Finite Group Databasebeta

← #641318

Subgroups — #641318

Order 512.

4112 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic, complemented
2C2×#2416C212yesmaximal, complemented
3C2×#2290C211yesmaximal, characteristic, complemented
4#16822C214yesmaximal, complemented
5#16819C214yesmaximal, complemented
6#15196C211yesmaximal, characteristic, complemented
7#15195C211yesmaximal, characteristic, complemented
8#15127C211yesmaximal, characteristic, complemented
9#15124C211yesmaximal, characteristic, complemented
10#14757C214yesmaximal, complemented
11#14747C214yesmaximal, complemented
12#14260C212yesmaximal, complemented
13#14257C212yesmaximal, complemented
14#14137C212yesmaximal, complemented
15C2×C4⋊#95C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2214yes—
20C22×#121C2214yes—
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C42C2211yescharacteristic
25#2856—28no—
26#2416—216no—
27C2×#396C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#392C2214yes—
31C2×#390C2212yes—
32C2×#390C2214yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yes—
37#2290—28no—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#386C2218yes—
43C2×#385C2212yes—
44C2×#384C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×#376C2211yescharacteristic
50C22×#95C2214yes—
51#1444C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1440C2218yes—
55#1439C2218yes—
56#1439C2214yes—
57#1439C2218yes—
58#1439C2218yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2218yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C23×D4C2314yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C23×D4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2314yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C2×#127—24no—
78C2×#127—44no—
79C2×#121—216no—
80C2×#121—216no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121—44no—
87C2×#121—416no—
88C2×#121—416no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4D418yescomplemented
92C2×#117—24no—
93C2×#117—44no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2318yes—
119C22×C42—44no—
120C22×C42D418yescomplemented
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#396—216no—
128#394—216no—
129#394—48no—
130#392—48no—
131#390—28no—
132#390—28no—
133#390—216no—
134#390—28no—
135#390—44no—
136#389—28no—
137#388—28no—
138#387—28no—
139#386—28no—
140#386—216no—
141#386—416no—
142#385—28no—
143#384—24no—
144#380—28no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148#376—24no—
149C2×#95—48no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C25C2414yes—
163C25C2411yescharacteristic
164C25C2411yescharacteristic
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—216no—
172C22×D4—216no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—48no—
186C22×D4—48no—
187C22×D4—44no—
188C22×D4—48no—
189C22×D4—48no—
190C22×D4C2×D414yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—216no—
202C23×C4—28no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—48no—
206C23×C4C2×D414yes—
207C23×C4C2×D418yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4#4114yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223#127—416no—
224#121—432no—
225#121—464no—
226#121—464no—
227#121—432no—
228C4×D4—232no—
229#117—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—432no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4#4118yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
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—216no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—416no—
298C2×C42—44no—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—48no—
313#95—48no—
314#95—48no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—28no—
333C24—24no—
334C24—28no—
335C24—24no—
336C24—28no—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C22×D412yes—
340C24C2×#4112yes—
341C24H(5,𝔽2)18yes—
342C24C2511yescharacteristic
343C2×D4—28no—
344C2×D4—216no—
345C2×D4—416no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—416no—
352C2×D4—432no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—432no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—816no—
365C2×D4—816no—
366C2×D4—816no—
367C2×D4—88no—
368C2×D4—816no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—432no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—88no—
430C22×C4—816no—
431C22×C4#121116yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C2×#4118yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+414yes—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—416no—
460C22⋊C4—48no—
461C22⋊C4—432no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—816no—
469C22⋊C4—816no—
470C42—28no—
471C42—28no—
472C42—28no—
473C42—48no—
474C42—44no—
475C42—44no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—48no—
508C23—44no—
509C23—44no—
510C23—48no—
511C23—48no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23C2×#12114yes—
528C23#53518yes—
529C23#53518yes—
530C23#54018yes—
531C23C23×D411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4112yes—
535C23C2×H(5,𝔽2)12yes—
536C23C2×H(5,𝔽2)14yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×2−1+412yes—
539D4—416no—
540D4—832no—
541D4—832no—
542D4—832no—
543D4—816no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—232no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—216no—
565C2×C4—432no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—416no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—832no—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C2×C4#53214yes—
603C2×C4#53414yes—
604C2×C4#53718yes—
605C2×C4#54618yes—
606C2×C4#55318yes—
607C22—24no—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—24no—
613C22—216no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—82no—
620C22—82no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—88no—
625C22—88no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—88no—
632C22—88no—
633C22#261918yes—
634C22C22×#12111yescharacteristic
635C22C2×#53211yescharacteristic
636C22C2×#53411yescharacteristic
637C22C2×#53512yes—
638C22C2×#53512yes—
639C22C2×#53712yes—
640C22C2×#54012yes—
641C22C2×#54612yes—
642C22C2×#55312yes—
643C22#349811yescharacteristic
644C22#349911yescharacteristic
645C22#351112yes—
646C22#351914yes—
647C22#352214yes—
648C22#353614yes—
649C22#353714yes—
650C4—24no—
651C4—28no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—48no—
656C4—42no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—84no—
661C4#254318yes—
662C2—28no—
663C2—82no—
664C2—164no—
665C2C2×#261911yescharacteristic
666C2C2×#254311yescharacteristic
667C2#3523812yes—
668C2#3527311yescharacteristic
669C2#3743912yes—
670C2#3807912yes—
671C2#3808012yes—
672C2#3812314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2856C2semidirect
2. C2×#2416C2semidirect
3. C2×#2290C2semidirect
4. #16822C2semidirect
5. #16819C2semidirect
6. #15196C2semidirect
7. #15195C2semidirect
8. #15127C2semidirect
9. #15124C2semidirect
10. #14757C2semidirect
11. #14747C2semidirect
12. #14260C2semidirect
13. #14257C2semidirect
14. #14137C2semidirect
15. C2×C4⋊#95C2semidirect
91. C2×C4×D4D4semidirect
120. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#24162
C2×#22902
#168222
#168192
#151962
#151952
#151272
#151242
#147572
#147472
#142602
#142572
#141372
C2×C4⋊#952