Finite Group Databasebeta

← #7624135

Subgroups — #7624135

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62249C211yesmaximal, characteristic, complemented
2C2×#3280C211yesmaximal, characteristic, complemented
3#33886C214yesmaximal, complemented
4#33878C214yesmaximal, complemented
5#33834C214yesmaximal, complemented
6#33833C214yesmaximal, complemented
7#33781C211yesmaximal, characteristic, complemented
8#33779C211yesmaximal, characteristic, complemented
9#33762C212yesmaximal, complemented
10#33754C212yesmaximal, complemented
11#33670C211yesmaximal, characteristic, complemented
12#33669C211yesmaximal, characteristic, complemented
13#33517C212yesmaximal, complemented
14#33516C212yesmaximal, complemented
15#33487C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C23×D8C2211yescharacteristic
18#3506—21no—
19C2×#546—21no—
20C2×#509C2214yescomplemented
21C2×#508C2214yescomplemented
22#3342C2212yescomplemented
23#3341C2212yescomplemented
24#3304—22no—
25#3303—22no—
26#3287C2212yescomplemented
27#3283C2212yescomplemented
28#3280—22no—
29#3280—22no—
30#3280C2214yes—
31#3280C2214yes—
32#3246—21no—
33#3191—21no—
34#3191—21no—
35#3170C2211yescharacteristic, complemented
36#3169C2211yescharacteristic, complemented
37C2×#495C2211yescharacteristic
38#3050C2212yes—
39#3050C2212yescomplemented
40C2×#467C2211yescharacteristic
41C2×#466C2211yescharacteristic
42C2×#463—21no—
43C2×#449—21no—
44C2×#450—21no—
45C2×C4×D8C2211yescharacteristic
46C2×C8×D4C2211yescharacteristic
47#2251C2212yescomplemented
48#2250C2212yescomplemented
49#2243C2212yescomplemented
50#2243C2212yescomplemented
51#2237C2214yescomplemented
52#2236C2212yescomplemented
53#2234C2214yescomplemented
54#2233C2212yescomplemented
55#2226C2212yescomplemented
56#2224C2212yescomplemented
57#2220C2214yescomplemented
58#2219C2214yescomplemented
59#2217C2214yescomplemented
60#2215C2214yescomplemented
61#2211C2214yescomplemented
62#2210C2214yescomplemented
63#2208C2212yescomplemented
64#2206C2212yescomplemented
65#2189C2214yescomplemented
66#2189C2214yescomplemented
67#2188C2212yescomplemented
68#2187C2212yes—
69#2187C2212yes—
70#2187C2214yescomplemented
71#2186C2214yescomplemented
72#2186C2214yescomplemented
73#2185C2212yescomplemented
74#2184C2212yes—
75#2184C2212yes—
76#2184C2214yescomplemented
77#2177C2214yescomplemented
78#2175C2214yescomplemented
79#2168C2214yescomplemented
80#2168C2214yescomplemented
81#2163C2211yescharacteristic, complemented
82#2163C2211yescharacteristic, complemented
83#2140C2214yescomplemented
84#2138C2214yescomplemented
85#2112C2212yes—
86C2×C22⋊C16C2212yescomplemented
87C22×#41—21no—
88C23×D4—21no—
89C2×#135—22no—
90C2×#135—44no—
91C2×#135C2314yes—
92C2×#135C2312yes—
93C2×#135C2312yescomplemented
94C22×Q16C2311yescharacteristic
95C22×SD16—21no—
96C22×SD16—21no—
97C22×D8—21no—
98C22×D8—22no—
99C22×D8—44no—
100C22×D8C2311yescharacteristic
101C22×D8C2314yes—
102C22×D8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C23×C8C2311yescharacteristic
105#559—42no—
106#556—22no—
107#555—22no—
108#553—42no—
109#550—22no—
110#547—22no—
111#546—24no—
112#546—24no—
113#546—42no—
114#546—42no—
115#538—41no—
116#538—41no—
117#535—41no—
118#533—21no—
119#532—21no—
120C2×#124—21no—
121#525—22no—
122#525—22no—
123C2×#123—41no—
124C2×C22⋊Q8—21no—
125C2×#121—21no—
126C2×#121—41no—
127C2×#120—41no—
128#520—41no—
129C2×C4×D4—21no—
130C2×C4×D4D411yescharacteristic
131C22×C22⋊C4—41no—
132#509—48no—
133#508—48no—
134C2×Q32C2314yescomplemented
135C2×SD32C2314yes—
136C2×SD32C2314yes—
137C2×SD32C2314yescomplemented
138C2×D16C2314yes—
139C2×D16C2314yes—
140C2×M32C2314yes—
141C22×C16C2314yescomplemented
142#500C2311yescharacteristic, complemented
143#500C2311yescharacteristic, complemented
144#495—21no—
145#495—21no—
146#495C2312yes—
147#495C2312yes—
148#494C2311yescharacteristic, complemented
149#493C2311yescharacteristic, complemented
150#487—21no—
151#486—21no—
152#485—22no—
153#485—22no—
154#482—22no—
155#482—22no—
156#476—22no—
157#475—22no—
158#467—21no—
159#467—22no—
160#467C2312yes—
161#467C2312yes—
162#467C2314yes—
163#467C2312yescomplemented
164#466—21no—
165#466—22no—
166#466C2312yes—
167#466C2312yes—
168#466C2314yes—
169#466C2312yescomplemented
170#465—22no—
171#465—22no—
172#465—22no—
173#465—22no—
174#463—24no—
175#463—24no—
176#461—22no—
177#460—22no—
178#452—22no—
179#452—22no—
180#450—24no—
181#450—24no—
182#449—22no—
183#449—22no—
184#449—24no—
185#449—24no—
186C4×Q16—21no—
187C4×Q16C2312yescomplemented
188C4×D8—21no—
189C4×D8—21no—
190C4×D8—21no—
191C4×D8C2312yes—
192C4×D8C2312yes—
193C4×D8C2312yescomplemented
194C8×D4D412yes—
195C8×D4D412yes—
196C8×D4C2312yes—
197C8×D4C2312yes—
198#427C2312yes—
199#427C2312yescomplemented
200C2×#107C2311yescharacteristic
201C2×C4⋊C8D411yescharacteristic
202#416—22no—
203#416—22no—
204C2×#103—21no—
205C2×#102—21no—
206C2×#102—21no—
207C2×C22⋊C8D411yescharacteristic
208C2×C22⋊C8D411yescharacteristic
209C2×C4×C8C2311yescharacteristic
210#367C2314yescomplemented
211#366C2314yescomplemented
212C4⋊C16C2312yescomplemented
213C4⋊C16C2312yescomplemented
214#358C2314yescomplemented
215#358C2314yescomplemented
216#358C2314yescomplemented
217#357C2314yes—
218#357C2314yes—
219#357C2314yes—
220#357C2314yes—
221#357C2314yescomplemented
222C22⋊C16C2312yes—
223C22⋊C16C2312yes—
224C22⋊C16C2314yes—
225C22⋊C16C2314yescomplemented
226#346C2312yescomplemented
227#346C2312yescomplemented
228C25—41no—
229C2×#41—24no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—41no—
233C2×#41—42no—
234C2×#41—44no—
235C2×#41—82no—
236C22×Q8—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—24no—
240C22×D4—44no—
241C22×D4—41no—
242C22×D4—41no—
243C22×D4—41no—
244C22×D4—42no—
245C22×D4—82no—
246C22×D4D811yescharacteristic
247C23×C4—41no—
248C23×C4—41no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251#135—28no—
252#135—44no—
253#135—44no—
254#135—44no—
255#135—48no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—42no—
259C2×Q16C2412yes—
260C2×Q16C2412yes—
261C2×SD16—24no—
262C2×SD16—22no—
263C2×SD16—22no—
264C2×SD16—22no—
265C2×SD16—22no—
266C2×SD16—44no—
267C2×SD16—82no—
268C2×D8—24no—
269C2×D8—28no—
270C2×D8—21no—
271C2×D8—24no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—44no—
275C2×D8—42no—
276C2×D8—48no—
277C2×D8—44no—
278C2×D8C2412yes—
279C2×D8C2412yes—
280C2×D8C2412yes—
281C22×C8—42no—
282C22×C8—42no—
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D412yes—
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D412yes—
287C22×C8C2×D411yescharacteristic
288C22×C8C2411yescharacteristic
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294#127—21no—
295#127—42no—
296#126—42no—
297#126—42no—
298#125—41no—
299#124—22no—
300#124—22no—
301#124—42no—
302#124—42no—
303#124—42no—
304#124—41no—
305#124—41no—
306#124—41no—
307#123—42no—
308#123—42no—
309#123—44no—
310#123—44no—
311#123—42no—
312#123—42no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—42no—
318C22⋊Q8—41no—
319#121—24no—
320#121—22no—
321#121—22no—
322#121—22no—
323#121—42no—
324#121—42no—
325#121—42no—
326#121—41no—
327#121—44no—
328#121—44no—
329#121—42no—
330#121—42no—
331#120—42no—
332#120—42no—
333#120—44no—
334#120—44no—
335C4×Q8—22no—
336C4×Q8—42no—
337C4×Q8—41no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—42no—
342C4×D4—42no—
343C4×D4—42no—
344C4×D4—42no—
345C4×D4—42no—
346C4×D4—41no—
347C4×D4—41no—
348C4×D4—41no—
349C4×D4D812yes—
350C4×D4D812yes—
351C4×D4C2×D412yes—
352C4×D4C2×D412yes—
353#117—22no—
354#117—22no—
355#117—41no—
356#117—42no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4D811yescharacteristic
361C2×C22⋊C4—41no—
362C2×C22⋊C4—41no—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4—44no—
367C2×C22⋊C4—44no—
368C2×C22⋊C4D811yescharacteristic
369C2×C22⋊C4D811yescharacteristic
370C2×C42C2×D411yescharacteristic
371Q32—44no—
372SD32—28no—
373SD32—44no—
374SD32—44no—
375D16—28no—
376D16—44no—
377M32—44no—
378C2×C16—44no—
379C2×C16C2414yes—
380C2×C16C2414yes—
381#107—21no—
382#107—21no—
383#107—21no—
384#107C2412yes—
385#107C2412yes—
386#107C2412yes—
387#106—22no—
388#106—22no—
389C4⋊C8C2×D412yes—
390C4⋊C8C2×D412yes—
391#103—24no—
392#103—22no—
393#103—22no—
394#103—22no—
395#103—22no—
396#102—22no—
397#102—22no—
398#102—24no—
399#102—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C22⋊C8C2×D412yes—
406C22⋊C8C2×D412yes—
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2411yescharacteristic
410C4×C8C2411yescharacteristic
411C24—41no—
412C24—41no—
413C24—44no—
414C24—84no—
415C24—84no—
416C24—162no—
417C24C2×D811yescharacteristic
418C24C2×D811yescharacteristic
419#41—48no—
420#41—84no—
421#41—84no—
422#41—82no—
423#41—82no—
424#41—82no—
425#41—82no—
426#41—88no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—41no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—48no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—44no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—84no—
456C2×D4—88no—
457C2×D4—84no—
458C2×D4—84no—
459C2×D4—162no—
460C2×D4C2×D812yes—
461C2×D4C2×D812yes—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—84no—
475C22×C4—162no—
476C22×C4C2×D811yescharacteristic
477C22×C4C2×D811yescharacteristic
478C22×C4C2×D811yescharacteristic
479C22×C4C2×D811yescharacteristic
480C22×C4C2×D812yes—
481C22×C4C2×D812yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485Q16—24no—
486Q16—24no—
487Q16—42no—
488Q16—81no—
489SD16—48no—
490SD16—82no—
491SD16—82no—
492SD16—82no—
493SD16—82no—
494D8—24no—
495D8—22no—
496D8—24no—
497D8—24no—
498D8—48no—
499D8—42no—
500D8—81no—
501D8H(5,𝔽2)14yes—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—42no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—42no—
509C2×C8—41no—
510C2×C8C22×D411yescharacteristic
511C2×C8C22×D411yescharacteristic
512C2×C8C22×D411yescharacteristic
513C2×C8C22×D412yes—
514C2×C8C22×D412yes—
515C2×C8C2511yescharacteristic
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—41no—
525C4⋊C4—42no—
526C4⋊C4—41no—
527C4⋊C4—42no—
528C4⋊C4—41no—
529C4⋊C4C2×D812yes—
530C4⋊C4C2×D812yes—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4C2×D812yes—
549C22⋊C4C2×D812yes—
550C22⋊C4C2×D812yes—
551C22⋊C4C2×D812yes—
552C42—42no—
553C42—42no—
554C42—42no—
555C42C2×D811yescharacteristic
556C42C2×D811yescharacteristic
557C42C22×D411yescharacteristic
558C42C22×D411yescharacteristic
559C16—24no—
560C16—24no—
561C23—22no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—88no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—88no—
576C23—82no—
577C23—82no—
578C23—164no—
579C23—1616no—
580C23C22×D812yes—
581C23C22×D812yes—
582C23C22×D811yescharacteristic
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—82no—
588Q8—82no—
589Q8—161no—
590D4—24no—
591D4—42no—
592D4—42no—
593D4—44no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—88no—
600D4—82no—
601D4—82no—
602D4—88no—
603D4—84no—
604D4—161no—
605D4—162no—
606D4—162no—
607D4—164no—
608D4—162no—
609D4—162no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4—162no—
634C2×C4—162no—
635C2×C4C22×D811yescharacteristic
636C2×C4C22×D812yes—
637C2×C4C22×D812yes—
638C2×C4C22×D811yescharacteristic
639C2×C4C22×D811yescharacteristic
640C2×C4C22×D811yescharacteristic
641C2×C4C22×D811yescharacteristic
642C2×C4C23×D411yescharacteristic
643C8—21no—
644C8—21no—
645C8—21no—
646C8—22no—
647C8—22no—
648C8C2×H(5,𝔽2)12yes—
649C22—21no—
650C22—21no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—44no—
658C22—44no—
659C22—82no—
660C22—88no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—164no—
670C22—1616no—
671C22#341412yes—
672C22C23×D811yescharacteristic
673C4—21no—
674C4—22no—
675C4—22no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#348411yescharacteristic
686C4#358511yescharacteristic
687C2—22no—
688C2—42no—
689C2—42no—
690C2—84no—
691C2—162no—
692C2—162no—
693C2#6155911yescharacteristic
694C2C2×#341411yescharacteristic
695C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62249C2semidirect
2. C2×#3280C2semidirect
3. #33886C2semidirect
4. #33878C2semidirect
5. #33834C2semidirect
6. #33833C2semidirect
7. #33781C2semidirect
8. #33779C2semidirect
9. #33762C2semidirect
10. #33754C2semidirect
11. #33670C2semidirect
12. #33669C2semidirect
13. #33517C2semidirect
14. #33516C2semidirect
15. #33487C2semidirect
20. C2×#509C22semidirect
21. C2×#508C22semidirect
22. #3342C22semidirect
23. #3341C22semidirect
26. #3287C22semidirect
27. #3283C22semidirect
35. #3170C22semidirect
36. #3169C22semidirect
39. #3050C22semidirect
47. #2251C22semidirect
48. #2250C22semidirect
49. #2243C22semidirect
50. #2243C22semidirect
51. #2237C22semidirect
52. #2236C22semidirect
53. #2234C22semidirect
54. #2233C22semidirect
55. #2226C22semidirect
56. #2224C22semidirect
57. #2220C22semidirect
58. #2219C22semidirect
59. #2217C22semidirect
60. #2215C22semidirect
61. #2211C22semidirect
62. #2210C22semidirect
63. #2208C22semidirect
64. #2206C22semidirect
65. #2189C22semidirect
66. #2189C22semidirect
67. #2188C22semidirect
70. #2187C22semidirect
71. #2186C22semidirect
72. #2186C22semidirect
73. #2185C22semidirect
76. #2184C22semidirect
77. #2177C22semidirect
78. #2175C22semidirect
79. #2168C22semidirect
80. #2168C22semidirect
81. #2163C22semidirect
82. #2163C22semidirect
83. #2140C22semidirect
84. #2138C22semidirect
86. C2×C22⋊C16C22semidirect
93. C2×#135C23semidirect
134. C2×Q32C23semidirect
137. C2×SD32C23semidirect
141. C22×C16C23semidirect
142. #500C23semidirect
143. #500C23semidirect
148. #494C23semidirect
149. #493C23semidirect
163. #467C23semidirect
169. #466C23semidirect
187. C4×Q16C23semidirect
193. C4×D8C23semidirect
199. #427C23semidirect
210. #367C23semidirect
211. #366C23semidirect
212. C4⋊C16C23semidirect
213. C4⋊C16C23semidirect
214. #358C23semidirect
215. #358C23semidirect
216. #358C23semidirect
221. #357C23semidirect
225. C22⋊C16C23semidirect
226. #346C23semidirect
227. #346C23semidirect

Maximal subgroups

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

subgroupindex
#622492
C2×#32802
#338862
#338782
#338342
#338332
#337812
#337792
#337622
#337542
#336702
#336692
#335172
#335162
#334872