Finite Group Databasebeta

← #7624360

Subgroups — #7624360

Order 512.

1407 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62251C211yesmaximal, characteristic, complemented
2C2×#3334C211yesmaximal, characteristic, complemented
3#33921C212yesmaximal, complemented
4#33912C214yesmaximal, complemented
5#33902C212yesmaximal, complemented
6#33899C212yesmaximal, complemented
7#33873C214yesmaximal, complemented
8#33861C212yesmaximal, complemented
9#33805C211yesmaximal, characteristic, complemented
10#33793C211yesmaximal, characteristic, complemented
11#33758C212yesmaximal, complemented
12#33754C212yesmaximal, complemented
13#33669C211yesmaximal, characteristic, complemented
14#33665C211yesmaximal, characteristic, complemented
15#33646C212yesmaximal, complemented
16C2×#2233C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18#3519—21no—
19C2×#547—21no—
20#3341C2211yescharacteristic, complemented
21#3339C2212yescomplemented
22#3338C2211yescharacteristic, complemented
23#3334—21no—
24#3334—22no—
25#3334—21no—
26#3334C2212yes—
27#3334C2214yescomplemented
28#3334C2212yescomplemented
29#3305—21no—
30#3297—22no—
31#3295—21no—
32#3287C2211yescharacteristic, complemented
33#3280C2212yescomplemented
34D4×D8C2211yescharacteristic, complemented
35#3247—21no—
36#3188—22no—
37#3169C2211yescharacteristic
38#3168C2211yescharacteristic
39C2×#500C2211yescharacteristic, complemented
40C2×#480—22no—
41C2×#474—21no—
42#3050C2212yes—
43C2×#466C2212yes—
44C2×#466C2212yescomplemented
45C2×C4×D8C2211yescharacteristic, complemented
46C2×C8×D4C2211yescharacteristic
47C22×#107C2212yescomplemented
48#2255C2212yescomplemented
49#2254C2212yescomplemented
50#2243C2212yescomplemented
51#2240C2212yescomplemented
52#2236C2212yescomplemented
53#2236C2211yescharacteristic, complemented
54#2235C2214yescomplemented
55#2234C2214yescomplemented
56#2233C2212yes—
57#2233C2211yescharacteristic
58#2233C2212yescomplemented
59#2233C2212yescomplemented
60#2233C2214yescomplemented
61#2233C2214yescomplemented
62#2233C2212yescomplemented
63#2233C2214yescomplemented
64#2232C2214yescomplemented
65#2228C2212yescomplemented
66#2226C2214yescomplemented
67#2226C2212yescomplemented
68#2225C2212yescomplemented
69#2224C2212yescomplemented
70#2218C2214yescomplemented
71#2218C2214yescomplemented
72#2217C2214yescomplemented
73#2215C2214yescomplemented
74#2210C2212yescomplemented
75#2206C2212yescomplemented
76#2187C2211yescharacteristic, complemented
77#2186C2214yescomplemented
78#2184C2211yescharacteristic
79#2184C2214yescomplemented
80#2184C2212yescomplemented
81#2177C2214yescomplemented
82C4×SD32C2212yescomplemented
83C4×D16C2212yescomplemented
84#2169C2212yescomplemented
85#2168C2214yescomplemented
86D4×C16C2212yescomplemented
87#2163C2211yescharacteristic, complemented
88#2162C2211yescharacteristic, complemented
89#2159C2214yescomplemented
90C2×#367C2214yescomplemented
91#2139C2214yescomplemented
92C2×#357C2214yescomplemented
93#2112C2212yescomplemented
94C2×C22⋊C16C2212yescomplemented
95C22×#41—41no—
96C2×#135C2312yes—
97C22×D8C2312yes—
98C22×D8C2311yescharacteristic
99C23×C8C2312yes—
100#555—21no—
101#554—21no—
102#553—22no—
103#553—41no—
104#550—21no—
105#547—22no—
106#547—24no—
107#547—22no—
108#547—41no—
109#547—42no—
110#547—41no—
111#546—22no—
112#546—42no—
113D42—21no—
114D42—41no—
115#540—41no—
116#538—42no—
117#532—21no—
118C2×C4⋊Q8—21no—
119#529—21no—
120#525—22no—
121C2×#123—42no—
122C2×C22⋊Q8—41no—
123C2×#121—22no—
124C2×#121—22no—
125#518—41no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C22×C4⋊C4—22no—
129C2×SD32C2312yes—
130C2×D16C2312yes—
131C2×D16C2314yes—
132C2×M32C2314yes—
133C22×C16C2314yes—
134#500—21no—
135#500—21no—
136#500C2312yes—
137#500C2312yes—
138#500C2311yescharacteristic
139#499C2311yescharacteristic
140#495C2311yescharacteristic
141#493C2311yescharacteristic
142#488—21no—
143#486—21no—
144#484—22no—
145#482—22no—
146#482—22no—
147#481—22no—
148#480—24no—
149#480—24no—
150#480—24no—
151#480—24no—
152#478—22no—
153#477—21no—
154#476—21no—
155#474—22no—
156#474—22no—
157#474—24no—
158#467C2312yes—
159#467C2311yescharacteristic
160#466—22no—
161#466—22no—
162#466—21no—
163#466—22no—
164#466—22no—
165#466C2314yes—
166#466C2314yes—
167#466C2312yes—
168#466C2312yes—
169#466C2312yes—
170#466C2311yescharacteristic
171#466C2312yes—
172#465—21no—
173#465—22no—
174#465—22no—
175#463—22no—
176#460—21no—
177#459—21no—
178#450—22no—
179#449—22no—
180#449—22no—
181#447—22no—
182C4×Q16C2311yescharacteristic
183C4×SD16—21no—
184C4×D8—21no—
185C4×D8—21no—
186C4×D8—21no—
187C4×D8C2312yes—
188C4×D8C2312yes—
189C4×D8C2312yes—
190C4×D8C2311yescharacteristic
191C8×D4D411yescharacteristic
192C8×D4D412yes—
193C8×D4D411yescharacteristic
194C8×D4C2311yescharacteristic
195C8×D4C2312yes—
196C8×D4C2311yescharacteristic
197#427C2312yes—
198C2×#107—22no—
199C2×#107—22no—
200C2×#107—21no—
201C2×#107—22no—
202C2×#107C2314yes—
203C2×#107C2314yes—
204C2×#107C2312yes—
205C2×#107C2312yes—
206C2×#107C2311yescharacteristic
207C2×C4⋊C8D411yescharacteristic
208#416—22no—
209C2×#102—21no—
210C2×#102—22no—
211C2×#102—22no—
212C2×C22⋊C8D412yes—
213C2×C4×C8C2311yescharacteristic
214#367C2312yes—
215#367C2314yes—
216#367C2314yes—
217#367C2314yes—
218#367C2314yes—
219#366C2312yes—
220#366C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223#358C2312yes—
224#358C2314yes—
225#357C2312yes—
226#357C2314yes—
227#357C2314yes—
228#357C2314yes—
229#357C2314yes—
230#357C2312yes—
231#357C2312yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2314yes—
238#346C2312yes—
239C4×C16C2312yes—
240C2×#41—22no—
241C2×#41—44no—
242C2×#41—42no—
243C2×#41—42no—
244C2×#41—41no—
245C2×#41—88no—
246C22×Q8—41no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—42no—
250C22×D4—42no—
251C22×D4—41no—
252C22×D4D811yescharacteristic
253C23×C4—42no—
254C23×C4—41no—
255C23×C4C2×D412yes—
256#135—44no—
257C2×Q16C2411yescharacteristic
258C2×SD16—22no—
259C2×SD16—21no—
260C2×D8—21no—
261C2×D8—22no—
262C2×D8—21no—
263C2×D8—42no—
264C2×D8—44no—
265C2×D8C2411yescharacteristic
266C2×D8C2412yes—
267C2×D8C2412yes—
268C22×C8—44no—
269C22×C8C2×D412yes—
270C22×C8C2×D412yes—
271C22×C8C2×D412yes—
272C22×C8C2×D411yescharacteristic
273C22×C8C2411yescharacteristic
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—21no—
279C4⋊Q8—41no—
280C4⋊Q8—41no—
281#127—21no—
282#127—41no—
283#126—42no—
284#126—41no—
285#125—21no—
286#125—41no—
287#124—21no—
288#124—42no—
289#124—41no—
290#123—42no—
291#123—41no—
292#123—42no—
293#123—44no—
294#123—44no—
295#123—44no—
296#123—44no—
297#123—42no—
298C22⋊Q8—21no—
299C22⋊Q8—22no—
300C22⋊Q8—42no—
301C22⋊Q8—42no—
302C22⋊Q8—44no—
303C22⋊Q8—42no—
304#121—21no—
305#121—22no—
306#121—22no—
307#121—24no—
308#121—24no—
309#121—22no—
310#121—22no—
311#121—42no—
312#121—41no—
313#121—42no—
314#121—42no—
315#121—42no—
316#121—42no—
317#121—41no—
318#121—41no—
319#121—42no—
320#121—42no—
321#120—42no—
322#120—42no—
323#120—42no—
324C4×Q8—21no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—21no—
328C4×D4—22no—
329C4×D4—41no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4—41no—
333C4×D4—41no—
334C4×D4—42no—
335C4×D4—41no—
336C4×D4—41no—
337C4×D4—41no—
338C4×D4D811yescharacteristic
339C4×D4D812yes—
340C4×D4D811yescharacteristic
341C4×D4C2×D411yescharacteristic
342C4×D4C2×D412yes—
343C4×D4C2×D411yescharacteristic
344#117—22no—
345#117—41no—
346#117—41no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4D811yescharacteristic
357C2×C22⋊C4—41no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—42no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4D812yes—
362C2×C42C2×D411yescharacteristic
363SD32—42no—
364D16—44no—
365D16—42no—
366M32—44no—
367C2×C16—44no—
368C2×C16C2412yes—
369C2×C16C2412yes—
370C2×C16C2414yes—
371#107—21no—
372#107—22no—
373#107—24no—
374#107—24no—
375#107—22no—
376#107—22no—
377#107—22no—
378#107—21no—
379#107C2412yes—
380#107C2414yes—
381#107C2411yescharacteristic
382#107C2411yescharacteristic
383#107C2412yes—
384#106—22no—
385#106—21no—
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D411yescharacteristic
388C4⋊C8C2×D412yes—
389#103—21no—
390#103—22no—
391#103—21no—
392#102—21no—
393#102—22no—
394#102—22no—
395#102—22no—
396#102—22no—
397#102—22no—
398#102—24no—
399#102—24no—
400#102—22no—
401#102—21no—
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—42no—
413C24C2×D812yes—
414#41—82no—
415#41—82no—
416#41—816no—
417#41—88no—
418#41—88no—
419#41—162no—
420C2×Q8—21no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—41no—
424C2×Q8—84no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—42no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4—88no—
446C2×D4—84no—
447C2×D4—164no—
448C2×D4C2×D812yes—
449C2×D4C2×D811yescharacteristic
450C2×D4C2×D811yescharacteristic
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—84no—
464C22×C4—88no—
465C22×C4C2×D811yescharacteristic
466C22×C4C2×D811yescharacteristic
467C22×C4C2×D812yes—
468C22×C4C2×D812yes—
469C22×C4C2×D812yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473Q16—22no—
474SD16—81no—
475SD16—82no—
476D8—22no—
477D8—22no—
478D8—22no—
479D8—24no—
480D8—42no—
481D8—81no—
482D8—82no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—41no—
490C2×C8C22×D411yescharacteristic
491C2×C8C22×D411yescharacteristic
492C2×C8C22×D411yescharacteristic
493C2×C8C22×D412yes—
494C2×C8C22×D412yes—
495C2×C8C2511yescharacteristic
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—41no—
514C4⋊C4C2×D811yescharacteristic
515C4⋊C4C2×D811yescharacteristic
516C4⋊C4C2×D812yes—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—42no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—41no—
530C22⋊C4—42no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4C2×D812yes—
534C22⋊C4C2×D812yes—
535C22⋊C4C2×D812yes—
536C22⋊C4C2×D812yes—
537C42—42no—
538C42—41no—
539C42C2×D811yescharacteristic
540C42C2×D811yescharacteristic
541C42C22×D411yescharacteristic
542C42C22×D411yescharacteristic
543C16—22no—
544C16—24no—
545C16—22no—
546C23—24no—
547C23—24no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23—84no—
556C23—162no—
557C23—162no—
558C23C22×D812yes—
559C23C22×D812yes—
560C23C22×D811yescharacteristic
561Q8—41no—
562Q8—41no—
563Q8—84no—
564Q8—84no—
565Q8—161no—
566D4—24no—
567D4—22no—
568D4—42no—
569D4—41no—
570D4—42no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—82no—
575D4—88no—
576D4—84no—
577D4—88no—
578D4—82no—
579D4—84no—
580D4—82no—
581D4—162no—
582D4—161no—
583D4—162no—
584D4—162no—
585D4—162no—
586D4#50914yes—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—84no—
605C2×C4—82no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4—82no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4C22×D812yes—
615C2×C4C22×D811yescharacteristic
616C2×C4C22×D811yescharacteristic
617C2×C4C22×D811yescharacteristic
618C2×C4C22×D811yescharacteristic
619C2×C4C22×D812yes—
620C2×C4C22×D811yescharacteristic
621C2×C4C23×D411yescharacteristic
622C8—21no—
623C8—21no—
624C8—21no—
625C8—22no—
626C8—22no—
627C8#58512yes—
628C22—21no—
629C22—22no—
630C22—21no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—48no—
636C22—82no—
637C22—81no—
638C22—82no—
639C22—82no—
640C22—81no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—81no—
645C22—164no—
646C22—324no—
647C22C2×#50914yes—
648C22#341612yes—
649C22C23×D811yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—22no—
656C4—82no—
657C4—84no—
658C4—81no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4C2×#50912yes—
663C4#348511yescharacteristic
664C4#358611yescharacteristic
665C2—22no—
666C2—24no—
667C2—42no—
668C2—162no—
669C2—162no—
670C2—161no—
671C2#6169411yescharacteristic
672C2C22×#50911yescharacteristic
673C2C2×#341611yescharacteristic
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. #62251C2semidirect
2. C2×#3334C2semidirect
3. #33921C2semidirect
4. #33912C2semidirect
5. #33902C2semidirect
6. #33899C2semidirect
7. #33873C2semidirect
8. #33861C2semidirect
9. #33805C2semidirect
10. #33793C2semidirect
11. #33758C2semidirect
12. #33754C2semidirect
13. #33669C2semidirect
14. #33665C2semidirect
15. #33646C2semidirect
16. C2×#2233C2semidirect
17. #33488C2semidirect
20. #3341C22semidirect
21. #3339C22semidirect
22. #3338C22semidirect
27. #3334C22semidirect
28. #3334C22semidirect
32. #3287C22semidirect
33. #3280C22semidirect
34. D4×D8C22semidirect
39. C2×#500C22semidirect
44. C2×#466C22semidirect
45. C2×C4×D8C22semidirect
47. C22×#107C22semidirect
48. #2255C22semidirect
49. #2254C22semidirect
50. #2243C22semidirect
51. #2240C22semidirect
52. #2236C22semidirect
53. #2236C22semidirect
54. #2235C22semidirect
55. #2234C22semidirect
58. #2233C22semidirect
59. #2233C22semidirect
60. #2233C22semidirect
61. #2233C22semidirect
62. #2233C22semidirect
63. #2233C22semidirect
64. #2232C22semidirect
65. #2228C22semidirect
66. #2226C22semidirect
67. #2226C22semidirect
68. #2225C22semidirect
69. #2224C22semidirect
70. #2218C22semidirect
71. #2218C22semidirect
72. #2217C22semidirect
73. #2215C22semidirect
74. #2210C22semidirect
75. #2206C22semidirect
76. #2187C22semidirect
77. #2186C22semidirect
79. #2184C22semidirect
80. #2184C22semidirect
81. #2177C22semidirect
82. C4×SD32C22semidirect
83. C4×D16C22semidirect
84. #2169C22semidirect
85. #2168C22semidirect
86. D4×C16C22semidirect
87. #2163C22semidirect
88. #2162C22semidirect
89. #2159C22semidirect
90. C2×#367C22semidirect
91. #2139C22semidirect
92. C2×#357C22semidirect
93. #2112C22semidirect
94. C2×C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#622512
C2×#33342
#339212
#339122
#339022
#338992
#338732
#338612
#338052
#337932
#337582
#337542
#336692
#336652
#336462
C2×#22332
#334882