Finite Group Databasebeta

← #920472

Subgroups — #920472

Order 512.

2856 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C22×#397C211yesmaximal, characteristic
3#17229C212yesmaximal, complemented
4#17201C212yesmaximal, complemented
5#17059C212yesmaximal, complemented
6#16831C212yesmaximal, complemented
7#16454C214yesmaximal, complemented
8#15485C214yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14914C212yesmaximal, complemented
11#14727C212yesmaximal
12#14263C212yesmaximal, complemented
13#14040C212yesmaximal, complemented
14#13877C212yesmaximal
15#13869C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21#2561—28no—
22C2×#397—28no—
23C2×#397C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#397C2214yes—
30C2×#393C2211yescharacteristic
31C2×#393C2211yescharacteristic
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2214yes—
40C2×#379C2212yes—
41C2×#379C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2211yescharacteristic
45#1446C2214yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1444C2214yescomplemented
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1443C2214yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1442C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1440C2212yes—
60#1440C2212yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1436C2214yescomplemented
67#1434C2214yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—42no—
86C2×C22⋊Q8—42no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—42no—
100C22×C22⋊C4—48no—
101C22×C22⋊C4—42no—
102C22×C22⋊C4—42no—
103C22×C22⋊C4D418yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42C2312yes—
122#397—24no—
123#397—24no—
124#397—28no—
125#397—28no—
126#397—28no—
127#397—28no—
128#397—48no—
129#393—24no—
130#393—24no—
131#388—28no—
132#388—28no—
133#385—28no—
134#385—44no—
135#385—44no—
136#381—28no—
137#381—28no—
138#380—48no—
139#379—28no—
140#379—42no—
141C4×C22⋊C4—24no—
142C2×#95—28no—
143C2×#95—24no—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25—24no—
172C25—24no—
173C25—22no—
174C25—22no—
175C25—24no—
176C25—42no—
177C25—44no—
178C25C2×D414yes—
179C25C2411yescharacteristic
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—44no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C22⋊Q8—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
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—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—416no—
268C2×C22⋊C4—416no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—432no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4C2×D414yes—
292C2×C22⋊C4#4118yes—
293C2×C42—24no—
294C2×C42—44no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—28no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—28no—
340C24—216no—
341C24—44no—
342C24—48no—
343C24—416no—
344C24—48no—
345C24—48no—
346C24—416no—
347C24—48no—
348C24—44no—
349C24—44no—
350C24—88no—
351C24—84no—
352C24—84no—
353C24—88no—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)14yes—
359C24H(5,𝔽2)12yes—
360C24H(5,𝔽2)12yes—
361C242−1+414yes—
362C24C2511yescharacteristic
363C2×Q8—44no—
364C2×Q8—84no—
365C2×Q8—84no—
366C2×Q8—84no—
367C22×C4—28no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
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—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—88no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—84no—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4118yes—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C22⋊C4—216no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C42—84no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—416no—
507C23—432no—
508C23—42no—
509C23—41no—
510C23—44no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—41no—
517C23—416no—
518C23—44no—
519C23—44no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—816no—
528C23—88no—
529C23—816no—
530C23—816no—
531C23—832no—
532C23#53414yes—
533C23#53514yes—
534C23#54218yes—
535C23#54718yes—
536C23#55114yes—
537C23#56314yes—
538C23C23×D411yescharacteristic
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)12yes—
543C23C2×2−1+412yes—
544C23#58512yes—
545C23#58514yes—
546Q8—88no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—416no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
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—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
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—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—88no—
615C2×C4—816no—
616C2×C4—816no—
617C2×C4—816no—
618C22—28no—
619C22—28no—
620C22—24no—
621C22—28no—
622C22—28no—
623C22—216no—
624C22—24no—
625C22—24no—
626C22—28no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—832no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22#271718yes—
643C22C2×#53411yescharacteristic
644C22C2×#53511yescharacteristic
645C22#344911yescharacteristic
646C22C2×#54212yes—
647C22C2×#54712yes—
648C22C2×#55111yescharacteristic
649C22#348512yes—
650C22#351314yes—
651C22#352114yes—
652C22C2×#56311yescharacteristic
653C22#353412yes—
654C22#353412yes—
655C22#353514yes—
656C22#354512yes—
657C22#354512yes—
658C22#356414yes—
659C4—48no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—82no—
668C4—84no—
669C4—84no—
670C4—82no—
671C2—28no—
672C2—44no—
673C2—82no—
674C2#3711911yescharacteristic
675C2#3744611yescharacteristic
676C2#3803912yes—
677C2#3891312yes—
678C2#4087014yes—
679C2#4330712yes—
680C2C2×#271711yescharacteristic
681C2#5260312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17229C2semidirect
4. #17201C2semidirect
5. #17059C2semidirect
6. #16831C2semidirect
7. #16454C2semidirect
8. #15485C2semidirect
9. #14919C2semidirect
10. #14914C2semidirect
12. #14263C2semidirect
13. #14040C2semidirect
45. #1446C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
66. #1436C22semidirect
67. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C22×#3972
#172292
#172012
#170592
#168312
#164542
#154852
#149192
#149142
#147272
#142632
#140402
#138772
#138692