Finite Group Databasebeta

← #205093

Subgroups — #205093

Order 512.

959 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic
2#21900C211yesmaximal, characteristic, complemented
3#21899C211yesmaximal, characteristic, complemented
4#21358C211yesmaximal, characteristic, complemented
5#17563C211yesmaximal, characteristic, complemented
6#17546C211yesmaximal, characteristic, complemented
7#17324C211yesmaximal, characteristic, complemented
8#9722C212yesmaximal, complemented
9#9722C212yesmaximal, complemented
10#9719C212yesmaximal, complemented
11#9718C212yesmaximal, complemented
12#3444C2211yescharacteristic
13#3444C2211yescharacteristic
14#3250—22no—
15#3231C411yescharacteristic, complemented
16#3231C411yescharacteristic, complemented
17#3228C411yescharacteristic, complemented
18#3228C411yescharacteristic, complemented
19#3210C411yescharacteristic, complemented
20#3210C411yescharacteristic, complemented
21#3207C411yescharacteristic, complemented
22#3207C411yescharacteristic, complemented
23#3191—22no—
24#3170C2211yescharacteristic
25#3169C2211yescharacteristic
26#3167C2211yescharacteristic
27#3167C2211yescharacteristic
28#3102—22no—
29#3102—22no—
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2285C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1986C2212yes—
36#1986C2212yes—
37#1980C2212yes—
38#1980C2212yes—
39#1947—22no—
40#1897—22no—
41#1896—22no—
42#1852—22no—
43#1806—22no—
44#1806—22no—
45C4×#107C2212yes—
46C4×#107C2212yes—
47#1513—22no—
48#1512—22no—
49#1509—22no—
50#1504—22no—
51#1503C2212yes—
52#1503C2212yes—
53#1502C2212yes—
54#1502C2212yes—
55#1454—22no—
56#1453—22no—
57#1451C2212yes—
58C8×M16C2212yes—
59#533C2×C411yescharacteristic
60#533C2×C411yescharacteristic
61#533C2×C411yescharacteristic
62#533C2×C411yescharacteristic
63#533C2311yescharacteristic
64#533C2311yescharacteristic
65#532C2×C411yescharacteristic
66#532C2×C411yescharacteristic
67#532C2×C411yescharacteristic
68#532C2×C411yescharacteristic
69#532C2311yescharacteristic
70#532C2311yescharacteristic
71#529—21no—
72#529—21no—
73#529—21no—
74#529—21no—
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83#525—21no—
84#525—21no—
85#525—21no—
86#525—21no—
87#525—21no—
88#525—21no—
89C4×#41—21no—
90C4×#41—21no—
91C4×#41—21no—
92C4×#41—21no—
93C4×#41—21no—
94C4×#41—21no—
95C2×#117—21no—
96C22×C4⋊C4—21no—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99#500C2×C411yescharacteristic, complemented
100#500C2×C411yescharacteristic, complemented
101#498C2×C411yescharacteristic, complemented
102#498C2×C411yescharacteristic, complemented
103#494C2×C411yescharacteristic, complemented
104#493C2×C411yescharacteristic, complemented
105#492C2×C411yescharacteristic, complemented
106#492C2×C411yescharacteristic, complemented
107#487C2×C411yescharacteristic, complemented
108#487C2×C411yescharacteristic, complemented
109#487C2×C411yescharacteristic, complemented
110#487C2×C411yescharacteristic, complemented
111#486C2×C411yescharacteristic, complemented
112#486C2×C411yescharacteristic, complemented
113#486C2×C411yescharacteristic, complemented
114#486C2×C411yescharacteristic, complemented
115#485—22no—
116#485—22no—
117#482—22no—
118#482—22no—
119#479—22no—
120#479—22no—
121#478—22no—
122#478—22no—
123#467—22no—
124#466—22no—
125#465—22no—
126#465—22no—
127#464—22no—
128#464—22no—
129#463—22no—
130#463—22no—
131#452—22no—
132#452—22no—
133#449—22no—
134#449—22no—
135C4×Q16—22no—
136C4×SD16—22no—
137C4×SD16—22no—
138C4×D8—22no—
139#432—21no—
140#432—21no—
141#432—21no—
142#432D412yes—
143#431—21no—
144#431—21no—
145#431—21no—
146#431C2×C411yescharacteristic
147#431C2×C411yescharacteristic
148#431C2×C411yescharacteristic
149#431C2×C411yescharacteristic
150#431D412yes—
151#431D412yes—
152#431C2312yes—
153C2×#107—22no—
154C2×#107—22no—
155#423—21no—
156#423D412yes—
157C2×C4⋊C8D412yes—
158C2×#104—24no—
159C2×#104—24no—
160#407D412yes—
161#407D412yes—
162C2×C22⋊C8D412yes—
163C4×M16—21no—
164C4×M16C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167#384—21no—
168#382—21no—
169#380—21no—
170#380—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C2×#95—21no—
178#340—22no—
179#340—22no—
180#339—24no—
181#339—24no—
182#328—24no—
183#328—24no—
184#327—24no—
185#327—24no—
186#322—22no—
187#322—24no—
188C82—22no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C23×C4—41no—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×Q16—22no—
200C2×SD16—22no—
201C2×SD16—22no—
202C2×D8—22no—
203C2×M16—22no—
204C2×M16C2×D414yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C4⋊Q8C22×C411yescharacteristic
208C4⋊Q8C22×C411yescharacteristic
209C4⋊Q8C22×C411yescharacteristic
210C4⋊Q8C22×C411yescharacteristic
211C4⋊Q8C22×C411yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213#127C22×C411yescharacteristic
214#127C22×C411yescharacteristic
215#126—41no—
216#126—41no—
217#126—41no—
218#126—41no—
219#125—21no—
220#125—21no—
221#125—21no—
222#125—21no—
223#125—21no—
224#125—21no—
225#124—21no—
226#124—21no—
227#124—21no—
228#124—21no—
229#124—21no—
230#124—21no—
231#123—41no—
232#123—41no—
233#123—41no—
234#123—41no—
235#123—41no—
236#123—41no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249#121—21no—
250#121—21no—
251#121—21no—
252#121—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—21no—
261#120—41no—
262#120—41no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—41no—
288C4×D4—41no—
289C4×D4—41no—
290C4×D4—41no—
291C4×D4—41no—
292C4×D4—41no—
293C4×D4—41no—
294C4×D4—41no—
295C4×D4—41no—
296C4×D4—41no—
297C4×D4—41no—
298C4×D4—41no—
299#117—21no—
300#117—21no—
301#117—21no—
302#117—21no—
303#117—21no—
304#117—21no—
305#117—41no—
306#117—41no—
307#117—41no—
308#117—41no—
309#117—41no—
310#117—41no—
311#117—82no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—41no—
316C2×C4⋊C4—41no—
317C2×C4⋊C4—41no—
318C2×C22⋊C4—41no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—41no—
322C2×C42—41no—
323C2×C42—41no—
324C2×C42—82no—
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42#4111yescharacteristic
333C2×C42#4111yescharacteristic
334C2×C42#4111yescharacteristic
335C2×C42#4111yescharacteristic
336C2×C42C2411yescharacteristic
337#107—22no—
338#107—22no—
339#107—44no—
340#107—44no—
341#106—22no—
342#106—22no—
343C4⋊C8—22no—
344C4⋊C8—22no—
345C4⋊C8—22no—
346C4⋊C8—22no—
347C4⋊C8C22⋊C412yes—
348C4⋊C8C22⋊C412yes—
349C4⋊C8C22⋊C412yes—
350C4⋊C8C22⋊C412yes—
351#104—84no—
352#104—84no—
353#103—22no—
354#103—22no—
355#103—22no—
356#103—22no—
357#102—22no—
358#102—22no—
359#102—22no—
360#102—22no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8C22⋊C412yes—
365C22⋊C8C22⋊C412yes—
366C22⋊C8C22⋊C412yes—
367C22⋊C8C22⋊C412yes—
368C22⋊C8C2×D414yes—
369#97—22no—
370#97—22no—
371C4×C8—22no—
372C4×C8—22no—
373C4×C8C22×C411yescharacteristic
374C4×C8C22×C411yescharacteristic
375C4×C8C22×C411yescharacteristic
376C4×C8C22×C411yescharacteristic
377C4×C8C2×D412yes—
378C4×C8C2×D412yes—
379#95—22no—
380#95—22no—
381#95—41no—
382#95—41no—
383C24#12711yescharacteristic
384#41—42no—
385#41—42no—
386#41—42no—
387#41—42no—
388#41—82no—
389#41—82no—
390#41—82no—
391#41—82no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—81no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4#12111yescharacteristic
435C22×C4#12411yescharacteristic
436C22×C4#12411yescharacteristic
437C22×C4#12711yescharacteristic
438C22×C4#12711yescharacteristic
439C22×C4#12711yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443Q16—82no—
444SD16—82no—
445SD16—82no—
446D8—82no—
447M16—24no—
448M16—24no—
449M16—82no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8C2×C22⋊C412yes—
455C2×C8C2×C22⋊C412yes—
456C2×C8C2×C22⋊C412yes—
457C2×C8C2×C22⋊C412yes—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—82no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—44no—
515C42—44no—
516C42—82no—
517C42C2×C22⋊C411yescharacteristic
518C42C2×C22⋊C411yescharacteristic
519C42C4×D411yescharacteristic
520C42C4×D411yescharacteristic
521C42C4×D411yescharacteristic
522C42C4×D411yescharacteristic
523C42#12111yescharacteristic
524C42#12111yescharacteristic
525C42#12111yescharacteristic
526C42#12111yescharacteristic
527C42#12411yescharacteristic
528C42#12411yescharacteristic
529C42C23×C411yescharacteristic
530C42C22×D411yescharacteristic
531C42C2×#4111yescharacteristic
532C42C2×#4111yescharacteristic
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23#39011yescharacteristic
540C23#39011yescharacteristic
541C23C2×#12711yescharacteristic
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—81no—
547Q8—81no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—81no—
553D4—81no—
554D4—82no—
555D4—82no—
556D4—82no—
557D4—82no—
558D4—81no—
559D4—81no—
560D4—81no—
561D4—81no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—88no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4#39011yescharacteristic
600C2×C4#39011yescharacteristic
601C2×C4#39011yescharacteristic
602C2×C4#39011yescharacteristic
603C2×C4#39011yescharacteristic
604C2×C4#39011yescharacteristic
605C2×C4C22×C22⋊C411yescharacteristic
606C2×C4C2×C4×D411yescharacteristic
607C2×C4C2×C4×D411yescharacteristic
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×#12411yescharacteristic
611C2×C4#57612yes—
612C2×C4#57812yes—
613C8—22no—
614C8—22no—
615C8—22no—
616C8—22no—
617C8—44no—
618C8—44no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22C2×#39011yescharacteristic
636C22#315511yescharacteristic
637C22#315611yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—84no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—162no—
655C4#290112yes—
656C4#295112yes—
657C4#306112yes—
658C2—21no—
659C2—21no—
660C2—41no—
661C2—41no—
662C2—161no—
663C2—161no—
664C2#2123011yescharacteristic
665C2#2124411yescharacteristic
666C2#2125311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60944C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21900C2semidirect
3. #21899C2semidirect
4. #21358C2semidirect
5. #17563C2semidirect
6. #17546C2semidirect
7. #17324C2semidirect
8. #9722C2semidirect
9. #9722C2semidirect
10. #9719C2semidirect
11. #9718C2semidirect
15. #3231C4semidirect
16. #3231C4semidirect
17. #3228C4semidirect
18. #3228C4semidirect
19. #3210C4semidirect
20. #3210C4semidirect
21. #3207C4semidirect
22. #3207C4semidirect
99. #500C2×C4semidirect
100. #500C2×C4semidirect
101. #498C2×C4semidirect
102. #498C2×C4semidirect
103. #494C2×C4semidirect
104. #493C2×C4semidirect
105. #492C2×C4semidirect
106. #492C2×C4semidirect
107. #487C2×C4semidirect
108. #487C2×C4semidirect
109. #487C2×C4semidirect
110. #487C2×C4semidirect
111. #486C2×C4semidirect
112. #486C2×C4semidirect
113. #486C2×C4semidirect
114. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609442
#219002
#218992
#213582
#175632
#175462
#173242
#97222
#97192
#97182