Finite Group Databasebeta

← #613439

Subgroups — #613439

Order 512.

5092 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#2486C211yesmaximal, characteristic
3C2×#2409C211yesmaximal, characteristic
4C2×C4×#123C211yesmaximal, characteristic
5#15073C212yesmaximal, complemented
6#15069C212yesmaximal
7#14882C214yesmaximal, complemented
8#14879C214yesmaximal, complemented
9#14706C212yesmaximal
10#14705C212yesmaximal, complemented
11#14685C211yesmaximal, characteristic, complemented
12#14017C214yesmaximal, complemented
13#13969C214yesmaximal, complemented
14#13922C211yesmaximal, characteristic, complemented
15#13904C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8—28no—
21C22×C4×Q8—28no—
22C22×C4×Q8—28no—
23C22×C4×Q8C2214yes—
24C22×C4×Q8C2214yes—
25C22×C4×D4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C42C2212yes—
29C23×C42C2211yescharacteristic
30#2486—28no—
31#2409—28no—
32C2×#398C2212yes—
33C2×#397C2212yes—
34C2×#395C2212yes—
35C2×#393C2212yes—
36C4×#123—28no—
37C2×#391C2214yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#388C2212yes—
41C2×#387C2212yes—
42C2×#386C2211yescharacteristic
43C2×#386C2212yes—
44C2×#385C2211yescharacteristic
45C2×#385C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#381C2211yescharacteristic
49C2×#381C2212yes—
50C2×#381C2212yes—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57#1446C2214yes—
58#1445C2214yes—
59#1444C2218yes—
60#1442C2218yes—
61#1441C2218yes—
62#1439C2218yes—
63#1437C2218yes—
64#1436C2218yes—
65#1436C2218yes—
66#1436C2218yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C23×Q8—216no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8D418yes—
77C23×Q8C2312yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×#123—24no—
87C2×#123—24no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—24no—
91C2×#123—44no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—216no—
94C2×C4×Q8—2128no—
95C2×C4×Q8—216no—
96C2×C4×Q8—216no—
97C2×C4×Q8—216no—
98C2×C4×Q8—216no—
99C2×C4×Q8—216no—
100C2×C4×Q8—216no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4D418yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2318yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C42—28no—
131C22×C42—28no—
132C22×C42—28no—
133C22×C42—24no—
134C22×C42—24no—
135C22×C42D418yes—
136C22×C42C2318yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141#398—28no—
142#397—28no—
143#395—28no—
144#393—28no—
145#391—28no—
146#391—216no—
147#389—28no—
148#388—28no—
149#387—28no—
150#386—28no—
151#386—24no—
152#385—28no—
153#385—24no—
154#381—28no—
155#381—24no—
156#381—28no—
157#381—44no—
158#381—44no—
159#380—24no—
160#380—24no—
161#380—28no—
162C4×C4⋊C4—28no—
163C4×C22⋊C4—24no—
164C4×C22⋊C4—28no—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2318yes—
181C2×#95C2318yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—264no—
185C22×Q8—264no—
186C22×Q8—264no—
187C22×Q8—232no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—216no—
193C22×Q8—28no—
194C22×Q8C2×D418yes—
195C22×Q8C2×D418yes—
196C22×Q8#41116yes—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4#4114yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—216no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—28no—
227C23×C4C2×D414yes—
228C23×C4C2×D418yes—
229C23×C4C2×D418yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247#123—232no—
248#123—432no—
249C22⋊Q8—432no—
250C4×Q8—2128no—
251C4×Q8—2128no—
252C4×D4—416no—
253C2×C4⋊C4—264no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—216no—
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—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
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—22no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—216no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4#4114yes—
309C2×C42—264no—
310C2×C42—232no—
311C2×C42—216no—
312C2×C42—216no—
313C2×C42—216no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—28no—
323C2×C42C2×D418yes—
324C2×C42C2×D418yes—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—216no—
337#95—216no—
338#95—216no—
339#95—216no—
340#95—48no—
341#95—48no—
342C24—24no—
343C24—24no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—24no—
351C24—21no—
352C24C22×D412yes—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C2×#4112yes—
356C24C2×#4112yes—
357C24C2×#4112yes—
358C24H(5,𝔽2)12yes—
359C242−1+412yes—
360C24C2511yescharacteristic
361C2×Q8—2256no—
362C2×Q8—232no—
363C2×Q8—232no—
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—232no—
367C2×Q8—216no—
368C2×Q8—232no—
369C2×Q8—232no—
370C2×Q8#123132yescomplemented
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—232no—
381C22×C4—232no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—232no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4#120116yes—
460C22×C4C22×D418yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D418yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#41116yes—
473C22×C4C2×#4118yes—
474C22×C42−1+418yes—
475C4⋊C4—264no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—416no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C22⋊C4—216no—
486C22⋊C4—28no—
487C22⋊C4—416no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C42—264no—
496C42—216no—
497C42—216no—
498C42—44no—
499C42—48no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—28no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23C2×#12014yes—
528C23#52514yes—
529C23#53714yes—
530C23#54614yes—
531C23#54714yes—
532C23#54814yes—
533C23#55014yes—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4114yes—
538C23C22×#4111yescharacteristic
539C23C22×#4111yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+414yes—
542C23C2×2−1+411yescharacteristic
543Q8—264no—
544Q8—264no—
545D4—416no—
546C2×C4—232no—
547C2×C4—264no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—216no—
555C2×C4—216no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—232no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—416no—
572C2×C4—432no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—88no—
612C2×C4C2×#12014yes—
613C2×C4C2×#12014yes—
614C2×C4C2×#123116yes—
615C2×C4C2×#12318yes—
616C2×C4#52518yes—
617C2×C4#52518yes—
618C2×C4#53618yes—
619C22—24no—
620C22—28no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—24no—
626C22—28no—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—44no—
631C22—81no—
632C22—82no—
633C22—82no—
634C22—81no—
635C22—82no—
636C22#240718yes—
637C22C22×#12011yescharacteristic
638C22C22×#12312yes—
639C22C2×#52512yes—
640C22C2×#52512yes—
641C22C2×#52511yescharacteristic
642C22C2×#53612yes—
643C22C2×#53711yescharacteristic
644C22C2×#54611yescharacteristic
645C22C2×#54711yescharacteristic
646C22C2×#54811yescharacteristic
647C22C2×#55011yescharacteristic
648C22#348214yes—
649C22#348214yes—
650C22#348612yes—
651C22#348612yes—
652C22#352214yes—
653C22#352314yes—
654C4—24no—
655C4—216no—
656C4—28no—
657C4—44no—
658C4—48no—
659C4—42no—
660C4—44no—
661C4—81no—
662C4—84no—
663C4—84no—
664C4—81no—
665C4—82no—
666C4#284918yes—
667C2—28no—
668C2—82no—
669C2C2×#240711yescharacteristic
670C2#3497112yes—
671C2#3524211yescharacteristic
672C2#3524712yes—
673C2#3642212yes—
674C2#3642312yes—
675C2#3648212yes—
676C2#3648312yes—
677C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×C4×Q8C2semidirect
5. #15073C2semidirect
7. #14882C2semidirect
8. #14879C2semidirect
10. #14705C2semidirect
11. #14685C2semidirect
12. #14017C2semidirect
13. #13969C2semidirect
14. #13922C2semidirect
15. #13904C2semidirect
370. C2×Q8#123semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Q82
C2×#24862
C2×#24092
C2×C4×#1232
#150732
#150692
#148822
#148792
#147062
#147052
#146852
#140172
#139692
#139222
#139042