Finite Group Databasebeta

← #554235

Subgroups — #554235

Order 512.

3478 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2406C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic, complemented
4#14706C212yesmaximal, complemented
5#14568C214yesmaximal, complemented
6#14561C214yesmaximal, complemented
7#14416C214yesmaximal, complemented
8#14398C214yesmaximal, complemented
9#14259C212yesmaximal, complemented
10#14128C211yesmaximal, characteristic, complemented
11#14020C212yesmaximal, complemented
12#13996C212yesmaximal, complemented
13#13956C211yesmaximal, characteristic
14#13870C211yesmaximal, characteristic, complemented
15#13776C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#123C2212yes—
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2406C4116yescomplemented
23C2×#395C2212yes—
24C2×#394C2212yes—
25#2311—28no—
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#387—28no—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#385C2212yes—
37C2×#385C2214yes—
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1446C2212yes—
49#1445C2214yes—
50#1444C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C22×#41—24no—
79C22×#41—24no—
80C22×#41—44no—
81C22×#41—44no—
82C22×#41C2×C418yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×#123—44no—
95C2×#123—44no—
96C2×#123C2×C4116yes—
97C2×#117—24no—
98C2×#117—44no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4C2×C418yes—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42—24no—
124C22×C42—44no—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128#395C2×C4116yescomplemented
129#394C2×C4116yescomplemented
130#391—28no—
131#390—28no—
132#387—216no—
133#387—216no—
134#386—28no—
135#386—28no—
136#386—44no—
137#386—44no—
138#386C2×C4116yes—
139#385—44no—
140#385C2×C4132yescomplemented
141#381—24no—
142#381—44no—
143#381C2×C418yes—
144#380—28no—
145#380—28no—
146C4×C22⋊C4—44no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C2×#41—232no—
180C2×#41—432no—
181C2×#41—416no—
182C2×#41—416no—
183C2×#41—432no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8C22×C414yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4C22×C418yes—
200C22×D4C22×C414yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C22×C414yes—
232C23×C4C22×C414yes—
233C23×C4C22×C414yes—
234C23×C4C22×C418yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252#123—432no—
253#117—416no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C22×C418yes—
274C2×C4⋊C4C22×C418yes—
275C2×C4⋊C4C22×C4116yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C22×C418yes—
305C2×C22⋊C4C22×C4116yes—
306C2×C22⋊C4C22×C4116yes—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—416no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C22×C418yes—
341#95C22×C418yes—
342#95C22×C4116yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24C23×C414yes—
358C24C23×C412yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24H(5,𝔽2)14yes—
362C24H(5,𝔽2)14yes—
363C24C2511yescharacteristic
364#41—4128no—
365C2×Q8—28no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—48no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
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—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
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—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—416no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4C4×D4116yes—
480C22×C4C23×C418yes—
481C22×C4C23×C414yes—
482C22×C4C23×C414yes—
483C22×C4C23×C414yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C42−1+414yes—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—416no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—88no—
501C22⋊C4—216no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C42—416no—
513C42—48no—
514C42—88no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23C2×C4×D414yes—
551C23#51818yes—
552C23#53414yes—
553C23#53514yes—
554C23#53818yes—
555C23C24×C411yescharacteristic
556C23C23×D411yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×2−1+412yes—
561C23#58512yes—
562C23#58512yes—
563C23#58512yes—
564Q8—416no—
565D4—416no—
566D4—432no—
567C2×C4—232no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—432no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4C2×C4×D414yes—
647C2×C4C2×C4×D414yes—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—216no—
653C22—24no—
654C22—216no—
655C22—24no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—84no—
666C22—84no—
667C22#232118yes—
668C22C22×C4×D411yescharacteristic
669C22C2×#51812yes—
670C22#342812yes—
671C22C4×2−1+412yes—
672C22C2×#53411yescharacteristic
673C22C2×#53511yescharacteristic
674C22C2×#53812yes—
675C22#348511yescharacteristic
676C22#348511yescharacteristic
677C22#348611yescharacteristic
678C22#348711yescharacteristic
679C22#349312yes—
680C22#350712yes—
681C22#353314yes—
682C22#353514yes—
683C22#355914yes—
684C22#356014yes—
685C4—24no—
686C4—24no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—48no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—84no—
698C2—28no—
699C2—84no—
700C2C2×#232111yescharacteristic
701C2#3420111yescharacteristic
702C2#3426611yescharacteristic
703C2#3446914yes—
704C2#3695511yescharacteristic
705C2#3695811yescharacteristic
706C2#3696111yescharacteristic
707C2#3699714yes—
708C2#3718611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#387C2semidirect
4. #14706C2semidirect
5. #14568C2semidirect
6. #14561C2semidirect
7. #14416C2semidirect
8. #14398C2semidirect
9. #14259C2semidirect
10. #14128C2semidirect
11. #14020C2semidirect
12. #13996C2semidirect
14. #13870C2semidirect
15. #13776C2semidirect
22. #2406C4semidirect
128. #395C2×C4semidirect
129. #394C2×C4semidirect
140. #385C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
C2×#23112
C22×#3872
#147062
#145682
#145612
#144162
#143982
#142592
#141282
#140202
#139962
#139562
#138702
#137762