Finite Group Databasebeta

← #4328655

Subgroups — #4328655

Order 512.

3566 conjugacy classes of subgroups.

The subgroup lattice has 788 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic, complemented
3#16698C212yesmaximal, complemented
4#16635C214yesmaximal, complemented
5#16395C212yesmaximal, complemented
6#16014C212yesmaximal, complemented
7#15375C214yesmaximal, complemented
8#15266C212yesmaximal, complemented
9#14888C212yesmaximal, complemented
10#14820C212yesmaximal, complemented
11#14776C212yesmaximal, complemented
12#14735C212yesmaximal, complemented
13#14513C212yesmaximal, complemented
14#14357C212yesmaximal, complemented
15#13956C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×#120C2211yescharacteristic
19C22×#117C2211yescharacteristic, complemented
20C23×C4⋊C4C2211yescharacteristic, complemented
21C23×C22⋊C4C2211yescharacteristic
22#2404—28no—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#394C2214yes—
26C2×#393C2212yes—
27C2×#392C2212yes—
28#2311—28no—
29C2×#391C2212yescomplemented
30C2×#390C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2211yescharacteristic
35C2×#386C2212yes—
36C2×#386C2211yescharacteristic
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#386C2212yescomplemented
40C2×#386C2212yescomplemented
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic, complemented
43C2×#380C2212yes—
44C2×#380C2212yescomplemented
45C2×#379C2214yes—
46C22×#95C2211yescharacteristic, complemented
47#1446C2212yescomplemented
48#1445C2212yescomplemented
49#1445C2214yescomplemented
50#1445C2212yescomplemented
51#1444C2212yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1443C2214yescomplemented
55#1442C2212yescomplemented
56#1442C2214yescomplemented
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yescomplemented
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yescomplemented
68#1438C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yescomplemented
75C22⋊#95C2214yescomplemented
76C22⋊#95C2212yescomplemented
77C26C2311yescharacteristic
78C22×#41—28no—
79C22×#41—42no—
80C22×#41—42no—
81C22×#41—44no—
82C22×#41D418yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94C24×C4C2312yes—
95C2×#120—24no—
96C2×#120—24no—
97C2×#120—28no—
98C2×#120—44no—
99C2×#117—24no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—42no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—42no—
110C22×C22⋊C4—42no—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130#397—44no—
131#397—44no—
132#394—48no—
133#393—28no—
134#392—28no—
135#391—28no—
136#390—28no—
137#386—28no—
138#386—28no—
139#386—24no—
140#386—24no—
141#386—28no—
142#386—28no—
143#386—28no—
144#386—48no—
145#386—48no—
146#386—44no—
147#381—24no—
148#381—42no—
149#380—28no—
150#380—28no—
151#379—48no—
152C2×#95—42no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C25—22no—
177C25—24no—
178C25—24no—
179C25—41no—
180C25—41no—
181C25—42no—
182C25—42no—
183C25—44no—
184C25C2412yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C2×#41—232no—
189C2×#41—416no—
190C2×#41—416no—
191C2×#41—432no—
192C2×#41—432no—
193C22×Q8—24no—
194C22×Q8—42no—
195C22×Q8—41no—
196C22×Q8—41no—
197C22×Q8C2×D414yes—
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—28no—
207C22×D4—28no—
208C22×D4—42no—
209C22×D4—41no—
210C22×D4—41no—
211C22×D4—41no—
212C22×D4—41no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—42no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—42no—
219C22×D4C2×D414yes—
220C22×D4C2×D414yes—
221C22×D4C2×D414yes—
222C22×D4#4118yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—41no—
239C23×C4—41no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—44no—
243C23×C4—42no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—42no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2414yes—
253C23×C4C2414yes—
254C23×C4C2414yes—
255C23×C4C2412yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263#120—416no—
264#120—416no—
265#117—416no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42—42no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—22no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—48no—
381C24—44no—
382C24—416no—
383C24—44no—
384C24—48no—
385C24—44no—
386C24—44no—
387C24—44no—
388C24—84no—
389C24—84no—
390C24—84no—
391C24—88no—
392C24—88no—
393C24—84no—
394C24—84no—
395C24—88no—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C22×D412yes—
399C24C2×#4114yes—
400C24C2×#4114yes—
401C24H(5,𝔽2)14yes—
402C24H(5,𝔽2)14yes—
403C24C2511yescharacteristic
404#41—4128no—
405C2×Q8—28no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—48no—
409C2×Q8—48no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
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—416no—
427C2×D4—416no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—88no—
433C2×D4—88no—
434C2×D4#123116yes—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—28no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—44no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—88no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—88no—
513C22×C4—88no—
514C22×C4—84no—
515C22×C4—84no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4—88no—
519C22×C4—84no—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C2×#4114yes—
525C22×C4H(5,𝔽2)12yes—
526C22×C4H(5,𝔽2)12yes—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—88no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—88no—
544C22⋊C4—88no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C42—48no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—28no—
569C23—28no—
570C23—28no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—24no—
575C23—28no—
576C23—28no—
577C23—24no—
578C23—24no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—41no—
588C23—42no—
589C23—42no—
590C23—41no—
591C23—41no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—44no—
599C23—44no—
600C23—416no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—44no—
606C23—44no—
607C23—42no—
608C23—42no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—42no—
614C23—44no—
615C23—42no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—816no—
620C23—88no—
621C23—816no—
622C23—832no—
623C23—816no—
624C23—88no—
625C23—816no—
626C23C2×#12314yes—
627C23C2×#12314yes—
628C23#53414yes—
629C23C23×D411yescharacteristic
630C23C22×#4112yes—
631C23C2×H(5,𝔽2)12yes—
632C23C2×H(5,𝔽2)11yescharacteristic
633C23C2×H(5,𝔽2)11yescharacteristic
634C23C2×H(5,𝔽2)12yes—
635C23#58512yes—
636C23#58514yes—
637Q8—416no—
638D4—416no—
639D4—416no—
640D4—416no—
641D4—816no—
642D4—816no—
643C2×C4—232no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—416no—
659C2×C4—416no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
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—88no—
708C2×C4—816no—
709C2×C4—816no—
710C2×C4C2×#12314yes—
711C2×C4#53414yes—
712C22—28no—
713C22—28no—
714C22—24no—
715C22—24no—
716C22—24no—
717C22—28no—
718C22—48no—
719C22—48no—
720C22—48no—
721C22—42no—
722C22—42no—
723C22—42no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—42no—
732C22—42no—
733C22—42no—
734C22—44no—
735C22—44no—
736C22—48no—
737C22—48no—
738C22—44no—
739C22—44no—
740C22—44no—
741C22—832no—
742C22—84no—
743C22—82no—
744C22—84no—
745C22—82no—
746C22—82no—
747C22—82no—
748C22C22×#12311yescharacteristic
749C22C2×#53411yescharacteristic
750C22C2×#53411yescharacteristic
751C22#348812yes—
752C22#349012yes—
753C22#349012yes—
754C22#350012yes—
755C22#351314yes—
756C22#352114yes—
757C22#353414yes—
758C22#353711yescharacteristic
759C22#354012yes—
760C22#354011yescharacteristic
761C22#354212yes—
762C22#355112yes—
763C22#356014yes—
764C4—24no—
765C4—24no—
766C4—42no—
767C4—42no—
768C4—84no—
769C4—82no—
770C4—84no—
771C4—82no—
772C4—84no—
773C4—84no—
774C4—82no—
775C2—44no—
776C2—44no—
777C2—44no—
778C2—84no—
779C2#3499211yescharacteristic
780C2#3597612yes—
781C2#3726911yescharacteristic
782C2#3741911yescharacteristic
783C2#4193612yes—
784C2#4660414yes—
785C2#5180312yes—
786C2#5181512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2311C2semidirect
3. #16698C2semidirect
4. #16635C2semidirect
5. #16395C2semidirect
6. #16014C2semidirect
7. #15375C2semidirect
8. #15266C2semidirect
9. #14888C2semidirect
10. #14820C2semidirect
11. #14776C2semidirect
12. #14735C2semidirect
13. #14513C2semidirect
14. #14357C2semidirect
15. #13956C2semidirect
19. C22×#117C22semidirect
20. C23×C4⋊C4C22semidirect
29. C2×#391C22semidirect
39. C2×#386C22semidirect
40. C2×#386C22semidirect
42. C2×#381C22semidirect
44. C2×#380C22semidirect
46. C22×#95C22semidirect
47. #1446C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1443C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
61. #1440C22semidirect
67. #1439C22semidirect
71. #1436C22semidirect
74. C22⋊#95C22semidirect
75. C22⋊#95C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
C2×#23112
#166982
#166352
#163952
#160142
#153752
#152662
#148882
#148202
#147762
#147352
#145132
#143572
#139562