Finite Group Databasebeta

← #4452847

Subgroups — #4452847

Order 512.

3492 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2617C211yesmaximal, characteristic
2C22×#396C211yesmaximal, characteristic, complemented
3#16832C214yesmaximal, complemented
4#16370C214yesmaximal, complemented
5#16367C214yesmaximal, complemented
6#16340C214yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#15015C211yesmaximal, characteristic, complemented
9#14913C211yesmaximal, characteristic, complemented
10#14822C211yesmaximal, characteristic, complemented
11#14776C212yesmaximal, complemented
12#14719C211yesmaximal, characteristic, complemented
13#14134C212yesmaximal, complemented
14#14134C212yesmaximal, complemented
15#13794C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#120C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2617—28no—
26C2×#400C2212yescomplemented
27C2×#397C2212yes—
28C2×#397C2214yescomplemented
29C2×#396—28no—
30C2×#396C2214yes—
31C2×#396C2214yes—
32C2×#394C2212yes—
33C2×#394C2212yes—
34C2×#394C2212yescomplemented
35C2×#392C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#380C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1445C2212yescomplemented
46#1445C2214yescomplemented
47#1443C2214yescomplemented
48#1442C2212yescomplemented
49#1442C2212yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2212yescomplemented
55#1441C2212yescomplemented
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1438C2214yes—
64#1438C2214yes—
65#1438C2212yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2314yes—
89C2×#121—24no—
90C2×#121—44no—
91C2×#120—28no—
92C2×#120—28no—
93C2×#120—28no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4D418yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133#400—44no—
134#397—28no—
135#397—48no—
136#396—216no—
137#396—216no—
138#394—28no—
139#394—28no—
140#394—44no—
141#392—28no—
142#386—28no—
143#386—48no—
144#386—48no—
145#386—44no—
146#386—44no—
147#386—44no—
148#386—44no—
149#380—24no—
150C2×#95—24no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25—24no—
175C25—22no—
176C25—22no—
177C25—42no—
178C25—42no—
179C25—42no—
180C25—42no—
181C25C2×D414yes—
182C25#4118yes—
183C25C2412yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4—42no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2412yes—
252#121—416no—
253#120—432no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—416no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—416no—
312C2×C22⋊C4—416no—
313C2×C22⋊C4—432no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D418yes—
327C2×C22⋊C4C2×D418yes—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—44no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—42no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—28no—
364C24—28no—
365C24—216no—
366C24—216no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—416no—
383C24—416no—
384C24—88no—
385C24—84no—
386C24—84no—
387C24—88no—
388C24—84no—
389C24—84no—
390C24#124116yes—
391C24C22×D414yes—
392C24C22×D412yes—
393C24C2×#4114yes—
394C24C2×#4114yes—
395C24C2×#4114yes—
396C24H(5,𝔽2)14yes—
397C24H(5,𝔽2)12yes—
398C242−1+412yes—
399C24C2511yescharacteristic
400C2×D4—44no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—88no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—416no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—416no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—416no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—88no—
506C22×C4—88no—
507C22×C4—88no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—88no—
511C22×C4—88no—
512C22×C4—88no—
513C22×C4—88no—
514C22×C4—88no—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D414yes—
518C22×C4H(5,𝔽2)14yes—
519C22×C4H(5,𝔽2)14yes—
520C4⋊C4—416no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—416no—
524C4⋊C4—44no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—416no—
530C22⋊C4—416no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—24no—
552C23—24no—
553C23—28no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—28no—
570C23—232no—
571C23—416no—
572C23—416no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—88no—
605C23—88no—
606C23—88no—
607C23—832no—
608C23—832no—
609C23—88no—
610C23C2×#12418yes—
611C23C2×#12414yes—
612C23#53414yes—
613C23#53714yes—
614C23#53818yes—
615C23#53918yes—
616C23C23×D411yescharacteristic
617C23C22×#4112yes—
618C23C2×H(5,𝔽2)12yes—
619C23C2×H(5,𝔽2)11yescharacteristic
620C23C2×H(5,𝔽2)12yes—
621C23C2×H(5,𝔽2)12yes—
622C23C2×2−1+411yescharacteristic
623C23#58512yes—
624C23#58512yes—
625D4—88no—
626D4—816no—
627D4—816no—
628D4—816no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—416no—
644C2×C4—416no—
645C2×C4—416no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
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—88no—
699C2×C4—88no—
700C2×C4—88no—
701C2×C4—832no—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—24no—
706C22—24no—
707C22—216no—
708C22—216no—
709C22—24no—
710C22—28no—
711C22—28no—
712C22—28no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—48no—
717C22—48no—
718C22—48no—
719C22—48no—
720C22—44no—
721C22—832no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—84no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—82no—
733C22#263918yes—
734C22C22×#12411yescharacteristic
735C22C2×#53411yescharacteristic
736C22C2×#53711yescharacteristic
737C22C2×#53812yes—
738C22C2×#53912yes—
739C22#348411yescharacteristic
740C22#348411yescharacteristic
741C22#348511yescharacteristic
742C22#348511yescharacteristic
743C22#350514yes—
744C22#350514yes—
745C22#353414yes—
746C22#353612yes—
747C22#353712yes—
748C22#354214yes—
749C22#356012yes—
750C22#356412yes—
751C4—44no—
752C4—44no—
753C4—44no—
754C4—42no—
755C4—42no—
756C4—82no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4—84no—
762C4—84no—
763C2—28no—
764C2—48no—
765C2—162no—
766C2C2×#263911yescharacteristic
767C2#3537711yescharacteristic
768C2#3537711yescharacteristic
769C2#3670411yescharacteristic
770C2#3670611yescharacteristic
771C2#3696011yescharacteristic
772C2#3711711yescharacteristic
773C2#4065914yes—
774C2#4066114yes—
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
2. C22×#396C2semidirect
3. #16832C2semidirect
4. #16370C2semidirect
5. #16367C2semidirect
6. #16340C2semidirect
7. #15091C2semidirect
8. #15015C2semidirect
9. #14913C2semidirect
10. #14822C2semidirect
11. #14776C2semidirect
12. #14719C2semidirect
13. #14134C2semidirect
14. #14134C2semidirect
15. #13794C2semidirect
26. C2×#400C22semidirect
28. C2×#397C22semidirect
34. C2×#394C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
58. #1440C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
65. #1438C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26172
C22×#3962
#168322
#163702
#163672
#163402
#150912
#150152
#149132
#148222
#147762
#147192
#141342
#137942