Finite Group Databasebeta

← #737662

Subgroups — #737662

Order 512.

4324 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#2545C211yesmaximal, characteristic, complemented
3C2×#2540C211yesmaximal, characteristic
4C2×#2640C211yesmaximal, characteristic, complemented
5#16682C214yesmaximal, complemented
6#15280C218yesmaximal, complemented
7#15215C212yesmaximal, complemented
8#15178C212yesmaximal, complemented
9#14795C214yesmaximal, complemented
10#14155C211yesmaximal, characteristic, complemented
11#14139C212yesmaximal, complemented
12#14137C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic, complemented
14#13786C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—28no—
17C2×#554—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2212yes—
21C22×#124C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2214yes—
25C22×C22⋊Q8C2212yes—
26C22×#121C2212yes—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2212yes—
29C22×C4×D4C2211yescharacteristic
30C23×C4⋊C4C2212yes—
31#2640—28no—
32#2545—28no—
33#2540—28no—
34C2×#399C2212yes—
35C2×#397C2214yes—
36C2×#396C2214yes—
37C2×#393C2214yes—
38C2×#392C2212yes—
39C2×#391C2212yes—
40C2×#391C2211yescharacteristic
41C2×#391C2212yes—
42C2×#391C2214yes—
43C2×#390C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2214yes—
46C2×#388C2214yes—
47C2×#386C2218yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1445C2212yes—
54#1445C2214yes—
55#1444C2218yes—
56#1441C2218yes—
57#1439C2214yes—
58#1439C2218yes—
59#1439C2218yes—
60#1439C2218yes—
61#1438C2212yes—
62#1438C2218yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68C22⋊#95C2218yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2311yescharacteristic
74C23×D4C2314yes—
75C24×C4C2312yes—
76C24×C4C2314yes—
77#554—2128no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—44no—
82C2×C4⋊Q8D418yescomplemented
83C2×#125—28no—
84C2×#125—28no—
85C2×#125—28no—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124—24no—
89C2×#124—24no—
90C2×#124—44no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—216no—
99C2×C22⋊Q8—44no—
100C2×#121—28no—
101C2×#121—28no—
102C2×#121—44no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8D418yescomplemented
106C2×C4×D4—28no—
107C2×C4×D4—28no—
108C2×C4×D4—28no—
109C2×C4×D4—24no—
110C2×C4×D4—24no—
111C2×C4×D4D418yes—
112C22×C4⋊C4—28no—
113C22×C4⋊C4—28no—
114C22×C4⋊C4—28no—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2318yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143#399—28no—
144#397—216no—
145#396—216no—
146#393—216no—
147#392—28no—
148#391—28no—
149#391—24no—
150#391—28no—
151#391—48no—
152#390—28no—
153#390—24no—
154#390—216no—
155#388—216no—
156#386—416no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C22⋊C4—44no—
160C4×C22⋊C4—44no—
161#376—24no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2318yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25C2412yes—
175C25C2412yes—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2×D414yes—
190C22×Q8C2×D418yes—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—216no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—28no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—44no—
203C22×D4—48no—
204C22×D4C2×D414yes—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—48no—
215C23×C4—48no—
216C23×C4C2×D418yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C4⋊Q8—232no—
230C4⋊Q8—416no—
231#125—264no—
232#124—416no—
233#124—416no—
234C22⋊Q8—264no—
235C22⋊Q8—2128no—
236C22⋊Q8—432no—
237#121—432no—
238C4×Q8—232no—
239C4×D4—264no—
240C4×D4—232no—
241C2×C4⋊C4—264no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4#4118yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—216no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4C2×D418yes—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—216no—
327#95—216no—
328#95—216no—
329#95—44no—
330#95—44no—
331#95—48no—
332#95—48no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—28no—
349C24—24no—
350C24C22×D414yes—
351C242−1+418yes—
352C24C2511yescharacteristic
353C2×Q8—28no—
354C2×Q8—216no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—416no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—88no—
366C2×Q8—84no—
367C2×Q8—84no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—28no—
372C2×D4—416no—
373C2×D4—432no—
374C2×D4—416no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—88no—
381C2×D4—816no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—232no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—816no—
437C22×C4—816no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C22×D414yes—
441C22×C4C22×D418yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4118yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)14yes—
457C4⋊C4—232no—
458C4⋊C4—232no—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C22⋊C4—232no—
469C22⋊C4—216no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—816no—
479C42—216no—
480C42—44no—
481C42—44no—
482C42—48no—
483C42—84no—
484C42#120116yes—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23#53618yes—
524C23C23×D411yescharacteristic
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C22×#4112yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+414yes—
531C23#58512yes—
532Q8—416no—
533Q8—816no—
534Q8—88no—
535D4—232no—
536D4—832no—
537D4—88no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—216no—
557C2×C4—216no—
558C2×C4—28no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—832no—
596C2×C4C2×#12014yes—
597C2×C4C2×#12014yes—
598C2×C4C2×#12014yes—
599C2×C4#53214yes—
600C2×C4#53514yes—
601C2×C4#54618yes—
602C2×C4#55018yes—
603C22—28no—
604C22—28no—
605C22—28no—
606C22—48no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—88no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22C22×#12011yescharacteristic
623C22C2×#53211yescharacteristic
624C22C2×#53511yescharacteristic
625C22C2×#53612yes—
626C22C2×#54612yes—
627C22C2×#55012yes—
628C22#348612yes—
629C22#348612yes—
630C22#348812yes—
631C22#349612yes—
632C22#349811yescharacteristic
633C22#350011yescharacteristic
634C22#351614yes—
635C22#352014yes—
636C22#353614yes—
637C22#354114yes—
638C4—24no—
639C4—24no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—48no—
645C4—48no—
646C4—84no—
647C4—82no—
648C4—82no—
649C4#243118yes—
650C2—48no—
651C2—162no—
652C2C2×#243111yescharacteristic
653C2#3524211yescharacteristic
654C2#3551311yescharacteristic
655C2#3590912yes—
656C2#3807812yes—
657C2#3808312yes—
658C2#4281012yes—
659C2#4511214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2540C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#554C2semidirect
2. C2×#2545C2semidirect
4. C2×#2640C2semidirect
5. #16682C2semidirect
6. #15280C2semidirect
7. #15215C2semidirect
8. #15178C2semidirect
9. #14795C2semidirect
10. #14155C2semidirect
11. #14139C2semidirect
12. #14137C2semidirect
13. #13847C2semidirect
14. #13786C2semidirect
82. C2×C4⋊Q8D4semidirect
105. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25452
C2×#25402
C2×#26402
#166822
#152802
#152152
#151782
#147952
#141552
#141392
#141372
#138472
#137862