Finite Group Databasebeta

← #560060

Subgroups — #560060

Order 512.

2456 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2#14682C212yesmaximal, complemented
3#14666C212yesmaximal
4#14507C212yesmaximal, complemented
5#14506C212yesmaximal
6#14500C214yesmaximal, complemented
7#14499C214yesmaximal, complemented
8#13911C212yesmaximal, complemented
9#13905C211yesmaximal, characteristic
10#13885C212yesmaximal, complemented
11#13879C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13875C212yesmaximal, complemented
14#13871C211yesmaximal, characteristic
15#13831C212yesmaximal, complemented
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C4×#123C4116yescomplemented
21C2×#388C2214yes—
22C2×#388C2212yes—
23C2×#387C2212yes—
24C2×#386C2211yescharacteristic
25C2×#385C2211yescharacteristic
26C2×#384C2212yes—
27C2×#382C2212yes—
28C2×#382C2212yes—
29C2×#382C2211yescharacteristic
30C2×#381C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40#1445C2214yes—
41#1445C2212yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1440C2212yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2212yes—
53#1438C2212yes—
54#1437C2212yes—
55#1437C2212yes—
56#1437C2214yes—
57#1437C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#123—28no—
82C2×#123C2×C418yes—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×C4×D4C2×C418yes—
86C2×#117—44no—
87C2×#117—42no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C42—42no—
109C22×C42—42no—
110C22×C42C2×C418yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118#388—216no—
119#388C2×C4116yes—
120#387C2×C4116yes—
121#386C2×C418yes—
122#385C2×C418yes—
123#384—44no—
124#382—42no—
125#382—44no—
126#382—44no—
127#381C2×C418yes—
128#380—44no—
129#380—44no—
130#380—44no—
131#380—42no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4—44no—
134C4×C4⋊C4C2×C4116yes—
135C4×C22⋊C4C2×C4116yes—
136C4×C22⋊C4C2×C418yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C25C2411yescharacteristic
165C25C2411yescharacteristic
166C22×D4—24no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—42no—
170C22×D4C22×C414yes—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4C22×C414yes—
188C23×C4C22×C414yes—
189C23×C4C22×C414yes—
190C23×C4C22×C414yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204#123—232no—
205C4×D4—416no—
206#117—416no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22×C418yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C418yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4C22×C418yes—
257C2×C22⋊C4C22×C414yes—
258C2×C22⋊C4C22×C414yes—
259C2×C22⋊C4C22×C414yes—
260C2×C22⋊C4C22×C418yes—
261C2×C22⋊C4C22×C414yes—
262C2×C22⋊C4#4118yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—416no—
271C2×C42—44no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22×C414yes—
280C2×C42C22×C418yes—
281C2×C42C22×C418yes—
282C2×C42C22×C418yes—
283#95—28no—
284#95—28no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—42no—
305#95C22×C418yes—
306#95C22×C418yes—
307#95C22×C418yes—
308#95C22×C418yes—
309#95C22×C418yes—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24C23×C412yes—
326C24C23×C412yes—
327C24C2×#4114yes—
328C24C2511yescharacteristic
329C2×D4—28no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—48no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
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—24no—
404C22×C4—24no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4C23×C414yes—
421C22×C4C23×C412yes—
422C22×C4C23×C414yes—
423C22×C4C23×C414yes—
424C22×C4C23×C412yes—
425C22×C4C23×C412yes—
426C22×C4C23×C412yes—
427C22×C4C23×C414yes—
428C22×C4C23×C412yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C42−1+414yes—
435C4⋊C4—216no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—84no—
453C22⋊C4—88no—
454C22⋊C4#117116yescomplemented
455C42—216no—
456C42—28no—
457C42—216no—
458C42—48no—
459C42—48no—
460C42—88no—
461C42—84no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23C2×#11714yes—
504C23C24×C411yescharacteristic
505C23C22×#4112yes—
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)12yes—
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58512yes—
512C23#58512yes—
513D4—416no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
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×C4C2×#11718yes—
594C2×C4#51814yes—
595C2×C4#52018yes—
596C2×C4#56014yes—
597C2×C4#56218yes—
598C2×C4#56314yes—
599C22—24no—
600C22—28no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22C22×#11711yescharacteristic
618C22C2×#51811yescharacteristic
619C22C2×#52012yes—
620C22#342812yes—
621C22#342811yescharacteristic
622C22#350612yes—
623C22#350612yes—
624C22#351412yes—
625C22#351412yes—
626C22#351512yes—
627C22#351512yes—
628C22C2×#56011yescharacteristic
629C22C2×#56212yes—
630C22C2×#56311yescharacteristic
631C22#353912yes—
632C22#354011yescharacteristic
633C22#354011yescharacteristic
634C22#356212yes—
635C22#356212yes—
636C22#356512yes—
637C22#356512yes—
638C4—28no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—84no—
647C4—84no—
648C4—84no—
649C4—84no—
650C4#237618yes—
651C2—44no—
652C2—82no—
653C2#3418611yescharacteristic
654C2#3419112yes—
655C2C2×#237611yescharacteristic
656C2#3436111yescharacteristic
657C2#3438512yes—
658C2#4570812yes—
659C2#4570912yes—
660C2#4571012yes—
661C2#4571112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14682C2semidirect
4. #14507C2semidirect
6. #14500C2semidirect
7. #14499C2semidirect
8. #13911C2semidirect
10. #13885C2semidirect
11. #13879C2semidirect
12. #13876C2semidirect
13. #13875C2semidirect
15. #13831C2semidirect
20. C4×#123C4semidirect
454. C22⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
#146822
#146662
#145072
#145062
#145002
#144992
#139112
#139052
#138852
#138792
#138762
#138752
#138712
#138312