Finite Group Databasebeta

← #4457761

Subgroups — #4457761

Order 512.

4630 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#536C211yesmaximal, characteristic
2C2×#2432C211yesmaximal, characteristic
3C2×#2434C211yesmaximal, characteristic, complemented
4C2×#2625C211yesmaximal, characteristic, complemented
5#16657C218yesmaximal, complemented
6#16606C214yesmaximal, complemented
7#16601C214yesmaximal, complemented
8#15243C212yesmaximal, complemented
9#15226C212yesmaximal, complemented
10#15190C212yesmaximal, complemented
11#15188C212yesmaximal, complemented
12#13937C211yesmaximal, characteristic
13#13922C211yesmaximal, characteristic, complemented
14#13853C211yesmaximal, characteristic, complemented
15C23×#41C2211yescharacteristic
16C24×Q8C2211yescharacteristic
17C2×#536—28no—
18C2×#536—28no—
19C2×#536—28no—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×#124C2212yes—
24C22×#124C2211yescharacteristic
25C22×#123C2214yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2212yes—
29C22×#117C2211yescharacteristic
30#2625—28no—
31#2434—28no—
32#2432—28no—
33C2×#400C2212yes—
34C2×#398C2212yes—
35C2×#397C2214yes—
36C2×#395C2214yes—
37C2×#394C2214yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#391C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2212yes—
46C2×#389C2212yes—
47C2×#385C2214yes—
48C2×#385C2214yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1446C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2218yes—
56#1444C2218yes—
57#1442C2218yes—
58#1441C2218yes—
59#1440C2214yes—
60#1440C2218yes—
61#1439C2218yes—
62#1439C2218yes—
63#1439C2218yes—
64#1438C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C4×#95C2212yes—
68C4×#95C2212yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—44no—
72C22×#41—44no—
73C22×#41D418yes—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—44no—
77C23×Q8—44no—
78C23×Q8D418yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C23×D4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88#536—2128no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8—44no—
97C2×#124—28no—
98C2×#124—28no—
99C2×#124—24no—
100C2×#124—28no—
101C2×#124—44no—
102C2×#123—216no—
103C2×#123—216no—
104C2×#123—216no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×#117—24no—
115C2×#117—24no—
116C2×#117D418yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2318yes—
126C22×C4⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2318yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2312yes—
141#400—28no—
142#398—28no—
143#397—216no—
144#395—216no—
145#394—216no—
146#391—24no—
147#391—28no—
148#391—28no—
149#391—44no—
150#391—44no—
151#391—44no—
152#390—24no—
153#390—44no—
154#389—28no—
155#385—216no—
156#385—216no—
157#380—24no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—24no—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C2×#41—232no—
178C2×#41—416no—
179C2×#41—416no—
180C2×#41—432no—
181C2×#41—432no—
182C22×Q8—232no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—432no—
203C22×Q8—416no—
204C22×Q8—416no—
205C22×Q8—432no—
206C22×Q8C2×D418yes—
207C22×Q8C2×D414yes—
208C22×Q8C2×D418yes—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4C2×D414yes—
218C22×D4C2×D418yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D418yes—
233C23×C4C2×D418yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C4⋊Q8—264no—
247C4⋊Q8—432no—
248C4⋊Q8—416no—
249#124—264no—
250#124—416no—
251#123—2128no—
252C22⋊Q8—264no—
253C22⋊Q8—264no—
254C22⋊Q8—416no—
255#117—232no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—216no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4C2×D418yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42C2×D418yes—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—216no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—48no—
334#95—48no—
335#95—48no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—28no—
347C24—24no—
348C24C22×D414yes—
349C24C22×D412yes—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C2511yescharacteristic
353#41—4128no—
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—28no—
357C2×Q8—216no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—28no—
361C2×Q8—416no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—416no—
365C2×Q8—4128no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×D4—216no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—48no—
386C2×D4—48no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—416no—
451C22×C4#120116yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D418yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C42−1+418yes—
465C22×C42−1+414yes—
466C4⋊C4—232no—
467C4⋊C4—232no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—416no—
471C4⋊C4—432no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—88no—
476C22⋊C4—216no—
477C22⋊C4—232no—
478C22⋊C4—232no—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—816no—
485C22⋊C4—84no—
486C42—216no—
487C42—44no—
488C42—44no—
489C42—48no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23C2×#12014yes—
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)12yes—
527C23C2×2−1+414yes—
528C23C2×2−1+412yes—
529C23#58512yes—
530Q8—432no—
531Q8—432no—
532Q8—416no—
533Q8—88no—
534D4—432no—
535D4—416no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—232no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—432no—
559C2×C4—432no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4C2×#12014yes—
598C2×C4C2×#12014yes—
599C2×C4#53414yes—
600C2×C4#53414yes—
601C2×C4#53418yes—
602C2×C4#53518yes—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—84no—
617C22—84no—
618C22—88no—
619C22C22×#12011yescharacteristic
620C22C2×#53412yes—
621C22C2×#53411yescharacteristic
622C22C2×#53411yescharacteristic
623C22C2×#53512yes—
624C22#348612yes—
625C22#348912yes—
626C22#349712yes—
627C22#349911yescharacteristic
628C22#349912yes—
629C22#349912yes—
630C22#350011yescharacteristic
631C22#355214yes—
632C22#355214yes—
633C22#355414yes—
634C22#356114yes—
635C4—24no—
636C4—48no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—48no—
642C4—84no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4#261318yes—
647C2—44no—
648C2—84no—
649C2C2×#261311yescharacteristic
650C2#3566011yescharacteristic
651C2#3566611yescharacteristic
652C2#3739012yes—
653C2#3739112yes—
654C2#4278714yes—
655C2#4291212yes—
656C2#4291712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2434C2semidirect
4. C2×#2625C2semidirect
5. #16657C2semidirect
6. #16606C2semidirect
7. #16601C2semidirect
8. #15243C2semidirect
9. #15226C2semidirect
10. #15190C2semidirect
11. #15188C2semidirect
13. #13922C2semidirect
14. #13853C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5362
C2×#24322
C2×#24342
C2×#26252
#166572
#166062
#166012
#152432
#152262
#151902
#151882
#139372
#139222
#138532