Finite Group Databasebeta

← #6492068

Subgroups — #6492068

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34127C211yesmaximal, characteristic
3#33109C212yesmaximal, complemented
4#33108C212yesmaximal, complemented
5#29926C212yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#26319C212yesmaximal, complemented
8#26317C212yesmaximal, complemented
9#26236C212yesmaximal, complemented
10#26234C212yesmaximal, complemented
11#24884C214yesmaximal, complemented
12#24880C214yesmaximal, complemented
13#21964C211yesmaximal, characteristic, complemented
14#21849C211yesmaximal, characteristic
15#21804C211yesmaximal, characteristic, complemented
16#21362C211yesmaximal, characteristic, complemented
17#21259C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#484C2214yes—
21C2×#483C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#419C2212yes—
25C2×#419C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2628—22no—
29#2568C2212yes—
30#2567—21no—
31#2563—21no—
32#2542C2212yes—
33#2487—22no—
34#2409C2214yes—
35#2355C2211yescharacteristic
36#2355C2211yescharacteristic
37#2352C2212yes—
38#2352C2212yes—
39D4×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4C2211yescharacteristic
41#2347—21no—
42#2343—22no—
43#2339—21no—
44#2317—21no—
45#2311—21no—
46#2295—21no—
47#2290—22no—
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50#2283C2212yes—
51C42×D4—21no—
52C4×#117C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54#2097C2214yes—
55#2097C2214yes—
56#2074C2214yes—
57#2074C2214yes—
58#2036C2214yes—
59#2036C2214yes—
60#2035C2214yes—
61#2035C2214yes—
62#1984C2212yes—
63#1981C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1956C2212yes—
70#1956C2212yes—
71#1954C2212yes—
72#1948C2211yescharacteristic
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1948C2212yes—
76#1947C2211yescharacteristic
77#1947C2212yes—
78#1947C2211yescharacteristic
79#1947C2212yes—
80#1928C2214yes—
81#1928C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1872C2214yes—
85#1872C2214yes—
86#1870C2214yes—
87#1854C2212yes—
88#1854C2212yes—
89#1809C2214yes—
90#1797C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×#107C2212yes—
94C4×#106C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8C2312yes—
97C22×M16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D412yes—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C2×#117—21no—
118C2×#117D411yescharacteristic
119C2×#117D412yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C42—22no—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#484—48no—
130#483—48no—
131#431—24no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138#423—24no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143#419—44no—
144#419—44no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157#400—24no—
158#399—24no—
159#395—24no—
160#393—24no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391C2314yes—
165#391C2314yes—
166#390—22no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#386—24no—
180#386—22no—
181#386—22no—
182#385—24no—
183#385—22no—
184#385—22no—
185#384—22no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#383—22no—
191#382—22no—
192#382—21no—
193#382—21no—
194#382C2312yes—
195#382C2312yes—
196#381—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226#376—21no—
227#376—22no—
228C2×#95—22no—
229C43—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—42no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8—42no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2414yes—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—42no—
263C2×M16—42no—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C4×Q8—24no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—44no—
287C4×D4—44no—
288#117—24no—
289#117—28no—
290#117—22no—
291#117—22no—
292#117—41no—
293#117—41no—
294#117—44no—
295#117—82no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—84no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—42no—
365#107—44no—
366#106—24no—
367#106—42no—
368#106—44no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373#103—28no—
374#103—28no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381C22⋊C8—24no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384C4×C8—24no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405C24C22×D411yescharacteristic
406C24C22×D412yes—
407#41—416no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—48no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8—84no—
423C2×Q8—84no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4#13714yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4#12314yes—
469C22×C4#12314yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481M16—84no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—81no—
522C4⋊C4—81no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4H(5,𝔽2)12yes—
526C4⋊C4H(5,𝔽2)12yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—82no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—42no—
547C42—42no—
548C42—44no—
549C42—42no—
550C42—82no—
551C42—82no—
552C42#12314yes—
553C42#12314yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C23—22no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—22no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—21no—
570C23C2×#12312yes—
571C23C2×#13714yes—
572C23C23×D411yescharacteristic
573Q8—44no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—44no—
578Q8—88no—
579D4—28no—
580D4—48no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4C2×#12312yes—
639C2×C4C2×#12312yes—
640C2×C4C2×#12312yes—
641C2×C4C2×#12312yes—
642C2×C4C2×#12312yes—
643C2×C4C2×#13712yes—
644C2×C4#57512yes—
645C2×C4#57712yes—
646C2×C4#57812yes—
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4#58511yescharacteristic
652C2×C4#58511yescharacteristic
653C8—42no—
654C8—42no—
655C8—44no—
656C22—24no—
657C22—24no—
658C22—22no—
659C22—24no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—42no—
665C22—41no—
666C22—41no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22C22×#12311yescharacteristic
671C22#348811yescharacteristic
672C22#348811yescharacteristic
673C22C22×#13711yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57711yescharacteristic
676C22C2×#57811yescharacteristic
677C4—22no—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—82no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4—81no—
690C4—82no—
691C4—84no—
692C4#309514yes—
693C4#309714yes—
694C4#346011yescharacteristic
695C4#346011yescharacteristic
696C4#350811yescharacteristic
697C4#351011yescharacteristic
698C2—44no—
699C2—42no—
700C2#3525411yescharacteristic
701C2C2×#309511yescharacteristic
702C2C2×#309711yescharacteristic
703C2#6198011yescharacteristic
704C2#6198411yescharacteristic
705C2#6199611yescharacteristic
706C2#6199811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33109C2semidirect
4. #33108C2semidirect
5. #29926C2semidirect
6. #29926C2semidirect
7. #26319C2semidirect
8. #26317C2semidirect
9. #26236C2semidirect
10. #26234C2semidirect
11. #24884C2semidirect
12. #24880C2semidirect
13. #21964C2semidirect
15. #21804C2semidirect
16. #21362C2semidirect
17. #21259C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#341272
#331092
#331082
#299262
#263192
#263172
#262362
#262342
#248842
#248802
#219642
#218492
#218042
#213622
#212592