Finite Group Databasebeta

← #4929606

Subgroups — #4929606

Order 512.

3036 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic, complemented
2C2×#2611C211yesmaximal, characteristic, complemented
3C2×#2598C211yesmaximal, characteristic
4C2×#2275C211yesmaximal, characteristic, complemented
5#15175C212yesmaximal, complemented
6#15156C212yesmaximal, complemented
7#14868C212yesmaximal, complemented
8#14840C212yesmaximal, complemented
9#14839C212yesmaximal, complemented
10#14826C212yesmaximal, complemented
11#14806C212yesmaximal
12#14735C212yesmaximal, complemented
13#14289C211yesmaximal, characteristic, complemented
14#14209C214yesmaximal, complemented
15#14182C214yesmaximal, complemented
16#13952C211yesmaximal, characteristic, complemented
17#13852C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2611—28no—
23#2598—28no—
24#2562—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2212yes—
27C2×#400C2211yescharacteristic
28C2×#399C2211yescharacteristic
29C2×#399C2211yescharacteristic
30C2×#398C2211yescharacteristic
31C2×#397C2214yes—
32C2×#397C2211yescharacteristic
33C2×#394C2211yescharacteristic
34C2×#394C2212yes—
35C2×#394C2214yes—
36C2×#392C2211yescharacteristic
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#387C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#383C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2212yes—
50#2275—28no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2214yes—
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×#376C2211yescharacteristic
58C22×#95C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C22⋊#95C2214yes—
86C4×#95C2212yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2314yes—
91C24×C4C2312yes—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—24no—
95C2×C4×D4—216no—
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D418yes—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—24no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135#400—28no—
136#400—24no—
137#400—24no—
138#399—24no—
139#399—24no—
140#398—24no—
141#397—216no—
142#397—24no—
143#394—28no—
144#394—24no—
145#394—216no—
146#392—24no—
147#389—28no—
148#388—216no—
149#388—28no—
150#388—28no—
151#387—28no—
152#386—28no—
153#386—28no—
154#386—48no—
155#385—28no—
156#385—28no—
157#383—24no—
158#380—28no—
159#380—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C22⋊C4—216no—
164C4×C22⋊C4—28no—
165C4×C22⋊C4—44no—
166#376D418yescomplemented
167C2×#95—28no—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2312yes—
196C25C2411yescharacteristic
197C25C2412yes—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4C2×D414yes—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—41no—
218C23×C4—41no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D418yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241#125—416no—
242C4×D4—264no—
243C4×D4—232no—
244#117—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
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—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D418yes—
314C2×C22⋊C4#41116yes—
315C2×C42—28no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C2×D418yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42#4114yes—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D414yes—
358#95C2×D418yes—
359#95C2×D414yes—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—24no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—24no—
373C24—21no—
374C24—21no—
375C24—24no—
376C24—21no—
377C24C22×D414yes—
378C24C2×#4118yes—
379C24H(5,𝔽2)12yes—
380C24C2511yescharacteristic
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—28no—
385C2×D4—48no—
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—84no—
389C2×D4—88no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—232no—
421C22×C4—216no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4118yes—
489C22×C42−1+412yes—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—88no—
502C4⋊C4—84no—
503C22⋊C4—216no—
504C22⋊C4—232no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—88no—
516C42—216no—
517C42—44no—
518C42—84no—
519C42—84no—
520C42—88no—
521C42#121116yes—
522C23—24no—
523C23—28no—
524C23—24no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—28no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23#53218yes—
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4114yes—
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+411yescharacteristic
568C23#58514yes—
569C23#58511yescharacteristic
570C23#58511yescharacteristic
571D4—232no—
572D4—816no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—816no—
652C2×C4—832no—
653C2×C4C2×#12114yes—
654C2×C4C2×#12114yes—
655C2×C4C2×#12114yes—
656C2×C4#52518yes—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—44no—
666C22—44no—
667C22—48no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22C22×#12111yescharacteristic
678C22C2×#52512yes—
679C22C2×#53212yes—
680C22#344412yes—
681C22#348512yes—
682C22#348811yescharacteristic
683C22#348811yescharacteristic
684C22#348911yescharacteristic
685C22#348911yescharacteristic
686C22#349211yescharacteristic
687C22#349311yescharacteristic
688C22#350012yes—
689C22#351012yes—
690C22#351314yes—
691C22#351514yes—
692C22#352114yes—
693C22#352114yes—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—28no—
698C4—44no—
699C4—48no—
700C4—44no—
701C4—42no—
702C4—84no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4—81no—
707C4—82no—
708C4—82no—
709C2—48no—
710C2—82no—
711C2#3540412yes—
712C2#3548512yes—
713C2#3551811yescharacteristic
714C2#3561711yescharacteristic
715C2#3585012yes—
716C2#3598212yes—
717C2#3604512yes—
718C2#3607012yes—
719C2#3662411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2562C2semidirect
2. C2×#2611C2semidirect
4. C2×#2275C2semidirect
5. #15175C2semidirect
6. #15156C2semidirect
7. #14868C2semidirect
8. #14840C2semidirect
9. #14839C2semidirect
10. #14826C2semidirect
12. #14735C2semidirect
13. #14289C2semidirect
14. #14209C2semidirect
15. #14182C2semidirect
16. #13952C2semidirect
17. #13852C2semidirect
166. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#26112
C2×#25982
C2×#22752
#151752
#151562
#148682
#148402
#148392
#148262
#148062
#147352
#142892
#142092
#141822
#139522
#138522