Finite Group Databasebeta

← #618592

Subgroups — #618592

Order 512.

4080 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic, complemented
2C2×#2591C211yesmaximal, characteristic
3C2×#2501C211yesmaximal, characteristic
4C2×#2618C211yesmaximal, characteristic
5#15882C214yesmaximal, complemented
6#15850C214yesmaximal, complemented
7#15846C218yesmaximal, complemented
8#15069C212yesmaximal, complemented
9#15069C212yesmaximal, complemented
10#14706C212yesmaximal, complemented
11#14256C212yesmaximal, complemented
12#13923C211yesmaximal, characteristic, complemented
13#13904C211yesmaximal, characteristic, complemented
14#13776C211yesmaximal, characteristic, complemented
15C22×#125—28no—
16C22×#125—28no—
17C22×#125—28no—
18C22×#125C2214yes—
19C22×#125C2214yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×#121C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2618—28no—
32#2591—28no—
33#2501—28no—
34C2×#400C2212yes—
35C2×#399C2214yes—
36C2×#398C2212yes—
37C2×#396C2214yes—
38C2×#394C2212yes—
39C2×#394C2212yes—
40C2×#390C2212yes—
41C2×#388C2214yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#384C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2214yes—
53#1444C2218yes—
54#1444C2218yes—
55#1441C2214yes—
56#1441C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1439C2218yes—
60#1437C2218yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2218yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×#125—2128no—
79C2×#125—216no—
80C2×#125—216no—
81C2×#125—216no—
82C2×#125—216no—
83C2×#125—216no—
84C2×#125—216no—
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—44no—
91C2×#123—44no—
92C2×#123—44no—
93C2×#123—44no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4D418yes—
113C22×C4⋊C4D418yes—
114C22×C4⋊C4D418yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C42—24no—
140C22×C42—24no—
141C22×C42—24no—
142C22×C42C2314yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145#400—28no—
146#399—216no—
147#398—28no—
148#396—216no—
149#394—28no—
150#394—28no—
151#390—28no—
152#388—216no—
153#386—28no—
154#386—28no—
155#386—28no—
156#384—28no—
157#381—28no—
158#381—44no—
159#381—44no—
160#380—24no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—44no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—28no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231#125—2128no—
232#125—2128no—
233#123—432no—
234#123—432no—
235#123—432no—
236#121—416no—
237C2×C4⋊C4—232no—
238C2×C4⋊C4—232no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D418yes—
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—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—216no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4#4118yes—
311C2×C42—232no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—44no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—216no—
326#95—216no—
327#95—48no—
328#95—48no—
329#95—48no—
330#95—48no—
331C24—24no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—28no—
341C24—28no—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)14yes—
347C24C2511yescharacteristic
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—84no—
358C2×D4—84no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4#120116yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C2×#4118yes—
447C22×C42−1+418yes—
448C4⋊C4—232no—
449C4⋊C4—232no—
450C4⋊C4—232no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—84no—
462C22⋊C4—216no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—416no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C42—232no—
473C42—48no—
474C42—84no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—216no—
482C23—216no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23C2×#12014yes—
501C23#53214yes—
502C23#53514yes—
503C23#54618yes—
504C23#55018yes—
505C23C23×D411yescharacteristic
506C23C23×D411yescharacteristic
507C23C23×D411yescharacteristic
508C23C22×#4112yes—
509C23C2×H(5,𝔽2)12yes—
510C23C2×2−1+414yes—
511C23#58514yes—
512D4—88no—
513D4—816no—
514C2×C4—232no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—216no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—416no—
549C2×C4—416no—
550C2×C4—416no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4C2×#12014yes—
577C2×C4C2×#12014yes—
578C2×C4#53618yes—
579C22—24no—
580C22—28no—
581C22—28no—
582C22—216no—
583C22—24no—
584C22—216no—
585C22—82no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—82no—
591C22—82no—
592C22—84no—
593C22—84no—
594C22#243118yes—
595C22C22×#12011yescharacteristic
596C22C2×#53211yescharacteristic
597C22C2×#53511yescharacteristic
598C22C2×#53612yes—
599C22#344912yes—
600C22C2×#54612yes—
601C22C2×#55012yes—
602C22#348512yes—
603C22#348512yes—
604C22#348612yes—
605C22#348612yes—
606C22#351518yes—
607C22#355918yes—
608C4—24no—
609C4—24no—
610C4—28no—
611C4—44no—
612C4—44no—
613C4—48no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4—84no—
620C2—28no—
621C2—162no—
622C2C2×#243111yescharacteristic
623C2#3508811yescharacteristic
624C2#3524211yescharacteristic
625C2#3546812yes—
626C2#3695812yes—
627C2#3802914yes—
628C2#3913414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#125C2semidirect
5. #15882C2semidirect
6. #15850C2semidirect
7. #15846C2semidirect
8. #15069C2semidirect
9. #15069C2semidirect
10. #14706C2semidirect
11. #14256C2semidirect
12. #13923C2semidirect
13. #13904C2semidirect
14. #13776C2semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#25912
C2×#25012
C2×#26182
#158822
#158502
#158462
#150692
#147062
#142562
#139232
#139042
#137762