Finite Group Databasebeta

← #4432447

Subgroups — #4432447

Order 512.

2976 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic, complemented
2C2×#2562C211yesmaximal, characteristic, complemented
3C2×#2631C211yesmaximal, characteristic
4C2×#2349C211yesmaximal, characteristic, complemented
5#16595C214yesmaximal, complemented
6#16401C212yesmaximal, complemented
7#16398C212yesmaximal, complemented
8#15426C214yesmaximal, complemented
9#15403C214yesmaximal, complemented
10#14836C212yesmaximal, complemented
11#14735C212yesmaximal, complemented
12#14732C214yesmaximal, complemented
13#14296C211yesmaximal, characteristic, complemented
14#14289C211yesmaximal, characteristic, complemented
15#14153C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20#2820—28no—
21#2631—28no—
22#2562—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#393C2212yes—
33C2×#392C2212yes—
34C2×#392C2212yes—
35#2349—28no—
36C2×#391C2212yes—
37C2×#390C2212yes—
38C2×#389C2212yes—
39C2×#388C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#383C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2214yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C22×#95C2212yes—
54#1445C2211yescharacteristic
55#1445C2211yescharacteristic
56#1445C2214yes—
57#1445C2211yescharacteristic
58#1445C2211yescharacteristic
59#1444C2212yes—
60#1443C2214yes—
61#1442C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1438C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2312yes—
83C23×D4C2312yes—
84C24×C4C2314yes—
85C24×C4C2312yes—
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#124—24no—
90C2×#124—24no—
91C2×#124—24no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—48no—
96C2×C4×D4D418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#400—28no—
126#399—28no—
127#397—28no—
128#397—28no—
129#396—216no—
130#395—28no—
131#394—28no—
132#394—28no—
133#394—48no—
134#393—28no—
135#392—28no—
136#392—28no—
137#391—44no—
138#390—44no—
139#389—28no—
140#388—216no—
141#386—28no—
142#386—28no—
143#386—48no—
144#386—48no—
145#385—216no—
146#383—24no—
147#382D418yescomplemented
148#380—28no—
149#380—28no—
150#380—28no—
151#380—48no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C2×#95—28no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2412yes—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—42no—
185C22×Q8—42no—
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—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4C2×D414yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C2×D418yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#125—416no—
232#124—416no—
233C4×D4—232no—
234C4×D4—432no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D418yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—44no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42#4114yes—
312#95—22no—
313#95—28no—
314#95—22no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95C2×D418yes—
335#95C2×D418yes—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24C22×D414yes—
356C24H(5,𝔽2)14yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—48no—
360C2×Q8—44no—
361C2×Q8—84no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—216no—
367C2×D4—216no—
368C2×D4—28no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—84no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—88no—
383C2×D4—88no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4C22×D414yes—
453C22×C4C22×D418yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)14yes—
464C4⋊C4—28no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C22⋊C4—216no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—88no—
489C42—48no—
490C42—84no—
491C42#121116yes—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23#53718yes—
538C23C23×D411yescharacteristic
539C23C23×D411yescharacteristic
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)12yes—
543C23C2×2−1+412yes—
544C23#58512yes—
545C23#58514yes—
546Q8—88no—
547D4—232no—
548D4—88no—
549D4—816no—
550D4—816no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—816no—
617C2×C4—832no—
618C2×C4C2×#12114yes—
619C2×C4C2×#12114yes—
620C2×C4C2×#12114yes—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C22×#12111yescharacteristic
644C22C2×#53712yes—
645C22#346012yes—
646C22#348411yescharacteristic
647C22#348512yes—
648C22#348611yescharacteristic
649C22#348812yes—
650C22#349212yes—
651C22#349811yescharacteristic
652C22#350012yes—
653C22#350011yescharacteristic
654C22#351012yes—
655C22#353314yes—
656C22#353414yes—
657C22#354114yes—
658C22#356414yes—
659C4—24no—
660C4—24no—
661C4—24no—
662C4—44no—
663C4—48no—
664C4—44no—
665C4—82no—
666C4—82no—
667C4—84no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C2—48no—
673C2—162no—
674C2#3522311yescharacteristic
675C2#3525611yescharacteristic
676C2#3561711yescharacteristic
677C2#4111014yes—
678C2#4136812yes—
679C2#4136912yes—
680C2#5107112yes—
681C2#5107212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2631C2
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×#2820C2semidirect
2. C2×#2562C2semidirect
4. C2×#2349C2semidirect
5. #16595C2semidirect
6. #16401C2semidirect
7. #16398C2semidirect
8. #15426C2semidirect
9. #15403C2semidirect
10. #14836C2semidirect
11. #14735C2semidirect
12. #14732C2semidirect
13. #14296C2semidirect
14. #14289C2semidirect
15. #14153C2semidirect
147. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#25622
C2×#26312
C2×#23492
#165952
#164012
#163982
#154262
#154032
#148362
#147352
#147322
#142962
#142892
#141532