Finite Group Databasebeta

← #5955421

Subgroups — #5955421

Order 512.

4634 conjugacy classes of subgroups.

The subgroup lattice has 787 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2325C211yesmaximal, characteristic, complemented
2C2×C4×#120C212yesmaximal, complemented
3C22×#381C211yesmaximal, characteristic, complemented
4#15014C211yesmaximal, characteristic, complemented
5#14974C214yesmaximal, complemented
6#14965C214yesmaximal, complemented
7#14921C212yesmaximal, complemented
8#14913C211yesmaximal, characteristic, complemented
9#14820C211yesmaximal, characteristic, complemented
10#14731C214yesmaximal, complemented
11#14730C214yesmaximal, complemented
12#14725C212yesmaximal, complemented
13#14716C211yesmaximal, characteristic
14#13784C212yesmaximal, complemented
15#13778C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2214yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#392C2212yes—
26#2325—28no—
27C4×#120—216no—
28C2×#388C2212yescomplemented
29C2×#387C2212yescomplemented
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2214yes—
35C2×#386C2212yescomplemented
36C2×#385C2212yes—
37C2×#385C2214yes—
38C2×#385C2212yescomplemented
39C2×#381—28no—
40C2×#381C2214yes—
41C2×#381C2214yes—
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1445C2212yescomplemented
52#1444C2212yescomplemented
53#1443C2214yescomplemented
54#1442C2212yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yescomplemented
60#1439C2212yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1438C2214yes—
66#1438C2214yes—
67#1438C2212yescomplemented
68#1437C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71#1436C2214yescomplemented
72#1434C2214yescomplemented
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C22⋊#95C2214yescomplemented
78C26C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2314yes—
81C23×D4C2311yescharacteristic
82C24×C4—24no—
83C24×C4—24no—
84C24×C4—44no—
85C24×C4—44no—
86C24×C4D418yescomplemented
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×#121—28no—
96C2×#121—24no—
97C2×#120—28no—
98C2×#120—28no—
99C2×#120—28no—
100C2×C4×D4—28no—
101C2×C4×D4—216no—
102C2×C4×D4—28no—
103C2×C4×D4—28no—
104C2×C4×D4—48no—
105C2×C4×D4—48no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4—44no—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2312yes—
133C22×C42C2314yes—
134#397—44no—
135#394—48no—
136#392—44no—
137#388—28no—
138#387—28no—
139#386—28no—
140#386—28no—
141#386—216no—
142#386—28no—
143#386—28no—
144#386—28no—
145#385—216no—
146#385—28no—
147#385—28no—
148#381—216no—
149#381—216no—
150#380—24no—
151#380—28no—
152#379—28no—
153C4×C22⋊C4—216no—
154C4×C22⋊C4—28no—
155C2×#95—24no—
156C2×#95—24no—
157C2×#95—24no—
158C2×#95D418yescomplemented
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25—22no—
183C25—22no—
184C25—24no—
185C25—24no—
186C25—24no—
187C25—24no—
188C25—42no—
189C25—42no—
190C25C2×D414yes—
191C25C2412yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×D4—28no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—28no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—28no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4#4118yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—232no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—432no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—416no—
246C23×C4—416no—
247C23×C4—432no—
248C23×C4C2×D4116yes—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2×D414yes—
253C23×C4C2×D414yes—
254C23×C4#4114yes—
255C23×C4C2412yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C23×C4C2412yes—
265C23×C4C2414yes—
266C23×C4C2412yes—
267#121—232no—
268#120—264no—
269C4×D4—464no—
270C4×D4—432no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—416no—
326C2×C22⋊C4—416no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4#4118yes—
334C2×C42—28no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42—44no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—44no—
354#95—42no—
355#95—42no—
356#95—44no—
357#95—42no—
358#95—44no—
359#95—42no—
360#95—44no—
361#95C2×D418yes—
362#95C2×D418yes—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—28no—
368C24—28no—
369C24—216no—
370C24—28no—
371C24—216no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—216no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—48no—
392C24—416no—
393C24—416no—
394C24—416no—
395C24—416no—
396C24—44no—
397C24—44no—
398C24—48no—
399C24—44no—
400C24—48no—
401C24—44no—
402C24C22×D414yes—
403C24C22×D412yes—
404C24C22×D412yes—
405C24C2×#4114yes—
406C24C2×#4112yes—
407C24C2×#4114yes—
408C24C2×#4114yes—
409C24H(5,𝔽2)14yes—
410C24H(5,𝔽2)14yes—
411C24C2511yescharacteristic
412C2×D4—232no—
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—216no—
416C2×D4—28no—
417C2×D4—416no—
418C2×D4—416no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—416no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—88no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—232no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—232no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—48no—
491C22×C4—416no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—416no—
501C22×C4—416no—
502C22×C4—416no—
503C22×C4—416no—
504C22×C4—416no—
505C22×C4—4128no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—48no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—48no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4—88no—
519C22×C4—88no—
520C22×C4—88no—
521C22×C4—88no—
522C22×C4—88no—
523C22×C4#121116yes—
524C22×C4C22×D414yes—
525C22×C4C22×D412yes—
526C22×C4C22×D414yes—
527C22×C4C22×D412yes—
528C22×C4C22×D414yes—
529C22×C4C22×D414yes—
530C22×C4C2×#4114yes—
531C22×C4C2×#4118yes—
532C22×C4C2×#4114yes—
533C4⋊C4—28no—
534C4⋊C4—416no—
535C4⋊C4—48no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—88no—
539C22⋊C4—232no—
540C22⋊C4—216no—
541C22⋊C4—216no—
542C22⋊C4—416no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—416no—
546C22⋊C4—416no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—416no—
550C22⋊C4—48no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—88no—
554C22⋊C4—88no—
555C42—416no—
556C42—48no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—24no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—232no—
570C23—24no—
571C23—28no—
572C23—24no—
573C23—28no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—28no—
579C23—28no—
580C23—28no—
581C23—28no—
582C23—24no—
583C23—24no—
584C23—24no—
585C23—28no—
586C23—28no—
587C23—464no—
588C23—416no—
589C23—416no—
590C23—464no—
591C23—416no—
592C23—416no—
593C23—416no—
594C23—416no—
595C23—42no—
596C23—44no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—44no—
610C23—44no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23C2×#12114yes—
624C23#52518yes—
625C23#52514yes—
626C23#52918yes—
627C23#53214yes—
628C23#53418yes—
629C23C23×D411yescharacteristic
630C23C23×D411yescharacteristic
631C23C22×#4111yescharacteristic
632C23C22×#4112yes—
633C23C22×#4112yes—
634C23C2×H(5,𝔽2)12yes—
635C23C2×H(5,𝔽2)12yes—
636C23#58512yes—
637C23#58512yes—
638D4—464no—
639D4—432no—
640D4—88no—
641C2×C4—28no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—28no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—28no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—28no—
661C2×C4—28no—
662C2×C4—416no—
663C2×C4—464no—
664C2×C4—416no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—816no—
708C2×C4—88no—
709C2×C4—88no—
710C2×C4—88no—
711C2×C4—88no—
712C2×C4C2×#12118yes—
713C22—24no—
714C22—24no—
715C22—216no—
716C22—24no—
717C22—24no—
718C22—24no—
719C22—24no—
720C22—216no—
721C22—28no—
722C22—216no—
723C22—464no—
724C22—44no—
725C22—44no—
726C22—48no—
727C22—48no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—44no—
732C22—48no—
733C22—44no—
734C22—44no—
735C22—48no—
736C22—44no—
737C22—48no—
738C22—84no—
739C22—82no—
740C22—82no—
741C22—82no—
742C22—82no—
743C22—84no—
744C22#239218yes—
745C22C22×#12111yescharacteristic
746C22C2×#52511yescharacteristic
747C22C2×#52512yes—
748C22C2×#52912yes—
749C22C2×#53211yescharacteristic
750C22C2×#53412yes—
751C22#344912yes—
752C22#348411yescharacteristic
753C22#348411yescharacteristic
754C22#348511yescharacteristic
755C22#348711yescharacteristic
756C22#349012yes—
757C22#349312yes—
758C22#351314yes—
759C22#351314yes—
760C22#351314yes—
761C22#351314yes—
762C4—28no—
763C4—42no—
764C4—42no—
765C4—44no—
766C4—44no—
767C4—44no—
768C4—48no—
769C4—82no—
770C4—82no—
771C4—82no—
772C4—82no—
773C4—84no—
774C2—28no—
775C2—48no—
776C2—84no—
777C2C2×#239211yescharacteristic
778C2#3498111yescharacteristic
779C2#3506314yes—
780C2#3515011yescharacteristic
781C2#3535411yescharacteristic
782C2#3535611yescharacteristic
783C2#3537314yes—
784C2#3539411yescharacteristic
785C2#3539611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14716C2
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×#2325C2semidirect
2. C2×C4×#120C2semidirect
3. C22×#381C2semidirect
4. #15014C2semidirect
5. #14974C2semidirect
6. #14965C2semidirect
7. #14921C2semidirect
8. #14913C2semidirect
9. #14820C2semidirect
10. #14731C2semidirect
11. #14730C2semidirect
12. #14725C2semidirect
14. #13784C2semidirect
15. #13778C2semidirect
28. C2×#388C22semidirect
29. C2×#387C22semidirect
35. C2×#386C22semidirect
38. C2×#385C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
59. #1440C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
67. #1438C22semidirect
68. #1437C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect
72. #1434C22semidirect
77. C22⋊#95C22semidirect
86. C24×C4D4semidirect
158. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23252
C2×C4×#1202
C22×#3812
#150142
#149742
#149652
#149212
#149132
#148202
#147312
#147302
#147252
#147162
#137842
#137782