Finite Group Databasebeta

← #6569368

Subgroups — #6569368

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36476C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#30327C212yesmaximal, complemented
4#30205C211yesmaximal, characteristic, complemented
5#29005C214yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28580C212yesmaximal, complemented
8#28370C211yesmaximal, characteristic, complemented
9#27861C212yesmaximal, complemented
10#27802C214yesmaximal, complemented
11#25393C212yesmaximal, complemented
12#25346C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23763C212yesmaximal, complemented
15#23100C212yesmaximal, complemented
16#23067C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#555—21no—
19C2×#500C2212yes—
20C2×#476C2212yes—
21C2×#468C2214yes—
22C2×#461C2212yes—
23C2×#436C2212yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2856C2212yes—
27#2855C2212yes—
28#2607—22no—
29#2589—22no—
30#2580—21no—
31#2570—21no—
32#2563—21no—
33#2563C2211yescharacteristic
34#2542C2211yescharacteristic
35#2537—21no—
36#2537C2212yes—
37#2500—22no—
38#2492C2212yes—
39#2491—22no—
40#2491C2214yes—
41#2423C2212yes—
42#2369C2211yescharacteristic
43#2365C2211yescharacteristic
44#2357—21no—
45#2355—21no—
46#2354—21no—
47#2354C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2211yescharacteristic
54#2099C2211yescharacteristic
55#2096C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2077C2214yes—
59#2072C2212yes—
60#2072C2212yes—
61#2071C2212yes—
62#2041C2211yescharacteristic
63#2041C2211yescharacteristic
64#2036C2214yes—
65#2033C2214yes—
66#2024C2214yes—
67#2023C2212yes—
68#2014C2212yes—
69#2000C2214yes—
70#1990C2212yes—
71#1990C2212yes—
72#1973C2214yes—
73#1968C2214yes—
74#1956C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1936C2214yes—
78#1928C2214yes—
79#1925C2214yes—
80#1919C2214yes—
81#1895C2214yes—
82#1894C2214yes—
83#1880C2214yes—
84#1809C2214yes—
85#1776C2214yes—
86#1770C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88#1763C2214yes—
89C4×#102C2212yes—
90C4×#102C2212yes—
91#1753C2214yes—
92#1749C2212yes—
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⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#124—24no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2314yes—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—22no—
122C22×C42C2312yes—
123#500—44no—
124#476—28no—
125#468—48no—
126#461—28no—
127#436—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#399—22no—
146#399—22no—
147#398—22no—
148#394—24no—
149#394—24no—
150#393—22no—
151#393—22no—
152#393—24no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2314yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—24no—
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#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382—21no—
194#382—22no—
195#382—21no—
196#382—21no—
197#381—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2312yes—
203#380C2314yes—
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#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8C2412yes—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C2×SD16—28no—
248C2×SD16—44no—
249C2×SD16—44no—
250C2×M16—28no—
251C2×M16—44no—
252C2×M16—44no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—41no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264#127—42no—
265#124—416no—
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—24no—
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—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—44no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—28no—
345#107—44no—
346#106—28no—
347C4⋊C8#4118yes—
348#103—28no—
349#103—44no—
350#103—44no—
351#102—28no—
352#102—24no—
353#102—42no—
354#102—42no—
355#102—42no—
356C22⋊C8—28no—
357#97—44no—
358C4×C8—24no—
359C4×C8—42no—
360#95—21no—
361#95—22no—
362#95—24no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
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#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24C22×D412yes—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—81no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—84no—
411C2×D4—84no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2511yescharacteristic
443SD16—88no—
444M16—88no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8C2×#4118yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4H(5,𝔽2)12yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—21no—
502C42—24no—
503C42—22no—
504C42—21no—
505C42—42no—
506C42—42no—
507C42—41no—
508C42—41no—
509C42—48no—
510C42—42no—
511C42—42no—
512C42—82no—
513C42—82no—
514C42AGL(1,ℤ/8)14yes—
515C42#13714yes—
516C42C2×#4114yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C23×D411yescharacteristic
532Q8—44no—
533Q8—44no—
534Q8—84no—
535Q8—164no—
536Q8—162no—
537D4—44no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—84no—
543D4—84no—
544D4—168no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
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—48no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4#53212yes—
594C2×C4#53212yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4#58512yes—
604C2×C4#58511yescharacteristic
605C8—28no—
606C8—42no—
607C8—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22C2×#53211yescharacteristic
623C22#348511yescharacteristic
624C22#348811yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22C22×#13711yescharacteristic
627C22#358511yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—84no—
639C4—81no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4#318214yes—
646C4#351412yes—
647C4#352412yes—
648C2—82no—
649C2—162no—
650C2#3605111yescharacteristic
651C2C2×#318211yescharacteristic
652C2#6072711yescharacteristic
653C2#6171711yescharacteristic
654C2#6172111yescharacteristic
655C2#6195911yescharacteristic
656C2#6196611yescharacteristic
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. #36476C2semidirect
3. #30327C2semidirect
4. #30205C2semidirect
5. #29005C2semidirect
6. #28689C2semidirect
7. #28580C2semidirect
8. #28370C2semidirect
9. #27861C2semidirect
10. #27802C2semidirect
11. #25393C2semidirect
12. #25346C2semidirect
13. #23815C2semidirect
14. #23763C2semidirect
15. #23100C2semidirect
16. #23067C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364762
#361872
#303272
#302052
#290052
#286892
#285802
#283702
#278612
#278022
#253932
#253462
#238152
#237632
#231002
#230672
#205682