Finite Group Databasebeta

← #556858

Subgroups — #556858

Order 512.

3878 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2#14669C214yesmaximal, complemented
3#14349C211yesmaximal, characteristic, complemented
4#14347C211yesmaximal, characteristic, complemented
5#14342C211yesmaximal, characteristic, complemented
6#14339C211yesmaximal, characteristic, complemented
7#14269C214yesmaximal, complemented
8#14265C214yesmaximal, complemented
9#14131C214yesmaximal, complemented
10#13882C212yesmaximal, complemented
11#13877C212yesmaximal
12#13874C212yesmaximal, complemented
13#13869C212yesmaximal
14C4×#379C211yesmaximal, characteristic
15C2×C22⋊#95C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C2×#387C2214yescomplemented
23C2×#385C4116yescomplemented
24C2×#385C2214yes—
25C2×#385C2212yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#379C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41#1445C2214yescomplemented
42#1445C2214yescomplemented
43#1445C2214yescomplemented
44#1443C2214yescomplemented
45#1443C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1434C2214yescomplemented
60C22⋊#95C2211yescharacteristic
61C22⋊#95C2211yescharacteristic
62C22⋊#95C2211yescharacteristic
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C26C2311yescharacteristic
77C24×C4—44no—
78C24×C4—44no—
79C24×C4—44no—
80C24×C4C2×C418yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2×C418yescomplemented
95C22×C4⋊C4C2×C418yescomplemented
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2×C418yes—
108C22×C22⋊C4C2×C418yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2314yes—
121#387—48no—
122#385—48no—
123#385C2×C4116yes—
124#385C2×C4116yes—
125#385C2×C4116yescomplemented
126#385C2×C4116yescomplemented
127#381—44no—
128#381—44no—
129#380—44no—
130#380—44no—
131#379—28no—
132C4×C22⋊C4—44no—
133C4×C22⋊C4—44no—
134C4×C22⋊C4—44no—
135C2×#95—44no—
136C2×#95—44no—
137C2×#95C2×C418yescomplemented
138C2×#95C2×C418yescomplemented
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25—22no—
169C25—24no—
170C25—24no—
171C25—24no—
172C25—22no—
173C25—42no—
174C25—42no—
175C25C22×C414yes—
176C25#4118yes—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—416no—
199C23×C4—42no—
200C23×C4—416no—
201C23×C4—416no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—464no—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C22×C414yes—
227C23×C4C22×C414yes—
228C23×C4C22×C414yes—
229C23×C4C22×C414yes—
230C23×C4C22×C414yes—
231C23×C4C22×C414yes—
232C23×C4C22×C414yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×C4⋊C4—28no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C22×C418yes—
263C2×C22⋊C4—232no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—416no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—432no—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C22×C418yes—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C418yes—
297C2×C42—28no—
298C2×C42—44no—
299C2×C42—44no—
300#95—28no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C22×C418yes—
326#95C22×C418yes—
327#95C22×C418yes—
328#95C22×C418yes—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—28no—
339C24—28no—
340C24—28no—
341C24—28no—
342C24—216no—
343C24—232no—
344C24—216no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—22no—
352C24—44no—
353C24—44no—
354C24—416no—
355C24—416no—
356C24—48no—
357C24—416no—
358C24—48no—
359C24—48no—
360C24—48no—
361C24#117116yes—
362C24C23×C412yes—
363C24C23×C412yes—
364C24C23×C412yes—
365C24C2×#4114yes—
366C24C2×#4114yes—
367C24C2×#4114yes—
368C24H(5,𝔽2)14yes—
369C24H(5,𝔽2)14yes—
370C24H(5,𝔽2)12yes—
371C242−1+414yes—
372C242−1+412yes—
373C24C2511yescharacteristic
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
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—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—4128no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4C23×C414yes—
482C22×C4C23×C414yes—
483C22×C4C23×C414yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C23×C412yes—
487C22×C4C23×C414yes—
488C22×C4C23×C412yes—
489C22×C4H(5,𝔽2)14yes—
490C4⋊C4—216no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—88no—
495C4⋊C4—88no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C42—88no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—24no—
527C23—28no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—232no—
534C23—232no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—28no—
544C23—28no—
545C23—416no—
546C23—416no—
547C23—416no—
548C23—416no—
549C23—416no—
550C23—464no—
551C23—464no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23C2×#11714yes—
559C23C2×#11714yes—
560C23C2×#11714yes—
561C23#51814yes—
562C23#51814yes—
563C23#51914yes—
564C23#52014yes—
565C23#53818yes—
566C23#539116yes—
567C23#53918yes—
568C23C24×C411yescharacteristic
569C23C22×#4112yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×2−1+412yes—
575C23C2×2−1+411yescharacteristic
576C23#58512yes—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
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—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—88no—
651C2×C4—88no—
652C2×C4—832no—
653C2×C4#51814yes—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—216no—
666C22—216no—
667C22—216no—
668C22—216no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—464no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—48no—
679C22—48no—
680C22#231318yes—
681C22#231818yes—
682C22C22×#11711yescharacteristic
683C22C2×#51811yescharacteristic
684C22C2×#51811yescharacteristic
685C22C2×#51811yescharacteristic
686C22C2×#51911yescharacteristic
687C22C2×#52011yescharacteristic
688C22#342811yescharacteristic
689C22C2×#53812yes—
690C22C2×#53912yes—
691C22C2×#53914yes—
692C22#345914yes—
693C22#353214yes—
694C22#353412yes—
695C22#353514yes—
696C22#353512yes—
697C22#353612yes—
698C22#353812yes—
699C4—24no—
700C4—44no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—44no—
706C4—44no—
707C4—84no—
708C4—84no—
709C4—84no—
710C4—84no—
711C2—28no—
712C2—28no—
713C2—44no—
714C2C2×#231311yescharacteristic
715C2C2×#231811yescharacteristic
716C2#3405711yescharacteristic
717C2#3435611yescharacteristic
718C2#3435911yescharacteristic
719C2#3437611yescharacteristic
720C2#3438011yescharacteristic
721C2#3748214yes—
722C2#3748614yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14669C2semidirect
3. #14349C2semidirect
4. #14347C2semidirect
5. #14342C2semidirect
6. #14339C2semidirect
7. #14269C2semidirect
8. #14265C2semidirect
9. #14131C2semidirect
10. #13882C2semidirect
12. #13874C2semidirect
22. C2×#387C22semidirect
23. C2×#385C4semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1443C22semidirect
45. #1443C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1434C22semidirect
94. C22×C4⋊C4C2×C4semidirect
95. C22×C4⋊C4C2×C4semidirect
125. #385C2×C4semidirect
126. #385C2×C4semidirect
137. C2×#95C2×C4semidirect
138. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
#146692
#143492
#143472
#143422
#143392
#142692
#142652
#141312
#138822
#138772
#138742
#138692
C4×#3792
C2×C22⋊#952