Finite Group Databasebeta

← #192777

Subgroups — #192777

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic
3#21984C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8896C211yesmaximal, characteristic, complemented
7#8895C211yesmaximal, characteristic, complemented
8#8892C211yesmaximal, characteristic, complemented
9#8891C211yesmaximal, characteristic, complemented
10#8888C211yesmaximal, characteristic, complemented
11#8887C211yesmaximal, characteristic, complemented
12#8884C211yesmaximal, characteristic, complemented
13#8883C211yesmaximal, characteristic, complemented
14#8038C211yesmaximal, characteristic
15#7894C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2355—21no—
21#2355—21no—
22#2355C2211yescharacteristic
23#2355C2211yescharacteristic
24Q8×C4⋊C4C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26#2344—22no—
27#2344—21no—
28Q8×C22⋊C4—21no—
29#2320—22no—
30#2320—21no—
31#2285—21no—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#2283—21no—
35#2283C2211yescharacteristic
36#2276—21no—
37#1949—22no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1854—21no—
43#1854—21no—
44#1854—22no—
45#1846—21no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1835—21no—
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1824—21no—
52#1803C2211yescharacteristic
53#1803C2211yescharacteristic
54#1803C2211yescharacteristic
55#1803C2211yescharacteristic
56#1787—21no—
57C4×#103—21no—
58C4×#103—21no—
59#1479C418yescomplemented
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C23×Q8D411yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8C2×C414yes—
80C2×C4×Q8D411yescharacteristic
81C2×C4×Q8D411yescharacteristic
82C2×C4×Q8Q814yes—
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×#117—22no—
86C2×#117—21no—
87C2×#117D412yes—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D411yescharacteristic
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#431—22no—
93#431—22no—
94#431—22no—
95#431—22no—
96#431C2×C414yes—
97#431C2×C414yes—
98C2×#107—22no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#106—22no—
102C2×#106—21no—
103C2×#106—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—22no—
121#391—22no—
122#391—21no—
123#391—21no—
124#389—21no—
125#389—21no—
126#389—22no—
127#389—22no—
128#387—22no—
129#387—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#384—21no—
135#384—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#383—22no—
141#383—21no—
142#382—22no—
143#382—21no—
144#382—21no—
145#382—21no—
146#382D412yes—
147#382D411yescharacteristic
148#382D411yescharacteristic
149#381—22no—
150#381—21no—
151#380—22no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380D411yescharacteristic
157#380D411yescharacteristic
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4C2311yescharacteristic
172C4×C4⋊C4C2311yescharacteristic
173C4×C22⋊C4—22no—
174C4×C22⋊C4—21no—
175#376—22no—
176#376—21no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—22no—
182#340—22no—
183#340—21no—
184#340—21no—
185#340—21no—
186#340—21no—
187Q8⋊C8C2×C414yes—
188Q8⋊C8C2×C414yes—
189Q8⋊C8C2×C414yes—
190Q8⋊C8C2×C414yes—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—41no—
196C22×Q8—41no—
197C22×Q8C4⋊C414yes—
198C22×Q8C2×D411yescharacteristic
199C22×Q8C2×D411yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225C4×Q8C4⋊C414yes—
226C4×Q8C4⋊C414yes—
227C4×Q8C22×C412yes—
228C4×Q8C22×C412yes—
229C4×Q8C2×Q812yes—
230C4×Q8C2×Q812yes—
231#117—84no—
232#117—81no—
233#117—81no—
234#117—82no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
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—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C4⋊C414yes—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—41no—
270C2×C42—24no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—24no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—41no—
284C2×C42C22×C412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×Q812yes—
289C2×C42#4112yes—
290C2×C42#4112yes—
291C2×C42C2411yescharacteristic
292#107—81no—
293#107—81no—
294#107—82no—
295#106—81no—
296#106—81no—
297#106—82no—
298C4⋊C8—24no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8C22×C412yes—
306C4⋊C8C22×C412yes—
307#103—42no—
308#103—42no—
309#103—42no—
310#103—42no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—42no—
314C4×C8—24no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—42no—
318C4×C8C22×C412yes—
319C4×C8C22×C412yes—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—21no—
329#95—22no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95—41no—
339#95—41no—
340C24#12011yescharacteristic
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—24no—
346C2×Q8—28no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8C2×C4⋊C412yes—
357C2×Q8C2×C4⋊C412yes—
358C2×Q8#13512yes—
359C2×Q8#13512yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4C2×C4⋊C412yes—
389C22×C4C4×D412yes—
390C22×C4C4×D412yes—
391C22×C4#12011yescharacteristic
392C22×C4#12011yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4C22⋊Q812yes—
395C22×C4C22⋊Q812yes—
396C22×C4#12312yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4C2×C4⋊C412yes—
460C4⋊C4C2×C4⋊C412yes—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—82no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—44no—
472C42—44no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—82no—
478C42—82no—
479C42—82no—
480C42—82no—
481C42C2×C4⋊C412yes—
482C42C4×D412yes—
483C42C4×D412yes—
484C42C22⋊Q812yes—
485C42C22⋊Q812yes—
486C42#12312yes—
487C42C23×C411yescharacteristic
488C42C22×Q811yescharacteristic
489C42C2×#4111yescharacteristic
490C42C2×#4111yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23#38512yes—
498C23C2×#12011yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—22no—
504Q8—48no—
505Q8—41no—
506Q8—41no—
507Q8#42714yes—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4#38512yes—
565C2×C4#38512yes—
566C2×C4#38512yes—
567C2×C4C22×C4⋊C411yescharacteristic
568C2×C4C2×C4×D411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×C22⋊Q811yescharacteristic
572C2×C4C2×#12311yescharacteristic
573C2×C4C2×#13511yescharacteristic
574C2×C4C2×#13511yescharacteristic
575C2×C4#57511yescharacteristic
576C2×C4#57511yescharacteristic
577C2×C4#57711yescharacteristic
578C2×C4#57711yescharacteristic
579C2×C4#57811yescharacteristic
580C2×C4#57811yescharacteristic
581C8—42no—
582C8—42no—
583C8—42no—
584C8—42no—
585C8—81no—
586C8—81no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22C2×#38511yescharacteristic
595C22#300211yescharacteristic
596C22#300211yescharacteristic
597C22#301011yescharacteristic
598C22#301111yescharacteristic
599C22#301911yescharacteristic
600C22#302011yescharacteristic
601C4—22no—
602C4—24no—
603C4—21no—
604C4—21no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—81no—
611C4—82no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—81no—
616C4—161no—
617C4—161no—
618C4C2×#42711yescharacteristic
619C4#291111yescharacteristic
620C4#294711yescharacteristic
621C4#294711yescharacteristic
622C4#308211yescharacteristic
623C4#308211yescharacteristic
624C4#309111yescharacteristic
625C4#309111yescharacteristic
626C2—42no—
627C2#1886811yescharacteristic
628C2#1886911yescharacteristic
629C2#1888011yescharacteristic
630C2#1888111yescharacteristic
631C2#2084311yescharacteristic
632C2#2086411yescharacteristic
633C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8896C2semidirect
7. #8895C2semidirect
8. #8892C2semidirect
9. #8891C2semidirect
10. #8888C2semidirect
11. #8887C2semidirect
12. #8884C2semidirect
13. #8883C2semidirect
59. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#341962
#219852
#219842
#207342
C2×#14792
#88962
#88952
#88922
#88912
#88882
#88872
#88842
#88832
#80382
#78942