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2212yes—
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22#2540—28no—
23C2×#399C2212yes—
24C2×#394C2214yes—
25C2×#392C2212yes—
26C2×#390C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#383C2211yescharacteristic
32#2275—28no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×#376C2211yescharacteristic
42C2×#376C2211yescharacteristic
43C22×#95C2212yes—
44#1446C2211yescharacteristic
45#1446C2212yes—
46#1446C2211yescharacteristic
47#1445C2212yes—
48#1444C2212yes—
49#1444C2212yes—
50#1444C2214yes—
51#1444C2212yes—
52#1442C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2214yes—
61#1438C2211yescharacteristic
62#1438C2212yes—
63#1438C2211yescharacteristic
64#1438C2212yes—
65#1437C2212yes—
66#1437C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2314yes—
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×#124—24no—
85C2×#124—24no—
86C2×#124—24no—
87C2×#123—216no—
88C2×#123—48no—
89C2×#123—48no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×#121—28no—
93C2×#121—48no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#399—28no—
129#394—216no—
130#392—28no—
131#390—216no—
132#388—48no—
133#387—48no—
134#386—48no—
135#385—48no—
136#383—24no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4—44no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—28no—
145#376—24no—
146#376—24no—
147C2×#95—28no—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2412yes—
173C25C2412yes—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—44no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—28no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4C2×D414yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D418yes—
208C23×C4#4118yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#124—416no—
223#123—464no—
224C22⋊Q8—432no—
225#121—432no—
226C4×D4—264no—
227C4×D4—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
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—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
286C2×C22⋊C4#4118yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306C2×C42#4114yes—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24C22×D414yes—
349C24C2×#4114yes—
350C24C2×#4114yes—
351C24C2511yescharacteristic
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4—28no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—44no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
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—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4H(5,𝔽2)18yes—
454C22×C42−1+414yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—232no—
458C4⋊C4—28no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—88no—
475C42—216no—
476C42—216no—
477C42—82no—
478C42—82no—
479C42—84no—
480C42—84no—
481C42—84no—
482C42#123116yescomplemented
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23#52518yes—
527C23C23×D411yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4112yes—
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)14yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×2−1+412yes—
535Q8—88no—
536D4—232no—
537D4—88no—
538D4—832no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
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—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—816no—
605C2×C4—816no—
606C2×C4—816no—
607C2×C4C2×#12314yes—
608C2×C4C2×#12314yes—
609C2×C4C2×#12314yes—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—82no—
629C22—82no—
630C22C22×#12311yescharacteristic
631C22C2×#52512yes—
632C22#344411yescharacteristic
633C22#344411yescharacteristic
634C22#344412yes—
635C22#348412yes—
636C22#349014yes—
637C22#349212yes—
638C22#349811yescharacteristic
639C22#349911yescharacteristic
640C22#350812yes—
641C22#350812yes—
642C22#351114yes—
643C22#351112yes—
644C22#351914yes—
645C22#352014yes—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—48no—
655C4—44no—
656C4—44no—
657C4—84no—
658C4—82no—
659C2—48no—
660C2—162no—
661C2#3541912yes—
662C2#3543612yes—
663C2#3553011yescharacteristic
664C2#3555411yescharacteristic
665C2#3569212yes—
666C2#3574012yes—
667C2#3574414yes—
668C2#3662611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2540C2
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×#2275C2semidirect
3. #15001C2semidirect
4. #14996C2semidirect
5. #14926C2semidirect
6. #14914C2semidirect
7. #14793C2semidirect
8. #14721C2semidirect
9. #14288C2semidirect
10. #14195C2semidirect
11. #14169C2semidirect
12. #14164C2semidirect
13. #14095C2semidirect
14. #13959C2semidirect
15. #13856C2semidirect
482. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25402
C2×#22752
#150012
#149962
#149262
#149142
#147932
#147212
#142882
#141952
#141692
#141642
#140952
#139592
#138562