Finite Group Databasebeta

← #377672

Subgroups — #377672

Order 512.

975 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2#21147C211yesmaximal, characteristic, complemented
3C2×#1938C211yesmaximal, characteristic
4C2×#1937C211yesmaximal, characteristic
5#21052C211yesmaximal, characteristic
6C2×#1773C211yesmaximal, characteristic, complemented
7C2×#1768C211yesmaximal, characteristic, complemented
8#11233C211yesmaximal, characteristic, complemented
9#11221C211yesmaximal, characteristic, complemented
10#11125C211yesmaximal, characteristic
11#10934C211yesmaximal, characteristic
12#10924C211yesmaximal, characteristic
13#10887C211yesmaximal, characteristic, complemented
14#10879C211yesmaximal, characteristic, complemented
15#10818C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2297C2211yescharacteristic
33#2296C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#2068C412yescomplemented
36#2067C412yescomplemented
37#2066C412yescomplemented
38#2065C412yescomplemented
39#1940—22no—
40#1939—22no—
41#1938—24no—
42#1937—24no—
43#1924—22no—
44#1923—22no—
45#1773—24no—
46#1768—24no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
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
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#108—22no—
79C2×#108—22no—
80C2×#108—22no—
81#427—22no—
82#427—22no—
83C2×#107C2×C412yes—
84C2×#106C2×C412yes—
85#424—22no—
86#424—22no—
87#424—22no—
88#424—41no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2×C412yes—
97C2×C4⋊C8C2×C412yes—
98C2×C4⋊C8D412yes—
99C2×C4⋊C8D412yes—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—41no—
106#416—22no—
107#416—22no—
108#416—22no—
109#416—22no—
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C412yes—
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C412yes—
115C2×#102C2×C411yescharacteristic
116#408—22no—
117#408—22no—
118#408—22no—
119#408—41no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—21no—
123#405—22no—
124#405—22no—
125#405—41no—
126#405—41no—
127C2×#97—22no—
128C2×#97—22no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97D414yes—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136C2×C4×C8—22no—
137#391C2×C412yes—
138#390C2×C412yes—
139#389C2×C412yes—
140#389C2×C412yes—
141#386—21no—
142#386—21no—
143#385—21no—
144#385—21no—
145C4×C4⋊C4—22no—
146C4×C22⋊C4—22no—
147#376—22no—
148#376—22no—
149C2×#95—21no—
150C2×#95—21no—
151#341—22no—
152#341—22no—
153#341—22no—
154#340C2×C412yes—
155#340C2×C412yes—
156#339—22no—
157#339—22no—
158#339—22no—
159#339—22no—
160#339—41no—
161#339D414yes—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C22×Q8C22×C411yescharacteristic
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4#4111yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4C2411yescharacteristic
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×M16—21no—
210C2×M16—21no—
211C2×M16C2×D412yes—
212C2×M16#4112yes—
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×C8—21no—
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×C8C22×C412yes—
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8#4112yes—
246C22×C8#4112yes—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117C2×D412yes—
258#117#4112yes—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C22×C412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42#4112yes—
290C2×C42#4112yes—
291#108—28no—
292#107—24no—
293#106—24no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—44no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301#104—44no—
302#104—44no—
303#103—22no—
304#103—22no—
305#103—24no—
306#102—22no—
307#102—22no—
308#102—24no—
309C22⋊C8—44no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312#97—28no—
313#97—42no—
314#97—42no—
315C4×C8—42no—
316C4×C8—42no—
317C4×C8—44no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324C24—21no—
325C24C2×#4111yescharacteristic
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
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×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C22×C42−1+411yescharacteristic
401C22×C42−1+411yescharacteristic
402C22×C42−1+411yescharacteristic
403M16—22no—
404M16—22no—
405M16—22no—
406M16—22no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8C4×D412yes—
462C2×C8C4×D412yes—
463C2×C8#12112yes—
464C2×C8#12112yes—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—82no—
474C4⋊C4C4×D414yes—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—44no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—82no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42#12114yes—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23C2×C4×D411yescharacteristic
509C23#52511yescharacteristic
510C23#52911yescharacteristic
511C23#54711yescharacteristic
512C23#55211yescharacteristic
513C23#55311yescharacteristic
514C23#55611yescharacteristic
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—81no—
524D4—81no—
525D4—81no—
526D4—81no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C4×#4111yescharacteristic
567C2×C4C4×#4111yescharacteristic
568C2×C4#51911yescharacteristic
569C2×C4#52011yescharacteristic
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4#52911yescharacteristic
574C2×C4#53211yescharacteristic
575C2×C4#53211yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53711yescharacteristic
578C2×C4#54111yescharacteristic
579C2×C4#54211yescharacteristic
580C2×C4D4211yescharacteristic
581C2×C4#54711yescharacteristic
582C2×C4#54711yescharacteristic
583C2×C4#54811yescharacteristic
584C2×C4#54811yescharacteristic
585C2×C4#54811yescharacteristic
586C2×C4#55111yescharacteristic
587C2×C4#55211yescharacteristic
588C2×C4#55211yescharacteristic
589C2×C4#55311yescharacteristic
590C2×C4#55611yescharacteristic
591C2×C4#55611yescharacteristic
592C8—22no—
593C8—22no—
594C8—22no—
595C8—22no—
596C8—42no—
597C8—42no—
598C8—42no—
599C8—42no—
600C8—81no—
601C8—81no—
602C22—22no—
603C22—22no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—22no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—82no—
615C22C4×#12111yescharacteristic
616C22#236111yescharacteristic
617C22#236211yescharacteristic
618C22#241311yescharacteristic
619C22#246911yescharacteristic
620C22#252511yescharacteristic
621C22#260111yescharacteristic
622C22#295512yes—
623C22#295712yes—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—82no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—82no—
633C4C4×#12111yescharacteristic
634C4#232311yescharacteristic
635C4#236211yescharacteristic
636C4#237311yescharacteristic
637C4#241611yescharacteristic
638C4#246111yescharacteristic
639C4#251911yescharacteristic
640C4#254511yescharacteristic
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—161no—
646C2#1420011yescharacteristic
647C2#2154711yescharacteristic
648C2#2155011yescharacteristic
649C2#2301712yes—
650C2#2308212yes—
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
2. #21147C2semidirect
6. C2×#1773C2semidirect
7. C2×#1768C2semidirect
8. #11233C2semidirect
9. #11221C2semidirect
13. #10887C2semidirect
14. #10879C2semidirect
15. #10818C2semidirect
35. #2068C4semidirect
36. #2067C4semidirect
37. #2066C4semidirect
38. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
#211472
C2×#19382
C2×#19372
#210522
C2×#17732
C2×#17682
#112332
#112212
#111252
#109342
#109242
#108872
#108792
#108182