Finite Group Databasebeta

← #920869

Subgroups — #920869

Order 512.

2380 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#2398C211yesmaximal, characteristic
3#17160C212yesmaximal, complemented
4#17106C212yesmaximal, complemented
5#16911C214yesmaximal, complemented
6#16309C214yesmaximal
7#15722C212yesmaximal, complemented
8#15622C212yesmaximal, complemented
9#15183C212yesmaximal, complemented
10#15158C212yesmaximal, complemented
11#15132C212yesmaximal, complemented
12#14266C212yesmaximal, complemented
13#14130C212yesmaximal, complemented
14#13951C212yesmaximal, complemented
15#13871C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2487—28no—
21#2398—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#397C2211yescharacteristic
26C2×#397C2211yescharacteristic
27C2×#397C2214yes—
28C2×#395C2214yes—
29C2×#394C2212yes—
30C2×#393C2211yescharacteristic
31C2×#393C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×#376C2211yescharacteristic
46#1445C2214yes—
47#1445C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2212yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1437C2212yes—
67#1436C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C4×Q8—24no—
83C2×C4×Q8—42no—
84C2×C4×Q8—42no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4D418yes—
88C2×#117—28no—
89C2×#117—42no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#400—28no—
121#400—28no—
122#399—28no—
123#397—24no—
124#397—24no—
125#397—48no—
126#395—48no—
127#394—28no—
128#393—24no—
129#393—24no—
130#389—28no—
131#389—28no—
132#389—44no—
133#389—44no—
134#386—28no—
135#385—28no—
136#385—48no—
137#380—28no—
138#380—24no—
139#380—28no—
140#380—42no—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—28no—
143#376—24no—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
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×#95C2312yes—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—42no—
178C22×D4—24no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4C2×D414yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C4×Q8—416no—
212C4×D4—416no—
213#117—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4#4118yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×D414yes—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—42no—
291C2×C42C2×D414yes—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24C22×D412yes—
335C24C22×D412yes—
336C242−1+414yes—
337C24C2511yescharacteristic
338C2×Q8—48no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—48no—
342C2×Q8—44no—
343C2×D4—28no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—48no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
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—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—88no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+414yes—
444C4⋊C4—216no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—88no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—84no—
467C42—28no—
468C42—48no—
469C42—48no—
470C42—48no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23#53614yes—
513C23C23×D411yescharacteristic
514C23C22×#4112yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)12yes—
518C23C2×2−1+412yes—
519C23C2×2−1+412yes—
520C23#58512yes—
521C23#58512yes—
522Q8—416no—
523D4—416no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—416no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4#53414yes—
604C2×C4#54218yes—
605C2×C4#54718yes—
606C2×C4#55114yes—
607C2×C4#56314yes—
608C22—24no—
609C22—28no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×#53411yescharacteristic
627C22C2×#53611yescharacteristic
628C22#344911yescharacteristic
629C22C2×#54212yes—
630C22C2×#54712yes—
631C22C2×#55111yescharacteristic
632C22#348812yes—
633C22#351612yes—
634C22#351712yes—
635C22#352012yes—
636C22#352112yes—
637C22C2×#56311yescharacteristic
638C22#353711yescharacteristic
639C22#354011yescharacteristic
640C22#354112yes—
641C22#354112yes—
642C22#354212yes—
643C22#354712yes—
644C22#355212yes—
645C22#356312yes—
646C22#356512yes—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—48no—
653C4—84no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4#271718yes—
660C2—44no—
661C2—82no—
662C2#3726911yescharacteristic
663C2#3743811yescharacteristic
664C2#3804012yes—
665C2#3804912yes—
666C2#4255412yes—
667C2#4274312yes—
668C2#4499212yes—
669C2C2×#271711yescharacteristic
670C2#5311912yes—
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. #17160C2semidirect
4. #17106C2semidirect
5. #16911C2semidirect
7. #15722C2semidirect
8. #15622C2semidirect
9. #15183C2semidirect
10. #15158C2semidirect
11. #15132C2semidirect
12. #14266C2semidirect
13. #14130C2semidirect
14. #13951C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#23982
#171602
#171062
#169112
#163092
#157222
#156222
#151832
#151582
#151322
#142662
#141302
#139512
#138712