Finite Group Databasebeta

← #554190

Subgroups — #554190

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic, complemented
3C22×#388C211yesmaximal, characteristic
4#14705C212yesmaximal, complemented
5#14579C214yesmaximal, complemented
6#14522C214yesmaximal, complemented
7#14434C214yesmaximal, complemented
8#14413C214yesmaximal, complemented
9#14262C212yesmaximal
10#14139C211yesmaximal, characteristic, complemented
11#13992C212yesmaximal, complemented
12#13991C212yesmaximal
13#13903C211yesmaximal, characteristic
14#13880C211yesmaximal, characteristic, complemented
15#13796C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2561C4116yescomplemented
23C2×#397C2212yes—
24C2×#393C2212yes—
25#2329—28no—
26C2×#391C2212yescomplemented
27C2×#391C2212yescomplemented
28C2×#389C2212yescomplemented
29C2×#388—28no—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2212yescomplemented
35C2×#387C2212yescomplemented
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#384C2212yescomplemented
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1444C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69C4⋊#95C2212yescomplemented
70C4⋊#95C2212yescomplemented
71#1434C2214yescomplemented
72#1434C2214yescomplemented
73#1434C2214yescomplemented
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C26C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2×C418yes—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2×C418yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42—24no—
121C22×C42—44no—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#397C2×C4116yescomplemented
128#393C2×C4116yescomplemented
129#391—28no—
130#391—44no—
131#389—28no—
132#388—216no—
133#388—216no—
134#388—48no—
135#388C2×C4132yescomplemented
136#387—28no—
137#387—28no—
138#385—28no—
139#385—44no—
140#385—44no—
141#384—44no—
142#381—44no—
143#381C2×C4116yes—
144#380—24no—
145#379—44no—
146#379C2×C4116yes—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4C2×C418yes—
150C2×#95—24no—
151C2×#95D418yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—42no—
181C25—42no—
182C25—42no—
183C25—42no—
184C25C22×C414yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8C22×C414yes—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
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×C4C22×C414yes—
213C23×C4C22×C414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4#4114yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C22⋊Q8—232no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C414yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—232no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—416no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C22×C418yes—
295C2×C22⋊C4C22×C4116yes—
296C2×C22⋊C4C22×C4116yes—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—48no—
303C2×C42—42no—
304C2×C42—416no—
305C2×C42—44no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42C22×C418yes—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C22×C418yes—
330#95C22×C4116yes—
331#95C22×C4116yes—
332#95C2×D418yes—
333#95C2×D418yes—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—28no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—216no—
350C24—48no—
351C24—48no—
352C24—44no—
353C24—44no—
354C24—44no—
355C24—44no—
356C24—48no—
357C24—48no—
358C24—44no—
359C24—44no—
360C24—48no—
361C24—88no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24C23×C414yes—
366C24C23×C412yes—
367C24C22×D412yes—
368C24C2×#4112yes—
369C24H(5,𝔽2)14yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—28no—
373C2×Q8—48no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—84no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
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—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
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—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—416no—
445C22×C4—44no—
446C22×C4—416no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4C4×D4116yes—
457C22×C4C23×C414yes—
458C22×C4C23×C418yes—
459C22×C4C23×C414yes—
460C22×C4C23×C414yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4H(5,𝔽2)14yes—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—48no—
493C42—416no—
494C42—84no—
495C42—84no—
496C42—84no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—416no—
515C23—416no—
516C23—416no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—832no—
536C23—88no—
537C23—88no—
538C23—88no—
539C23—88no—
540C23—832no—
541C23C2×C4×D414yes—
542C23#52018yes—
543C23#53514yes—
544C23#53614yes—
545C23#54218yes—
546C23C24×C411yescharacteristic
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553C23#58512yes—
554C23#58512yes—
555Q8—88no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
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—88no—
626C2×C4—88no—
627C2×C4C2×C4×D414yes—
628C2×C4C2×C4×D414yes—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—216no—
634C22—24no—
635C22—216no—
636C22—24no—
637C22—44no—
638C22—44no—
639C22—48no—
640C22—48no—
641C22—44no—
642C22—832no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22#232618yes—
650C22C22×C4×D411yescharacteristic
651C22C2×#52012yes—
652C22#342812yes—
653C22C4×H(5,𝔽2)12yes—
654C22C2×#53511yescharacteristic
655C22C2×#53611yescharacteristic
656C22C2×#54212yes—
657C22#348411yescharacteristic
658C22#348511yescharacteristic
659C22#348511yescharacteristic
660C22#348711yescharacteristic
661C22#349312yes—
662C22#350512yes—
663C22#353314yes—
664C22#353414yes—
665C22#356114yes—
666C22#356414yes—
667C4—24no—
668C4—24no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—48no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—84no—
680C2—28no—
681C2—84no—
682C2C2×#232611yescharacteristic
683C2#3420111yescharacteristic
684C2#3426411yescharacteristic
685C2#3445814yes—
686C2#3695611yescharacteristic
687C2#3696211yescharacteristic
688C2#3696611yescharacteristic
689C2#3698814yes—
690C2#3718511yescharacteristic
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. C2×#2329C2semidirect
4. #14705C2semidirect
5. #14579C2semidirect
6. #14522C2semidirect
7. #14434C2semidirect
8. #14413C2semidirect
10. #14139C2semidirect
11. #13992C2semidirect
14. #13880C2semidirect
15. #13796C2semidirect
22. #2561C4semidirect
26. C2×#391C22semidirect
27. C2×#391C22semidirect
28. C2×#389C22semidirect
34. C2×#387C22semidirect
35. C2×#387C22semidirect
39. C2×#384C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. C4⋊#95C22semidirect
70. C4⋊#95C22semidirect
71. #1434C22semidirect
72. #1434C22semidirect
73. #1434C22semidirect
127. #397C2×C4semidirect
128. #393C2×C4semidirect
135. #388C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#23292
C22×#3882
#147052
#145792
#145222
#144342
#144132
#142622
#141392
#139922
#139912
#139032
#138802
#137962