Finite Group Databasebeta

← #625866

Subgroups — #625866

Order 512.

4128 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2412C214yesmaximal, complemented
2C2×#2617C211yesmaximal, characteristic
3C2×#2315C211yesmaximal, characteristic, complemented
4#15549C214yesmaximal, complemented
5#15539C214yesmaximal, complemented
6#15384C214yesmaximal, complemented
7#15370C214yesmaximal, complemented
8#14836C214yesmaximal, complemented
9#14698C211yesmaximal, characteristic, complemented
10#14686C211yesmaximal, characteristic, complemented
11#14353C211yesmaximal, characteristic, complemented
12#14271C211yesmaximal, characteristic, complemented
13C2×#1437C211yesmaximal, characteristic, complemented
14C22×#123C2214yes—
15C22×#121C2211yescharacteristic
16C22×#120C2212yes—
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C42C2211yescharacteristic
21#2617—28no—
22#2412—232no—
23C2×#399C2214yescomplemented
24C2×#397C2214yes—
25C2×#397C2214yes—
26C2×#397C2212yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#392C2212yes—
31#2315—28no—
32C2×#390C2212yescomplemented
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#386C2212yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2212yescomplemented
41C2×#381C2214yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2214yes—
47C2×#380C2214yes—
48C22×#95C2211yescharacteristic
49C22×#95C2212yes—
50#1445C2214yescomplemented
51#1445C2212yescomplemented
52#1445C2214yescomplemented
53#1445C2214yescomplemented
54#1444C2214yescomplemented
55#1444C2214yescomplemented
56#1444C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2212yescomplemented
62#1437C2214yes—
63#1437C2212yes—
64#1437C2212yes—
65#1437C2212yescomplemented
66#1437C2212yescomplemented
67C4⋊#95C2214yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2212yescomplemented
70C26C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2314yes—
74C24×C4C2312yes—
75C24×C4C2314yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×#123—216no—
80C2×#123—48no—
81C2×#123—48no—
82C2×#121—24no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#120—28no—
86C2×#120—216no—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—216no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4D418yes—
103C22×C22⋊C4D4116yescomplemented
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42—24no—
120C22×C42—44no—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124#399—48no—
125#397—28no—
126#397—216no—
127#397—216no—
128#394—216no—
129#394—216no—
130#394—216no—
131#392—28no—
132#390—44no—
133#386—28no—
134#386—216no—
135#386—216no—
136#386—216no—
137#385—216no—
138#385—216no—
139#385—216no—
140#384—44no—
141#381—216no—
142#380—28no—
143#380—24no—
144#380—24no—
145#380—24no—
146#380—216no—
147#380—216no—
148C2×#95—24no—
149C2×#95—44no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25—22no—
168C25—22no—
169C25—24no—
170C25—42no—
171C25—42no—
172C25—42no—
173C25—42no—
174C25C2×D414yes—
175C25C2×D418yes—
176C25C2412yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—42no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4C2×D418yes—
222C23×C4C2×D414yes—
223C23×C4C2×D418yes—
224C23×C4C2×D414yes—
225C23×C4#4114yes—
226C23×C4#4114yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237#123—464no—
238#121—416no—
239#120—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C22⋊C4—232no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—432no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—432no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×D418yes—
309C2×C22⋊C4C2×D418yes—
310C2×C22⋊C4C2×D4116yes—
311C2×C22⋊C4C2×D4116yes—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—416no—
318C2×C42—42no—
319C2×C42—42no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—44no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—28no—
349C24—28no—
350C24—216no—
351C24—216no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—82no—
368C24—82no—
369C24—84no—
370C24—84no—
371C24—82no—
372C24—82no—
373C24—84no—
374C24—84no—
375C24—84no—
376C24—84no—
377C24#127116yes—
378C24C22×D414yes—
379C24C22×D418yes—
380C24C22×D412yes—
381C24C22×D414yes—
382C24C2×#4112yes—
383C24C2×#4112yes—
384C242−1+414yes—
385C24C2511yescharacteristic
386C2×D4—48no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—44no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—216no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—82no—
475C22×C4—84no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—88no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4#123116yescomplemented
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D418yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C4H(5,𝔽2)14yes—
496C4⋊C4—416no—
497C4⋊C4—44no—
498C4⋊C4—416no—
499C4⋊C4—84no—
500C4⋊C4—88no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—88no—
504C4⋊C4—88no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—232no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C42—28no—
521C42—48no—
522C42—44no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—232no—
550C23—432no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—88no—
583C23—816no—
584C23—816no—
585C23—816no—
586C23—88no—
587C23—816no—
588C23—816no—
589C23C2×#12314yes—
590C23C2×#12718yes—
591C23C2×#12714yes—
592C23#53614yes—
593C23#54718yes—
594C23#54718yes—
595C23C23×D411yescharacteristic
596C23C23×D412yes—
597C23C22×#4111yescharacteristic
598C23C22×#4111yescharacteristic
599C23C2×H(5,𝔽2)12yes—
600C23C2×H(5,𝔽2)12yes—
601C23C2×2−1+412yes—
602C23#58512yes—
603C23#58512yes—
604D4—88no—
605D4—832no—
606D4—816no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—416no—
630C2×C4—432no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—816no—
669C2×C4—88no—
670C2×C4—832no—
671C2×C4C2×#12314yes—
672C2×C4C2×#12314yes—
673C2×C4#53418yes—
674C22—24no—
675C22—28no—
676C22—28no—
677C22—28no—
678C22—28no—
679C22—24no—
680C22—28no—
681C22—28no—
682C22—216no—
683C22—28no—
684C22—44no—
685C22—48no—
686C22—48no—
687C22—48no—
688C22—44no—
689C22—48no—
690C22—48no—
691C22—832no—
692C22—82no—
693C22—84no—
694C22—84no—
695C22—84no—
696C22—84no—
697C22—82no—
698C22—82no—
699C22—84no—
700C22—84no—
701C22#250118yes—
702C22C22×#12311yescharacteristic
703C22C22×#12711yescharacteristic
704C22C2×#53412yes—
705C22C2×#53611yescharacteristic
706C22#344912yes—
707C22C2×#54712yes—
708C22C2×#54712yes—
709C22#349011yescharacteristic
710C22#349111yescharacteristic
711C22#349412yes—
712C22#349612yes—
713C22#349612yes—
714C22#349612yes—
715C22#349612yes—
716C22#350512yes—
717C22#350811yescharacteristic
718C22#351011yescharacteristic
719C22#353512yes—
720C22#353612yes—
721C22#356112yes—
722C22#356412yes—
723C4—24no—
724C4—24no—
725C4—28no—
726C4—48no—
727C4—44no—
728C4—44no—
729C4—84no—
730C4—84no—
731C4—84no—
732C4—82no—
733C2—28no—
734C2—48no—
735C2—162no—
736C2C2×#250111yescharacteristic
737C2#3500112yes—
738C2#3536011yescharacteristic
739C2#3554111yescharacteristic
740C2#3674012yes—
741C2#3786012yes—
742C2#3786612yes—
743C2#3948212yes—
744C2#3948612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2412C2semidirect
3. C2×#2315C2semidirect
4. #15549C2semidirect
5. #15539C2semidirect
6. #15384C2semidirect
7. #15370C2semidirect
8. #14836C2semidirect
9. #14698C2semidirect
10. #14686C2semidirect
11. #14353C2semidirect
12. #14271C2semidirect
13. C2×#1437C2semidirect
23. C2×#399C22semidirect
32. C2×#390C22semidirect
40. C2×#384C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1438C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
103. C22×C22⋊C4D4semidirect
483. C22×C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24122
C2×#26172
C2×#23152
#155492
#155392
#153842
#153702
#148362
#146982
#146862
#143532
#142712
C2×#14372