Finite Group Databasebeta

← #373213

Subgroups — #373213

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21147C211yesmaximal, characteristic, complemented
2C2×#1937C211yesmaximal, characteristic
3#21086C211yesmaximal, characteristic
4C2×#1866C211yesmaximal, characteristic
5#20838C211yesmaximal, characteristic
6C2×#1771C211yesmaximal, characteristic, complemented
7C2×#1765C211yesmaximal, characteristic
8#11329C211yesmaximal, characteristic, complemented
9#11226C211yesmaximal, characteristic, complemented
10#11125C211yesmaximal, characteristic
11#11075C211yesmaximal, characteristic
12#10910C211yesmaximal, characteristic, complemented
13#10883C211yesmaximal, characteristic, complemented
14#10882C211yesmaximal, characteristic, complemented
15#10865C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2297C2211yescharacteristic
33#2296C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1940—22no—
36#1939—22no—
37#1937—24no—
38#1936C412yescomplemented
39#1936C412yescomplemented
40#1935C412yescomplemented
41#1935C412yescomplemented
42#1866—24no—
43#1851—22no—
44#1850—22no—
45#1771—24no—
46#1765—24no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1728C2211yescharacteristic
60#1728C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—41no—
69C2×#131—41no—
70C22×M16—21no—
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8—21no—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—22no—
91C2×#108—41no—
92C2×#108—41no—
93#424—22no—
94#424—22no—
95#424—22no—
96#424—22no—
97#424—41no—
98#424—41no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8D412yes—
106C2×C4⋊C8D412yes—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—41no—
113#416—22no—
114#416—22no—
115#416D414yes—
116#416Q814yes—
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C412yes—
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C412yes—
123#408—22no—
124#408—41no—
125C2×C22⋊C8—22no—
126#405—22no—
127#405—41no—
128C2×#97—22no—
129C2×#97—22no—
130C2×#97—22no—
131C2×#97—22no—
132C2×#97—21no—
133C2×#97—41no—
134C2×#97C2×C412yes—
135C2×#97C2×C412yes—
136C2×C4×C8—22no—
137C2×C4×C8—21no—
138C2×C4×C8—22no—
139C2×C4×C8—41no—
140#391C2×C412yes—
141#390C2×C412yes—
142#389C2×C412yes—
143#389C2×C412yes—
144#386—21no—
145#386—21no—
146#385—21no—
147#385—21no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C22⋊C4—22no—
152C2×#95—21no—
153#341—22no—
154#341—22no—
155#340C2×C412yes—
156#340C2×C412yes—
157#340C2×C412yes—
158#340C2×C412yes—
159#339—22no—
160#339—22no—
161#339—22no—
162#339—22no—
163#339—22no—
164#339—41no—
165#336—21no—
166#336—21no—
167#336—22no—
168#336—22no—
169#336—22no—
170#336—22no—
171#336D414yes—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41C2×D412yes—
183C2×#41C2×Q812yes—
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4#4111yescharacteristic
191C23×C4#4111yescharacteristic
192C23×C4C2411yescharacteristic
193#131—28no—
194#131—44no—
195#131—44no—
196#131—44no—
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×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—42no—
220C2×M16—42no—
221C2×M16—42no—
222C2×M16#4112yes—
223C2×M16#4112yes—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8C22×C412yes—
250C22×C8C22×C411yescharacteristic
251C22×C8C22×C411yescharacteristic
252C22×C8C2×D412yes—
253C22×C8C2×D412yes—
254C22×C8C2×D412yes—
255C22×C8C2×D412yes—
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8C2×Q812yes—
259C22×C8#4112yes—
260C22×C8#4112yes—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117C2×D412yes—
272#117C2×Q812yes—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C412yes—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C22×C412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42#4112yes—
307#108—44no—
308#108—44no—
309C4⋊C8—24no—
310C4⋊C8—24no—
311C4⋊C8—44no—
312C4⋊C8—44no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315#104—28no—
316#104—44no—
317#103—22no—
318#103—22no—
319#103C4⋊C418yescomplemented
320#102—22no—
321#102—22no—
322#102C4⋊C418yescomplemented
323C22⋊C8—44no—
324#97—24no—
325#97—24no—
326#97—42no—
327#97—44no—
328C4×C8—44no—
329C4×C8—42no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335C24—21no—
336C24C2×#4111yescharacteristic
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8C2×C4⋊C412yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4C2×C4⋊C412yes—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×Q811yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413C22×C42−1+411yescharacteristic
414M16—22no—
415M16—22no—
416M16—22no—
417M16—22no—
418M16—48no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431C2×C8—22no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—48no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8C2×C4⋊C414yes—
474C2×C8C4×D412yes—
475C2×C8C4×D412yes—
476C2×C8#12112yes—
477C2×C8#12112yes—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—82no—
488C4⋊C4C2×C4⋊C414yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—44no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—82no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23C2×C4×D411yescharacteristic
523C23#52511yescharacteristic
524C23#52911yescharacteristic
525C23#54711yescharacteristic
526C23#55411yescharacteristic
527C23#55611yescharacteristic
528C23#55911yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—41no—
532Q8—41no—
533D4—22no—
534D4—22no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—41no—
540D4—41no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4C22×C4⋊C411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C4×#4111yescharacteristic
581C2×C4C4×#4111yescharacteristic
582C2×C4#52011yescharacteristic
583C2×C4#52011yescharacteristic
584C2×C4C2×#12111yescharacteristic
585C2×C4#52511yescharacteristic
586C2×C4#52511yescharacteristic
587C2×C4#53211yescharacteristic
588C2×C4#53311yescharacteristic
589C2×C4#53311yescharacteristic
590C2×C4#53811yescharacteristic
591C2×C4#53911yescharacteristic
592C2×C4#54111yescharacteristic
593C2×C4#54211yescharacteristic
594C2×C4D4211yescharacteristic
595C2×C4D4211yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4#54811yescharacteristic
601C2×C4D4×Q811yescharacteristic
602C2×C4D4×Q811yescharacteristic
603C2×C4#55111yescharacteristic
604C2×C4#55211yescharacteristic
605C2×C4#55411yescharacteristic
606C8—24no—
607C8—22no—
608C8—22no—
609C8—22no—
610C8—22no—
611C8—42no—
612C8—42no—
613C8—42no—
614C8—81no—
615C8—81no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—44no—
626C22—81no—
627C22—81no—
628C22—82no—
629C22C4×#12111yescharacteristic
630C22D4×C4⋊C411yescharacteristic
631C22#236711yescharacteristic
632C22#245511yescharacteristic
633C22#246511yescharacteristic
634C22#247811yescharacteristic
635C22#258911yescharacteristic
636C22#295512yes—
637C22#295612yes—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4—81no—
645C4—81no—
646C4—82no—
647C4#229611yescharacteristic
648C4#233411yescharacteristic
649C4D4×C4⋊C411yescharacteristic
650C4#235911yescharacteristic
651C4#242411yescharacteristic
652C4#242911yescharacteristic
653C4#246111yescharacteristic
654C4#255611yescharacteristic
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—82no—
660C2#1411211yescharacteristic
661C2#2154911yescharacteristic
662C2#2155011yescharacteristic
663C2#2268712yes—
664C2#2323612yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21147C2semidirect
6. C2×#1771C2semidirect
8. #11329C2semidirect
9. #11226C2semidirect
12. #10910C2semidirect
13. #10883C2semidirect
14. #10882C2semidirect
15. #10865C2semidirect
38. #1936C4semidirect
39. #1936C4semidirect
40. #1935C4semidirect
41. #1935C4semidirect
319. #103C4⋊C4semidirect
322. #102C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#211472
C2×#19372
#210862
C2×#18662
#208382
C2×#17712
C2×#17652
#113292
#112262
#111252
#110752
#109102
#108832
#108822
#108652