Finite Group Databasebeta

← #618173

Subgroups — #618173

Order 512.

5584 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2591C211yesmaximal, characteristic, complemented
3C2×#2436C211yesmaximal, characteristic, complemented
4C2×#2617C211yesmaximal, characteristic
5#15465C214yesmaximal, complemented
6#15275C218yesmaximal, complemented
7#15264C214yesmaximal, complemented
8#15091C212yesmaximal, complemented
9#15091C212yesmaximal, complemented
10#14694C212yesmaximal, complemented
11#14255C212yesmaximal, complemented
12#13923C211yesmaximal, characteristic, complemented
13#13922C211yesmaximal, characteristic, complemented
14#13775C211yesmaximal, characteristic, complemented
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×#124C2212yescomplemented
18C22×C22⋊Q8—28no—
19C22×C22⋊Q8—28no—
20C22×C22⋊Q8—28no—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2214yes—
23C22×#121C2211yescharacteristic
24C22×#120C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30#2617—28no—
31#2591—28no—
32#2436—28no—
33C2×#400C2212yescomplemented
34C2×#397C2212yes—
35C2×#394C2214yes—
36C2×#394C2212yescomplemented
37C2×#393C2214yescomplemented
38C2×#392C2212yes—
39C2×#391C2212yescomplemented
40C2×#391C2214yescomplemented
41C2×#388C2214yescomplemented
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#386C2218yes—
48C2×#380C2211yescharacteristic
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1446C2214yescomplemented
53#1442C2218yescomplemented
54#1442C2218yescomplemented
55#1441C2214yescomplemented
56#1440C2218yes—
57#1439C2218yescomplemented
58#1439C2218yescomplemented
59#1438C2218yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2218yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—48no—
73C23×Q8D418yescomplemented
74C23×Q8C2312yes—
75C23×Q8C2312yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C23×D4C2314yes—
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—24no—
82C24×C4—48no—
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#124—28no—
91C2×#124—48no—
92C2×C22⋊Q8—2128no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8—216no—
97C2×C22⋊Q8—216no—
98C2×C22⋊Q8—216no—
99C2×#121—24no—
100C2×#121—24no—
101C2×#121—24no—
102C2×#120—28no—
103C2×#120—28no—
104C2×#120—28no—
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⋊C4—24no—
113C22×C4⋊C4D418yescomplemented
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—24no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4—24no—
125C22×C22⋊C4D418yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2318yes—
140C22×C42C2312yes—
141C22×C42C2312yes—
142#400—28no—
143#397—28no—
144#394—28no—
145#394—216no—
146#393—216no—
147#392—28no—
148#391—28no—
149#391—48no—
150#388—216no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—28no—
155#386—28no—
156#386—416no—
157#380—24no—
158#379—28no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C25—22no—
175C25—22no—
176C25—24no—
177C25—44no—
178C25—44no—
179C25C2×D414yes—
180C25#4118yes—
181C25C2412yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—232no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—28no—
190C22×Q8—28no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—432no—
194C22×Q8—432no—
195C22×Q8—432no—
196C22×Q8C2×D418yes—
197C22×Q8C2×D418yes—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—28no—
207C22×D4—28no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—44no—
212C22×D4—48no—
213C22×D4—48no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—28no—
217C23×C4—232no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—28no—
227C23×C4—28no—
228C23×C4—28no—
229C23×C4—216no—
230C23×C4—48no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—432no—
237C23×C4—432no—
238C23×C4—432no—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2414yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258#124—432no—
259C22⋊Q8—2128no—
260C22⋊Q8—2128no—
261#121—416no—
262#120—432no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—232no—
265C2×C4⋊C4—232no—
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—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C22⋊C4—232no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—232no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
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—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—216no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4C2×D418yes—
321C2×C22⋊C4C2×D418yes—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42—44no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—216no—
335#95—48no—
336#95—48no—
337#95—48no—
338#95—48no—
339C24—24no—
340C24—24no—
341C24—28no—
342C24—28no—
343C24—216no—
344C24—216no—
345C24—22no—
346C24—22no—
347C24—28no—
348C24—24no—
349C24—28no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—22no—
357C24—22no—
358C24—28no—
359C24—48no—
360C24—48no—
361C24—432no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—432no—
367C24#124116yescomplemented
368C24C22×D414yes—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C22×D412yes—
372C24C2×#4118yes—
373C24C2×#4114yes—
374C242−1+414yes—
375C24C2511yescharacteristic
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—216no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—416no—
383C2×Q8—416no—
384C2×Q8—416no—
385C2×Q8—416no—
386C2×Q8—4128no—
387C2×D4—48no—
388C2×D4—432no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—816no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—216no—
412C22×C4—232no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—432no—
461C22×C4—432no—
462C22×C4—4128no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4#120116yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4H(5,𝔽2)18yes—
486C4⋊C4—232no—
487C4⋊C4—232no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—44no—
491C4⋊C4—84no—
492C22⋊C4—232no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—816no—
506C42—48no—
507C42—84no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—216no—
515C23—216no—
516C23—24no—
517C23—24no—
518C23—232no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—216no—
524C23—216no—
525C23—432no—
526C23—416no—
527C23—464no—
528C23—464no—
529C23—416no—
530C23—432no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—416no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23C2×#12014yes—
554C23C2×#12418yes—
555C23C2×#12414yes—
556C23#53614yes—
557C23#54818yes—
558C23#54818yes—
559C23C23×D411yescharacteristic
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)14yes—
564C23C2×2−1+412yes—
565C23#58514yes—
566Q8—432no—
567Q8—432no—
568D4—88no—
569D4—832no—
570D4—816no—
571C2×C4—232no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—416no—
593C2×C4—464no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—416no—
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—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4C2×#12014yes—
629C2×C4C2×#12014yes—
630C2×C4#53418yes—
631C22—216no—
632C22—216no—
633C22—28no—
634C22—24no—
635C22—28no—
636C22—24no—
637C22—216no—
638C22—28no—
639C22—464no—
640C22—416no—
641C22—416no—
642C22—44no—
643C22—48no—
644C22—44no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—84no—
652C22—82no—
653C22—82no—
654C22#243618yes—
655C22C22×#12011yescharacteristic
656C22C22×#12411yescharacteristic
657C22C2×#53412yes—
658C22C2×#53611yescharacteristic
659C22#344912yes—
660C22C2×#54812yes—
661C22C2×#54812yes—
662C22#348412yes—
663C22#348412yes—
664C22#348512yes—
665C22#348712yes—
666C22#350518yes—
667C22#354118yes—
668C4—28no—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—84no—
676C4—84no—
677C4—84no—
678C4—82no—
679C2—28no—
680C2—48no—
681C2—162no—
682C2C2×#243611yescharacteristic
683C2#3498011yescharacteristic
684C2#3514911yescharacteristic
685C2#3537712yes—
686C2#3670512yes—
687C2#3971614yes—
688C2#3972314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
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
2. C2×#2591C2semidirect
3. C2×#2436C2semidirect
5. #15465C2semidirect
6. #15275C2semidirect
7. #15264C2semidirect
8. #15091C2semidirect
9. #15091C2semidirect
10. #14694C2semidirect
11. #14255C2semidirect
12. #13923C2semidirect
13. #13922C2semidirect
14. #13775C2semidirect
17. C22×#124C22semidirect
33. C2×#400C22semidirect
36. C2×#394C22semidirect
37. C2×#393C22semidirect
39. C2×#391C22semidirect
40. C2×#391C22semidirect
41. C2×#388C22semidirect
52. #1446C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
62. #1436C22semidirect
73. C23×Q8D4semidirect
113. C22×C4⋊C4D4semidirect
367. C24#124semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#25912
C2×#24362
C2×#26172
#154652
#152752
#152642
#150912
#146942
#142552
#139232
#139222
#137752