Finite Group Databasebeta

← #4308772

Subgroups — #4308772

Order 512.

4284 conjugacy classes of subgroups.

The subgroup lattice has 794 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2849C212yesmaximal, complemented
2C2×#2617C211yesmaximal, characteristic
3C22×#380C211yesmaximal, characteristic, complemented
4#16397C214yesmaximal, complemented
5#16396C214yesmaximal, complemented
6#16257C214yesmaximal, complemented
7#16256C214yesmaximal, complemented
8#15014C211yesmaximal, characteristic, complemented
9#14921C212yesmaximal, complemented
10#14913C211yesmaximal, characteristic, complemented
11#14821C211yesmaximal, characteristic, complemented
12#14729C212yesmaximal, complemented
13#14719C211yesmaximal, characteristic, complemented
14#14338C211yesmaximal, characteristic, complemented
15#14255C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2214yes—
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2849—216no—
25#2617—28no—
26C2×#398C2212yescomplemented
27C2×#397C2212yes—
28C2×#396C2214yescomplemented
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#394C2212yescomplemented
32C2×#392C2212yes—
33C2×#386C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#381C2214yes—
43C2×#380—28no—
44C2×#380C2214yes—
45C2×#380C2214yes—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#379C2212yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1445C2212yescomplemented
52#1444C2212yescomplemented
53#1443C2214yescomplemented
54#1442C2212yescomplemented
55#1442C2214yescomplemented
56#1441C2212yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2212yescomplemented
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yescomplemented
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1439C2214yescomplemented
70#1439C2214yescomplemented
71#1438C2214yes—
72#1438C2214yes—
73#1438C2212yescomplemented
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C26C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C23×D4C2311yescharacteristic
83C24×C4—24no—
84C24×C4—24no—
85C24×C4—44no—
86C24×C4—44no—
87C24×C4D418yescomplemented
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×#123—216no—
96C2×#123—48no—
97C2×#123—48no—
98C2×#121—24no—
99C2×#121—44no—
100C2×#120—28no—
101C2×#120—28no—
102C2×#120—28no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—44no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—44no—
118C22×C22⋊C4—44no—
119C22×C22⋊C4D418yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137#398—44no—
138#397—28no—
139#396—48no—
140#394—28no—
141#394—28no—
142#394—44no—
143#392—28no—
144#386—28no—
145#386—216no—
146#386—28no—
147#386—28no—
148#386—44no—
149#386—44no—
150#385—28no—
151#385—216no—
152#385—28no—
153#381—216no—
154#380—24no—
155#380—216no—
156#380—216no—
157#380—44no—
158#379—28no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25—24no—
183C25—24no—
184C25—42no—
185C25—42no—
186C25—42no—
187C25—42no—
188C25C2×D414yes—
189C25C2×D414yes—
190C25#4114yes—
191C25C2412yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—42no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—42no—
214C22×D4—42no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—232no—
220C23×C4—22no—
221C23×C4—28no—
222C23×C4—28no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—28no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—432no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—416no—
246C23×C4—416no—
247C23×C4—432no—
248C23×C4C2×D414yes—
249C23×C4C2×D4116yes—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2×D414yes—
253C23×C4C2414yes—
254C23×C4C2414yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2412yes—
265#123—464no—
266#121—416no—
267#120—432no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—416no—
288C2×C4⋊C4—416no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—416no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—416no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4C2×D418yes—
344C2×C22⋊C4C2×D418yes—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—24no—
355#95—42no—
356#95—42no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—42no—
363#95—44no—
364#95—42no—
365#95—44no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—28no—
371C24—28no—
372C24—216no—
373C24—216no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24—22no—
379C24—22no—
380C24—24no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—24no—
393C24—48no—
394C24—44no—
395C24—44no—
396C24—48no—
397C24—48no—
398C24—44no—
399C24—44no—
400C24—48no—
401C24—416no—
402C24—416no—
403C24—416no—
404C24—416no—
405C24#121116yes—
406C24C22×D414yes—
407C24C22×D414yes—
408C24C22×D412yes—
409C24C22×D412yes—
410C24C2×#4114yes—
411C24C2×#4112yes—
412C24H(5,𝔽2)14yes—
413C242−1+414yes—
414C24C2511yescharacteristic
415C2×D4—44no—
416C2×D4—416no—
417C2×D4—416no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—88no—
434C2×D4—88no—
435C2×D4—88no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—216no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—216no—
445C22×C4—232no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—232no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—416no—
506C22×C4—48no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—416no—
512C22×C4—416no—
513C22×C4—416no—
514C22×C4—416no—
515C22×C4—4128no—
516C22×C4—44no—
517C22×C4—44no—
518C22×C4—48no—
519C22×C4—44no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—44no—
523C22×C4—44no—
524C22×C4—44no—
525C22×C4—88no—
526C22×C4—88no—
527C22×C4—88no—
528C22×C4—88no—
529C22×C4C22×D414yes—
530C22×C4C22×D414yes—
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D414yes—
534C22×C4H(5,𝔽2)14yes—
535C22×C4H(5,𝔽2)14yes—
536C4⋊C4—416no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—416no—
541C4⋊C4—416no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—88no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—416no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C4—416no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—416no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C22⋊C4—88no—
561C22⋊C4—88no—
562C23—24no—
563C23—24no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—24no—
570C23—28no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—232no—
575C23—24no—
576C23—28no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—28no—
581C23—24no—
582C23—24no—
583C23—24no—
584C23—24no—
585C23—24no—
586C23—24no—
587C23—28no—
588C23—28no—
589C23—416no—
590C23—416no—
591C23—464no—
592C23—416no—
593C23—416no—
594C23—416no—
595C23—416no—
596C23—464no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—44no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—44no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—44no—
626C23—44no—
627C23—44no—
628C23C2×#12118yes—
629C23C2×#12114yes—
630C23#53414yes—
631C23#53614yes—
632C23#53718yes—
633C23#53918yes—
634C23C23×D411yescharacteristic
635C23C23×D411yescharacteristic
636C23C22×#4111yescharacteristic
637C23C2×H(5,𝔽2)12yes—
638C23C2×H(5,𝔽2)12yes—
639C23C2×H(5,𝔽2)12yes—
640C23C2×2−1+412yes—
641C23#58512yes—
642C23#58512yes—
643D4—88no—
644D4—816no—
645D4—816no—
646D4—816no—
647C2×C4—232no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—28no—
659C2×C4—24no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—24no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—464no—
666C2×C4—416no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—44no—
711C2×C4—44no—
712C2×C4—44no—
713C2×C4—44no—
714C2×C4—44no—
715C2×C4—88no—
716C2×C4—88no—
717C2×C4—88no—
718C2×C4—88no—
719C2×C4#53418yes—
720C22—24no—
721C22—216no—
722C22—24no—
723C22—24no—
724C22—24no—
725C22—24no—
726C22—216no—
727C22—24no—
728C22—28no—
729C22—28no—
730C22—28no—
731C22—464no—
732C22—48no—
733C22—48no—
734C22—44no—
735C22—44no—
736C22—44no—
737C22—44no—
738C22—48no—
739C22—48no—
740C22—82no—
741C22—84no—
742C22—84no—
743C22—82no—
744C22—82no—
745C22—82no—
746C22—82no—
747C22—84no—
748C22—84no—
749C22—82no—
750C22—82no—
751C22#261818yes—
752C22C22×#12111yescharacteristic
753C22C2×#53412yes—
754C22C2×#53411yescharacteristic
755C22C2×#53611yescharacteristic
756C22C2×#53712yes—
757C22#344912yes—
758C22C2×#53912yes—
759C22#348411yescharacteristic
760C22#348411yescharacteristic
761C22#348511yescharacteristic
762C22#348511yescharacteristic
763C22#349012yes—
764C22#349112yes—
765C22#354114yes—
766C22#354114yes—
767C22#354214yes—
768C22#354214yes—
769C4—28no—
770C4—44no—
771C4—42no—
772C4—42no—
773C4—44no—
774C4—82no—
775C4—84no—
776C4—84no—
777C4—82no—
778C4—82no—
779C4—82no—
780C4—82no—
781C2—28no—
782C2—48no—
783C2—162no—
784C2C2×#261811yescharacteristic
785C2#3498111yescharacteristic
786C2#3508911yescharacteristic
787C2#3670411yescharacteristic
788C2#3670511yescharacteristic
789C2#3671511yescharacteristic
790C2#3671611yescharacteristic
791C2#3672614yes—
792C2#3744414yes—
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×#2849C2semidirect
3. C22×#380C2semidirect
4. #16397C2semidirect
5. #16396C2semidirect
6. #16257C2semidirect
7. #16256C2semidirect
8. #15014C2semidirect
9. #14921C2semidirect
10. #14913C2semidirect
11. #14821C2semidirect
12. #14729C2semidirect
13. #14719C2semidirect
14. #14338C2semidirect
15. #14255C2semidirect
26. C2×#398C22semidirect
28. C2×#396C22semidirect
31. C2×#394C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
64. #1440C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
69. #1439C22semidirect
70. #1439C22semidirect
73. #1438C22semidirect
87. C24×C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
C2×#26172
C22×#3802
#163972
#163962
#162572
#162562
#150142
#149212
#149132
#148212
#147292
#147192
#143382
#142552