Finite Group Databasebeta

← #561413

Subgroups — #561413

Order 512.

3954 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C211yesmaximal, characteristic
2C2×#2339C211yesmaximal, characteristic, complemented
3C2×#2312C211yesmaximal, characteristic, complemented
4#14560C214yesmaximal, complemented
5#14374C214yesmaximal, complemented
6#14357C214yesmaximal, complemented
7#14337C211yesmaximal, characteristic, complemented
8#14295C211yesmaximal, characteristic, complemented
9#14187C214yesmaximal, complemented
10#13999C214yesmaximal, complemented
11#13984C214yesmaximal, complemented
12#13903C211yesmaximal, characteristic, complemented
13#13828C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#120C2212yes—
16C22×C4×D4C2212yescomplemented
17C23×C22⋊C4C2211yescharacteristic
18#2404C4116yescomplemented
19C2×#393C2212yes—
20C2×#392C2212yes—
21#2339—28no—
22#2312—28no—
23C2×#391C2212yescomplemented
24C2×#390C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#386C2212yes—
32C2×#381C2212yes—
33C2×#381C2211yescharacteristic
34C2×#381C2212yescomplemented
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2214yescomplemented
39C2×#379C2212yes—
40C2×#379C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2214yescomplemented
44C22×#95C2212yescomplemented
45#1446C2212yescomplemented
46#1444C2212yescomplemented
47#1443C2214yescomplemented
48#1442C2212yescomplemented
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2214yescomplemented
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yescomplemented
59#1437C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1434C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2214yescomplemented
70C26C2311yescharacteristic
71C22×#41—24no—
72C22×#41—24no—
73C22×#41—48no—
74C22×#41C2×C418yes—
75C23×Q8C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2314yes—
79C24×C4C2312yes—
80C24×C4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#120—48no—
84C2×#120C2×C4116yescomplemented
85C2×C4×D4—28no—
86C2×C4×D4—48no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4C2×C418yescomplemented
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2312yes—
111C22×C42C2314yes—
112C22×C42C2314yes—
113#393C2×C4116yescomplemented
114#392C2×C4116yes—
115#391—28no—
116#390—48no—
117#388—216no—
118#388—216no—
119#386—216no—
120#386—48no—
121#386—48no—
122#386C2×C4132yescomplemented
123#386C2×C4116yescomplemented
124#381—28no—
125#381—44no—
126#381C2×C418yescomplemented
127#380—24no—
128#380—24no—
129#380—48no—
130#380D418yescomplemented
131#379—28no—
132#379—44no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—48no—
136C2×#95—28no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C25—22no—
159C25—22no—
160C25—42no—
161C25—42no—
162C25—44no—
163C25—42no—
164C25—42no—
165C25C22×C414yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C25C2412yes—
169C2×#41—232no—
170C2×#41—432no—
171C2×#41—432no—
172C2×#41—432no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—44no—
176C22×Q8C22×C414yes—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4C22×C414yes—
191C22×D4C22×C418yes—
192C22×D4C22×C4116yes—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4C22×C418yes—
210C23×C4C22×C414yes—
211C23×C4C22×C414yes—
212C23×C4C2×D414yes—
213C23×C4#4114yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224#120—432no—
225C4×D4—432no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D418yes—
235C2×C22⋊C4—232no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
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—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
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⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C4116yes—
281C2×C22⋊C4C22×C4116yes—
282C2×C22⋊C4C2×D418yes—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291#95—22no—
292#95—28no—
293#95—22no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95C22×C418yes—
309#95C22×C418yes—
310#95C22×C4116yes—
311#95C2×D418yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—28no—
335C24—48no—
336C24—48no—
337C24—48no—
338C24—44no—
339C24—44no—
340C24—48no—
341C24—84no—
342C24—84no—
343C24—84no—
344C24—82no—
345C24—82no—
346C24—88no—
347C24—88no—
348C24—88no—
349C24—88no—
350C24C23×C414yes—
351C24C23×C412yes—
352C24C23×C414yes—
353C24C22×D412yes—
354C24C2×#4112yes—
355C24H(5,𝔽2)18yes—
356C24C2511yescharacteristic
357#41—4128no—
358C2×Q8—28no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—216no—
367C2×D4—232no—
368C2×D4—28no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—88no—
375C2×D4—88no—
376C2×D4—88no—
377C2×D4—88no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—216no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
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—22no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—84no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C4×D4116yes—
444C22×C4C23×C418yes—
445C22×C4C23×C414yes—
446C22×C4C23×C414yes—
447C22×C4C23×C412yes—
448C22×C4C23×C412yes—
449C22×C4C22×D418yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—48no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—84no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—48no—
476C42—88no—
477C42—88no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—24no—
483C23—28no—
484C23—24no—
485C23—28no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—416no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—88no—
538C23—88no—
539C23—832no—
540C23—816no—
541C23—816no—
542C23—816no—
543C23—816no—
544C23C2×C4×D414yes—
545C23#51818yes—
546C23C24×C411yescharacteristic
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)14yes—
550C23C2×H(5,𝔽2)12yes—
551C23#58512yes—
552C23#58512yes—
553C23#58512yes—
554Q8—416no—
555D4—232no—
556D4—432no—
557D4—832no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—416no—
578C2×C4—432no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
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—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
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—832no—
616C2×C4C2×C4×D414yes—
617C2×C4C2×C4×D414yes—
618C22—28no—
619C22—24no—
620C22—24no—
621C22—216no—
622C22—24no—
623C22—48no—
624C22—44no—
625C22—48no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—44no—
630C22—48no—
631C22—48no—
632C22—48no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—44no—
640C22—832no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C22×C4×D411yescharacteristic
648C22C2×#51812yes—
649C22#342812yes—
650C22C4×H(5,𝔽2)12yes—
651C22#348412yes—
652C22#348511yescharacteristic
653C22#348711yescharacteristic
654C22#348812yes—
655C22#349012yes—
656C22#349312yes—
657C22#350811yescharacteristic
658C22#351011yescharacteristic
659C22#353414yes—
660C22#354214yes—
661C22#354514yes—
662C22#356014yes—
663C4—24no—
664C4—24no—
665C4—48no—
666C4—44no—
667C4—44no—
668C4—48no—
669C4—44no—
670C4—82no—
671C4—84no—
672C4—84no—
673C2—48no—
674C2—44no—
675C2—84no—
676C2#3419811yescharacteristic
677C2#3420111yescharacteristic
678C2#3426911yescharacteristic
679C2#3449214yes—
680C2#4955012yes—
681C2#4955212yes—
682C2#4959512yes—
683C2#4959712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC2×#2404C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2339C2semidirect
3. C2×#2312C2semidirect
4. #14560C2semidirect
5. #14374C2semidirect
6. #14357C2semidirect
7. #14337C2semidirect
8. #14295C2semidirect
9. #14187C2semidirect
10. #13999C2semidirect
11. #13984C2semidirect
12. #13903C2semidirect
13. #13828C2semidirect
16. C22×C4×D4C22semidirect
18. #2404C4semidirect
23. C2×#391C22semidirect
34. C2×#381C22semidirect
38. C2×#380C22semidirect
43. C2×C4×C22⋊C4C22semidirect
44. C22×#95C22semidirect
45. #1446C22semidirect
46. #1444C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
55. #1439C22semidirect
58. #1438C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
69. C22⋊#95C22semidirect
84. C2×#120C2×C4semidirect
93. C22×C22⋊C4C2×C4semidirect
113. #393C2×C4semidirect
122. #386C2×C4semidirect
123. #386C2×C4semidirect
126. #381C2×C4semidirect
130. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
C2×#23392
C2×#23122
#145602
#143742
#143572
#143372
#142952
#141872
#139992
#139842
#139032
#138282