Finite Group Databasebeta

← #6461384

Subgroups — #6461384

Order 512.

2368 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic, complemented
2#34259C211yesmaximal, characteristic
3#29120C214yesmaximal, complemented
4#29118C214yesmaximal, complemented
5#29113C212yesmaximal, complemented
6#29111C212yesmaximal, complemented
7#29107C212yesmaximal, complemented
8#29105C212yesmaximal, complemented
9#23629C211yesmaximal, characteristic, complemented
10#23627C211yesmaximal, characteristic, complemented
11#23452C211yesmaximal, characteristic, complemented
12C2×#2024C211yesmaximal, characteristic, complemented
13#21090C212yesmaximal, complemented
14#21087C212yesmaximal, complemented
15#20703C212yesmaximal, complemented
16#20603C212yesmaximal
17#20378C211yesmaximal, characteristic, complemented
18C22×2−1+4C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#417C2212yes—
23C2×#419C2212yes—
24C2×#416C2212yes—
25C22×#103C2212yes—
26C2×#408C2212yes—
27C2×#405C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2628C2214yes—
31#2568—21no—
32#2567—21no—
33#2563—21no—
34#2542—21no—
35#2487—22no—
36#2409—22no—
37Q8×C22⋊C4—21no—
38Q8×C22⋊C4—21no—
39#2339—21no—
40#2339—22no—
41#2339—22no—
42#2339—21no—
43#2317C2212yes—
44#2311C2211yescharacteristic
45#2311C2212yes—
46#2311C2212yes—
47#2310C2211yescharacteristic
48#2310C2212yes—
49#2297C2211yescharacteristic
50#2297C2212yes—
51C2×#391C2211yescharacteristic
52C2×#391C2212yes—
53#2285C2211yescharacteristic
54#2074C2212yes—
55#2074C2214yes—
56#2073C2212yes—
57#2072C2211yescharacteristic
58#2072C2211yescharacteristic
59#2072C2212yes—
60#2072C2212yes—
61#2071C2211yescharacteristic
62#2071C2211yescharacteristic
63#2024C2212yes—
64#2024C2212yes—
65#2024C2214yes—
66#2024C2212yes—
67#2024C2212yes—
68#2023C2212yes—
69#2023C2212yes—
70#1936C2214yes—
71#1935C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1934C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1825C2214yes—
79#1825C2214yes—
80#1824C2214yes—
81#1823C2214yes—
82#1788C2214yes—
83#1788C2214yes—
84#1787C2214yes—
85#1787C2214yes—
86#1750C2214yes—
87#1749C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C2×2−1+4—24no—
96C2×2−1+4—24no—
97C2×2−1+4—28no—
98C2×2−1+4—44no—
99C22×#41—22no—
100C22×#41—21no—
101C22×#41C2311yescharacteristic
102C22×#41C2312yes—
103C22×#41C2311yescharacteristic
104C23×Q8—21no—
105C23×Q8—21no—
106C23×Q8C2311yescharacteristic
107C23×Q8C2312yes—
108C22×M16C2312yes—
109C23×C8C2312yes—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—44no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117#520—28no—
118C4×#41—44no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D412yes—
126C2×C4×D4D412yes—
127C2×C4×D4D411yescharacteristic
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#419—28no—
140#417—44no—
141#416—28no—
142C2×#103—44no—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153#408—28no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156#405—44no—
157C2×#97C2314yes—
158C2×C4×C8C2314yes—
159#400—24no—
160#399—24no—
161#395—24no—
162#393—24no—
163#391—22no—
164#391—21no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—21no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391C2311yescharacteristic
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#391C2311yescharacteristic
180#391C2312yes—
181#390—22no—
182#390C2311yescharacteristic
183#390C2312yes—
184#390C2311yescharacteristic
185#389—22no—
186#389—22no—
187#389C2314yes—
188#389C2314yes—
189#386—24no—
190#386—21no—
191#386—21no—
192#386—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#385—24no—
198#384—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—22no—
202#382C2312yes—
203#381—22no—
204#381—22no—
205#381—22no—
206#380—22no—
207#380—22no—
208#380—21no—
209#380—21no—
210#380—21no—
211#380—21no—
212#380—22no—
213#380—22no—
214#380C2314yes—
215#380C2314yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95—22no—
225C2×#95—22no—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2311yescharacteristic
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240#336C2312yes—
2412−1+4—432no—
2422−1+4—432no—
243C2×#41—216no—
244C2×#41—28no—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—28no—
253C2×#41—24no—
254C2×#41—44no—
255C2×#41—48no—
256C2×#41—44no—
257C2×#41—44no—
258C2×#41—48no—
259C2×#41#4114yes—
260C2×#41#4118yes—
261C22×Q8—28no—
262C22×Q8—24no—
263C22×Q8—24no—
264C22×Q8—22no—
265C22×Q8—21no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8—24no—
270C22×Q8—28no—
271C22×Q8—24no—
272C22×Q8—44no—
273C22×Q8—44no—
274C22×Q8—44no—
275C22×Q8—44no—
276C22×Q8—82no—
277C22×Q8—82no—
278C22×Q8C2×D411yescharacteristic
279C22×Q8#4114yes—
280C22×Q8C2412yes—
281C22×Q8C2411yescharacteristic
282C22×D4—22no—
283C22×D4—21no—
284C22×D4C2×D412yes—
285C22×D4C2×D411yescharacteristic
286C22×D4C2411yescharacteristic
287C23×C4—22no—
288C23×C4—21no—
289C23×C4—22no—
290C23×C4—21no—
291C23×C4—22no—
292C23×C4—22no—
293C23×C4—21no—
294C23×C4C2×D411yescharacteristic
295C23×C4C2×D411yescharacteristic
296C23×C4C2×D412yes—
297C23×C4C2×D412yes—
298C23×C4C2412yes—
299C23×C4C2411yescharacteristic
300C2×M16—24no—
301C2×M16—24no—
302C2×M16—24no—
303C22×C8—24no—
304C22×C8—24no—
305C22×C8—24no—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2414yes—
309C4⋊Q8—44no—
310C4⋊Q8—44no—
311C4⋊Q8—44no—
312C4⋊Q8—44no—
313C22⋊Q8—84no—
314C22⋊Q8—84no—
315C4×Q8—42no—
316C4×Q8—42no—
317C4×Q8—44no—
318C4×D4—42no—
319C4×D4—42no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—48no—
323C4×D4—44no—
324#117—24no—
325#117—28no—
326#117—24no—
327#117—24no—
328#117—24no—
329#117—44no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42—44no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382#103—24no—
383#103—24no—
384#103—28no—
385#103—28no—
386#103—44no—
387#103—44no—
388#103—44no—
389#102—24no—
390#102—24no—
391#102—44no—
392C22⋊C8—28no—
393C22⋊C8—28no—
394#97—44no—
395C4×C8—44no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416C24—21no—
417C24C22×D412yes—
418C24C22×D411yescharacteristic
419#41—216no—
420#41—216no—
421#41—216no—
422#41—416no—
423#41—416no—
424#41—464no—
425#41—432no—
426#41—48no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—216no—
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—416no—
439C2×Q8—416no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—432no—
445C2×Q8—44no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—48no—
449C2×Q8—44no—
450C2×Q8—82no—
451C2×Q8—82no—
452C2×Q8—816no—
453C2×Q8#12414yes—
454C2×Q8C2×#4112yes—
455C2×Q8C2×#4114yes—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—28no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4#12418yes—
470C2×D4#12414yes—
471C2×D4C2×#4112yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—82no—
511C22×C4—82no—
512C22×C4—81no—
513C22×C4—82no—
514C22×C4—81no—
515C22×C4—82no—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C2×#4118yes—
525C22×C4C2×#4112yes—
526C22×C4C2×#4112yes—
527C22×C4C2×#4114yes—
528C22×C4H(5,𝔽2)14yes—
529C22×C4C2511yescharacteristic
530M16—48no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—24no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—24no—
537C2×C8—22no—
538C2×C8—24no—
539C2×C8—22no—
540C2×C8—48no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—44no—
559C4⋊C4—42no—
560C4⋊C4—44no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—84no—
564C4⋊C4—84no—
565C22⋊C4—24no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—82no—
574C22⋊C4—81no—
575C22⋊C4—81no—
576C22⋊C4—82no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C42—22no—
580C42—22no—
581C42—24no—
582C42—22no—
583C42—22no—
584C42—42no—
585C42—42no—
586C42—44no—
587C42—44no—
588C42—42no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23#52512yes—
607C23C2×#12414yes—
608C23C2×#12412yes—
609C23#53412yes—
610C23C23×D411yescharacteristic
611Q8—28no—
612Q8—28no—
613Q8—24no—
614Q8—416no—
615Q8—48no—
616Q8—44no—
617Q8—42no—
618Q8—88no—
619Q8—88no—
620D4—216no—
621D4—28no—
622D4—24no—
623D4—416no—
624D4—48no—
625D4—42no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—22no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—416no—
651C2×C4—416no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—44no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—88no—
679C2×C4—162no—
680C2×C4—164no—
681C2×C4#52512yes—
682C2×C4C2×#12414yes—
683C2×C4C2×#12412yes—
684C2×C4#53412yes—
685C2×C4#55118yes—
686C2×C4#55118yes—
687C2×C4#57712yes—
688C2×C4#57812yes—
689C2×C4C22×#4111yescharacteristic
690C2×C4C22×#4111yescharacteristic
691C2×C4C22×#4112yes—
692C2×C4C2×H(5,𝔽2)12yes—
693C8—42no—
694C8—42no—
695C8—44no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22#256114yes—
717C22C2×#52511yescharacteristic
718C22C22×#12411yescharacteristic
719C22C2×#53411yescharacteristic
720C22C2×#57711yescharacteristic
721C22C2×#57811yescharacteristic
722C22#358511yescharacteristic
723C22#358611yescharacteristic
724C4—24no—
725C4—22no—
726C4—22no—
727C4—44no—
728C4—42no—
729C4—41no—
730C4—42no—
731C4—41no—
732C4—42no—
733C4—84no—
734C4—82no—
735C4—82no—
736C4—81no—
737C4—82no—
738C4—81no—
739C4#256114yes—
740C4C2×#55112yes—
741C4C2×#55112yes—
742C2—24no—
743C2—44no—
744C2—82no—
745C2C2×#256111yescharacteristic
746C2#6059911yescharacteristic
747C2#6060011yescharacteristic
748C2#6062311yescharacteristic
749C2#6062511yescharacteristic
750C2#6075711yescharacteristic
751C2#6075911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
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. #29120C2semidirect
4. #29118C2semidirect
5. #29113C2semidirect
6. #29111C2semidirect
7. #29107C2semidirect
8. #29105C2semidirect
9. #23629C2semidirect
10. #23627C2semidirect
11. #23452C2semidirect
12. C2×#2024C2semidirect
13. #21090C2semidirect
14. #21087C2semidirect
15. #20703C2semidirect
17. #20378C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#342592
#291202
#291182
#291132
#291112
#291072
#291052
#236292
#236272
#234522
C2×#20242
#210902
#210872
#207032
#206032
#203782