Finite Group Databasebeta

← #626522

Subgroups — #626522

Order 512.

3654 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2311C211yesmaximal, characteristic
4C22×#389C211yesmaximal, characteristic, complemented
5#15860C214yesmaximal, complemented
6#15679C214yesmaximal, complemented
7#15458C214yesmaximal, complemented
8#15436C214yesmaximal, complemented
9#14927C212yesmaximal, complemented
10#14729C212yesmaximal, complemented
11#14696C212yesmaximal, complemented
12#14339C211yesmaximal, characteristic, complemented
13#14256C212yesmaximal, complemented
14#13958C211yesmaximal, characteristic, complemented
15#13956C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×#123C2212yes—
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2484—28no—
25#2406—28no—
26C2×#400C2212yes—
27C2×#398C2212yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#394C2212yes—
31#2311—28no—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#389—28no—
35C2×#389C2214yes—
36C2×#389C2214yes—
37C2×#388C2214yes—
38C2×#387C2214yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#385C2214yes—
45C2×#385C2212yes—
46C2×#381C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C22×#95C2211yescharacteristic
53C22×#95C2211yescharacteristic
54#1446C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1436C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22×#41—28no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—24no—
83C22×#41—44no—
84C23×Q8C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—44no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—44no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4—44no—
108C22×C4⋊C4D418yes—
109C22×C4⋊C4D418yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C42—24no—
133C22×C42—44no—
134C22×C42C2312yes—
135C22×C42C2314yes—
136#400—28no—
137#398—28no—
138#396—48no—
139#395—28no—
140#394—28no—
141#391—28no—
142#390—28no—
143#389—216no—
144#389—216no—
145#388—48no—
146#387—216no—
147#386—28no—
148#386—28no—
149#386—28no—
150#386—28no—
151#386—44no—
152#385—28no—
153#385—216no—
154#381—24no—
155#381—24no—
156#381—28no—
157#380—28no—
158#380—28no—
159#380—44no—
160C2×#95—24no—
161C2×#95—24no—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C2×#41—232no—
189C2×#41—432no—
190C2×#41—432no—
191C2×#41—416no—
192C2×#41—416no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—42no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—42no—
209C22×D4#4118yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4#4114yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261#123—432no—
262#117—416no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—416no—
292C2×C4⋊C4—416no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×D418yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—44no—
335C2×C42—416no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C2×#4114yes—
377C24C2×#4112yes—
378C24C2×#4114yes—
379C24H(5,𝔽2)14yes—
380C24C2511yescharacteristic
381#41—4128no—
382C2×Q8—28no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×D4—28no—
388C2×D4—216no—
389C2×D4—216no—
390C2×D4—216no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—216no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—416no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—44no—
484C22×C4—416no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—88no—
494C22×C4—88no—
495C22×C4#121116yes—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D414yes—
504C22×C4C2×#4114yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)14yes—
508C22×C42−1+414yes—
509C22×C42−1+414yes—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—416no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—416no—
520C4⋊C4—88no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—416no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—88no—
532C42—416no—
533C42—48no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—28no—
550C23—28no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23C2×#12114yes—
570C23#52514yes—
571C23#53414yes—
572C23#54618yes—
573C23#55118yes—
574C23C23×D411yescharacteristic
575C23C23×D411yescharacteristic
576C23C22×#4111yescharacteristic
577C23C22×#4112yes—
578C23C2×H(5,𝔽2)12yes—
579C23C2×H(5,𝔽2)12yes—
580C23C2×2−1+412yes—
581C23C2×2−1+412yes—
582C23#58512yes—
583Q8—416no—
584D4—416no—
585D4—432no—
586C2×C4—232no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—232no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—432no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—416no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4C2×#12114yes—
665C2×C4C2×#12114yes—
666C2×C4#53718yes—
667C22—24no—
668C22—216no—
669C22—216no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—44no—
676C22—44no—
677C22—48no—
678C22—48no—
679C22—44no—
680C22—84no—
681C22—84no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22#253418yes—
687C22C22×#12111yescharacteristic
688C22C2×#52511yescharacteristic
689C22C2×#53411yescharacteristic
690C22C2×#53712yes—
691C22#346112yes—
692C22C2×#54612yes—
693C22C2×#55112yes—
694C22#348411yescharacteristic
695C22#348511yescharacteristic
696C22#348611yescharacteristic
697C22#348611yescharacteristic
698C22#349212yes—
699C22#349712yes—
700C22#351514yes—
701C22#352014yes—
702C22#354214yes—
703C22#354414yes—
704C4—24no—
705C4—24no—
706C4—28no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—84no—
713C4—84no—
714C4—82no—
715C4—82no—
716C4—82no—
717C2—28no—
718C2—84no—
719C2C2×#253411yescharacteristic
720C2#3515311yescharacteristic
721C2#3522311yescharacteristic
722C2#3535711yescharacteristic
723C2#3545311yescharacteristic
724C2#3670411yescharacteristic
725C2#3725811yescharacteristic
726C2#3819714yes—
727C2#3895014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2484C2semidirect
4. C22×#389C2semidirect
5. #15860C2semidirect
6. #15679C2semidirect
7. #15458C2semidirect
8. #15436C2semidirect
9. #14927C2semidirect
10. #14729C2semidirect
11. #14696C2semidirect
12. #14339C2semidirect
13. #14256C2semidirect
14. #13958C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#24062
C2×#23112
C22×#3892
#158602
#156792
#154582
#154362
#149272
#147292
#146962
#143392
#142562
#139582
#139562