Finite Group Databasebeta

← #385417

Subgroups — #385417

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21173C211yesmaximal, characteristic, complemented
2C2×#1937C211yesmaximal, characteristic
3#21067C211yesmaximal, characteristic
4#20881C211yesmaximal, characteristic
5C2×#1859C211yesmaximal, characteristic
6#20847C211yesmaximal, characteristic
7C2×#1845C211yesmaximal, characteristic, complemented
8#11340C211yesmaximal, characteristic, complemented
9#11331C211yesmaximal, characteristic, complemented
10#11270C211yesmaximal, characteristic, complemented
11#11239C211yesmaximal, characteristic, complemented
12#11107C211yesmaximal, characteristic
13#11078C211yesmaximal, characteristic
14#10891C211yesmaximal, characteristic, complemented
15#10818C211yesmaximal, characteristic, complemented
16C2×#520C2211yescharacteristic
17C2×#428C2211yescharacteristic
18C2×#428C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C2×#417C2211yescharacteristic
26C2×#417C2211yescharacteristic
27C2×#416C2211yescharacteristic
28C2×#408C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2317C2211yescharacteristic
31#2311C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34#2274C2211yescharacteristic
35#1949—22no—
36#1947—22no—
37#1937C418yescomplemented
38#1928—22no—
39#1925—22no—
40#1863—22no—
41#1862—22no—
42#1859—24no—
43#1854—22no—
44#1852—22no—
45#1845—24no—
46C2×#343C2211yescharacteristic
47C2×#343C2211yescharacteristic
48C2×#343C2211yescharacteristic
49C2×#343C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#341C2211yescharacteristic
52#1736C2211yescharacteristic
53#1736C2211yescharacteristic
54#1736C2211yescharacteristic
55#1736C2211yescharacteristic
56#1735C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62C22×#41C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69#520—22no—
70#520—41no—
71#520—41no—
72#520—41no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—41no—
76C2×#117—21no—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—41no—
82C22×C4⋊C4—41no—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#428—22no—
89#428—41no—
90#428—41no—
91#428—41no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—21no—
101C2×#107—21no—
102C2×#107—41no—
103C2×#107—41no—
104C2×#107—41no—
105C2×#107—41no—
106C2×#106—21no—
107C2×#106—21no—
108#424—22no—
109#424C2×C414yescomplemented
110C2×C4⋊C8—22no—
111C2×#104—22no—
112C2×#104—22no—
113C2×#104C2×C414yes—
114#417—22no—
115#417—41no—
116#417—41no—
117#417—41no—
118#416C2×C414yes—
119#416Q814yes—
120C2×#103—21no—
121C2×#103—21no—
122C2×#103D412yes—
123C2×#102—21no—
124C2×#102—21no—
125C2×#102D412yes—
126#408—22no—
127#408C2×C414yes—
128C2×C22⋊C8—22no—
129C2×C4×C8—22no—
130C2×C4×C8—22no—
131C2×C4×C8C2×C414yescomplemented
132#391—21no—
133#390—21no—
134#389—22no—
135#386—41no—
136#385—41no—
137#384—21no—
138#384—22no—
139#384—41no—
140#384—41no—
141#384D414yes—
142#382—21no—
143#380—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C22⋊C4—41no—
147#343—22no—
148#343—41no—
149#343—41no—
150#343—41no—
151#341—22no—
152#341C2×C414yescomplemented
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340D412yes—
158#340D412yes—
159#339—22no—
160#339—22no—
161#339C2×C414yescomplemented
162#336—22no—
163#336—22no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41C22×C412yes—
175C2×#41C2×Q812yes—
176C22×Q8C2×D411yescharacteristic
177C22×D4—21no—
178C22×D4C2×D411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—41no—
186C23×C4C2411yescharacteristic
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16C22×C412yes—
210C2×M16C22×C412yes—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8C22×C412yes—
224C22×C8C22×C412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×Q812yes—
228C22×C8#4112yes—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×D4—42no—
232C4×D4—44no—
233C4×D4—42no—
234C4×D4—44no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—42no—
246#117—42no—
247#117C22×C412yes—
248#117C2×Q812yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—84no—
278C2×C4⋊C4—84no—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—42no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—41no—
297C2×C42—42no—
298C2×C42C22×C412yes—
299C2×C42C2×D412yes—
300C2×C42#4112yes—
301#107—42no—
302#107—44no—
303#107—44no—
304#107—44no—
305#107—44no—
306#107—42no—
307#106—42no—
308#106—42no—
309C4⋊C8—28no—
310#104—28no—
311#103—42no—
312#103—42no—
313#103C4⋊C418yescomplemented
314#102—42no—
315#102—42no—
316#102C4⋊C418yescomplemented
317C22⋊C8—28no—
318C4×C8—28no—
319#95—42no—
320#95—42no—
321#95—42no—
322C24—21no—
323C24H(5,𝔽2)11yescharacteristic
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8C2×C4⋊C412yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4C2×C4⋊C412yes—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—162no—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C42−1+411yescharacteristic
403C22×C42−1+411yescharacteristic
404C22×C42−1+411yescharacteristic
405C22×C42−1+411yescharacteristic
406M16—22no—
407M16—22no—
408M16—22no—
409M16—22no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8C2×C4⋊C414yes—
447C2×C8C4×D414yes—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—44no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C4⋊C4—162no—
475C4⋊C4C2×C4⋊C414yes—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—44no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—162no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—44no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—162no—
495C42C4×D414yes—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23#52011yescharacteristic
504C23#53411yescharacteristic
505C23#53911yescharacteristic
506C23D4211yescharacteristic
507C23#55411yescharacteristic
508C23#56011yescharacteristic
509C23#56311yescharacteristic
510Q8—22no—
511Q8—22no—
512Q8—41no—
513Q8—41no—
514D4—22no—
515D4—22no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—41no—
521D4—41no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—161no—
557C2×C4—161no—
558C2×C4—164no—
559C2×C4—161no—
560C2×C4—161no—
561C2×C4C22×C4⋊C411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C2×C4×D411yescharacteristic
564C2×C4#52011yescharacteristic
565C2×C4#52011yescharacteristic
566C2×C4#52011yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#53511yescharacteristic
569C2×C4#53611yescharacteristic
570C2×C4#53611yescharacteristic
571C2×C4#53611yescharacteristic
572C2×C4#53911yescharacteristic
573C2×C4#54211yescharacteristic
574C2×C4#54211yescharacteristic
575C2×C4#54411yescharacteristic
576C2×C4#54411yescharacteristic
577C2×C4D4211yescharacteristic
578C2×C4D4211yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4#54711yescharacteristic
581C2×C4#54711yescharacteristic
582C2×C4#54811yescharacteristic
583C2×C4D4×Q811yescharacteristic
584C2×C4D4×Q811yescharacteristic
585C2×C4#55311yescharacteristic
586C2×C4#55411yescharacteristic
587C2×C4#56311yescharacteristic
588C2×C4#56311yescharacteristic
589C8—24no—
590C8—24no—
591C8—42no—
592C8—42no—
593C8—81no—
594C8—81no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—44no—
605C22—81no—
606C22—81no—
607C22—82no—
608C22#232211yescharacteristic
609C22D4×C4⋊C411yescharacteristic
610C22#238311yescharacteristic
611C22#266911yescharacteristic
612C22#268011yescharacteristic
613C22#272911yescharacteristic
614C22#279011yescharacteristic
615C22#329212yes—
616C22#329312yes—
617C4—28no—
618C4—42no—
619C4—44no—
620C4—81no—
621C4—82no—
622C4—82no—
623C4—81no—
624C4—82no—
625C4—161no—
626C4—161no—
627C4#231711yescharacteristic
628C4#232611yescharacteristic
629C4D4×C4⋊C411yescharacteristic
630C4#236111yescharacteristic
631C4#267411yescharacteristic
632C4#267711yescharacteristic
633C4#271811yescharacteristic
634C4#272011yescharacteristic
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—82no—
640C2#1440511yescharacteristic
641C2#2269212yes—
642C2#3048411yescharacteristic
643C2#3048811yescharacteristic
644C2#3111012yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21173C2semidirect
7. C2×#1845C2semidirect
8. #11340C2semidirect
9. #11331C2semidirect
10. #11270C2semidirect
11. #11239C2semidirect
14. #10891C2semidirect
15. #10818C2semidirect
37. #1937C4semidirect
109. #424C2×C4semidirect
131. C2×C4×C8C2×C4semidirect
152. #341C2×C4semidirect
161. #339C2×C4semidirect
313. #103C4⋊C4semidirect
316. #102C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#211732
C2×#19372
#210672
#208812
C2×#18592
#208472
C2×#18452
#113402
#113312
#112702
#112392
#111072
#110782
#108912
#108182