Finite Group Databasebeta

← #769183

Subgroups — #769183

Order 512.

2918 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2537C211yesmaximal, characteristic
3#15882C214yesmaximal, complemented
4#15788C214yesmaximal, complemented
5#15777C214yesmaximal, complemented
6#15319C214yesmaximal, complemented
7#15183C212yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#13998C212yesmaximal, complemented
10#13941C211yesmaximal, characteristic, complemented
11#13905C211yesmaximal, characteristic, complemented
12#13876C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic, complemented
14#13785C211yesmaximal, characteristic, complemented
15#13779C211yesmaximal, characteristic, complemented
16C22×#125C2212yes—
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2552—28no—
22#2537—28no—
23C2×#398C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2214yes—
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#384C2212yes—
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2211yescharacteristic
47#1445C2214yes—
48#1445C2212yes—
49#1445C2212yes—
50#1443C2214yes—
51#1442C2214yes—
52#1440C2214yes—
53#1439C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1438C2212yes—
63#1438C2212yes—
64#1438C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2314yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C2×#125—28no—
83C2×#125—48no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—48no—
88C2×#121—44no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128#398—216no—
129#394—216no—
130#394—48no—
131#394—48no—
132#390—24no—
133#390—24no—
134#390—44no—
135#390—48no—
136#389—28no—
137#389—28no—
138#388—48no—
139#384—28no—
140#384D418yes—
141#382—24no—
142#380—28no—
143#380—28no—
144#380—44no—
145#380—44no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—44no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—44no—
151#376—24no—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
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×#95C2312yes—
172C2×#95C2312yes—
173C25C2412yes—
174C25C2412yes—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—28no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4#4114yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#125—432no—
220#121—432no—
221#121—432no—
222C4×D4—416no—
223C4×D4—416no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D418yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
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×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C2×D414yes—
315C2×C42#4114yes—
316C2×C42#4114yes—
317C2×C42#4114yes—
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—44no—
329#95—44no—
330#95—42no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24C2×#4114yes—
364C24H(5,𝔽2)14yes—
365C24C2511yescharacteristic
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
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—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4C22×D418yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+414yes—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—416no—
484C4⋊C4—416no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—28no—
506C42—48no—
507C42—48no—
508C42—44no—
509C42—88no—
510C42#124116yes—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23C23×D411yescharacteristic
570C23C22×#4111yescharacteristic
571C23C22×#4111yescharacteristic
572C23C22×#4111yescharacteristic
573C23C22×#4111yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23C2×2−1+412yes—
579C23#58512yes—
580D4—416no—
581D4—416no—
582D4—832no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—816no—
656C2×C4—816no—
657C2×C4C2×#12414yes—
658C2×C4C2×#12414yes—
659C2×C4C2×#12414yes—
660C2×C4#52914yes—
661C2×C4#54014yes—
662C2×C4#54614yes—
663C2×C4#55014yes—
664C2×C4#55314yes—
665C2×C4#55314yes—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—48no—
677C22—44no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—84no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22—82no—
692C22C22×#12411yescharacteristic
693C22C2×#52911yescharacteristic
694C22C2×#54011yescharacteristic
695C22C2×#54611yescharacteristic
696C22C2×#55011yescharacteristic
697C22C2×#55311yescharacteristic
698C22C2×#55311yescharacteristic
699C22#348411yescharacteristic
700C22#348411yescharacteristic
701C22#348611yescharacteristic
702C22#348711yescharacteristic
703C22#349212yes—
704C22#349412yes—
705C22#349412yes—
706C22#349412yes—
707C22#351112yes—
708C22#351112yes—
709C22#351512yes—
710C22#351912yes—
711C22#353612yes—
712C22#354212yes—
713C22#354312yes—
714C22#355412yes—
715C4—24no—
716C4—24no—
717C4—44no—
718C4—44no—
719C4—48no—
720C4—44no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—84no—
726C4—82no—
727C4#249018yes—
728C2—84no—
729C2—84no—
730C2C2×#249011yescharacteristic
731C2#3555112yes—
732C2#3558411yescharacteristic
733C2#3559211yescharacteristic
734C2#3734812yes—
735C2#4493311yescharacteristic
736C2#4493611yescharacteristic
737C2#4497412yes—
738C2#4503512yes—
739C2#4533511yescharacteristic
740C2#4535611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2552C2semidirect
3. #15882C2semidirect
4. #15788C2semidirect
5. #15777C2semidirect
6. #15319C2semidirect
7. #15183C2semidirect
8. #15130C2semidirect
9. #13998C2semidirect
10. #13941C2semidirect
11. #13905C2semidirect
12. #13876C2semidirect
13. #13847C2semidirect
14. #13785C2semidirect
15. #13779C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#25372
#158822
#157882
#157772
#153192
#151832
#151302
#139982
#139412
#139052
#138762
#138472
#137852
#137792