Finite Group Databasebeta

← #635998

Subgroups — #635998

Order 512.

5753 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C2×#2413C212yesmaximal, complemented
4C2×#2614C211yesmaximal, characteristic, complemented
5C2×#2280C211yesmaximal, characteristic, complemented
6#16265C214yesmaximal, complemented
7#16264C214yesmaximal, complemented
8#14918C212yesmaximal, complemented
9#14911C212yesmaximal
10#14765C214yesmaximal, complemented
11#14763C214yesmaximal, complemented
12#14336C211yesmaximal, characteristic, complemented
13#14263C211yesmaximal, characteristic, complemented
14#14256C211yesmaximal, characteristic, complemented
15#13870C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#123C2211yescharacteristic
20C22×#123C2212yes—
21C22×C22⋊Q8—28no—
22C22×C22⋊Q8—216no—
23C22×C22⋊Q8C2214yes—
24C22×C22⋊Q8C2214yes—
25C22×#120C2211yescharacteristic
26C22×#117C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29C23×C22⋊C4C2212yes—
30#2849—28no—
31#2614—28no—
32#2413—216no—
33C2×#398C2214yescomplemented
34C2×#396C2214yescomplemented
35C2×#395C2214yescomplemented
36C2×#393C2214yescomplemented
37C2×#389C2214yescomplemented
38C2×#388C2214yescomplemented
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45#2280—28no—
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2211yescharacteristic
49C2×#381C2211yescharacteristic
50C2×#381C2212yes—
51C2×#381C2212yes—
52C2×#381C2211yescharacteristic
53C2×#380C2214yes—
54C2×#379C2211yescharacteristic
55C2×#379C2212yes—
56C2×C4×C22⋊C4C2214yes—
57#1446C2214yescomplemented
58#1446C2214yescomplemented
59#1444C2214yescomplemented
60#1442C2214yescomplemented
61#1441C2218yescomplemented
62#1441C2214yescomplemented
63#1440C2218yes—
64#1439C2218yes—
65#1439C2218yes—
66#1439C2214yescomplemented
67#1438C2218yes—
68#1436C2218yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C26C2311yescharacteristic
72C23×Q8—28no—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—44no—
77C23×Q8C2314yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4—28no—
81C24×C4—24no—
82C24×C4—24no—
83C24×C4—44no—
84C24×C4D418yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—24no—
97C2×#123—28no—
98C2×#123—44no—
99C2×C22⋊Q8—2128no—
100C2×C22⋊Q8—216no—
101C2×C22⋊Q8—216no—
102C2×C22⋊Q8—232no—
103C2×C22⋊Q8—232no—
104C2×#120—24no—
105C2×#120—44no—
106C2×#117—28no—
107C2×#117—28no—
108C2×#117—44no—
109C22×C4⋊C4—28no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4D4116yescomplemented
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C22⋊C4—28no—
124C22×C22⋊C4—28no—
125C22×C22⋊C4—28no—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2318yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2318yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C42C2314yes—
142#398—216no—
143#396—216no—
144#395—216no—
145#393—216no—
146#389—216no—
147#388—216no—
148#386—28no—
149#386—28no—
150#386—28no—
151#385—28no—
152#385—28no—
153#385—28no—
154#381—24no—
155#381—24no—
156#381—28no—
157#381—24no—
158#381—28no—
159#381—44no—
160#381D418yescomplemented
161#380—216no—
162#379—24no—
163#379—28no—
164C4×C22⋊C4—216no—
165C2×#95C2318yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C2×#95C2318yes—
178C2×#95C2314yes—
179C25—24no—
180C25—28no—
181C25—22no—
182C25—22no—
183C25—24no—
184C25—44no—
185C25C2×D414yes—
186C25C2411yescharacteristic
187C25C2412yes—
188C25C2411yescharacteristic
189C22×Q8—232no—
190C22×Q8—216no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—28no—
194C22×Q8—48no—
195C22×Q8—432no—
196C22×Q8—432no—
197C22×Q8—416no—
198C22×Q8—416no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—28no—
206C22×D4—44no—
207C22×D4—44no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—232no—
222C23×C4—232no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—416no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—48no—
233C23×C4—432no—
234C23×C4—432no—
235C23×C4C2×D418yes—
236C23×C4C2×D418yes—
237C23×C4C2×D418yes—
238C23×C4C2×D414yes—
239C23×C4C2×D418yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D418yes—
243C23×C4#4114yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2414yes—
252C23×C4C2414yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256#123—432no—
257#123—416no—
258#123—416no—
259C22⋊Q8—2128no—
260C22⋊Q8—2128no—
261#120—416no—
262#117—432no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—264no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D4116yes—
287C2×C4⋊C4C2×D4116yes—
288C2×C22⋊C4—264no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—216no—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—216no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4C2×D418yes—
326C2×C42—28no—
327C2×C42—28no—
328C2×C42—44no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—216no—
337#95—28no—
338#95—28no—
339#95—48no—
340#95—48no—
341#95—48no—
342#95C2×D4116yes—
343C24—24no—
344C24—24no—
345C24—232no—
346C24—28no—
347C24—216no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—28no—
356C24—28no—
357C24—22no—
358C24—21no—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—48no—
365C24—416no—
366C24—416no—
367C24—432no—
368C24—416no—
369C24—48no—
370C24—48no—
371C24C22×D414yes—
372C24C22×D412yes—
373C24C22×D414yes—
374C24C22×D412yes—
375C24C2×#4112yes—
376C24H(5,𝔽2)14yes—
377C242−1+418yes—
378C24C2511yescharacteristic
379C2×Q8—232no—
380C2×Q8—432no—
381C2×Q8—432no—
382C2×Q8—4128no—
383C2×Q8—416no—
384C2×Q8—416no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—816no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—4128no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4#120116yes—
472C22×C4#121116yes—
473C22×C4C22×D418yes—
474C22×C4C22×D418yes—
475C22×C4C22×D418yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C2×#4114yes—
483C4⋊C4—232no—
484C4⋊C4—232no—
485C4⋊C4—232no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C22⋊C4—264no—
493C22⋊C4—216no—
494C22⋊C4—416no—
495C22⋊C4—432no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C42—416no—
508C23—24no—
509C23—28no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—216no—
515C23—232no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—216no—
523C23—28no—
524C23—28no—
525C23—216no—
526C23—432no—
527C23—432no—
528C23—464no—
529C23—416no—
530C23—464no—
531C23—416no—
532C23—48no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—48no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—48no—
550C23C2×#12014yes—
551C23C2×#12114yes—
552C23#53414yes—
553C23#53618yes—
554C23D4218yes—
555C23#54618yes—
556C23C23×D411yescharacteristic
557C23C23×D412yes—
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+414yes—
562C23#58514yes—
563Q8—464no—
564D4—816no—
565D4—88no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—232no—
570C2×C4—232no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—216no—
588C2×C4—432no—
589C2×C4—432no—
590C2×C4—432no—
591C2×C4—44no—
592C2×C4—432no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4C2×#12014yes—
627C2×C4C2×#12014yes—
628C2×C4C2×#12118yes—
629C22—216no—
630C22—28no—
631C22—24no—
632C22—24no—
633C22—216no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—464no—
639C22—416no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—416no—
644C22—44no—
645C22—84no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—84no—
651C22—82no—
652C22#241018yes—
653C22C22×#12011yescharacteristic
654C22C22×#12111yescharacteristic
655C22C2×#53411yescharacteristic
656C22C2×#53612yes—
657C22#344914yes—
658C22#344912yes—
659C22C2×D4212yes—
660C22C2×#54612yes—
661C22#348614yes—
662C22#348914yes—
663C22#349514yes—
664C22#353518yes—
665C4—24no—
666C4—24no—
667C4—28no—
668C4—48no—
669C4—48no—
670C4—48no—
671C4—84no—
672C4—82no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—82no—
677C2—28no—
678C2—48no—
679C2—161no—
680C2#3463211yescharacteristic
681C2C2×#241011yescharacteristic
682C2#3524211yescharacteristic
683C2#3530612yes—
684C2#3744112yes—
685C2#3765314yes—
686C2#3784114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×C22⋊Q8C2semidirect
3. C2×#2413C2semidirect
4. C2×#2614C2semidirect
5. C2×#2280C2semidirect
6. #16265C2semidirect
7. #16264C2semidirect
8. #14918C2semidirect
10. #14765C2semidirect
11. #14763C2semidirect
12. #14336C2semidirect
13. #14263C2semidirect
14. #14256C2semidirect
15. #13870C2semidirect
33. C2×#398C22semidirect
34. C2×#396C22semidirect
35. C2×#395C22semidirect
36. C2×#393C22semidirect
37. C2×#389C22semidirect
38. C2×#388C22semidirect
57. #1446C22semidirect
58. #1446C22semidirect
59. #1444C22semidirect
60. #1442C22semidirect
61. #1441C22semidirect
62. #1441C22semidirect
66. #1439C22semidirect
68. #1436C22semidirect
113. C22×C4⋊C4D4semidirect
160. #381D4semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28492
C2×#24132
C2×#26142
C2×#22802
#162652
#162642
#149182
#149112
#147652
#147632
#143362
#142632
#142562
#138702