Finite Group Databasebeta

← #7510596

Subgroups — #7510596

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34955C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32647C212yesmaximal, complemented
4#32645C212yesmaximal, complemented
5#32638C214yesmaximal, complemented
6#32636C214yesmaximal, complemented
7#32631C212yesmaximal, complemented
8#32629C212yesmaximal, complemented
9#23617C212yesmaximal, complemented
10#23508C212yesmaximal
11#23452C212yesmaximal, complemented
12#21303C211yesmaximal, characteristic, complemented
13#21093C212yesmaximal, complemented
14#21059C212yesmaximal, complemented
15#21034C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#428C2212yes—
20C2×#423C2211yescharacteristic
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#417C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2592C2212yes—
27#2592C2212yes—
28#2591—21no—
29#2591—21no—
30#2568C2211yescharacteristic
31#2568C2211yescharacteristic
32#2567—21no—
33#2567—21no—
34#2567—21no—
35#2567—21no—
36#2567C2212yes—
37#2567C2212yes—
38#2567C2212yes—
39#2567C2212yes—
40#2548C2211yescharacteristic
41#2548C2211yescharacteristic
42#2504—21no—
43#2436—21no—
44#2398—21no—
45#2398—21no—
46C2×#395—21no—
47C2×#395—21no—
48C4×#124—21no—
49C4×#124—21no—
50#2297C2212yes—
51#2297C2212yes—
52C2×#391—21no—
53C2×#389C2212yes—
54C2×#389C2212yes—
55C2×#384—21no—
56C4×#117C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2068C2214yes—
59#2067C2214yes—
60#2066C2214yes—
61#2065C2214yes—
62#2043C2214yes—
63#2042C2214yes—
64#2042C2214yes—
65#2041C2214yes—
66#2024C2214yes—
67#2024C2214yes—
68#2023C2214yes—
69#2023C2214yes—
70#1985C2212yes—
71#1985C2212yes—
72#1984C2212yes—
73#1984C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1934C2214yes—
77#1933C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1926C2214yes—
81#1925C2214yes—
82#1921C2214yes—
83#1920C2214yes—
84#1919C2214yes—
85#1918C2214yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1728C2212yes—
90#1728C2212yes—
91C22×#41C2312yes—
92C23×Q8—21no—
93C23×Q8D411yescharacteristic
94C22×M16C2312yes—
95C22×M16C2312yes—
96#532—42no—
97#532—42no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#124—21no—
107C2×#124—21no—
108C2×#124C2312yes—
109C2×#121—21no—
110C2×#121—21no—
111C2×C4×Q8—42no—
112C2×C4×Q8—42no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#428—28no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139#423—42no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142#417—28no—
143#417—28no—
144#417—44no—
145#417—44no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C4×M16—42no—
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157#400—21no—
158#400—21no—
159#400—21no—
160#400—21no—
161#397—22no—
162#397—22no—
163#395—22no—
164#395—22no—
165#395—22no—
166#395—22no—
167#395—22no—
168#395—22no—
169#395—22no—
170#395—22no—
171#394—21no—
172#394—21no—
173#394—21no—
174#394—21no—
175#393—22no—
176#393—22no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391—21no—
182#391—21no—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389C2314yes—
198#389C2314yes—
199#389C2314yes—
200#389C2314yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#386—22no—
206#386—22no—
207#386—22no—
208#386—22no—
209#384—22no—
210#384—22no—
211#384—22no—
212#384—22no—
213#382—21no—
214#382—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228#376—21no—
229C2×#95—21no—
230C2×#95—22no—
231C2×#95—22no—
232C43—21no—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—24no—
248C2×#41—24no—
249C22×Q8—24no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—42no—
253C22×Q8—42no—
254C22×Q8—44no—
255C22×Q8—84no—
256C22×Q8C2×D411yescharacteristic
257C22×Q8C2×D411yescharacteristic
258C22×Q8C2×D411yescharacteristic
259C22×Q8C2×D411yescharacteristic
260C22×Q8C2412yes—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4C2412yes—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2411yescharacteristic
275C2×M16—24no—
276C2×M16—24no—
277C2×M16—24no—
278C2×M16—24no—
279C2×M16—24no—
280C2×M16—24no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C4⋊Q8—42no—
286C4⋊Q8—42no—
287C4⋊Q8—41no—
288C4⋊Q8—41no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291#127—41no—
292#127—41no—
293#124—42no—
294#124—42no—
295#124—44no—
296#124—44no—
297#121—44no—
298#121—44no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—42no—
302C4×Q8—42no—
303C4×Q8—41no—
304C4×Q8—41no—
305C4×Q8—41no—
306C4×Q8—41no—
307C4×Q8—41no—
308C4×Q8—41no—
309C4×Q8—41no—
310C4×Q8—41no—
311C4×D4—44no—
312C4×D4—44no—
313#117—24no—
314#117—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4—84no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4#4118yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2414yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—44no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#106—28no—
389C4⋊C8—42no—
390C4⋊C8—42no—
391#103—28no—
392#103—28no—
393#103—44no—
394#103—44no—
395#102—28no—
396#102—28no—
397#102—44no—
398#102—44no—
399#97—42no—
400C4×C8—42no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—21no—
409#95—21no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418C24—22no—
419C24C22×D411yescharacteristic
420#41—48no—
421#41—48no—
422#41—48no—
423#41—48no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—41no—
437C2×Q8—41no—
438C2×Q8—41no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×Q8—416no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—84no—
445C2×Q8—84no—
446C2×Q8—816no—
447C2×Q8#13714yes—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—82no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—28no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4114yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C42−1+414yes—
508C22×C4C2511yescharacteristic
509M16—48no—
510M16—48no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C4⋊C4—28no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4C2×#4118yes—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—21no—
576C42—44no—
577C42—42no—
578C42—44no—
579C42—42no—
580C42—44no—
581C42—82no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23#53412yes—
593C23#54814yes—
594C23C23×D411yescharacteristic
595Q8—44no—
596Q8—44no—
597Q8—44no—
598Q8—44no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—88no—
604Q8—88no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—84no—
610D4—84no—
611D4—88no—
612C2×C4—28no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—28no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—416no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—82no—
659C2×C4—82no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4#53412yes—
663C2×C4#54218yes—
664C2×C4#54814yes—
665C2×C4#55514yes—
666C2×C4#56314yes—
667C2×C4C2×#13712yes—
668C2×C4C2×#13712yes—
669C2×C4C2×#13712yes—
670C2×C4#57512yes—
671C2×C4C22×#4112yes—
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×2−1+412yes—
675C8—44no—
676C8—44no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—24no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#273114yes—
689C22C2×#53411yescharacteristic
690C22C2×#54812yes—
691C22C22×#13711yescharacteristic
692C22C2×#57511yescharacteristic
693C22#358512yes—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—48no—
707C4—84no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#273114yes—
712C4C2×#54212yes—
713C4C2×#55511yescharacteristic
714C4C2×#56311yescharacteristic
715C2—24no—
716C2—162no—
717C2C2×#273111yescharacteristic
718C2#6075111yescharacteristic
719C2#6075311yescharacteristic
720C2#6131812yes—
721C2#6131912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. #32647C2semidirect
4. #32645C2semidirect
5. #32638C2semidirect
6. #32636C2semidirect
7. #32631C2semidirect
8. #32629C2semidirect
9. #23617C2semidirect
11. #23452C2semidirect
12. #21303C2semidirect
13. #21093C2semidirect
14. #21059C2semidirect
15. #21034C2semidirect

Maximal subgroups

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

subgroupindex
#349552
C2×#25672
#326472
#326452
#326382
#326362
#326312
#326292
#236172
#235082
#234522
#213032
#210932
#210592
#210342