Finite Group Databasebeta

← #228162

Subgroups — #228162

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22983C211yesmaximal, characteristic, complemented
2#22505C211yesmaximal, characteristic
3#18523C211yesmaximal, characteristic
4#17995C211yesmaximal, characteristic, complemented
5#17512C211yesmaximal, characteristic
6#9850C211yesmaximal, characteristic, complemented
7#9589C211yesmaximal, characteristic, complemented
8#9583C211yesmaximal, characteristic, complemented
9#9231C211yesmaximal, characteristic, complemented
10#9209C211yesmaximal, characteristic, complemented
11#9056C211yesmaximal, characteristic, complemented
12#8908C211yesmaximal, characteristic, complemented
13#8904C211yesmaximal, characteristic, complemented
14#8196C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic, complemented
16#3475C2211yescharacteristic
17#3170C2211yescharacteristic
18#3101C2211yescharacteristic
19#3076C2211yescharacteristic
20#2965C2211yescharacteristic
21#2962C2211yescharacteristic
22D4×M16C2211yescharacteristic
23#2906C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1994C2211yescharacteristic
26#1994C2211yescharacteristic
27#1986C2211yescharacteristic
28#1950C2211yescharacteristic
29#1946—21no—
30#1922—21no—
31#1896—21no—
32#1871—21no—
33#1866—21no—
34#1826—21no—
35#1806C2211yescharacteristic
36#1767C2211yescharacteristic
37#1765—21no—
38C4×#104C2211yescharacteristic
39C2×#339—21no—
40C4×#97C2211yescharacteristic
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#1590—21no—
50#1583—21no—
51#1582—21no—
52C8×SD16—21no—
53#1570C2211yescharacteristic
54#1566C2211yescharacteristic
55#1542C2211yescharacteristic
56#1503C2211yescharacteristic
57#1495—21no—
58#1493C2211yescharacteristic
59#1493C2211yescharacteristic
60#1489—21no—
61#1486C2211yescharacteristic
62#1483—21no—
63#1481C2211yescharacteristic
64#1477—21no—
65#1476C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C8×M16C2211yescharacteristic
69#1303C2211yescharacteristic
70#1301C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#131—21no—
76C22×M16—21no—
77#556C2×C411yescharacteristic
78#555C2×C411yescharacteristic
79#554C2×C411yescharacteristic
80#553—21no—
81#552—21no—
82#552—21no—
83#551—41no—
84#547C2×C411yescharacteristic
85#546—21no—
86#533C2311yescharacteristic
87#533C2311yescharacteristic
88#529—21no—
89#529—41no—
90#525—21no—
91C4×#41—21no—
92C4×#41C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4C2311yescharacteristic
95C2×#117—21no—
96C22×C42—21no—
97#500C2×C411yescharacteristic, complemented
98#494C2×C411yescharacteristic, complemented
99#487C2×C411yescharacteristic, complemented
100#487C2×C411yescharacteristic, complemented
101#483—21no—
102#479—21no—
103#478—21no—
104#477C2×C411yescharacteristic, complemented
105#476C2×C411yescharacteristic
106#476C2×C411yescharacteristic
107#476C2×C411yescharacteristic, complemented
108#475C2×C411yescharacteristic, complemented
109#474C2×C411yescharacteristic, complemented
110#467—41no—
111#461C2×C411yescharacteristic
112#461C2×C411yescharacteristic
113#450—21no—
114#446—21no—
115#446—21no—
116#446—41no—
117C8×Q8—21no—
118C4×Q16—41no—
119C4×SD16—21no—
120C4×SD16—21no—
121#436—21no—
122#436—21no—
123#435—41no—
124#435—41no—
125C8×D4—21no—
126#431—21no—
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129C2×#108—21no—
130#427—21no—
131#424—21no—
132#424—21no—
133#424—41no—
134#423—21no—
135#423C2311yescharacteristic
136#423C2311yescharacteristic
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×#104—21no—
141C2×#104D411yescharacteristic
142C2×#104D411yescharacteristic
143#416—21no—
144#407—21no—
145#405—21no—
146C4×M16—21no—
147C4×M16C2311yescharacteristic
148C2×#97—21no—
149C2×#97—41no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—41no—
156C43C2311yescharacteristic
157#341—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#339—21no—
165#339D411yescharacteristic
166#339D411yescharacteristic
167#336—21no—
168#334—21no—
169#334—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—
197C82—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—
206#131—42no—
207#131—81no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—41no—
211C2×M16—41no—
212C2×M16—41no—
213C2×M16—42no—
214C2×M16—81no—
215C2×M16C2×D411yescharacteristic
216C2×M16C2×D411yescharacteristic
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—41no—
222C22×C8—41no—
223C4⋊Q8C22×C411yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226C4⋊Q8C22×C411yescharacteristic
227#126—41no—
228#126—41no—
229#126—41no—
230#125—21no—
231#125—21no—
232#125—21no—
233#125—41no—
234#124—21no—
235#124—41no—
236#123—41no—
237#123—41no—
238#123—41no—
239#123—41no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—41no—
248#121—21no—
249#121—21no—
250#121—21no—
251#121—41no—
252#120—41no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8C22×C411yescharacteristic
259C4×Q8#4111yescharacteristic
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—41no—
266C4×D4—42no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—42no—
270C4×D4C22×C411yescharacteristic
271C4×D4C22×C411yescharacteristic
272C4×D4C22×C411yescharacteristic
273C4×D4C2×D411yescharacteristic
274C4×D4C2×D411yescharacteristic
275C4×D4#4111yescharacteristic
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—41no—
283#117—41no—
284#117—41no—
285#117C2×D411yescharacteristic
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4#4111yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—41no—
298C2×C42—42no—
299C2×C42—41no—
300C2×C42—81no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2411yescharacteristic
304#108—81no—
305#107—21no—
306#107—21no—
307#107—41no—
308#106—21no—
309#106—21no—
310#106—21no—
311#106—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—41no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8—41no—
323C4⋊C8—41no—
324C4⋊C8C22×C411yescharacteristic
325C4⋊C8C22×C411yescharacteristic
326C4⋊C8C22×C411yescharacteristic
327C4⋊C8C22×C411yescharacteristic
328C4⋊C8C2×D411yescharacteristic
329C4⋊C8C2×D411yescharacteristic
330#104—22no—
331#104—41no—
332#104—81no—
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337#103—41no—
338#103—41no—
339#102—21no—
340#102—21no—
341#102—21no—
342#102—21no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—41no—
346C22⋊C8—41no—
347C22⋊C8—41no—
348#97—21no—
349#97—41no—
350#97—41no—
351#97—44no—
352#97—81no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—41no—
359C4×C8C22×C411yescharacteristic
360C4×C8C22×C411yescharacteristic
361C4×C8C2×D411yescharacteristic
362C4×C8C2×D411yescharacteristic
363#95—21no—
364#95—41no—
365C24—21no—
366#41—21no—
367#41—21no—
368#41—21no—
369#41—21no—
370#41—42no—
371#41—81no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8C4×D411yescharacteristic
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4C4×D411yescharacteristic
384C2×D4C4×D411yescharacteristic
385C2×D4C4×D411yescharacteristic
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—41no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409Q16—82no—
410SD16—42no—
411SD16—81no—
412M16—21no—
413M16—21no—
414M16—21no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419M16—82no—
420M16—82no—
421M16—82no—
422M16#13712yes—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—82no—
445C2×C8—82no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4C4×D411yescharacteristic
468C4⋊C4C4×D411yescharacteristic
469C4⋊C4C4×D411yescharacteristic
470C4⋊C4C4×D411yescharacteristic
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—44no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—82no—
504C42C23×C411yescharacteristic
505C42C22×D411yescharacteristic
506C42C2×#4111yescharacteristic
507C42H(5,𝔽2)11yescharacteristic
508C23—21no—
509C23—21no—
510C23—21no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—81no—
515C23#54611yescharacteristic
516Q8—21no—
517Q8—21no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—81no—
523Q8—81no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—42no—
530D4—41no—
531D4—41no—
532D4—82no—
533D4—161no—
534D4#44412yes—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
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—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—88no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—161no—
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4#51811yescharacteristic
577C2×C4D4211yescharacteristic
578C2×C4#54611yescharacteristic
579C2×C4#55111yescharacteristic
580C2×C4C2×#13711yescharacteristic
581C2×C4#57511yescharacteristic
582C2×C4#57711yescharacteristic
583C2×C4#57711yescharacteristic
584C8—22no—
585C8—22no—
586C8—22no—
587C8—41no—
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C8—41no—
593C8—41no—
594C8—41no—
595C8—41no—
596C8—81no—
597C8—81no—
598C8—81no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—82no—
611C22—82no—
612C22#235811yescharacteristic
613C22#332311yescharacteristic
614C22#333111yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4C4×#13711yescharacteristic
631C4#295211yescharacteristic
632C4C2×#44411yescharacteristic
633C4#295611yescharacteristic
634C4#326611yescharacteristic
635C4#326711yescharacteristic
636C2—21no—
637C2—21no—
638C2—42no—
639C2—81no—
640C2—81no—
641C2#2288211yescharacteristic
642C2#2288611yescharacteristic
643C2#2289911yescharacteristic
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. #22983C2semidirect
4. #17995C2semidirect
6. #9850C2semidirect
7. #9589C2semidirect
8. #9583C2semidirect
9. #9231C2semidirect
10. #9209C2semidirect
11. #9056C2semidirect
12. #8908C2semidirect
13. #8904C2semidirect
14. #8196C2semidirect
15. #8172C2semidirect
41. #1650C4semidirect
42. #1647C4semidirect
43. #1646C4semidirect
44. #1643C4semidirect
45. #1638C4semidirect
46. #1635C4semidirect
47. #1634C4semidirect
48. #1632C4semidirect
97. #500C2×C4semidirect
98. #494C2×C4semidirect
99. #487C2×C4semidirect
100. #487C2×C4semidirect
104. #477C2×C4semidirect
107. #476C2×C4semidirect
108. #475C2×C4semidirect
109. #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
#229832
#225052
#185232
#179952
#175122
#98502
#95892
#95832
#92312
#92092
#90562
#89082
#89042
#81962
#81722