Finite Group Databasebeta

← #6513860

Subgroups — #6513860

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34976C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#27682C212yesmaximal, complemented
4#27646C212yesmaximal, complemented
5#27629C212yesmaximal, complemented
6#27624C212yesmaximal, complemented
7#25063C212yesmaximal, complemented
8#25011C212yesmaximal, complemented
9#24993C212yesmaximal, complemented
10#24970C212yesmaximal, complemented
11#24963C214yesmaximal, complemented
12#24889C214yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21964C211yesmaximal, characteristic, complemented
15#21849C211yesmaximal, characteristic, complemented
16#20737C211yesmaximal, characteristic, complemented
17#20648C211yesmaximal, characteristic, complemented
18C2×C4×#41C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#482C2214yes—
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417C2212yes—
24#2609—22no—
25#2582C2211yescharacteristic
26#2538C2211yescharacteristic
27#2538C2211yescharacteristic
28#2537C2211yescharacteristic
29#2485—21no—
30#2485C2212yes—
31#2407—21no—
32#2407C2212yes—
33#2392—22no—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40#2314—22no—
41#2311—21no—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8C2212yes—
44C4×#121—21no—
45C4×#121C2212yes—
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2211yescharacteristic
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51#2274—21no—
52#2101C2214yes—
53#2097C2214yes—
54#2085C2214yes—
55#2076C2214yes—
56#2075C2214yes—
57#2073C2214yes—
58#2036C2214yes—
59#2033C2214yes—
60#1984C2212yes—
61#1954C2212yes—
62#1948C2212yes—
63#1948C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1928C2214yes—
69#1925C2214yes—
70#1921C2214yes—
71#1918C2214yes—
72#1873C2214yes—
73#1870C2214yes—
74#1854C2214yes—
75#1852C2214yes—
76#1845C2212yes—
77#1834C2212yes—
78#1825C2212yes—
79#1825C2214yes—
80#1809C2214yes—
81#1803C2211yescharacteristic
82#1803C2211yescharacteristic
83#1803C2212yes—
84#1803C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1797C2212yes—
90#1797C2212yes—
91#1788C2212yes—
92C4×#103C2212yes—
93C4×#102C2212yes—
94#1736C2212yes—
95C22×#41D411yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×#121—21no—
101C2×#121C2312yes—
102C4×#41—24no—
103C4×#41—24no—
104C4×#41—24no—
105C4×#41—24no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—22no—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117D412yes—
124C2×#117D411yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2314yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C42—21no—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#482—48no—
134#431—24no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139#423—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144#417—44no—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155#398—22no—
156#397—22no—
157#396—22no—
158#396—22no—
159#394—22no—
160#394—22no—
161#394—22no—
162#392—22no—
163#391—22no—
164#391—21no—
165#391C2312yes—
166#390—22no—
167#390—21no—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389C2312yes—
172#389C2312yes—
173#388—24no—
174#388—24no—
175#388—22no—
176#388—22no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—22no—
181#386—24no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—21no—
195#382—22no—
196#382—21no—
197#382C2314yes—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—24no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41—41no—
244C2×#41—41no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—22no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D412yes—
251C22×D4C2411yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—42no—
264C2×M16—42no—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22⋊Q8—24no—
269C22⋊Q8—42no—
270C22⋊Q8—44no—
271#121—24no—
272#121—42no—
273#121—44no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—84no—
294#117—24no—
295#117—28no—
296#117—22no—
297#117—22no—
298#117—44no—
299#117—44no—
300#117—44no—
301#117—44no—
302#117—82no—
303#117—84no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—44no—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42#4114yes—
368C2×C42#4114yes—
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—44no—
373#107—44no—
374#107—44no—
375#106—44no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378C4⋊C8—22no—
379C4⋊C8—22no—
380#103—24no—
381#103—42no—
382#103—44no—
383#102—28no—
384#102—24no—
385#102—42no—
386C22⋊C8—24no—
387C22⋊C8—44no—
388C22⋊C8—44no—
389C4×C8—24no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—22no—
404C24—21no—
405C24C22×D412yes—
406C24C22×D411yescharacteristic
407#41—28no—
408#41—44no—
409#41—44no—
410#41—416no—
411#41—44no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q82−1+412yes—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—82no—
438C2×D4H(5,𝔽2)12yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4#12314yes—
480C22×C4#12314yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2511yescharacteristic
492M16—84no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C4⋊C4—82no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—82no—
550C22⋊C4—84no—
551C22⋊C4—82no—
552C22⋊C4—84no—
553C22⋊C4—82no—
554C42—24no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—42no—
559C42—42no—
560C42—42no—
561C42—44no—
562C42—44no—
563C42—84no—
564C42—84no—
565C42#12314yes—
566C42#12314yes—
567C42C2×#4112yes—
568C42C2×#4112yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23C2×#12312yes—
592C23C23×D411yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—44no—
596Q8—41no—
597Q8—41no—
598D4—22no—
599D4—22no—
600D4—48no—
601D4—48no—
602D4—44no—
603D4—41no—
604D4—41no—
605D4—88no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—48no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—164no—
661C2×C4C2×#12312yes—
662C2×C4C2×#12312yes—
663C2×C4C2×#12312yes—
664C2×C4C2×#12312yes—
665C2×C4C2×#12312yes—
666C2×C4#57512yes—
667C2×C4#57512yes—
668C2×C4#57612yes—
669C2×C4#57812yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C2×C4#58511yescharacteristic
675C2×C4#58511yescharacteristic
676C8—42no—
677C8—42no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22C22×#12311yescharacteristic
701C22#348811yescharacteristic
702C22#348911yescharacteristic
703C22C2×#57511yescharacteristic
704C22C2×#57511yescharacteristic
705C22C2×#57611yescharacteristic
706C22C2×#57811yescharacteristic
707C4—22no—
708C4—22no—
709C4—22no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—82no—
715C4—84no—
716C4—82no—
717C4—81no—
718C4—81no—
719C4—82no—
720C4—82no—
721C4#309314yes—
722C4#309814yes—
723C4#349211yescharacteristic
724C4#349311yescharacteristic
725C4#349411yescharacteristic
726C4#349511yescharacteristic
727C2—42no—
728C2—82no—
729C2—82no—
730C2#3551611yescharacteristic
731C2C2×#309311yescharacteristic
732C2C2×#309811yescharacteristic
733C2#6198511yescharacteristic
734C2#6199511yescharacteristic
735C2#6213611yescharacteristic
736C2#6214611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34976C2
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
2. #34188C2semidirect
3. #27682C2semidirect
4. #27646C2semidirect
5. #27629C2semidirect
6. #27624C2semidirect
7. #25063C2semidirect
8. #25011C2semidirect
9. #24993C2semidirect
10. #24970C2semidirect
11. #24963C2semidirect
12. #24889C2semidirect
13. #21982C2semidirect
14. #21964C2semidirect
15. #21849C2semidirect
16. #20737C2semidirect
17. #20648C2semidirect

Maximal subgroups

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

subgroupindex
#349762
#341882
#276822
#276462
#276292
#276242
#250632
#250112
#249932
#249702
#249632
#248892
#219822
#219642
#218492
#207372
#206482