Finite Group Databasebeta

← #6433536

Subgroups — #6433536

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic, complemented
2#34267C211yesmaximal, characteristic
3#30183C214yesmaximal, complemented
4#30177C214yesmaximal, complemented
5#30176C212yesmaximal, complemented
6#30172C212yesmaximal, complemented
7#30169C212yesmaximal, complemented
8#30165C212yesmaximal, complemented
9#23657C211yesmaximal, characteristic, complemented
10#23639C212yesmaximal, complemented
11#23627C211yesmaximal, characteristic, complemented
12C2×#2057C211yesmaximal, characteristic, complemented
13#23485C212yesmaximal
14#23452C211yesmaximal, characteristic, complemented
15#21035C212yesmaximal, complemented
16#20881C212yesmaximal, complemented
17#20684C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C22×#107C2212yes—
24C2×#417C2212yes—
25C2×#416C2212yes—
26C2×#408C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2628C2214yes—
30#2568—21no—
31#2567—21no—
32#2563—21no—
33#2542—21no—
34#2487—22no—
35#2409—22no—
36#2364—22no—
37#2363—21no—
38#2362—21no—
39#2361—22no—
40#2361—21no—
41#2360—21no—
42#2324C2212yes—
43#2323C2212yes—
44#2317C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×#121C2212yes—
49#2297C2211yescharacteristic
50C2×#389C2211yescharacteristic
51#2285C2211yescharacteristic
52#2274C2211yescharacteristic
53#2085C2214yes—
54#2084C2212yes—
55#2083C2212yes—
56#2076C2214yes—
57#2076C2214yes—
58#2075C2214yes—
59#2075C2214yes—
60#2074C2212yes—
61#2073C2212yes—
62#2072C2211yescharacteristic
63#2072C2211yescharacteristic
64#2071C2211yescharacteristic
65#2071C2211yescharacteristic
66#2057C2212yes—
67#2057C2212yes—
68#2057C2214yes—
69#2036C2214yes—
70#2035C2214yes—
71#2034C2214yes—
72#2033C2214yes—
73#2024C2212yes—
74#2024C2212yes—
75#2023C2212yes—
76#2023C2212yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1818C2214yes—
86#1817C2212yes—
87#1817C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8—21no—
98C22×M16C2312yes—
99C23×C8C2312yes—
100#537—24no—
101#537—24no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121C2312yes—
114#520—44no—
115C4×#41—44no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#428—28no—
140C2×#107—44no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#417—44no—
146#416—28no—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153#408—44no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156#400—24no—
157#399—24no—
158#395—24no—
159#393—24no—
160#391—22no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391C2311yescharacteristic
165#391C2311yescharacteristic
166#390—22no—
167#390—21no—
168#390—22no—
169#390C2311yescharacteristic
170#390C2311yescharacteristic
171#389—22no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#389C2314yes—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—24no—
188#386—21no—
189#386—22no—
190#386—21no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—24no—
195#384—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382—22no—
199#382C2312yes—
200#380—22no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380C2312yes—
205#380C2312yes—
206#380C2314yes—
207#380C2314yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41#4114yes—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—44no—
249C22×Q8—82no—
250C22×Q8—82no—
251C22×Q8C2×D411yescharacteristic
252C22×Q8C2411yescharacteristic
253C22×D4—21no—
254C22×D4—22no—
255C22×D4C2×D412yes—
256C22×D4C2×D411yescharacteristic
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D412yes—
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C22⋊Q8—28no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—84no—
286C22⋊Q8—84no—
287#121—216no—
288#121—28no—
289#121—24no—
290#121—24no—
291C4×Q8—42no—
292C4×Q8—42no—
293C4×Q8—44no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—48no—
300C4×D4—84no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—44no—
306#117—44no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4#4114yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—28no—
372#107—44no—
373#107—44no—
374#106—28no—
375#103—24no—
376#103—24no—
377#103—44no—
378#102—24no—
379#102—24no—
380#102—44no—
381C22⋊C8—44no—
382C22⋊C8—44no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399C24—21no—
400C24C22×D412yes—
401C24C22×D411yescharacteristic
402#41—216no—
403#41—48no—
404#41—48no—
405#41—48no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8—816no—
419C2×Q8C2×#4112yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—28no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4C2×#4112yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C42−1+414yes—
487C22×C4C2511yescharacteristic
488M16—48no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—48no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C2×#4114yes—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—24no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—81no—
540C22⋊C4—81no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—24no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—44no—
554C42—42no—
555C42—82no—
556C42—82no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23#52512yes—
576C23#53412yes—
577C23#53612yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—44no—
581Q8—42no—
582Q8—88no—
583Q8—88no—
584D4—24no—
585D4—48no—
586D4—48no—
587D4—44no—
588D4—42no—
589D4—88no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—88no—
642C2×C4—164no—
643C2×C4—162no—
644C2×C4#52512yes—
645C2×C4#53412yes—
646C2×C4#53612yes—
647C2×C4#53918yes—
648C2×C4#54218yes—
649C2×C4#57612yes—
650C2×C4#57812yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4C2×2−1+412yes—
655C8—42no—
656C8—42no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22#263814yes—
679C22C2×#52511yescharacteristic
680C22C2×#53411yescharacteristic
681C22C2×#53611yescharacteristic
682C22C2×#57611yescharacteristic
683C22C2×#57811yescharacteristic
684C22#358511yescharacteristic
685C22#358611yescharacteristic
686C4—22no—
687C4—22no—
688C4—42no—
689C4—42no—
690C4—41no—
691C4—42no—
692C4—41no—
693C4—42no—
694C4—44no—
695C4—84no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—81no—
700C4—81no—
701C4#263814yes—
702C4C2×#53912yes—
703C4C2×#54212yes—
704C2—24no—
705C2—82no—
706C2—82no—
707C2C2×#263811yescharacteristic
708C2#6059611yescharacteristic
709C2#6060011yescharacteristic
710C2#6075611yescharacteristic
711C2#6075911yescharacteristic
712C2#6088011yescharacteristic
713C2#6088711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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
1. #35279C2semidirect
3. #30183C2semidirect
4. #30177C2semidirect
5. #30176C2semidirect
6. #30172C2semidirect
7. #30169C2semidirect
8. #30165C2semidirect
9. #23657C2semidirect
10. #23639C2semidirect
11. #23627C2semidirect
12. C2×#2057C2semidirect
14. #23452C2semidirect
15. #21035C2semidirect
16. #20881C2semidirect
17. #20684C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#342672
#301832
#301772
#301762
#301722
#301692
#301652
#236572
#236392
#236272
C2×#20572
#234852
#234522
#210352
#208812
#206842