Finite Group Databasebeta

← #6557105

Subgroups — #6557105

Order 512.

1830 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36416C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#30190C212yesmaximal, complemented
5#29332C212yesmaximal, complemented
6#29121C211yesmaximal, characteristic, complemented
7#28748C211yesmaximal, characteristic, complemented
8#27768C212yesmaximal, complemented
9#27748C212yesmaximal, complemented
10#27261C214yesmaximal, complemented
11#27255C214yesmaximal, complemented
12#25858C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23504C212yesmaximal, complemented
15#22417C212yesmaximal, complemented
16#22417C212yesmaximal, complemented
17#20555C211yesmaximal, characteristic, complemented
18C2×#547C2212yes—
19C2×#532C2212yes—
20C2×#488C2214yes—
21C2×#487C2212yes—
22C2×C8×D4C2212yes—
23C2×#416C2214yes—
24C2×#416C2214yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2851C2212yes—
29#2589—22no—
30#2587—22no—
31#2580—21no—
32#2571C2211yescharacteristic
33#2566—21no—
34#2552C2211yescharacteristic
35#2547C2212yes—
36#2537—21no—
37#2537—21no—
38#2441C2212yes—
39#2439C2212yes—
40#2432C2214yes—
41#2422—22no—
42#2422—22no—
43#2370—21no—
44#2355—21no—
45#2354C2211yescharacteristic
46#2354C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51#2296C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2211yescharacteristic
54#2099C2211yescharacteristic
55#2099C2212yes—
56#2074C2214yes—
57#2071C2212yes—
58#2043C2212yes—
59#2043C2212yes—
60#2043C2214yes—
61#2043C2212yes—
62#2043C2212yes—
63#2042C2214yes—
64#2042C2214yes—
65#2041C2211yescharacteristic
66#2041C2212yes—
67#2041C2211yescharacteristic
68#2026C2214yes—
69#2023C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1958C2214yes—
73#1956C2212yes—
74#1956C2212yes—
75#1956C2212yes—
76#1934C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1873C2214yes—
82#1873C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86#1788C2214yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×#102C2212yes—
90C4×#102C2212yes—
91C8×C22⋊C4C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41C2314yes—
96C23×C8C2314yes—
97#547—28no—
98#547—28no—
99#532—48no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#125—21no—
106C2×#125C2312yes—
107C2×#124C2314yes—
108C2×#123—24no—
109C2×C22⋊Q8C2314yes—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121—22no—
113C2×C4×Q8—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—22no—
125C22×C42C2312yes—
126#488—48no—
127#487—44no—
128C8×D4—28no—
129C2×C4⋊C8C2312yes—
130#416—48no—
131#416—48no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#399—22no—
152#398—22no—
153#397—24no—
154#396—22no—
155#396—22no—
156#394—24no—
157#394—24no—
158#393—22no—
159#391—21no—
160#391—22no—
161#391C2314yes—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—21no—
203#382—21no—
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380C2312yes—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2311yescharacteristic
236C2×#41—28no—
237C2×#41—28no—
238C2×#41—28no—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—44no—
242C22×Q8—21no—
243C22×Q8C2414yes—
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—24no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C22×C8—28no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—41no—
264C4⋊Q8—44no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267#127—24no—
268#125—44no—
269#125—44no—
270#124—28no—
271#124—44no—
272#123—416no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275#121—48no—
276#121—48no—
277#121—48no—
278C4×Q8—44no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—48no—
284C4×D4—48no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4#4118yes—
288#117—24no—
289#117—42no—
290#117—44no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C4⋊C8—24no—
355#103—28no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#102—24no—
364#102—28no—
365#102—24no—
366#102—24no—
367#102—42no—
368#102—44no—
369C22⋊C8—28no—
370C4×C8—24no—
371C4×C8—42no—
372C4×C8—44no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—22no—
391C24C22×D412yes—
392#41—416no—
393#41—416no—
394#41—88no—
395#41—88no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—81no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—84no—
421C2×D4#12414yes—
422C2×D4H(5,𝔽2)14yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—84no—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4118yes—
456C22×C4C2511yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—88no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#12414yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4#12418yes—
518C42—21no—
519C42—24no—
520C42—22no—
521C42—21no—
522C42—42no—
523C42—42no—
524C42—41no—
525C42—41no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42—82no—
530C42#13514yes—
531C42#13514yes—
532C42C2×#4114yes—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C2×#12414yes—
547C23C23×D411yescharacteristic
548Q8—44no—
549Q8—44no—
550Q8—44no—
551Q8—44no—
552Q8—84no—
553D4—28no—
554D4—24no—
555D4—24no—
556D4—48no—
557D4—48no—
558D4—42no—
559D4—42no—
560D4—816no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—816no—
608C2×C4—88no—
609C2×C4C2×#12412yes—
610C2×C4C2×#12414yes—
611C2×C4C2×#12412yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13512yes—
618C2×C4C22×#4112yes—
619C2×C4C2×H(5,𝔽2)12yes—
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4#58511yescharacteristic
622C8—42no—
623C8—44no—
624C8—42no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22C22×#12411yescharacteristic
640C22#348411yescharacteristic
641C22#348811yescharacteristic
642C22C22×#13511yescharacteristic
643C22C22×#13511yescharacteristic
644C22#358611yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4—84no—
657C4—81no—
658C4—82no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4#313114yes—
663C4#351312yes—
664C4#351912yes—
665C2—44no—
666C2—84no—
667C2#3590311yescharacteristic
668C2C2×#313111yescharacteristic
669C2#6062711yescharacteristic
670C2#6156511yescharacteristic
671C2#6156511yescharacteristic
672C2#6195011yescharacteristic
673C2#6195411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
1. #36416C2semidirect
3. #30220C2semidirect
4. #30190C2semidirect
5. #29332C2semidirect
6. #29121C2semidirect
7. #28748C2semidirect
8. #27768C2semidirect
9. #27748C2semidirect
10. #27261C2semidirect
11. #27255C2semidirect
12. #25858C2semidirect
13. #23815C2semidirect
14. #23504C2semidirect
15. #22417C2semidirect
16. #22417C2semidirect
17. #20555C2semidirect

Maximal subgroups

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

subgroupindex
#364162
#361272
#302202
#301902
#293322
#291212
#287482
#277682
#277482
#272612
#272552
#258582
#238152
#235042
#224172
#205552