Finite Group Databasebeta

← #722886

Subgroups — #722886

Order 512.

3758 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic, complemented
2C2×#2626C211yesmaximal, characteristic
3C22×#393C211yesmaximal, characteristic
4C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
5#16628C214yesmaximal, complemented
6#15499C214yesmaximal, complemented
7#15322C214yesmaximal, complemented
8#14920C212yesmaximal, complemented
9#14877C214yesmaximal, complemented
10#14726C212yesmaximal
11#14159C211yesmaximal, characteristic, complemented
12#14149C211yesmaximal, characteristic, complemented
13#14044C212yesmaximal, complemented
14#13971C212yesmaximal
15#13775C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yescomplemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yescomplemented
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22#2626—28no—
23#2434—28no—
24C2×#399C2212yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#395C2212yescomplemented
28C2×#395C2212yescomplemented
29C2×#393—28no—
30C2×#393C2212yes—
31C2×#393C2214yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34Q8×C22⋊C4—28no—
35C2×#391C2214yescomplemented
36C2×#391C2212yescomplemented
37C2×#391C2212yescomplemented
38C2×#388C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×#379C2212yes—
47C2×#379C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2211yescharacteristic
52C22×#95C2212yes—
53#1446C2212yescomplemented
54#1445C2214yescomplemented
55#1445C2212yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yescomplemented
71#1436C2214yescomplemented
72#1436C2214yescomplemented
73C4⋊#95C2214yescomplemented
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C26C2311yescharacteristic
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—48no—
83C23×Q8D418yes—
84C23×Q8C2311yescharacteristic
85C23×Q8C2312yes—
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—48no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×Q8—48no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—48no—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131#399—28no—
132#396—216no—
133#396—48no—
134#395—28no—
135#395—28no—
136#393—28no—
137#393—216no—
138#393—28no—
139#393—28no—
140#391—28no—
141#391—28no—
142#391—216no—
143#388—48no—
144#385—28no—
145#385—28no—
146#385—28no—
147#385—28no—
148#380—28no—
149#380D418yes—
150#379—28no—
151#379—24no—
152#379Q818yescomplemented
153C4×C22⋊C4—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—28no—
156C2×#95—28no—
157C2×#95—24no—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C25—22no—
182C25—22no—
183C25—22no—
184C25—22no—
185C25—22no—
186C25—22no—
187C25—44no—
188C25—44no—
189C25C2×Q814yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×Q8—232no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—28no—
203C22×Q8—28no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—432no—
208C22×Q8—432no—
209C22×Q8—432no—
210C22×Q8C2×D414yes—
211C22×Q8C2×D418yes—
212C22×Q8C2×D414yes—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D418yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C4⋊Q8—432no—
245C22⋊Q8—416no—
246C4×Q8—432no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D418yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
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—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—432no—
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⋊C4C2×D418yes—
312C2×C22⋊C4C2×Q814yes—
313C2×C22⋊C4C2×Q814yes—
314C2×C22⋊C4C2×Q8116yescomplemented
315C2×C22⋊C4#4114yes—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D418yes—
346C24—24no—
347C24—24no—
348C24—28no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—28no—
360C24—28no—
361C24—416no—
362C24—48no—
363C24—48no—
364C24—416no—
365C24—44no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24—84no—
370C24—84no—
371C24—84no—
372C24—84no—
373C24—88no—
374C24—88no—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×Q814yes—
378C24C22×Q812yes—
379C24C2×#4112yes—
380C24C2×#4112yes—
381C242−1+412yes—
382C242−1+412yes—
383C24C2511yescharacteristic
384C2×Q8—216no—
385C2×Q8—216no—
386C2×Q8—28no—
387C2×Q8—216no—
388C2×Q8—28no—
389C2×Q8—216no—
390C2×Q8—216no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—416no—
394C2×Q8—416no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—4128no—
399C2×Q8—44no—
400C2×Q8—84no—
401C2×Q8—84no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—216no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4#121116yes—
474C22×C4C22×D414yes—
475C22×C4C22×D418yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q814yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)18yes—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C22⋊C4—216no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—416no—
512C42—48no—
513C42—48no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—432no—
530C23—42no—
531C23—44no—
532C23—416no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—816no—
550C23—816no—
551C23—88no—
552C23—816no—
553C23—816no—
554C23—88no—
555C23—816no—
556C23C2×#12114yes—
557C23#53314yes—
558C23#54114yes—
559C23#54714yes—
560C23#54814yes—
561C23D4×Q814yes—
562C23D4×Q814yes—
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C23×Q811yescharacteristic
566C23C22×#4111yescharacteristic
567C23C22×#4111yescharacteristic
568C23C2×H(5,𝔽2)14yes—
569C23C2×2−1+411yescharacteristic
570C23C2×2−1+411yescharacteristic
571C23#58514yes—
572Q8—232no—
573Q8—432no—
574Q8—416no—
575Q8—88no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—416no—
599C2×C4—432no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—88no—
643C2×C4—816no—
644C2×C4C2×#12114yes—
645C2×C4C2×#12114yes—
646C2×C4#53418yes—
647C22—24no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—28no—
653C22—28no—
654C22—28no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22—832no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22—82no—
666C22—84no—
667C22#242918yes—
668C22C22×#12111yescharacteristic
669C22C2×#53311yescharacteristic
670C22C2×#53412yes—
671C22C2×#54111yescharacteristic
672C22C2×#54711yescharacteristic
673C22C2×#54811yescharacteristic
674C22C2×D4×Q811yescharacteristic
675C22C2×D4×Q811yescharacteristic
676C22#348412yes—
677C22#348712yes—
678C22#349012yes—
679C22#350012yes—
680C22#350114yes—
681C22#351012yes—
682C22#351414yes—
683C22#354114yes—
684C22#356114yes—
685C4—24no—
686C4—24no—
687C4—28no—
688C4—48no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—84no—
698C2—28no—
699C2—84no—
700C2C2×#242911yescharacteristic
701C2#3498111yescharacteristic
702C2#3567711yescharacteristic
703C2#3570512yes—
704C2#3912312yes—
705C2#3959512yes—
706C2#4054112yes—
707C2#4300312yes—
708C2#4389212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2434C2semidirect
4. C2×Q8×C22⋊C4C2semidirect
5. #16628C2semidirect
6. #15499C2semidirect
7. #15322C2semidirect
8. #14920C2semidirect
9. #14877C2semidirect
11. #14159C2semidirect
12. #14149C2semidirect
13. #14044C2semidirect
17. C22×C4⋊Q8C22semidirect
19. C22×C4×Q8C22semidirect
27. C2×#395C22semidirect
28. C2×#395C22semidirect
35. C2×#391C22semidirect
36. C2×#391C22semidirect
37. C2×#391C22semidirect
53. #1446C22semidirect
54. #1445C22semidirect
55. #1445C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect
72. #1436C22semidirect
73. C4⋊#95C22semidirect
152. #379Q8semidirect
314. C2×C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#26262
C22×#3932
C2×Q8×C22⋊C42
#166282
#154992
#153222
#149202
#148772
#147262
#141592
#141492
#140442
#139712
#137752