Finite Group Databasebeta

← #192745

Subgroups — #192745

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#117C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#20740C211yesmaximal, 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#8046C211yesmaximal, characteristic
15#7902C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2355—21no—
21#2355—21no—
22#2355C2211yescharacteristic
23#2355C2211yescharacteristic
24Q8×C4⋊C4C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26#2345—22no—
27#2345—21no—
28Q8×C22⋊C4—21no—
29#2295—22no—
30#2295—21no—
31#2285—21no—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34C42×Q8—21no—
35C4×#117—21no—
36C4×#117C2211yescharacteristic
37#1949—22no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1854—22no—
43#1854—21no—
44#1854—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1823—21no—
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1803C2211yescharacteristic
54#1787—21no—
55C4×#107—21no—
56C4×#106—21no—
57#1763—21no—
58#1763—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—21no—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#432—22no—
94#432—22no—
95#431—22no—
96#431—22no—
97#431C2×C414yes—
98#431C2×C414yes—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—22no—
102C2×#106—21no—
103C2×#106—21no—
104C2×#106—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—22no—
122#391—22no—
123#391—21no—
124#391—21no—
125#389—21no—
126#389—21no—
127#389—22no—
128#389—22no—
129#387—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#387—22no—
135#384—21no—
136#384—21no—
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#382—21no—
142#382—21no—
143#382—21no—
144#382—21no—
145#380—22no—
146#380—22no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380D412yes—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D412yes—
167C4×C4⋊C4C2311yescharacteristic
168C4×C4⋊C4C2311yescharacteristic
169C4×C22⋊C4—22no—
170C4×C22⋊C4—22no—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173#376—22no—
174#376—21no—
175C43—22no—
176C43—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—22no—
232#117—22no—
233#117—41no—
234#117—41no—
235#117—42no—
236#117—42no—
237#117—82no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
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—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C4⋊C414yes—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—41no—
273C2×C42—24no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—24no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—41no—
291C2×C42C22×C412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×Q812yes—
296C2×C42#4112yes—
297C2×C42#4112yes—
298C2×C42C2411yescharacteristic
299#107—42no—
300#107—42no—
301#107—82no—
302#106—42no—
303#106—42no—
304#106—82no—
305C4⋊C8—24no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8C22×C412yes—
313C4⋊C8C22×C412yes—
314#103—42no—
315#103—42no—
316#103—42no—
317#103—42no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320C22⋊C8—42no—
321#97—42no—
322C4×C8—24no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8C22×C412yes—
326C4×C8C22×C412yes—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—21no—
336#95—21no—
337#95—22no—
338#95—22no—
339#95—21no—
340#95—21no—
341#95—41no—
342#95—41no—
343C24#12011yescharacteristic
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—24no—
349C2×Q8—28no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8C2×C4⋊C412yes—
360C2×Q8C2×C4⋊C412yes—
361C2×Q8#13512yes—
362C2×Q8#13512yes—
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—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4C2×C4⋊C412yes—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4#12011yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4C22⋊Q812yes—
398C22×C4C22⋊Q812yes—
399C22×C4#12312yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
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—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
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—41no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4C2×C4⋊C412yes—
461C4⋊C4C2×C4⋊C412yes—
462C4⋊C4#13712yes—
463C4⋊C4#13712yes—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—82no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—24no—
475C42—42no—
476C42—44no—
477C42—44no—
478C42—44no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—81no—
484C42—81no—
485C42—82no—
486C42—82no—
487C42—82no—
488C42C2×C4⋊C412yes—
489C42C4×D412yes—
490C42C4×D412yes—
491C42C22⋊Q812yes—
492C42C22⋊Q812yes—
493C42#12312yes—
494C42C23×C411yescharacteristic
495C42C22×Q811yescharacteristic
496C42C2×#4111yescharacteristic
497C42C2×#4111yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#38512yes—
505C23C2×#12011yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8—22no—
511Q8—48no—
512Q8—41no—
513Q8—41no—
514Q8#42714yes—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4#38512yes—
572C2×C4#38512yes—
573C2×C4#38512yes—
574C2×C4C22×C4⋊C411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C22⋊Q811yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4C2×#12311yescharacteristic
580C2×C4C2×#13511yescharacteristic
581C2×C4C2×#13511yescharacteristic
582C2×C4C2×#13711yescharacteristic
583C2×C4C2×#13711yescharacteristic
584C2×C4#57511yescharacteristic
585C2×C4#57511yescharacteristic
586C2×C4#57511yescharacteristic
587C2×C4#57511yescharacteristic
588C8—42no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—81no—
593C8—81no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22C2×#38511yescharacteristic
602C22#300211yescharacteristic
603C22#300211yescharacteristic
604C22#300411yescharacteristic
605C22#300511yescharacteristic
606C22#302111yescharacteristic
607C22#302211yescharacteristic
608C4—22no—
609C4—24no—
610C4—21no—
611C4—21no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—82no—
621C4—81no—
622C4—82no—
623C4—161no—
624C4—161no—
625C4C2×#42711yescharacteristic
626C4#291111yescharacteristic
627C4C4×#13711yescharacteristic
628C4C4×#13711yescharacteristic
629C4#307811yescharacteristic
630C4#307811yescharacteristic
631C4#309111yescharacteristic
632C4#309111yescharacteristic
633C2—42no—
634C2#1879611yescharacteristic
635C2#1879711yescharacteristic
636C2#1880811yescharacteristic
637C2#1880911yescharacteristic
638C2#2084311yescharacteristic
639C2#2086511yescharacteristic
640C2#2087411yescharacteristic
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
Q8×#1172
#219842
#213622
#207402
C2×#14792
#88962
#88952
#88922
#88912
#88882
#88872
#88842
#88832
#80462
#79022