Finite Group Databasebeta

← #6792524

Subgroups — #6792524

Order 512.

1753 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic
4#31647C214yesmaximal, complemented
5#31531C214yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#28402C212yesmaximal, complemented
8#27816C212yesmaximal, complemented
9#25903C211yesmaximal, characteristic
10#25325C212yesmaximal, complemented
11#24818C212yesmaximal, complemented
12#24503C212yesmaximal, complemented
13#24301C211yesmaximal, characteristic, complemented
14#23937C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21861C212yesmaximal, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#486C2212yes—
22C2×#475C2212yes—
23C2×#460C2212yes—
24C2×#451C2211yescharacteristic
25C2×#448C2212yes—
26C2×C4×Q16C2212yes—
27C2×#431C2212yes—
28#2856C2211yescharacteristic
29#2854C2212yes—
30#2568—21no—
31#2568—22no—
32#2568—21no—
33#2568C2214yes—
34#2568C2212yes—
35#2568C2212yes—
36#2552C2211yescharacteristic
37#2548C2211yescharacteristic
38#2512—21no—
39#2501—21no—
40#2479—22no—
41#2454C2212yes—
42#2448C2212yes—
43#2425—22no—
44C2×#400—22no—
45#2366C2211yescharacteristic
46#2361—21no—
47#2354—21no—
48Q8×C4⋊C4C2211yescharacteristic
49C4×C22⋊Q8C2211yescharacteristic
50C2×#391—21no—
51C2×#391—21no—
52C2×#391C2212yes—
53C2×#382C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2101C2214yes—
56#2092C2214yes—
57#2085C2214yes—
58#2073C2214yes—
59#2066C2214yes—
60#2058C2212yes—
61#2055C2214yes—
62#2026C2214yes—
63#2024C2212yes—
64#2023C2214yes—
65#2015C2214yes—
66#1991C2214yes—
67#1990C2212yes—
68#1988C2212yes—
69#1968C2214yes—
70#1958C2214yes—
71#1939C2212yes—
72#1933C2214yes—
73#1916C2214yes—
74#1880C2214yes—
75#1878C2214yes—
76#1878C2212yes—
77#1877C2214yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2212yes—
81#1872C2211yescharacteristic
82#1872C2212yes—
83#1863C2212yes—
84#1853C2212yes—
85#1852C2211yescharacteristic
86#1817C2212yes—
87#1805C2214yes—
88#1804C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105#529—28no—
106#529—44no—
107C2×#126—22no—
108C2×#125C2312yes—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—44no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#486—44no—
137#475—44no—
138#460—44no—
139#451—24no—
140#448—44no—
141C4×Q16—44no—
142#431—44no—
143C2×#107C2312yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2314yes—
161#400—24no—
162#400—24no—
163#400—24no—
164#400—24no—
165#399—22no—
166#399—22no—
167#398—22no—
168#397—22no—
169#396—22no—
170#394—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—24no—
176#391—24no—
177#391—22no—
178#391—22no—
179#391C2314yes—
180#391C2314yes—
181#391C2311yescharacteristic
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390C2312yes—
188#390C2312yes—
189#390C2312yes—
190#389—21no—
191#389—22no—
192#389C2311yescharacteristic
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—21no—
200#386—22no—
201#386—22no—
202#385—21no—
203#385—22no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#382—22no—
208#382—22no—
209#382C2314yes—
210#382C2314yes—
211#380—22no—
212#380C2311yescharacteristic
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C2×#95—22no—
224C2×#95—22no—
225C2×#95—22no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8C2411yescharacteristic
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2412yes—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—42no—
267C2×SD16—28no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#127—24no—
279#126—48no—
280#125—24no—
281#125—42no—
282#124—24no—
283#124—42no—
284#123—48no—
285#123—84no—
286C22⋊Q8—22no—
287C22⋊Q8#4114yes—
288#121—24no—
289#121—24no—
290#121—44no—
291#121—82no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—42no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—24no—
302C4×D4—42no—
303C4×D4—48no—
304C4×D4—44no—
305#117—48no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—44no—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369#107—42no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—44no—
373#103—24no—
374#103—24no—
375#103—42no—
376#103—42no—
377#103—44no—
378#103—44no—
379#102—44no—
380#102—44no—
381#102—44no—
382C22⋊C8—22no—
383C22⋊C8—22no—
384C22⋊C8—24no—
385C4×C8—24no—
386C4×C8—44no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—22no—
407C24C22×D411yescharacteristic
408C2×Q8—24no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—48no—
418C2×Q8—48no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—816no—
428C2×Q8C2×#4112yes—
429C2×Q82−1+414yes—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—81no—
442C2×D4—81no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—82no—
476C22×C4#13514yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4C2511yescharacteristic
489Q16—48no—
490SD16—88no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—81no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4C2×#4112yes—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42H(5,𝔽2)14yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23#54612yes—
571C23C2×#13512yes—
572C23#57712yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—24no—
576Q8—24no—
577Q8—48no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—88no—
582Q8—88no—
583D4—44no—
584D4—44no—
585D4—88no—
586D4—84no—
587D4—162no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—88no—
641C2×C4—84no—
642C2×C4—88no—
643C2×C4#53512yes—
644C2×C4#53512yes—
645C2×C4#54612yes—
646C2×C4C2×#13512yes—
647C2×C4C2×#13512yes—
648C2×C4#57512yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+412yes—
653C8—42no—
654C8—42no—
655C8—44no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22#330414yes—
670C22C2×#53511yescharacteristic
671C22C2×#54611yescharacteristic
672C22#348611yescharacteristic
673C22C22×#13511yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57711yescharacteristic
676C22#358611yescharacteristic
677C4—22no—
678C4—22no—
679C4—22no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—84no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4#321114yes—
694C4#352012yes—
695C4#353612yes—
696C2—24no—
697C2—162no—
698C2#3890511yescharacteristic
699C2C2×#321111yescharacteristic
700C2#6081311yescharacteristic
701C2C2×#330411yescharacteristic
702C2#6116511yescharacteristic
703C2#6182211yescharacteristic
704C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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×#2568C2semidirect
4. #31647C2semidirect
5. #31531C2semidirect
6. #29912C2semidirect
7. #28402C2semidirect
8. #27816C2semidirect
10. #25325C2semidirect
11. #24818C2semidirect
12. #24503C2semidirect
13. #24301C2semidirect
14. #23937C2semidirect
15. #22599C2semidirect
16. #21861C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#361172
C2×#25682
#329032
#316472
#315312
#299122
#284022
#278162
#259032
#253252
#248182
#245032
#243012
#239372
#225992
#218612
C4×#4512