Finite Group Databasebeta

← #6569366

Subgroups — #6569366

Order 512.

1602 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#36454C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#30319C212yesmaximal, complemented
4#30205C211yesmaximal, characteristic, complemented
5#28998C214yesmaximal, complemented
6#28694C211yesmaximal, characteristic, complemented
7#28580C212yesmaximal, complemented
8#28382C211yesmaximal, characteristic, complemented
9#27864C212yesmaximal, complemented
10#27796C214yesmaximal, complemented
11#25393C212yesmaximal, complemented
12#25347C212yesmaximal, complemented
13#23819C211yesmaximal, characteristic, complemented
14#23757C212yesmaximal, complemented
15#23094C212yesmaximal, complemented
16#23060C212yesmaximal, complemented
17#20567C211yesmaximal, characteristic, complemented
18C2×#555—21no—
19C2×#501C2212yes—
20C2×#474C2212yes—
21C2×#468C2214yes—
22C2×#461C2212yes—
23C2×#436C2212yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2856C2212yes—
27#2855C2212yes—
28#2575—22no—
29#2563—21no—
30#2547C2211yescharacteristic
31#2542C2211yescharacteristic
32#2537—21no—
33#2537C2212yes—
34#2517—22no—
35#2511—21no—
36#2500—22no—
37#2491—22no—
38#2491C2214yes—
39#2489C2212yes—
40#2447—21no—
41#2421C2212yes—
42#2367—21no—
43#2365C2211yescharacteristic
44#2365C2211yescharacteristic
45#2355—21no—
46#2354—21no—
47#2354C2212yes—
48C4×#127C2211yescharacteristic
49C4×#125C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2095C2212yes—
56#2084C2212yes—
57#2083C2212yes—
58#2076C2214yes—
59#2073C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2041C2211yescharacteristic
63#2041C2211yescharacteristic
64#2036C2214yes—
65#2034C2214yes—
66#2025C2212yes—
67#2024C2214yes—
68#2014C2212yes—
69#2000C2214yes—
70#1993C2212yes—
71#1990C2212yes—
72#1973C2214yes—
73#1968C2214yes—
74#1956C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1934C2214yes—
78#1927C2214yes—
79#1925C2214yes—
80#1918C2214yes—
81#1895C2214yes—
82#1894C2214yes—
83#1880C2214yes—
84#1809C2214yes—
85C4×#107C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2214yes—
89C4×#102C2212yes—
90C4×#102C2212yes—
91#1753C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×SD16C2314yes—
96C22×M16C2314yes—
97#555—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#125C2311yescharacteristic
105C2×#124—21no—
106C2×#124—24no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121C2314yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C42C2312yes—
124#501—44no—
125#474—28no—
126#468—48no—
127#461—28no—
128#436—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#400—22no—
147#399—22no—
148#397—22no—
149#397—22no—
150#397—22no—
151#394—24no—
152#394—24no—
153#393—24no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—22no—
165#390—22no—
166#390C2312yes—
167#390C2314yes—
168#389—21no—
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—24no—
175#388—24no—
176#387—21no—
177#387—21no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—21no—
194#382—21no—
195#382—22no—
196#382C2311yescharacteristic
197#381—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2412yes—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×SD16—28no—
247C2×SD16—44no—
248C2×SD16—44no—
249C2×M16—28no—
250C2×M16—44no—
251C2×M16—44no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261#127—42no—
262#125—22no—
263#125—41no—
264#124—416no—
265#124—44no—
266C22⋊Q8—24no—
267C22⋊Q8—42no—
268C22⋊Q8—44no—
269#121—48no—
270#121—48no—
271#121—44no—
272#121—44no—
273#121—84no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—42no—
281C4×D4—44no—
282C4×D4—44no—
283#117—48no—
284#117—48no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—28no—
344#107—28no—
345#106—44no—
346C4⋊C8#4118yes—
347#103—28no—
348#103—44no—
349#103—44no—
350#102—28no—
351#102—24no—
352#102—42no—
353#102—42no—
354#102—42no—
355C22⋊C8—28no—
356#97—42no—
357C4×C8—24no—
358C4×C8—44no—
359#95—21no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—24no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24C22×D412yes—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—84no—
409C2×D4—84no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2511yescharacteristic
441SD16—88no—
442M16—88no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8C2×#4118yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4AGL(1,ℤ/8)14yes—
483C4⋊C4H(5,𝔽2)12yes—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—82no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—21no—
503C42—24no—
504C42—22no—
505C42—21no—
506C42—42no—
507C42—42no—
508C42—48no—
509C42—42no—
510C42—44no—
511C42—42no—
512C42—82no—
513C42C2×#4114yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C23×D411yescharacteristic
529Q8—44no—
530Q8—44no—
531Q8—84no—
532Q8—164no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—84no—
539D4—84no—
540D4—168no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—48no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—84no—
589C2×C4—88no—
590C2×C4#53212yes—
591C2×C4#53212yes—
592C2×C4C2×AGL(1,ℤ/8)14yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4#57512yes—
595C2×C4#57712yes—
596C2×C4#57812yes—
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C2×C4#58512yes—
600C2×C4#58511yescharacteristic
601C8—28no—
602C8—42no—
603C8—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#53211yescharacteristic
619C22#348511yescharacteristic
620C22#348811yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57711yescharacteristic
624C22C2×#57811yescharacteristic
625C4—22no—
626C4—22no—
627C4—24no—
628C4—22no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—84no—
640C4—82no—
641C4#317114yes—
642C4#351412yes—
643C4#352412yes—
644C2—82no—
645C2—162no—
646C2#3605111yescharacteristic
647C2C2×#317111yescharacteristic
648C2#6072311yescharacteristic
649C2#6173111yescharacteristic
650C2#6173711yescharacteristic
651C2#6199811yescharacteristic
652C2#6199911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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. #36454C2semidirect
3. #30319C2semidirect
4. #30205C2semidirect
5. #28998C2semidirect
6. #28694C2semidirect
7. #28580C2semidirect
8. #28382C2semidirect
9. #27864C2semidirect
10. #27796C2semidirect
11. #25393C2semidirect
12. #25347C2semidirect
13. #23819C2semidirect
14. #23757C2semidirect
15. #23094C2semidirect
16. #23060C2semidirect
17. #20567C2semidirect

Maximal subgroups

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

subgroupindex
#364542
#361872
#303192
#302052
#289982
#286942
#285802
#283822
#278642
#277962
#253932
#253472
#238192
#237572
#230942
#230602
#205672