Finite Group Databasebeta

← #559913

Subgroups — #559913

Order 512.

3608 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#120C211yesmaximal, characteristic
2#14680C212yesmaximal, complemented
3#14657C212yesmaximal
4#14416C214yesmaximal, complemented
5#14415C214yesmaximal, complemented
6#14410C212yesmaximal, complemented
7#14409C212yesmaximal, complemented
8#13959C211yesmaximal, characteristic, complemented
9#13947C212yesmaximal, complemented
10#13940C211yesmaximal, characteristic, complemented
11#13919C212yesmaximal, complemented
12#13912C212yesmaximal, complemented
13#13898C212yesmaximal, complemented
14#13890C212yesmaximal, complemented
15#13827C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C4×#120C4116yescomplemented
21C2×#388C2214yescomplemented
22C2×#387C2214yescomplemented
23C2×#386C2211yescharacteristic
24C2×#386C2212yes—
25C2×#385C2211yescharacteristic
26C2×#385C2212yes—
27C2×#383C2211yescharacteristic, complemented
28C2×#381C2211yescharacteristic, complemented
29C2×#380C2212yes—
30C2×#380C2214yes—
31C2×#380C2212yes—
32C2×#380C2214yes—
33C2×#379C2214yes—
34C2×#379C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic, complemented
38C2×#376C2211yescharacteristic, complemented
39#1445C2212yescomplemented
40#1443C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1441C2214yescomplemented
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2212yescomplemented
48#1439C2214yescomplemented
49#1438C2214yes—
50#1438C2212yes—
51#1438C2212yes—
52#1438C2214yes—
53#1437C2214yes—
54#1437C2212yes—
55#1437C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2212yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2212yescomplemented
64#1434C2212yescomplemented
65#1434C2212yescomplemented
66#1434C2212yescomplemented
67#1434C2212yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yescomplemented
75C4×#95C2212yescomplemented
76C26C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4—48no—
80C24×C4—42no—
81C24×C4—42no—
82C24×C4C2×C418yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×#120—28no—
89C2×#120C2×C418yes—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4—48no—
93C2×C4×D4C2×C418yes—
94C2×C4×D4C2×C4116yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121#388—48no—
122#387—48no—
123#386C2×C418yes—
124#386C2×C4116yes—
125#385C2×C418yes—
126#385C2×C4116yes—
127#383—42no—
128#381—42no—
129#380—44no—
130#380—44no—
131#380—48no—
132#380—48no—
133#379—216no—
134#379C2×C418yes—
135C4×C22⋊C4—42no—
136C4×C22⋊C4C2×C4116yes—
137C4×C22⋊C4C2×C418yes—
138#376—42no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C25—41no—
165C25—41no—
166C25—44no—
167C25—44no—
168C25—42no—
169C25C22×C414yes—
170C25#4118yes—
171C25C2411yescharacteristic
172C25C2412yes—
173C25C2411yescharacteristic
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4C22×C414yes—
183C22×D4C22×C418yes—
184C23×C4—28no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—432no—
194C23×C4—416no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—432no—
204C23×C4—432no—
205C23×C4C22×C418yes—
206C23×C4C22×C414yes—
207C23×C4C22×C414yes—
208C23×C4C22×C414yes—
209C23×C4C22×C418yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222#120—232no—
223C4×D4—416no—
224C4×D4—432no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—42no—
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—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C22×C418yes—
273C2×C22⋊C4C22×C418yes—
274C2×C22⋊C4C22×C414yes—
275C2×C22⋊C4C22×C414yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C414yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C414yes—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42—42no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42C22×C414yes—
291C2×C42C22×C418yes—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C22×C418yes—
315#95C22×C418yes—
316#95C22×C418yes—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—21no—
322C24—21no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—24no—
332C24—21no—
333C24—21no—
334C24—24no—
335C24—22no—
336C24—44no—
337C24—48no—
338C24—48no—
339C24—44no—
340C24—48no—
341C24—416no—
342C24—416no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—416no—
349C24—416no—
350C24#117116yes—
351C24C23×C412yes—
352C24C23×C414yes—
353C24C23×C412yes—
354C24C2×#4114yes—
355C24C2×#4114yes—
356C24C2×#4114yes—
357C24C2511yescharacteristic
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—48no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
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—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—4128no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4C23×C412yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C23×C414yes—
448C22×C4C23×C412yes—
449C22×C4C23×C412yes—
450C22×C4C23×C414yes—
451C22×C4C23×C412yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)18yes—
455C4⋊C4—216no—
456C4⋊C4—44no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—28no—
461C22⋊C4—216no—
462C22⋊C4—48no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—84no—
475C22⋊C4—88no—
476C42—28no—
477C42—48no—
478C42—48no—
479C42—84no—
480C42—88no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—416no—
509C23—432no—
510C23—416no—
511C23—416no—
512C23—432no—
513C23—464no—
514C23—464no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23C2×#11718yes—
540C23C2×#11714yes—
541C23C24×C411yescharacteristic
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)14yes—
546C23#58514yes—
547C23#58512yes—
548D4—416no—
549D4—432no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—432no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—432no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
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—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—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—88no—
617C2×C4#51814yes—
618C2×C4#51818yes—
619C22—24no—
620C22—28no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—464no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—42no—
643C22—84no—
644C22—84no—
645C22C22×#11711yescharacteristic
646C22C2×#51811yescharacteristic
647C22C2×#51812yes—
648C22#342812yes—
649C22#342811yescharacteristic
650C22#350512yes—
651C22#350514yes—
652C22#350514yes—
653C22#350512yes—
654C22#352912yes—
655C22#352912yes—
656C22#353212yes—
657C22#353212yes—
658C22#353914yes—
659C22#354014yes—
660C4—24no—
661C4—28no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—84no—
669C4—84no—
670C4—84no—
671C4—84no—
672C2—44no—
673C2—48no—
674C2—82no—
675C2#3418511yescharacteristic
676C2#3418512yes—
677C2#3434211yescharacteristic
678C2#3435611yescharacteristic
679C2#3439112yes—
680C2#4550614yes—
681C2#4550714yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14680C2semidirect
4. #14416C2semidirect
5. #14415C2semidirect
6. #14410C2semidirect
7. #14409C2semidirect
8. #13959C2semidirect
9. #13947C2semidirect
10. #13940C2semidirect
11. #13919C2semidirect
12. #13912C2semidirect
13. #13898C2semidirect
14. #13890C2semidirect
15. #13827C2semidirect
20. C4×#120C4semidirect
21. C2×#388C22semidirect
22. C2×#387C22semidirect
27. C2×#383C22semidirect
28. C2×#381C22semidirect
37. C2×C4×C22⋊C4C22semidirect
38. C2×#376C22semidirect
39. #1445C22semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
60. #1437C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
68. #1434C22semidirect
74. C22⋊#95C22semidirect
75. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1202
#146802
#146572
#144162
#144152
#144102
#144092
#139592
#139472
#139402
#139192
#139122
#138982
#138902
#138272