Finite Group Databasebeta

← #760515

Subgroups — #760515

Order 512.

3734 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2607C211yesmaximal, characteristic, complemented
2C22×#381C211yesmaximal, characteristic
3#16668C212yesmaximal, complemented
4#16666C212yesmaximal, complemented
5#14772C214yesmaximal, complemented
6#14768C214yesmaximal, complemented
7#14734C212yesmaximal
8#14717C212yesmaximal
9#14165C211yesmaximal, characteristic, complemented
10#14052C214yesmaximal, complemented
11#13980C214yesmaximal, complemented
12#13953C212yesmaximal, complemented
13C4×#379C212yesmaximal
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2607—28no—
19C2×#399C2212yes—
20C2×#399C2212yes—
21C2×#393C2212yes—
22C2×#393C2212yes—
23C2×#391C2214yescomplemented
24C2×#387C2214yescomplemented
25C2×#385C2214yes—
26C2×#381—28no—
27C2×#381C2211yescharacteristic
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#381C2214yes—
31C2×#381C2211yescharacteristic
32C2×#380C2212yes—
33C2×#379C2214yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2214yes—
41#1446C2214yescomplemented
42#1445C2214yescomplemented
43#1445C2214yescomplemented
44#1443C2214yescomplemented
45#1442C2214yescomplemented
46#1441C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1439C2212yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61C4⋊#95C2214yescomplemented
62#1434C2214yescomplemented
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×Q8C2312yes—
71C24×C4—24no—
72C24×C4—24no—
73C24×C4—42no—
74C24×C4—42no—
75C24×C4—42no—
76C24×C4—42no—
77C24×C4Q818yescomplemented
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—44no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—42no—
95C22×C22⋊C4—42no—
96C22×C22⋊C4—42no—
97C22×C22⋊C4—42no—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114#399—28no—
115#399—44no—
116#393—28no—
117#393—44no—
118#391—216no—
119#387—216no—
120#385—216no—
121#381—24no—
122#381—24no—
123#381—24no—
124#381—216no—
125#381D418yes—
126#380—44no—
127#379—216no—
128C4×C22⋊C4—28no—
129C4×C22⋊C4—28no—
130C4×C22⋊C4—44no—
131C4×C22⋊C4—44no—
132C4×C22⋊C4—44no—
133C4×C22⋊C4—44no—
134C2×#95—216no—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25—22no—
159C25—22no—
160C25—22no—
161C25—22no—
162C25—22no—
163C25—22no—
164C25—28no—
165C25—44no—
166C25C2×Q814yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—28no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×Q8—42no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—232no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—416no—
193C23×C4—416no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—416no—
211C23×C4—416no—
212C23×C4—416no—
213C23×C4—416no—
214C23×C4C2×D414yes—
215C23×C4C2×Q818yes—
216C23×C4C2×Q818yes—
217C23×C4C2×Q814yes—
218C23×C4C2×Q814yes—
219C23×C4#4114yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C22⋊Q8—432no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—416no—
281C2×C22⋊C4—416no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4#41116yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95C2×D4116yes—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—24no—
325C24—28no—
326C24—28no—
327C24—28no—
328C24—28no—
329C24—28no—
330C24—28no—
331C24—28no—
332C24—21no—
333C24—24no—
334C24—24no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—24no—
343C24—24no—
344C24—216no—
345C24—48no—
346C24—416no—
347C24—416no—
348C24—416no—
349C24—416no—
350C24—48no—
351C24—48no—
352C24—48no—
353C24—48no—
354C24C22×D412yes—
355C24C22×Q812yes—
356C24C22×Q812yes—
357C24C22×Q812yes—
358C24C2×#4112yes—
359C24C2×#4118yes—
360C24H(5,𝔽2)12yes—
361C24H(5,𝔽2)12yes—
362C24H(5,𝔽2)12yes—
363C24H(5,𝔽2)12yes—
364C24C2511yescharacteristic
365C2×Q8—48no—
366C2×Q8—416no—
367C2×Q8—84no—
368C2×Q8—88no—
369C2×Q8—84no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—4128no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4C22⋊Q8116yescomplemented
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×Q814yes—
467C22×C4C22×Q814yes—
468C22×C4C2×#4118yes—
469C22×C4C2×#4118yes—
470C22×C4C2×#4114yes—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—84no—
477C22⋊C4—232no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—416no—
489C42—84no—
490C42—84no—
491C42—84no—
492C42—84no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—232no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—28no—
526C23—28no—
527C23—432no—
528C23—432no—
529C23—432no—
530C23—432no—
531C23—416no—
532C23—432no—
533C23—432no—
534C23—416no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23C2×C22⋊Q814yes—
548C23#53318yes—
549C23#54314yes—
550C23#54314yes—
551C23#55118yes—
552C23#55918yes—
553C23C23×D411yescharacteristic
554C23C23×Q811yescharacteristic
555C23C22×#4111yescharacteristic
556C23C22×#4114yes—
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23#58514yes—
562Q8—816no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—416no—
583C2×C4—432no—
584C2×C4—432no—
585C2×C4—416no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—816no—
629C2×C4—816no—
630C2×C4C2×C22⋊Q818yes—
631C22—28no—
632C22—28no—
633C22—24no—
634C22—24no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—28no—
643C22—464no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22#260718yescomplemented
659C22C22×C22⋊Q811yescharacteristic
660C22C2×#53312yes—
661C22C2×#54311yescharacteristic
662C22C2×#54311yescharacteristic
663C22#345812yes—
664C22D4×#4114yes—
665C22C2×#55112yes—
666C22#347514yes—
667C22C2×#55912yes—
668C22#348411yescharacteristic
669C22#348411yescharacteristic
670C22#349011yescharacteristic
671C22#349011yescharacteristic
672C22#351314yes—
673C22#351314yes—
674C22#353312yes—
675C22#353312yes—
676C4—28no—
677C4—44no—
678C4—44no—
679C4—48no—
680C4—44no—
681C4—48no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4—84no—
686C2—28no—
687C2—44no—
688C2—44no—
689C2#3473912yes—
690C2C2×#260711yescharacteristic
691C2#3498211yescharacteristic
692C2#3506512yes—
693C2#3747911yescharacteristic
694C2#5918912yes—
695C2#5919112yes—
696C2#5936712yes—
697C2#5936912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2607C2semidirect
3. #16668C2semidirect
4. #16666C2semidirect
5. #14772C2semidirect
6. #14768C2semidirect
9. #14165C2semidirect
10. #14052C2semidirect
11. #13980C2semidirect
12. #13953C2semidirect
23. C2×#391C22semidirect
24. C2×#387C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. C4⋊#95C22semidirect
62. #1434C22semidirect
77. C24×C4Q8semidirect
462. C22×C4C22⋊Q8semidirect
658. C22#2607semidirect

Maximal subgroups

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

subgroupindex
C2×#26072
C22×#3812
#166682
#166662
#147722
#147682
#147342
#147172
#141652
#140522
#139802
#139532
C4×#3792