Finite Group Databasebeta

← #6848495

Subgroups — #6848495

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32166C214yesmaximal, complemented
5#32082C214yesmaximal, complemented
6#29624C212yesmaximal
7#28400C212yesmaximal, complemented
8#27749C212yesmaximal, complemented
9#27583C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25276C212yesmaximal, complemented
12#24630C212yesmaximal, complemented
13#24503C212yesmaximal
14#24209C211yesmaximal, characteristic
15#22599C211yesmaximal, characteristic, complemented
16#21804C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C2×#552—21no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#487C2212yes—
23C2×#475C2212yes—
24C2×#461C2212yes—
25C2×#448C2211yescharacteristic
26C2×#448C2212yes—
27C2×C4×SD16C2212yes—
28C2×#431C2212yes—
29#2857C2211yescharacteristic
30#2854C2212yes—
31#2552C2211yescharacteristic
32#2546—21no—
33#2546—21no—
34#2546—22no—
35#2546C2212yes—
36#2546C2214yes—
37#2546C2212yes—
38#2542C2211yescharacteristic
39#2519—22no—
40#2512C2212yes—
41#2506C2212yes—
42#2498—22no—
43#2489—21no—
44#2417—21no—
45C2×#398—22no—
46#2365C2211yescharacteristic
47#2362—21no—
48#2354C2211yescharacteristic
49C4×#121C2211yescharacteristic
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53#2285C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2096C2214yes—
56#2072C2214yes—
57#2066C2214yes—
58#2055C2214yes—
59#2043C2214yes—
60#2043C2212yes—
61#2043C2214yes—
62#2043C2214yes—
63#2013C2214yes—
64#2013C2212yes—
65#2013C2214yes—
66#1981C2214yes—
67#1981C2214yes—
68#1981C2212yes—
69#1968C2214yes—
70#1956C2212yes—
71#1956C2214yes—
72#1956C2214yes—
73#1939C2212yes—
74#1933C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1916C2214yes—
78#1880C2212yes—
79#1874C2211yescharacteristic
80#1865C2212yes—
81#1865C2214yes—
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1817C2212yes—
85#1803C2212yes—
86#1797C2211yescharacteristic
87#1787C2212yes—
88#1787C2212yes—
89#1787C2211yescharacteristic
90#1787C2212yes—
91#1787C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99#552—48no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8—44no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×#126—22no—
113C2×#125—21no—
114C2×#125—22no—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×#121C2311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—44no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141#487—44no—
142#475—44no—
143#461—44no—
144#448—24no—
145#448—44no—
146C4×SD16—44no—
147#431—44no—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#102C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2311yescharacteristic
167C2×C22⋊C8C2311yescharacteristic
168C2×C4×C8C2312yes—
169C2×C4×C8C2314yes—
170#400—22no—
171#399—22no—
172#398—22no—
173#398—24no—
174#398—24no—
175#398—24no—
176#398—24no—
177#398—22no—
178#397—22no—
179#396—22no—
180#391—21no—
181#391—21no—
182#391—22no—
183#391—22no—
184#391C2314yes—
185#391C2314yes—
186#390C2312yes—
187#390C2312yes—
188#390C2311yescharacteristic
189#389—21no—
190#389—22no—
191#389—24no—
192#389—22no—
193#389—22no—
194#389—24no—
195#389—22no—
196#389—21no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2311yescharacteristic
201#388—22no—
202#388—22no—
203#388—21no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#386—22no—
209#386—21no—
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2312yes—
213#382—22no—
214#380—21no—
215#380—22no—
216#380—21no—
217#380C2312yes—
218#380C2311yescharacteristic
219C4×C4⋊C4—22no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D412yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—21no—
230C2×#95—22no—
231C2×#95—22no—
232C2×#95D412yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239#336C2312yes—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—42no—
247C22×Q8—42no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8#4114yes—
251C22×Q8C2414yes—
252C22×Q8C2412yes—
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×SD16—24no—
265C2×SD16—42no—
266C2×SD16—42no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—28no—
274C4⋊Q8—28no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—48no—
282C4⋊Q8—48no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285#126—48no—
286#125—48no—
287#125—44no—
288#123—48no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C22⋊Q8—44no—
292#121—22no—
293#121—22no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—24no—
302C4×D4—44no—
303#117—48no—
304#117—44no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4SD1618yescomplemented
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374#106—42no—
375#106—44no—
376C4⋊C8—24no—
377C4⋊C8—44no—
378#103—24no—
379#103—42no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—42no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389#103—44no—
390#102—42no—
391C22⋊C8—22no—
392C22⋊C8—22no—
393C22⋊C8—24no—
394C4×C8—24no—
395C4×C8—44no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95C2×D412yes—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413C24—21no—
414C24C22×D411yescharacteristic
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—48no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—88no—
438C2×Q8—88no—
439C2×Q8C2×#4114yes—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4C2×SD1618yes—
480C22×C4C2×SD1614yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4C2511yescharacteristic
493SD16—48no—
494SD16—88no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—82no—
535C4⋊C4—81no—
536C4⋊C4C2×SD1614yes—
537C4⋊C4C2×SD1614yes—
538C4⋊C4C2×SD1618yes—
539C4⋊C4#13714yes—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—81no—
549C22⋊C4—82no—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—42no—
564C42H(5,𝔽2)14yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23#54612yes—
577C23C22×SD1612yes—
578C23#57812yes—
579C23C23×D411yescharacteristic
580Q8—24no—
581Q8—24no—
582Q8—48no—
583Q8—42no—
584Q8—42no—
585Q8—44no—
586Q8—44no—
587Q8—44no—
588Q8—44no—
589Q8—88no—
590Q8—84no—
591D4—42no—
592D4—42no—
593D4—84no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—216no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—84no—
644C2×C4—88no—
645C2×C4—84no—
646C2×C4#53512yes—
647C2×C4#53512yes—
648C2×C4#54612yes—
649C2×C4C22×SD1614yes—
650C2×C4C22×SD1612yes—
651C2×C4C22×SD1614yes—
652C2×C4C22×SD1612yes—
653C2×C4C2×#13714yes—
654C2×C4C2×#13712yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4#58512yes—
659C8—42no—
660C8—42no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22#329814yes—
676C22C2×#53511yescharacteristic
677C22C2×#54611yescharacteristic
678C22#348711yescharacteristic
679C22C23×SD1611yescharacteristic
680C22C22×#13711yescharacteristic
681C22C2×#57811yescharacteristic
682C22#358611yescharacteristic
683C4—22no—
684C4—22no—
685C4—22no—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—44no—
690C4—41no—
691C4—42no—
692C4—84no—
693C4—81no—
694C4—82no—
695C4—82no—
696C4—81no—
697C4—82no—
698C4—81no—
699C4#320614yes—
700C4#350512yes—
701C4#354212yes—
702C2—24no—
703C2—161no—
704C2#3946711yescharacteristic
705C2C2×#320611yescharacteristic
706C2#6081411yescharacteristic
707C2C2×#329811yescharacteristic
708C2#6115711yescharacteristic
709C2#6188311yescharacteristic
710C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
3. #32954C2semidirect
4. #32166C2semidirect
5. #32082C2semidirect
7. #28400C2semidirect
8. #27749C2semidirect
9. #27583C2semidirect
10. #25856C2semidirect
11. #25276C2semidirect
12. #24630C2semidirect
15. #22599C2semidirect
16. #21804C2semidirect
334. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#25462
#329542
#321662
#320822
#296242
#284002
#277492
#275832
#258562
#252762
#246302
#245032
#242092
#225992
#218042
C4×#4482