Finite Group Databasebeta

← #228159

Subgroups — #228159

Order 512.

775 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23081C211yesmaximal, characteristic, complemented
2D4×#104C211yesmaximal, characteristic
3#18517C211yesmaximal, characteristic
4#17995C211yesmaximal, characteristic, complemented
5#17511C211yesmaximal, characteristic
6#9851C211yesmaximal, characteristic, complemented
7#9590C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9231C211yesmaximal, characteristic, complemented
10#9209C211yesmaximal, characteristic, complemented
11#9054C211yesmaximal, characteristic, complemented
12#8906C211yesmaximal, characteristic, complemented
13#8903C211yesmaximal, characteristic, complemented
14#8196C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic, complemented
16D4×#41C2211yescharacteristic
17#3167C2211yescharacteristic
18#3101C2211yescharacteristic
19#3076C2211yescharacteristic
20#2965C2211yescharacteristic
21#2962C2211yescharacteristic, complemented
22D4×M16C2211yescharacteristic
23#2925C2211yescharacteristic
24C22×#104—21no—
25C42×D4C2211yescharacteristic
26#1994C2211yescharacteristic, complemented
27#1980C2211yescharacteristic
28#1962C2211yescharacteristic
29#1950C2211yescharacteristic, complemented
30#1938—21no—
31#1929—21no—
32#1922—21no—
33#1896—21no—
34#1871—21no—
35#1806C2211yescharacteristic
36#1789—21no—
37#1773—21no—
38#1767C2211yescharacteristic, complemented
39C4×#104C2211yescharacteristic
40C4×#97C2211yescharacteristic, complemented
41#1654C411yescharacteristic, complemented
42#1651C411yescharacteristic, complemented
43#1642C411yescharacteristic, complemented
44#1639C411yescharacteristic, complemented
45#1637C411yescharacteristic, complemented
46#1635C411yescharacteristic, complemented
47#1631C411yescharacteristic, complemented
48#1629C411yescharacteristic, complemented
49#1590—21no—
50#1583—21no—
51#1582—21no—
52C8×SD16—21no—
53#1570C2211yescharacteristic, complemented
54#1567C2211yescharacteristic, complemented
55#1540C2211yescharacteristic
56#1503C2211yescharacteristic, complemented
57#1496—21no—
58#1493C2211yescharacteristic, complemented
59#1492C2211yescharacteristic
60#1490—21no—
61#1486C2211yescharacteristic
62#1484—21no—
63#1480C2211yescharacteristic
64#1478—21no—
65#1476C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C8×M16C2211yescharacteristic, complemented
69#1303C2211yescharacteristic, complemented
70#1301C2211yescharacteristic, complemented
71#1279C2211yescharacteristic, complemented
72#1279C2211yescharacteristic, complemented
73#1278C2211yescharacteristic, complemented
74#1276C2211yescharacteristic, complemented
75C22×#41—21no—
76C22×M16—21no—
77#550C2×C411yescharacteristic
78D4×Q8C2×C411yescharacteristic
79#548—21no—
80#547—21no—
81#547C2×C411yescharacteristic
82#546—21no—
83#546—41no—
84D42—21no—
85D42C2×C411yescharacteristic
86#533C2311yescharacteristic
87#532—21no—
88#532C2311yescharacteristic
89#525—21no—
90#525—41no—
91C4×#41C2311yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4C2311yescharacteristic
95C22×C42—21no—
96#498C2×C411yescharacteristic, complemented
97#492C2×C411yescharacteristic, complemented
98#487C2×C411yescharacteristic, complemented
99#486C2×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#465—41no—
110#461C2×C411yescharacteristic
111#461C2×C411yescharacteristic
112#450—21no—
113#446—21no—
114#446—21no—
115#446—41no—
116C8×Q8—21no—
117C4×SD16—21no—
118C4×SD16—21no—
119C4×SD16—41no—
120#436—21no—
121#436—21no—
122#435—41no—
123#435—41no—
124C8×D4—21no—
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×#104—21no—
140C2×#104—21no—
141C2×#104—21no—
142C2×#104—21no—
143C2×#104—44no—
144C2×#104D411yescharacteristic
145C2×#104D411yescharacteristic
146#416—21no—
147#408—21no—
148#407—21no—
149#405—21no—
150#405—21no—
151C4×M16—21no—
152C4×M16C2311yescharacteristic
153C2×#97—21no—
154C2×#97—41no—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2311yescharacteristic
159C4×C22⋊C4—21no—
160C4×C22⋊C4—41no—
161C43C2311yescharacteristic
162#341—21no—
163#339—21no—
164#339—21no—
165#339—21no—
166#339D411yescharacteristic
167#339D411yescharacteristic
168#336—21no—
169#335—41no—
170#334—21no—
171#334—21no—
172#332C2×C411yescharacteristic, complemented
173#332C2×C411yescharacteristic, complemented
174#331C2×C411yescharacteristic, complemented
175#331C2×C411yescharacteristic, complemented
176#328—41no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8C2×C411yescharacteristic, complemented
180Q8⋊C8C2×C411yescharacteristic, complemented
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8C2×C411yescharacteristic
190D4⋊C8C2×C411yescharacteristic
191D4⋊C8C2×C411yescharacteristic, complemented
192D4⋊C8C2×C411yescharacteristic, complemented
193D4⋊C8D411yescharacteristic
194D4⋊C8D411yescharacteristic
195D4⋊C8D411yescharacteristic
196D4⋊C8D411yescharacteristic
197#322—21no—
198C82—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—81no—
207C2×#41—81no—
208C2×#41C2×D411yescharacteristic
209C22×Q8—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—41no—
213C22×D4#4111yescharacteristic
214C23×C4—21no—
215C23×C4—41no—
216C23×C4—41no—
217C2×SD16—21no—
218C2×SD16—21no—
219C2×SD16—41no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—41no—
223C2×M16—41no—
224C2×M16—42no—
225C2×M16—81no—
226C2×M16C2×D411yescharacteristic
227C2×M16C2×D411yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—41no—
233C4⋊Q8—21no—
234C4⋊Q8C22×C411yescharacteristic
235C4⋊Q8C22×C411yescharacteristic
236C4⋊Q8C22×C411yescharacteristic
237#127—21no—
238#127C22×C411yescharacteristic
239#125—21no—
240#124—21no—
241#124—41no—
242#123—41no—
243#123—41no—
244#123—41no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—41no—
250C22⋊Q8—41no—
251#121—21no—
252#121—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—41no—
258#121—41no—
259#121—41no—
260#120—41no—
261#120—41no—
262#120—41no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8C22×C411yescharacteristic
266C4×Q8#4111yescharacteristic
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—42no—
279C4×D4—42no—
280C4×D4C22×C411yescharacteristic
281C4×D4C22×C411yescharacteristic
282C4×D4C22×C411yescharacteristic
283C4×D4C2×D411yescharacteristic
284C4×D4C2×D411yescharacteristic
285C4×D4#4111yescharacteristic
286#117—21no—
287#117—21no—
288#117C2×D411yescharacteristic
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4#4111yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—42no—
300C2×C42—41no—
301C2×C42—81no—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305#108—42no—
306#107—21no—
307#107—21no—
308#106—21no—
309#106—21no—
310#106—21no—
311#106—21no—
312#106—41no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8—41no—
323C4⋊C8C22×C411yescharacteristic
324C4⋊C8C22×C411yescharacteristic
325C4⋊C8C22×C411yescharacteristic
326C4⋊C8C22×C411yescharacteristic
327C4⋊C8C2×D411yescharacteristic
328C4⋊C8C2×D411yescharacteristic
329#104—22no—
330#104—41no—
331#104—42no—
332#104—42no—
333#104—42no—
334#104—81no—
335#104—81no—
336#103—21no—
337#103—21no—
338#103—21no—
339#103—21no—
340#103—41no—
341#102—21no—
342#102—21no—
343#102—21no—
344#102—21no—
345#102—41no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—41no—
349C22⋊C8—41no—
350C22⋊C8—41no—
351C22⋊C8—41no—
352C22⋊C8—41no—
353#97—21no—
354#97—41no—
355#97—41no—
356#97—42no—
357#97—41no—
358#97—44no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—41no—
365C4×C8—41no—
366C4×C8C22×C411yescharacteristic
367C4×C8C22×C411yescharacteristic
368C4×C8C2×D411yescharacteristic
369C4×C8C2×D411yescharacteristic
370#95—21no—
371#95—41no—
372C24—21no—
373C24—41no—
374#41—21no—
375#41—21no—
376#41—21no—
377#41—21no—
378#41—42no—
379#41—42no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—82no—
385#41—82no—
386#41—81no—
387#41—81no—
388#41—82no—
389#41—82no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—41no—
394C2×Q8—42no—
395C2×Q8—81no—
396C2×Q8C4×D411yescharacteristic
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4C4×D411yescharacteristic
417C2×D4C4×D411yescharacteristic
418C2×D4C4×D411yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—44no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446SD16—42no—
447SD16—82no—
448SD16—81no—
449M16—21no—
450M16—21no—
451M16—21no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—82no—
457M16—82no—
458M16AGL(1,ℤ/8)12yes—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—82no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4C4×D411yescharacteristic
497C4⋊C4C4×D411yescharacteristic
498C4⋊C4C4×D411yescharacteristic
499C4⋊C4C4×D411yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—44no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—42no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—82no—
529C42C23×C411yescharacteristic
530C42C22×D411yescharacteristic
531C42C2×#4111yescharacteristic
532C42H(5,𝔽2)11yescharacteristic
533C23—21no—
534C23—21no—
535C23—21no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—81no—
541C23—82no—
542C23—82no—
543C23—161no—
544C23#54611yescharacteristic
545Q8—21no—
546Q8—21no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—81no—
552Q8—82no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—21no—
557D4—21no—
558D4—42no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—81no—
566D4—82no—
567D4—82no—
568D4—82no—
569D4—82no—
570D4—81no—
571D4—81no—
572D4—161no—
573D4#44412yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—88no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4#51811yescharacteristic
618C2×C4D4211yescharacteristic
619C2×C4#54611yescharacteristic
620C2×C4#55111yescharacteristic
621C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
622C2×C4#57511yescharacteristic
623C2×C4#57711yescharacteristic
624C2×C4#57711yescharacteristic
625C8—22no—
626C8—22no—
627C8—22no—
628C8—41no—
629C8—41no—
630C8—41no—
631C8—41no—
632C8—41no—
633C8—41no—
634C8—41no—
635C8—41no—
636C8—41no—
637C8—81no—
638C8—81no—
639C8—81no—
640C22—21no—
641C22—21no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—82no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—164no—
657C22#235811yescharacteristic
658C22#332311yescharacteristic
659C22#332411yescharacteristic
660C4—21no—
661C4—21no—
662C4—21no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—81no—
670C4—82no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4C4×AGL(1,ℤ/8)11yescharacteristic
675C4#294911yescharacteristic
676C4C2×#44411yescharacteristic
677C4#295611yescharacteristic
678C4#326511yescharacteristic
679C4#326711yescharacteristic
680C2—21no—
681C2—21no—
682C2—42no—
683C2—81no—
684C2—81no—
685C2—161no—
686C2#2288411yescharacteristic
687C2#2288611yescharacteristic
688C2#2289911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radicalD4×#41C22
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. #23081C2semidirect
4. #17995C2semidirect
6. #9851C2semidirect
7. #9590C2semidirect
8. #9582C2semidirect
9. #9231C2semidirect
10. #9209C2semidirect
11. #9054C2semidirect
12. #8906C2semidirect
13. #8903C2semidirect
14. #8196C2semidirect
15. #8172C2semidirect
21. #2962C22semidirect
26. #1994C22semidirect
29. #1950C22semidirect
38. #1767C22semidirect
40. C4×#97C22semidirect
41. #1654C4semidirect
42. #1651C4semidirect
43. #1642C4semidirect
44. #1639C4semidirect
45. #1637C4semidirect
46. #1635C4semidirect
47. #1631C4semidirect
48. #1629C4semidirect
53. #1570C22semidirect
54. #1567C22semidirect
56. #1503C22semidirect
58. #1493C22semidirect
68. C8×M16C22semidirect
69. #1303C22semidirect
70. #1301C22semidirect
71. #1279C22semidirect
72. #1279C22semidirect
73. #1278C22semidirect
74. #1276C22semidirect
96. #498C2×C4semidirect
97. #492C2×C4semidirect
98. #487C2×C4semidirect
99. #486C2×C4semidirect
103. #477C2×C4semidirect
106. #476C2×C4semidirect
107. #475C2×C4semidirect
108. #474C2×C4semidirect
172. #332C2×C4semidirect
173. #332C2×C4semidirect
174. #331C2×C4semidirect
175. #331C2×C4semidirect
179. Q8⋊C8C2×C4semidirect
180. Q8⋊C8C2×C4semidirect
191. D4⋊C8C2×C4semidirect
192. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#230812
D4×#1042
#185172
#179952
#175112
#98512
#95902
#95822
#92312
#92092
#90542
#89062
#89032
#81962
#81722