Finite Group Databasebeta

← #2926336

Subgroups — #2926336

Order 512.

3192 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2541C211yesmaximal, characteristic
3#16799C214yesmaximal, complemented
4#16596C212yesmaximal, complemented
5#15674C214yesmaximal, complemented
6#15377C214yesmaximal, complemented
7#15261C212yesmaximal, complemented
8#15261C212yesmaximal, complemented
9#15140C212yesmaximal, complemented
10#14824C212yesmaximal, complemented
11#14361C214yesmaximal, complemented
12#14242C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×#123C2212yes—
18C22×#121C2214yes—
19C22×C4×D4C2211yescharacteristic
20#2724—28no—
21#2541—28no—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#397C2212yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#392C2214yes—
29C2×#390C2211yescharacteristic
30C2×#390C2214yes—
31C2×#390C2214yes—
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41#1445C2212yes—
42#1442C2212yes—
43#1441C2212yes—
44#1441C2212yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2212yes—
51#1438C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2212yes—
58#1438C2214yes—
59#1438C2212yes—
60#1438C2212yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×D4C2311yescharacteristic
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2314yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#127—24no—
79C2×#127—24no—
80C2×#127—24no—
81C2×#127—42no—
82C2×#127—42no—
83C2×#127—44no—
84C2×#127—44no—
85C2×#123—28no—
86C2×#123—48no—
87C2×#121—216no—
88C2×#121—216no—
89C2×#121—48no—
90C2×C4×D4—24no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120#400—28no—
121#398—28no—
122#397—216no—
123#397—44no—
124#394—48no—
125#394—48no—
126#392—216no—
127#390—24no—
128#390—48no—
129#390—48no—
130#390D418yes—
131#389—28no—
132#388—48no—
133#386—44no—
134#386—44no—
135#386—48no—
136#385—216no—
137#380—44no—
138C4×C4⋊C4—24no—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25C2412yes—
157C25C2412yes—
158C25C2412yes—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4C2×D414yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218#127—416no—
219#127—416no—
220#127—416no—
221#123—432no—
222#121—464no—
223C4×D4—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4#4118yes—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
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⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D4116yes—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—28no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24C22×D414yes—
343C24H(5,𝔽2)14yes—
344C24C2511yescharacteristic
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—416no—
353C2×D4—416no—
354C2×D4—48no—
355C2×D4—416no—
356C2×D4—48no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—416no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—416no—
367C2×D4—48no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—88no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
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—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
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—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C4⋊C4—216no—
452C4⋊C4—44no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C22⋊C4—232no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—84no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—84no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—28no—
475C42—44no—
476C42—44no—
477C42—88no—
478C23—24no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C23×D411yescharacteristic
544C23C22×#4112yes—
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23#58514yes—
551D4—416no—
552D4—88no—
553D4—816no—
554D4—832no—
555D4—88no—
556D4—832no—
557D4—816no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
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—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—816no—
613C2×C4—816no—
614C2×C4#53514yes—
615C2×C4#54618yes—
616C2×C4#55914yes—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—48no—
622C22—48no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—82no—
628C22—84no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—82no—
639C22—82no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—82no—
645C22—84no—
646C22C2×#53511yescharacteristic
647C22C2×#54612yes—
648C22C2×#55911yescharacteristic
649C22#348411yescharacteristic
650C22#348411yescharacteristic
651C22#350012yes—
652C22#350812yes—
653C22#351314yes—
654C22#351714yes—
655C22#353312yes—
656C22#353414yes—
657C22#353712yes—
658C22#353712yes—
659C22#354311yescharacteristic
660C22#354614yes—
661C22#355112yes—
662C4—24no—
663C4—24no—
664C4—28no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—84no—
669C4—84no—
670C4—84no—
671C4—84no—
672C2—84no—
673C2—84no—
674C2—162no—
675C2#3671711yescharacteristic
676C2#3895612yes—
677C2#4300512yes—
678C2#4531512yes—
679C2#5078414yes—
680C2#5197811yescharacteristic
681C2#5379412yes—
682C2#5914311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×#2724C2semidirect
3. #16799C2semidirect
4. #16596C2semidirect
5. #15674C2semidirect
6. #15377C2semidirect
7. #15261C2semidirect
8. #15261C2semidirect
9. #15140C2semidirect
10. #14824C2semidirect
11. #14361C2semidirect
12. #14242C2semidirect
13. #14157C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25412
#167992
#165962
#156742
#153772
#152612
#151402
#148242
#143612
#142422
#141572