Finite Group Databasebeta

← #618702

Subgroups — #618702

Order 512.

5590 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic
2C2×#2561C211yesmaximal, characteristic, complemented
3C2×#2502C211yesmaximal, characteristic, complemented
4C2×#2626C211yesmaximal, characteristic, complemented
5#15668C214yesmaximal, complemented
6#15612C218yesmaximal, complemented
7#15384C214yesmaximal, complemented
8#14918C212yesmaximal, complemented
9#14724C212yesmaximal
10#14693C212yesmaximal, complemented
11#14263C212yesmaximal, complemented
12#13924C211yesmaximal, characteristic, complemented
13#13903C211yesmaximal, characteristic, complemented
14#13775C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×#123—28no—
18C22×#123—28no—
19C22×#123—28no—
20C22×#123C2214yes—
21C22×#123C2214yes—
22C22×C22⋊Q8C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28#2626—28no—
29#2561—28no—
30#2502—28no—
31C2×#399C2212yescomplemented
32C2×#398C2214yescomplemented
33C2×#397C2212yescomplemented
34C2×#396C2214yescomplemented
35C2×#394C2218yes—
36C2×#393C2212yescomplemented
37C2×#393C2212yescomplemented
38C2×#391C2212yescomplemented
39C2×#388C2214yescomplemented
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2218yescomplemented
44C2×#384C2212yescomplemented
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×#379C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#1445C2214yescomplemented
55#1444C2218yescomplemented
56#1442C2218yescomplemented
57#1440C2218yescomplemented
58#1439C2218yescomplemented
59#1439C2214yescomplemented
60#1438C2218yes—
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64C4⋊#95C2214yescomplemented
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2218yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8C2311yescharacteristic
72C23×D4—24no—
73C23×D4—24no—
74C23×D4—24no—
75C23×D4—48no—
76C23×D4C2314yes—
77C24×C4—24no—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—48no—
81C24×C4C2311yescharacteristic
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#123—2128no—
88C2×#123—216no—
89C2×#123—216no—
90C2×#123—216no—
91C2×#123—216no—
92C2×#123—216no—
93C2×#123—216no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4D418yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4D418yes—
120C22×C22⋊C4D418yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2318yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138#399—28no—
139#398—216no—
140#397—28no—
141#396—216no—
142#394—416no—
143#393—28no—
144#393—28no—
145#391—28no—
146#388—216no—
147#385—28no—
148#385—28no—
149#385—28no—
150#385—416no—
151#384—28no—
152#381—28no—
153#381—28no—
154#380—24no—
155#379—28no—
156#379—28no—
157#379—28no—
158#379—28no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C25—22no—
174C25—22no—
175C25—28no—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—24no—
181C25—24no—
182C25—44no—
183C25—44no—
184C25—44no—
185C25C2×D414yes—
186C25C2×D414yes—
187C25C2412yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—44no—
194C22×D4—232no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—216no—
198C22×D4—28no—
199C22×D4—48no—
200C22×D4—432no—
201C22×D4—432no—
202C22×D4—432no—
203C23×C4—232no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—216no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—48no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4—432no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#123—2128no—
245#123—2128no—
246C22⋊Q8—416no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—232no—
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—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C22⋊C4—232no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—232no—
277C2×C22⋊C4—232no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
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—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—216no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×D418yes—
313C2×C22⋊C4C2×D418yes—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4#4118yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42—44no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—216no—
329#95—48no—
330#95—48no—
331#95—48no—
332C24—24no—
333C24—24no—
334C24—216no—
335C24—216no—
336C24—28no—
337C24—216no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—28no—
344C24—28no—
345C24—216no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—28no—
350C24—24no—
351C24—28no—
352C24—28no—
353C24—28no—
354C24—48no—
355C24—48no—
356C24—48no—
357C24—432no—
358C24—432no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—44no—
366C24—44no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—816no—
371C24—816no—
372C24C22×D414yes—
373C24C22×D414yes—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C2×#4114yes—
378C24H(5,𝔽2)18yes—
379C242−1+414yes—
380C24C2511yescharacteristic
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×D4—232no—
386C2×D4—432no—
387C2×D4—432no—
388C2×D4—4128no—
389C2×D4—416no—
390C2×D4—416no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—232no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—432no—
441C22×C4—432no—
442C22×C4—4128no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4#120116yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4118yes—
466C4⋊C4—232no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C22⋊C4—232no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—432no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—816no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C42—48no—
491C42—84no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—216no—
499C23—216no—
500C23—232no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—216no—
508C23—216no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—464no—
515C23—416no—
516C23—432no—
517C23—432no—
518C23—464no—
519C23—416no—
520C23—416no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—416no—
525C23—44no—
526C23—44no—
527C23—48no—
528C23—48no—
529C23—44no—
530C23—44no—
531C23—416no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—48no—
538C23—416no—
539C23—816no—
540C23—816no—
541C23—832no—
542C23—832no—
543C23C2×#12014yes—
544C23#53214yes—
545C23#53518yes—
546C23#53614yes—
547C23#54718yes—
548C23#54818yes—
549C23C23×D411yescharacteristic
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C22×#4112yes—
553C23C2×H(5,𝔽2)14yes—
554C23C2×2−1+412yes—
555C23#58514yes—
556Q8—88no—
557D4—464no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—216no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—464no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—416no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4C2×#12014yes—
611C2×C4C2×#12014yes—
612C22—24no—
613C22—28no—
614C22—28no—
615C22—24no—
616C22—216no—
617C22—216no—
618C22—28no—
619C22—216no—
620C22—464no—
621C22—44no—
622C22—48no—
623C22—48no—
624C22—48no—
625C22—44no—
626C22—416no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—44no—
631C22—44no—
632C22—832no—
633C22—88no—
634C22—88no—
635C22#243218yes—
636C22C22×#12011yescharacteristic
637C22C2×#53211yescharacteristic
638C22C2×#53512yes—
639C22C2×#53611yescharacteristic
640C22#344912yes—
641C22C2×#54712yes—
642C22C2×#54812yes—
643C22#348412yes—
644C22#348412yes—
645C22#348512yes—
646C22#348512yes—
647C22#351318yes—
648C22#353518yes—
649C4—24no—
650C4—24no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—48no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4—84no—
660C2—28no—
661C2—48no—
662C2—84no—
663C2C2×#243211yescharacteristic
664C2#3508811yescharacteristic
665C2#3522211yescharacteristic
666C2#3539412yes—
667C2#3672012yes—
668C2#3803914yes—
669C2#3911614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2561C2semidirect
3. C2×#2502C2semidirect
4. C2×#2626C2semidirect
5. #15668C2semidirect
6. #15612C2semidirect
7. #15384C2semidirect
8. #14918C2semidirect
10. #14693C2semidirect
11. #14263C2semidirect
12. #13924C2semidirect
13. #13903C2semidirect
14. #13775C2semidirect
31. C2×#399C22semidirect
32. C2×#398C22semidirect
33. C2×#397C22semidirect
34. C2×#396C22semidirect
36. C2×#393C22semidirect
37. C2×#393C22semidirect
38. C2×#391C22semidirect
39. C2×#388C22semidirect
43. C2×#385C22semidirect
44. C2×#384C22semidirect
54. #1445C22semidirect
55. #1444C22semidirect
56. #1442C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. C4⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1232
C2×#25612
C2×#25022
C2×#26262
#156682
#156122
#153842
#149182
#147242
#146932
#142632
#139242
#139032
#137752