Finite Group Databasebeta

← #637068

Subgroups — #637068

Order 512.

3642 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2549C211yesmaximal, characteristic, complemented
2C22×#381C211yesmaximal, characteristic
3#16302C214yesmaximal, complemented
4#16299C214yesmaximal, complemented
5#14769C214yesmaximal, complemented
6#14768C214yesmaximal, complemented
7#14726C212yesmaximal
8#14717C212yesmaximal
9#14680C211yesmaximal, characteristic
10#14281C212yesmaximal, complemented
11#14265C212yesmaximal, complemented
12#14149C211yesmaximal, characteristic, complemented
13#13966C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C22⋊C4C2212yes—
19#2549—28no—
20C2×#398C2212yescomplemented
21C2×#396C2214yes—
22C2×#395C2212yescomplemented
23C2×#393C2214yes—
24C2×#391C2212yescomplemented
25C2×#389C2212yescomplemented
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#381—28no—
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#381C2214yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×#379C2211yescharacteristic
37C2×#379C2211yescharacteristic
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42C22×#95C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2211yescharacteristic
45#1446C2214yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1444C2212yescomplemented
49#1443C2214yescomplemented
50#1442C2212yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1434C2212yescomplemented
66#1434C2212yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—48no—
82C24×C4Q818yescomplemented
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—44no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—48no—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117#398—28no—
118#396—48no—
119#395—28no—
120#393—48no—
121#391—28no—
122#389—28no—
123#385—28no—
124#385—28no—
125#381—216no—
126#381—28no—
127#381—28no—
128#380—24no—
129#380—24no—
130#380—44no—
131#380—48no—
132#379—24no—
133#379—24no—
134#379—44no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—48no—
138C2×#95—28no—
139C2×#95—24no—
140C2×#95D418yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C25—22no—
170C25—22no—
171C25—28no—
172C25—22no—
173C25—22no—
174C25—22no—
175C25—22no—
176C25—44no—
177C25C2×Q814yes—
178C25C2412yes—
179C25C2411yescharacteristic
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—42no—
183C22×Q8—44no—
184C22×Q8—44no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—232no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—432no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—432no—
209C23×C4—432no—
210C23×C4C2×D418yes—
211C23×C4C2×D414yes—
212C23×C4C2×Q818yes—
213C23×C4C2×Q818yes—
214C23×C4C2×Q814yes—
215C23×C4C2×Q814yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C22⋊Q8—432no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—432no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4#4114yes—
290C2×C22⋊C4#4114yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—44no—
294C2×C42—44no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
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—44no—
318#95—44no—
319#95—44no—
320#95C2×D418yes—
321#95C2×D414yes—
322#95C2×D414yes—
323C24—24no—
324C24—24no—
325C24—216no—
326C24—216no—
327C24—28no—
328C24—216no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—28no—
339C24—28no—
340C24—48no—
341C24—416no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—48no—
346C24—48no—
347C24—44no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24C22×D412yes—
352C24C22×Q814yes—
353C24C22×Q812yes—
354C24C2×#4112yes—
355C24C2×#4112yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)18yes—
358C24H(5,𝔽2)12yes—
359C242−1+412yes—
360C24C2511yescharacteristic
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—88no—
365C2×Q8—88no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—4128no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4C22⋊Q8116yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q812yes—
457C22×C4C22×Q812yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—84no—
484C42—84no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—232no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—432no—
509C23—432no—
510C23—432no—
511C23—416no—
512C23—432no—
513C23—432no—
514C23—432no—
515C23—44no—
516C23—44no—
517C23—416no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C2×C22⋊Q814yes—
527C23#53314yes—
528C23#54318yes—
529C23#54414yes—
530C23#54614yes—
531C23#54814yes—
532C23#55514yes—
533C23#55614yes—
534C23C23×D411yescharacteristic
535C23C23×Q811yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C2×H(5,𝔽2)14yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23#58514yes—
543Q8—816no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—432no—
565C2×C4—432no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
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—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—816no—
614C2×C4—816no—
615C2×C4C2×C22⋊Q814yes—
616C2×C4C2×C22⋊Q814yes—
617C22—28no—
618C22—24no—
619C22—28no—
620C22—24no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—464no—
629C22—48no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—44no—
637C22#243918yes—
638C22C22×C22⋊Q811yescharacteristic
639C22C2×#53311yescharacteristic
640C22#344912yes—
641C22#345512yes—
642C22C2×#54312yes—
643C22C2×#54411yescharacteristic
644C22#345812yes—
645C22C2×#54611yescharacteristic
646C22C2×#54811yescharacteristic
647C22C2×#55511yescharacteristic
648C22C2×#55611yescharacteristic
649C22#348412yes—
650C22#349012yes—
651C22#351314yes—
652C22#351314yes—
653C22#353414yes—
654C22#353514yes—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—82no—
663C4—84no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—82no—
668C2—28no—
669C2—48no—
670C2#3463411yescharacteristic
671C2C2×#243911yescharacteristic
672C2#3498211yescharacteristic
673C2#3506512yes—
674C2#3747812yes—
675C2#3890012yes—
676C2#3893412yes—
677C2#3911612yes—
678C2#3915012yes—
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×#2549C2semidirect
3. #16302C2semidirect
4. #16299C2semidirect
5. #14769C2semidirect
6. #14768C2semidirect
10. #14281C2semidirect
11. #14265C2semidirect
12. #14149C2semidirect
13. #13966C2semidirect
14. #13908C2semidirect
20. C2×#398C22semidirect
22. C2×#395C22semidirect
24. C2×#391C22semidirect
25. C2×#389C22semidirect
45. #1446C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1444C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
82. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25492
C22×#3812
#163022
#162992
#147692
#147682
#147262
#147172
#146802
#142812
#142652
#141492
#139662
#139082
C4×#3792