Finite Group Databasebeta

← #6559996

Subgroups — #6559996

Order 512.

1745 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36415C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30188C212yesmaximal, complemented
5#29347C211yesmaximal, characteristic, complemented
6#29139C211yesmaximal, characteristic, complemented
7#29107C212yesmaximal, complemented
8#27814C212yesmaximal, complemented
9#27768C212yesmaximal, complemented
10#27700C214yesmaximal, complemented
11#27259C214yesmaximal, complemented
12#25906C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23451C212yesmaximal, complemented
15#22489C212yesmaximal, complemented
16#22417C212yesmaximal, complemented
17#20555C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#533C2212yes—
21C2×#488C2214yes—
22C2×C8×D4C2212yes—
23C2×#416C2214yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2850C2212yes—
28#2563C2211yescharacteristic
29#2557—22no—
30#2557—22no—
31#2552C2212yes—
32#2547C2211yescharacteristic
33#2542—21no—
34#2538—21no—
35#2535—21no—
36#2535—21no—
37#2499C2212yes—
38#2441C2214yes—
39#2431C2212yes—
40#2429—22no—
41#2423—22no—
42#2365—21no—
43#2354C2212yes—
44#2352—21no—
45#2352—21no—
46Q8×C4⋊C4—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49#2297C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2099C2212yes—
53#2072C2214yes—
54#2072C2212yes—
55#2071C2211yescharacteristic
56#2071C2211yescharacteristic
57#2042C2214yes—
58#2041C2212yes—
59#2026C2214yes—
60#2025C2214yes—
61#2024C2212yes—
62#2024C2212yes—
63#2024C2212yes—
64#2023C2211yescharacteristic
65#2023C2211yescharacteristic
66#2023C2212yes—
67#2023C2212yes—
68#1991C2214yes—
69#1990C2212yes—
70#1987C2212yes—
71#1981C2212yes—
72#1957C2212yes—
73#1956C2212yes—
74#1934C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1875C2214yes—
80#1873C2214yes—
81#1834C2212yes—
82#1825C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86C8×C4⋊C4C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#102C2212yes—
89C4×#102C2212yes—
90C8×C22⋊C4C2214yes—
91C4×C22⋊C8C2212yes—
92#1744C2211yescharacteristic
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41C2312yes—
96C23×C8C2314yes—
97#555—28no—
98#547—28no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125C2314yes—
108C2×#124—22no—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117C2312yes—
130C2×#117C2314yes—
131C22×C4⋊C4—21no—
132C22×C42C2312yes—
133#488—48no—
134C8×D4—28no—
135C2×C4⋊C8C2312yes—
136#416—48no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#399—24no—
152#398—24no—
153#397—22no—
154#397—22no—
155#397—22no—
156#396—24no—
157#393—22no—
158#392—22no—
159#392—22no—
160#391—21no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#390—22no—
168#390—21no—
169#390—21no—
170#390C2311yescharacteristic
171#390C2311yescharacteristic
172#390C2312yes—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—24no—
177#389C2312yes—
178#389C2314yes—
179#388—24no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—21no—
199#382—21no—
200#382C2312yes—
201#380—22no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380C2314yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—28no—
236C2×#41—42no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C22×C8—28no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267#127—22no—
268#127—41no—
269#125—28no—
270#125—44no—
271#124—44no—
272#124—48no—
273#123—48no—
274C22⋊Q8—28no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—84no—
278#121—28no—
279#121—44no—
280#121—44no—
281C4×Q8—48no—
282C4×Q8—48no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4#4118yes—
291#117—28no—
292#117—28no—
293#117—44no—
294#117—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C4⋊C8—24no—
356#103—28no—
357#103—44no—
358#103—44no—
359#103—44no—
360#102—24no—
361#102—24no—
362#102—24no—
363#102—42no—
364#102—44no—
365#102—44no—
366C22⋊C8—28no—
367C4×C8—24no—
368C4×C8—42no—
369C4×C8—44no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24—21no—
387C24C22×D412yes—
388#41—416no—
389#41—88no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4#12414yes—
425C2×D42−1+412yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4118yes—
461C22×C4C2511yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—88no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4#12414yes—
501C4⋊C4H(5,𝔽2)14yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C22⋊C4#12418yes—
520C42—24no—
521C42—22no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42—41no—
528C42—48no—
529C42—44no—
530C42—44no—
531C42#13514yes—
532C42#13514yes—
533C42C2×#4114yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×#12414yes—
555C23C23×D411yescharacteristic
556Q8—48no—
557Q8—44no—
558Q8—44no—
559Q8—88no—
560Q8—162no—
561D4—28no—
562D4—24no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—84no—
567D4—84no—
568D4—88no—
569D4—164no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4C2×#12414yes—
617C2×C4C2×#12412yes—
618C2×C4C2×#12412yes—
619C2×C4C2×#13512yes—
620C2×C4C2×#13512yes—
621C2×C4C2×#13512yes—
622C2×C4C2×#13512yes—
623C2×C4C2×#13512yes—
624C2×C4C2×#13512yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×2−1+411yescharacteristic
628C2×C4#58511yescharacteristic
629C8—44no—
630C8—42no—
631C8—42no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C22×#12411yescharacteristic
654C22#348411yescharacteristic
655C22#348911yescharacteristic
656C22C22×#13511yescharacteristic
657C22C22×#13511yescharacteristic
658C22#358511yescharacteristic
659C22#358611yescharacteristic
660C4—24no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4#313114yes—
676C4#351412yes—
677C4#352012yes—
678C2—44no—
679C2—82no—
680C2—161no—
681C2#3593911yescharacteristic
682C2C2×#313111yescharacteristic
683C2#6062511yescharacteristic
684C2#6156511yescharacteristic
685C2#6156611yescharacteristic
686C2#6209711yescharacteristic
687C2#6209811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36415C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36134C2semidirect
3. #30205C2semidirect
4. #30188C2semidirect
5. #29347C2semidirect
6. #29139C2semidirect
7. #29107C2semidirect
8. #27814C2semidirect
9. #27768C2semidirect
10. #27700C2semidirect
11. #27259C2semidirect
12. #25906C2semidirect
13. #23815C2semidirect
14. #23451C2semidirect
15. #22489C2semidirect
16. #22417C2semidirect
17. #20555C2semidirect

Maximal subgroups

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

subgroupindex
#364152
#361342
#302052
#301882
#293472
#291392
#291072
#278142
#277682
#277002
#272592
#259062
#238152
#234512
#224892
#224172
#205552