Finite Group Databasebeta

← #228243

Subgroups — #228243

Order 512.

689 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23080C211yesmaximal, characteristic, complemented
2#22505C211yesmaximal, characteristic
3#18523C211yesmaximal, characteristic
4#18000C211yesmaximal, characteristic, complemented
5#17508C211yesmaximal, characteristic
6#9853C211yesmaximal, characteristic, complemented
7#9591C211yesmaximal, characteristic, complemented
8#9580C211yesmaximal, characteristic, complemented
9#9231C211yesmaximal, characteristic, complemented
10#9207C211yesmaximal, characteristic, complemented
11#9056C211yesmaximal, characteristic, complemented
12#8908C211yesmaximal, characteristic, complemented
13#8905C211yesmaximal, characteristic, complemented
14#8193C211yesmaximal, characteristic, complemented
15#8170C211yesmaximal, characteristic, complemented
16#3475C2211yescharacteristic
17#3170C2211yescharacteristic
18#3101C2211yescharacteristic
19#3076C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2962C2211yescharacteristic
22D4×M16C2211yescharacteristic
23#2925C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1994C2211yescharacteristic
26#1994C2211yescharacteristic
27#1986C2211yescharacteristic
28#1950C2211yescharacteristic
29#1937—21no—
30#1929—21no—
31#1922—21no—
32#1896—21no—
33#1871—21no—
34#1826—21no—
35#1806C2211yescharacteristic
36#1772—21no—
37C4×C4⋊C8C2211yescharacteristic
38C4×#104C2211yescharacteristic
39#1744C2211yescharacteristic
40C2×#339—21no—
41#1650C411yescharacteristic, complemented
42#1647C411yescharacteristic, complemented
43#1646C411yescharacteristic, complemented
44#1643C411yescharacteristic, complemented
45#1638C411yescharacteristic, complemented
46#1635C411yescharacteristic, complemented
47#1634C411yescharacteristic, complemented
48#1632C411yescharacteristic, complemented
49#1592—21no—
50#1588—21no—
51#1585—21no—
52SD16⋊C8—21no—
53#1571C2211yescharacteristic
54#1566C2211yescharacteristic
55#1542C2211yescharacteristic
56#1503C2211yescharacteristic
57#1494—21no—
58#1493C2211yescharacteristic
59#1493C2211yescharacteristic
60#1488—21no—
61#1486C2211yescharacteristic
62#1482—21no—
63#1481C2211yescharacteristic
64#1476—21no—
65#1476C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68#1451C2211yescharacteristic
69#1303C2211yescharacteristic
70#1301C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1276C2211yescharacteristic
75C22×M16—21no—
76#556C2×C411yescharacteristic
77#555C2×C411yescharacteristic
78#554C2×C411yescharacteristic
79#553—21no—
80#552—21no—
81#552—21no—
82#551—41no—
83#547C2×C411yescharacteristic
84#546—21no—
85#533C2311yescharacteristic
86#533C2311yescharacteristic
87#529—21no—
88#529—41no—
89#525—21no—
90C4×#41—21no—
91C4×#41C2311yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4C2311yescharacteristic
94C2×#117—21no—
95C22×C42—21no—
96#500C2×C411yescharacteristic, complemented
97#494C2×C411yescharacteristic, complemented
98#487C2×C411yescharacteristic, complemented
99#487C2×C411yescharacteristic, complemented
100#483—21no—
101#479—21no—
102#478—21no—
103#477C2×C411yescharacteristic, complemented
104#476C2×C411yescharacteristic
105#476C2×C411yescharacteristic
106#476C2×C411yescharacteristic, complemented
107#475C2×C411yescharacteristic, complemented
108#474C2×C411yescharacteristic, complemented
109#467—41no—
110#461C2×C411yescharacteristic
111#461C2×C411yescharacteristic
112#450—21no—
113#446—21no—
114#446—41no—
115C8×Q8—21no—
116C8×Q8—21no—
117C4×Q16—41no—
118C4×SD16—21no—
119C4×SD16—21no—
120#436—21no—
121#435—41no—
122C8×D4—21no—
123C8×D4—21no—
124C8×D4—41no—
125#433—41no—
126#432—21no—
127#431—21no—
128#431—21no—
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131C2×#108—21no—
132#427—21no—
133#424—21no—
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×#104—21no—
139C2×#104D411yescharacteristic
140C2×#104D411yescharacteristic
141#416—21no—
142#408—21no—
143#407—21no—
144#405—21no—
145#405—21no—
146C4×M16C2311yescharacteristic
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2311yescharacteristic
153C4×C22⋊C4—21no—
154C4×C22⋊C4—41no—
155C43C2311yescharacteristic
156#341—21no—
157#339—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#339D411yescharacteristic
165#339D411yescharacteristic
166#336—21no—
167#336—41no—
168#335—21no—
169#335—21no—
170#334—41no—
171#333C2×C411yescharacteristic, complemented
172#333C2×C411yescharacteristic, complemented
173#332C2×C411yescharacteristic, complemented
174#332C2×C411yescharacteristic, complemented
175#328—41no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8—21no—
181Q8⋊C8—21no—
182Q8⋊C8C2×C411yescharacteristic, complemented
183Q8⋊C8C2×C411yescharacteristic, complemented
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8C2×C411yescharacteristic
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8C2×C411yescharacteristic
190D4⋊C8C2×C411yescharacteristic, complemented
191D4⋊C8C2×C411yescharacteristic, complemented
192D4⋊C8D411yescharacteristic
193D4⋊C8D411yescharacteristic
194D4⋊C8D411yescharacteristic
195D4⋊C8D411yescharacteristic
196#322—21no—
197#322—21no—
198C2×#41—21no—
199C2×#41C2×D411yescharacteristic
200C22×D4#4111yescharacteristic
201C23×C4—21no—
202C23×C4—41no—
203C2×Q16—41no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—41no—
209C2×M16—42no—
210C2×M16—81no—
211C2×M16C2×D411yescharacteristic
212C2×M16C2×D411yescharacteristic
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—41no—
218C22×C8—41no—
219C4⋊Q8C22×C411yescharacteristic
220C4⋊Q8C22×C411yescharacteristic
221C4⋊Q8C22×C411yescharacteristic
222C4⋊Q8C22×C411yescharacteristic
223#126—41no—
224#126—41no—
225#126—41no—
226#125—21no—
227#125—21no—
228#125—21no—
229#125—41no—
230#124—21no—
231#124—41no—
232#123—41no—
233#123—41no—
234#123—41no—
235#123—41no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—41no—
244#121—21no—
245#121—21no—
246#121—21no—
247#121—41no—
248#120—41no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C22×C411yescharacteristic
255C4×Q8#4111yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—24no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—42no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4C22×C411yescharacteristic
267C4×D4C22×C411yescharacteristic
268C4×D4C22×C411yescharacteristic
269C4×D4C2×D411yescharacteristic
270C4×D4C2×D411yescharacteristic
271C4×D4#4111yescharacteristic
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—41no—
279#117—41no—
280#117—41no—
281#117C2×D411yescharacteristic
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4#4111yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—41no—
294C2×C42—42no—
295C2×C42—41no—
296C2×C42—81no—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300#108—42no—
301#107—21no—
302#107—21no—
303#107—41no—
304#106—21no—
305#106—21no—
306#106—21no—
307#106—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317C4⋊C8—41no—
318C4⋊C8C22×C411yescharacteristic
319C4⋊C8C22×C411yescharacteristic
320C4⋊C8C22×C411yescharacteristic
321C4⋊C8C22×C411yescharacteristic
322C4⋊C8C2×D411yescharacteristic
323C4⋊C8C2×D411yescharacteristic
324#104—41no—
325#104—41no—
326#104—81no—
327#103—21no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—41no—
332#103—41no—
333#102—21no—
334#102—21no—
335#102—21no—
336#102—21no—
337C22⋊C8—21no—
338C22⋊C8—21no—
339C22⋊C8—41no—
340C22⋊C8—41no—
341C22⋊C8—41no—
342C22⋊C8—41no—
343C22⋊C8—41no—
344#97—21no—
345#97—41no—
346#97—41no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—41no—
353C4×C8—41no—
354C4×C8—42no—
355C4×C8—41no—
356C4×C8C22×C411yescharacteristic
357C4×C8C22×C411yescharacteristic
358C4×C8C2×D411yescharacteristic
359C4×C8C2×D411yescharacteristic
360#95—21no—
361#95—41no—
362C24—21no—
363#41—21no—
364#41—21no—
365#41—21no—
366#41—21no—
367#41—42no—
368#41—81no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8C4×D411yescharacteristic
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4C4×D411yescharacteristic
381C2×D4C4×D411yescharacteristic
382C2×D4C4×D411yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—41no—
394C22×C4—44no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406Q16—82no—
407SD16—42no—
408SD16—81no—
409M16—21no—
410M16—21no—
411M16—21no—
412M16—21no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—82no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—81no—
438C2×C8—81no—
439C2×C8—82no—
440C2×C8—82no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4C4×D411yescharacteristic
463C4⋊C4C4×D411yescharacteristic
464C4⋊C4C4×D411yescharacteristic
465C4⋊C4C4×D411yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—44no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C42—22no—
483C42—22no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—81no—
498C42—82no—
499C42C23×C411yescharacteristic
500C42C22×D411yescharacteristic
501C42C2×#4111yescharacteristic
502C42H(5,𝔽2)11yescharacteristic
503C23—21no—
504C23—21no—
505C23—21no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—81no—
510C23#54611yescharacteristic
511Q8—21no—
512Q8—21no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—81no—
518Q8—81no—
519D4—21no—
520D4—21no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—42no—
525D4—41no—
526D4—41no—
527D4—82no—
528D4—161no—
529D4#44312yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—88no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—161no—
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4#51811yescharacteristic
572C2×C4D4211yescharacteristic
573C2×C4#54611yescharacteristic
574C2×C4#55111yescharacteristic
575C2×C4#57511yescharacteristic
576C2×C4#57511yescharacteristic
577C2×C4#57711yescharacteristic
578C2×C4#57811yescharacteristic
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—41no—
584C8—41no—
585C8—41no—
586C8—41no—
587C8—41no—
588C8—41no—
589C8—41no—
590C8—41no—
591C8—81no—
592C8—81no—
593C8—81no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—82no—
606C22—82no—
607C22#235811yescharacteristic
608C22#331111yescharacteristic
609C22#331511yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—82no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4#294311yescharacteristic
626C4#294711yescharacteristic
627C4C2×#44311yescharacteristic
628C4#295611yescharacteristic
629C4#326711yescharacteristic
630C4#326911yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2—81no—
635C2—81no—
636C2#2285611yescharacteristic
637C2#2288811yescharacteristic
638C2#2289811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#3475C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23080C2semidirect
4. #18000C2semidirect
6. #9853C2semidirect
7. #9591C2semidirect
8. #9580C2semidirect
9. #9231C2semidirect
10. #9207C2semidirect
11. #9056C2semidirect
12. #8908C2semidirect
13. #8905C2semidirect
14. #8193C2semidirect
15. #8170C2semidirect
41. #1650C4semidirect
42. #1647C4semidirect
43. #1646C4semidirect
44. #1643C4semidirect
45. #1638C4semidirect
46. #1635C4semidirect
47. #1634C4semidirect
48. #1632C4semidirect
96. #500C2×C4semidirect
97. #494C2×C4semidirect
98. #487C2×C4semidirect
99. #487C2×C4semidirect
103. #477C2×C4semidirect
106. #476C2×C4semidirect
107. #475C2×C4semidirect
108. #474C2×C4semidirect
171. #333C2×C4semidirect
172. #333C2×C4semidirect
173. #332C2×C4semidirect
174. #332C2×C4semidirect
182. Q8⋊C8C2×C4semidirect
183. Q8⋊C8C2×C4semidirect
190. D4⋊C8C2×C4semidirect
191. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#230802
#225052
#185232
#180002
#175082
#98532
#95912
#95802
#92312
#92072
#90562
#89082
#89052
#81932
#81702