Finite Group Databasebeta

← #1420849

Subgroups — #1420849

Order 512.

3612 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2431C211yesmaximal, characteristic
2C2×#2366C211yesmaximal, characteristic, complemented
3#15984C212yesmaximal, complemented
4#15784C212yesmaximal, complemented
5#15450C214yesmaximal, complemented
6#15285C214yesmaximal, complemented
7#15089C212yesmaximal, complemented
8#14795C214yesmaximal, complemented
9#14513C212yesmaximal, complemented
10#14419C211yesmaximal, characteristic, complemented
11#14225C212yesmaximal, complemented
12#13956C212yesmaximal, complemented
13#13894C214yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×C4×D4C2214yes—
19#2431—28no—
20C2×#397C2212yes—
21C2×#397C2214yes—
22C2×#392C2214yes—
23#2366—28no—
24C2×#391C2211yescharacteristic
25C2×#390C2214yes—
26C2×#390C2211yescharacteristic
27C2×#388C2214yes—
28C2×#386C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C22×#95C2212yes—
41#1446C2214yes—
42#1445C2212yes—
43#1445C2214yes—
44#1444C2212yes—
45#1442C2214yes—
46#1442C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2212yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2214yes—
58#1438C2214yes—
59#1436C2214yes—
60#1434C2214yes—
61#1434C2214yes—
62#1434C2212yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×#127—28no—
86C2×#127D418yes—
87C2×#124—24no—
88C2×#124—44no—
89C2×#121—24no—
90C2×#121—42no—
91C2×#121—42no—
92C2×C4×D4—216no—
93C2×C4×D4—216no—
94C2×C4×D4—48no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#397—28no—
118#397—48no—
119#392—216no—
120#391—24no—
121#390—216no—
122#390—24no—
123#388—48no—
124#386—28no—
125#386—28no—
126#386—216no—
127#386—28no—
128#386—48no—
129#386—44no—
130#386—44no—
131#386—44no—
132#381—44no—
133#381—44no—
134C4×C4⋊C4—28no—
135C4×C22⋊C4—24no—
136C2×#95—44no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25C2412yes—
158C25C2411yescharacteristic
159C25C2412yes—
160C2×#41—416no—
161C2×#41—432no—
162C2×#41—432no—
163C2×#41—416no—
164C2×#41—416no—
165C22×Q8—22no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—42no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4C2×D4116yes—
196C22×D4C2×D414yes—
197C22×D4C2×D414yes—
198C22×D4#4118yes—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236#127—232no—
237#124—416no—
238#121—416no—
239C4×D4—464no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4#4118yescomplemented
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
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—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—42no—
299C2×C42C2×D414yes—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—28no—
310#95—28no—
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—
320C24—24no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—21no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24C22×D414yes—
344C24C22×D414yes—
345C24C2×#4114yes—
346C24C2×#4114yes—
347C24H(5,𝔽2)12yes—
348C24C2511yescharacteristic
349#41—4128no—
350C2×Q8—28no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×D4—216no—
356C2×D4—232no—
357C2×D4—28no—
358C2×D4—232no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—88no—
381C2×D4—84no—
382C2×D4—88no—
383C2×D4#123116yes—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
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—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—88no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4—84no—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C4H(5,𝔽2)14yes—
471C4⋊C4—216no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—88no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—84no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—84no—
490C42—28no—
491C42—44no—
492C42—416no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—24no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—41no—
552C23C2×#12318yes—
553C23C23×D411yescharacteristic
554C23C22×#4112yes—
555C23C22×#4112yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)11yescharacteristic
560C23#58514yes—
561Q8—416no—
562D4—416no—
563D4—464no—
564D4—432no—
565D4—88no—
566D4—816no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—416no—
584C2×C4—432no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
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—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—816no—
628C2×C4—816no—
629C2×C4—816no—
630C2×C4C2×#12314yes—
631C2×C4#54618yes—
632C22—28no—
633C22—216no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—48no—
638C22—44no—
639C22—48no—
640C22—48no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—48no—
646C22—42no—
647C22—44no—
648C22—48no—
649C22—44no—
650C22—42no—
651C22—48no—
652C22—82no—
653C22—84no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—84no—
658C22—82no—
659C22—84no—
660C22C22×#12311yescharacteristic
661C22C2×#54612yes—
662C22#347514yes—
663C22#348111yescharacteristic
664C22#348411yescharacteristic
665C22#348411yescharacteristic
666C22#348512yes—
667C22#349012yes—
668C22#349012yes—
669C22#351314yes—
670C22#351714yes—
671C22#351912yes—
672C22#353312yes—
673C22#354012yes—
674C22#354614yes—
675C22#354911yescharacteristic
676C4—24no—
677C4—28no—
678C4—42no—
679C4—44no—
680C4—48no—
681C4—82no—
682C4—84no—
683C4—82no—
684C4—84no—
685C4—84no—
686C2—48no—
687C2—82no—
688C2—84no—
689C2#3535011yescharacteristic
690C2#3615512yes—
691C2#4321414yes—
692C2#4493512yes—
693C2#4963812yes—
694C2#5205411yescharacteristic
695C2#5904111yescharacteristic
696C2#5997312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
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×#2366C2semidirect
3. #15984C2semidirect
4. #15784C2semidirect
5. #15450C2semidirect
6. #15285C2semidirect
7. #15089C2semidirect
8. #14795C2semidirect
9. #14513C2semidirect
10. #14419C2semidirect
11. #14225C2semidirect
12. #13956C2semidirect
13. #13894C2semidirect
250. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#24312
C2×#23662
#159842
#157842
#154502
#152852
#150892
#147952
#145132
#144192
#142252
#139562
#138942