Finite Group Databasebeta

← #6630508

Subgroups — #6630508

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35292C211yesmaximal, characteristic
2C4×#520C211yesmaximal, characteristic, complemented
3#33153C212yesmaximal, complemented
4#33149C212yesmaximal, complemented
5#33113C212yesmaximal, complemented
6#33106C212yesmaximal, complemented
7#33103C214yesmaximal, complemented
8#32986C214yesmaximal, complemented
9#21590C214yesmaximal, complemented
10#21586C214yesmaximal, complemented
11#21395C211yesmaximal, characteristic, complemented
12#21390C211yesmaximal, characteristic, complemented
13#21364C211yesmaximal, characteristic, complemented
14#21358C211yesmaximal, characteristic, complemented
15#20721C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#520C2212yes—
18C2×C4×#41C2211yescharacteristic
19C2×#482C2214yes—
20C2×#480C2214yes—
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2857C2212yes—
24#2856C2212yes—
25#2624C2212yes—
26#2615C2212yes—
27#2592—21no—
28#2580—21no—
29#2540—21no—
30#2535—21no—
31#2414—22no—
32#2407—22no—
33#2324C2214yes—
34#2285C2211yescharacteristic
35#2276—21no—
36#2275—22no—
37#2275—21no—
38#2274C2212yes—
39#2274C2212yes—
40#2274C2212yes—
41C42×Q8—21no—
42C42×D4—22no—
43C42×D4—21no—
44C4×#117C2212yes—
45C4×#117C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48#1987C2211yescharacteristic
49#1987C2212yes—
50#1987C2211yescharacteristic
51#1982C2211yescharacteristic
52#1982C2212yes—
53#1982C2211yescharacteristic
54#1948C2212yes—
55#1947C2212yes—
56#1947C2212yes—
57#1947C2212yes—
58#1926C2214yes—
59#1925C2214yes—
60#1925C2214yes—
61#1925C2214yes—
62#1921C2214yes—
63#1921C2214yes—
64#1919C2214yes—
65#1918C2214yes—
66#1875C2214yes—
67#1875C2214yes—
68#1874C2214yes—
69#1874C2214yes—
70#1854C2214yes—
71#1854C2214yes—
72#1852C2214yes—
73#1852C2214yes—
74#1825C2214yes—
75#1824C2214yes—
76#1818C2212yes—
77#1797C2212yes—
78#1797C2212yes—
79#1780C2212yes—
80#1776C2214yes—
81C4×#107C2212yes—
82C4×#107C2212yes—
83#1762C2212yes—
84#1762C2212yes—
85C4×#102C2214yes—
86#1750C2214yes—
87C22×#41—21no—
88C22×#41C2311yescharacteristic
89#532—24no—
90#532—24no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#124—21no—
94C2×C22⋊Q8—22no—
95C2×#121—21no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121C2314yes—
99C2×#121C2312yes—
100#520—28no—
101#520—28no—
102C4×#41—24no—
103C4×#41—24no—
104C2×C4×Q8—22no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4D412yes—
112C2×C4×D4D412yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C2×#117C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C42—22no—
126C22×C42—21no—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#482—216no—
130#480—216no—
131#432—24no—
132#431—24no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#399—22no—
151#397—22no—
152#394—24no—
153#393—22no—
154#392—24no—
155#392—22no—
156#391—22no—
157#390—24no—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#389—22no—
162#389C2314yes—
163#388—24no—
164#386—24no—
165#385—24no—
166#384C2312yes—
167#382—21no—
168#380—22no—
169#380—21no—
170#380C2314yes—
171#380C2312yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2311yescharacteristic
190C4×C4⋊C4C2311yescharacteristic
191C4×C4⋊C4C2311yescharacteristic
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204#376—22no—
205#376—21no—
206C2×#95—22no—
207C2×#95—21no—
208C43—22no—
209C43—21no—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—24no—
223C2×#41—24no—
224C2×#41—44no—
225C2×#41—84no—
226C22×Q8—21no—
227C22×Q8C2×D411yescharacteristic
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2×D411yescharacteristic
233C22×D4C2×D412yes—
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250#127—22no—
251#127—22no—
252#124—44no—
253C22⋊Q8—84no—
254#121—24no—
255#121—24no—
256#121—28no—
257#121—28no—
258#121—44no—
259#121—84no—
260#121—84no—
261C4×Q8—216no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×D4—216no—
265C4×D4—216no—
266C4×D4—216no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—44no—
274#117—22no—
275#117—22no—
276#117—24no—
277#117—24no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—82no—
284#117#4118yescomplemented
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4#4118yescomplemented
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—82no—
338C2×C42—84no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42#4114yes—
344C2×C42#4114yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—28no—
349#107—28no—
350#107—28no—
351#106—28no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—24no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—28no—
362C22⋊C8—28no—
363#97—24no—
364C4×C8—24no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379C24—22no—
380C24C22×D411yescharacteristic
381C24C22×D412yes—
382#41—28no—
383#41—216no—
384#41—28no—
385#41—816no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—48no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—84no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—84no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4#12314yes—
442C22×C4#12314yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2511yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—28no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—84no—
482C4⋊C4C2×#4114yes—
483C4⋊C4C2×#4114yes—
484C4⋊C4C2×#4114yes—
485C4⋊C4C2×#4114yes—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—24no—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—82no—
498C22⋊C4—84no—
499C42—28no—
500C42—28no—
501C42—24no—
502C42—24no—
503C42—28no—
504C42—22no—
505C42—22no—
506C42—44no—
507C42—42no—
508C42—42no—
509C42—82no—
510C42—82no—
511C42—82no—
512C42—81no—
513C42—81no—
514C42—82no—
515C42—82no—
516C42—82no—
517C42#12314yes—
518C42#12314yes—
519C42C2×#4112yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23C2×#12312yes—
539C23#52514yes—
540C23C23×D411yescharacteristic
541Q8—28no—
542Q8—84no—
543D4—28no—
544D4—216no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—84no—
549D4—88no—
550D4—84no—
551D4—168no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—84no—
593C2×C4—88no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4C2×#12312yes—
598C2×C4C2×#12312yes—
599C2×C4C2×#12312yes—
600C2×C4C2×#12312yes—
601C2×C4C2×#12312yes—
602C2×C4#52514yes—
603C2×C4#57612yes—
604C2×C4#57812yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4112yes—
608C2×C4C22×#4112yes—
609C8—42no—
610C8—42no—
611C8—44no—
612C22—22no—
613C22—24no—
614C22—22no—
615C22—22no—
616C22—24no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—84no—
628C22—84no—
629C22C22×#12311yescharacteristic
630C22C2×#52512yes—
631C22C2×#57611yescharacteristic
632C22C2×#57811yescharacteristic
633C22#358512yes—
634C4—22no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—82no—
646C4—84no—
647C4#309814yes—
648C4#344411yescharacteristic
649C4#344411yescharacteristic
650C4#344412yes—
651C2—42no—
652C2—44no—
653C2—162no—
654C2#3662611yescharacteristic
655C2C2×#309811yescharacteristic
656C2#6051811yescharacteristic
657C2#6059012yes—
658C2#6060012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35292C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#520C2semidirect
3. #33153C2semidirect
4. #33149C2semidirect
5. #33113C2semidirect
6. #33106C2semidirect
7. #33103C2semidirect
8. #32986C2semidirect
9. #21590C2semidirect
10. #21586C2semidirect
11. #21395C2semidirect
12. #21390C2semidirect
13. #21364C2semidirect
14. #21358C2semidirect
15. #20721C2semidirect
284. #117#41semidirect
301. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
#352922
C4×#5202
#331532
#331492
#331132
#331062
#331032
#329862
#215902
#215862
#213952
#213902
#213642
#213582
#207212