Finite Group Databasebeta

← #6489410

Subgroups — #6489410

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2C4×#536C211yesmaximal, characteristic
3#32958C212yesmaximal
4#32957C212yesmaximal, complemented
5#30089C212yesmaximal, complemented
6#30086C212yesmaximal, complemented
7#28725C211yesmaximal, characteristic, complemented
8#28721C211yesmaximal, characteristic, complemented
9#28519C211yesmaximal, characteristic, complemented
10#28511C211yesmaximal, characteristic, complemented
11#28206C212yesmaximal, complemented
12#28204C212yesmaximal, complemented
13#27958C212yesmaximal, complemented
14#27596C212yesmaximal, complemented
15#27595C212yesmaximal
16#25868C212yesmaximal
17#22231C212yesmaximal, complemented
18#22228C212yesmaximal, complemented
19#20786C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C2×#491C2212yes—
24C2×#488C2212yes—
25C2×#477C2212yes—
26C2×#477C2212yes—
27C2×#477C2212yes—
28C2×#476C2212yes—
29#2868C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2212yes—
32#2647C2212yes—
33#2629C2212yes—
34#2507—22no—
35#2497—22no—
36#2446—22no—
37#2429—22no—
38#2330C2212yes—
39#2327C2211yescharacteristic
40#2327C2211yescharacteristic
41#2326C2211yescharacteristic
42#2326C2211yescharacteristic
43#2323—22no—
44#2322—22no—
45#2317C2211yescharacteristic
46#2314C2212yes—
47#2311C2211yescharacteristic
48#2310C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#124C2211yescharacteristic
53C4×#124C2211yescharacteristic
54C4×C22⋊Q8—22no—
55C4×#123—22no—
56C4×#117C2211yescharacteristic
57#2096C2212yes—
58#2095C2212yes—
59#2095C2212yes—
60#2093C2212yes—
61#2085C2214yes—
62#2083C2214yes—
63#2074C2212yes—
64#2074C2212yes—
65#2074C2212yes—
66#2073C2212yes—
67#2043C2214yes—
68#2042C2214yes—
69#1992C2212yes—
70#1991C2212yes—
71#1959C2212yes—
72#1956C2212yes—
73#1928C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1925C2214yes—
77#1865C2214yes—
78#1863C2214yes—
79#1863C2214yes—
80#1862C2214yes—
81#1854C2214yes—
82#1852C2214yes—
83#1847C2212yes—
84#1845C2212yes—
85#1840C2212yes—
86#1831C2212yes—
87#1818C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93#1788C2214yes—
94#1787C2214yes—
95#1767C2211yescharacteristic
96C4×C4⋊C8C2211yescharacteristic
97#1750C2212yes—
98#1746C2212yes—
99C22×#41C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101#536—24no—
102#536—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#125C2312yes—
108C2×#124C2311yescharacteristic
109C2×#124C2311yescharacteristic
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C4×#41—44no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#491—28no—
138#488—28no—
139#477—44no—
140#477—44no—
141#477—44no—
142#476—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#400—24no—
161#398—24no—
162#398—24no—
163#396—24no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2311yescharacteristic
168#391C2311yescharacteristic
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#390C2311yescharacteristic
174#390C2311yescharacteristic
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—24no—
183#389C2314yes—
184#389C2314yes—
185#388—24no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—24no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—24no—
196#386—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—22no—
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2312yes—
206#382C2312yes—
207#382C2311yescharacteristic
208#382C2311yescharacteristic
209#381—22no—
210#380—22no—
211#380—22no—
212#380C2314yes—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223#376C2311yescharacteristic
224#376C2311yescharacteristic
225C43C2311yescharacteristic
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—24no—
239C2×#41—42no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—84no—
267C4⋊Q8#4114yes—
268#125—24no—
269#125—24no—
270#124—22no—
271#124—22no—
272#124—22no—
273#124#4114yes—
274#123—216no—
275C22⋊Q8—216no—
276C22⋊Q8—84no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—84no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290#117—24no—
291#117—22no—
292#117—22no—
293#117—44no—
294#117—42no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2414yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#107—44no—
359#107—44no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—44no—
371C22⋊C8—28no—
372#97—24no—
373C4×C8—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24C22×D411yescharacteristic
393#41—48no—
394#41—48no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8C2×#4112yes—
413C2×Q8C2×#4112yes—
414C2×Q8C2×#4112yes—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C2×#4112yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C42−1+414yes—
460C22×C4C2511yescharacteristic
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C22⋊C4—28no—
497C22⋊C4—24no—
498C22⋊C4—28no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—84no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—21no—
511C42—21no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—84no—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C42H(5,𝔽2)14yes—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23#53612yes—
538C23C23×D411yescharacteristic
539Q8—24no—
540Q8—24no—
541Q8—24no—
542Q8—48no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—88no—
547D4—24no—
548D4—48no—
549D4—42no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4#52514yes—
601C2×C4#53512yes—
602C2×C4#53512yes—
603C2×C4#53612yes—
604C2×C4#57814yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4C2×2−1+412yes—
609C8—44no—
610C8—44no—
611C22—22no—
612C22—24no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22C2×#52511yescharacteristic
622C22C2×#53511yescharacteristic
623C22C2×#53611yescharacteristic
624C22C2×#57812yes—
625C22#358512yes—
626C4—22no—
627C4—22no—
628C4—24no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—84no—
642C4—84no—
643C4#322314yes—
644C4#352012yes—
645C4#352012yes—
646C2—42no—
647C2—82no—
648C2#3524011yescharacteristic
649C2#6060012yes—
650C2C2×#322311yescharacteristic
651C2#6080811yescharacteristic
652C2#6089312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
4. #32957C2semidirect
5. #30089C2semidirect
6. #30086C2semidirect
7. #28725C2semidirect
8. #28721C2semidirect
9. #28519C2semidirect
10. #28511C2semidirect
11. #28206C2semidirect
12. #28204C2semidirect
13. #27958C2semidirect
14. #27596C2semidirect
17. #22231C2semidirect
18. #22228C2semidirect
19. #20786C2semidirect

Maximal subgroups

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

subgroupindex
#352842
C4×#5362
#329582
#329572
#300892
#300862
#287252
#287212
#285192
#285112
#282062
#282042
#279582
#275962
#275952
#258682
#222312
#222282
#207862