Finite Group Databasebeta

← #7625151

Subgroups — #7625151

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62473C211yesmaximal, characteristic, complemented
2#62419C211yesmaximal, characteristic
3#33932C212yesmaximal, complemented
4#33931C212yesmaximal, complemented
5#33895C212yesmaximal, complemented
6#33894C214yesmaximal, complemented
7#33891C212yesmaximal, complemented
8#33890C214yesmaximal, complemented
9#33806C212yesmaximal, complemented
10#33805C211yesmaximal, characteristic, complemented
11#33782C211yesmaximal, characteristic, complemented
12#33781C212yesmaximal, complemented
13#33738C211yesmaximal, characteristic, complemented
14#33738C212yesmaximal, complemented
15#33707C211yesmaximal, characteristic, complemented
16#33706C212yesmaximal, complemented
17#33485C211yesmaximal, characteristic, complemented
18#3514—21no—
19#3509—21no—
20#3396C2211yescharacteristic
21#3394—21no—
22#3393C2211yescharacteristic
23#3383—21no—
24#3383C2212yescomplemented
25#3342—21no—
26#3342C2212yes—
27#3336—22no—
28#3336C2214yescomplemented
29D4×Q16C2211yescharacteristic
30#3285C2212yes—
31#3282—22no—
32D4×SD16—21no—
33#3280C2212yes—
34D4×D8C2211yescharacteristic
35#3231—22no—
36#3229—21no—
37#3212—21no—
38#3211—22no—
39#3170C2212yescomplemented
40#3170C2211yescharacteristic, complemented
41#3168C2211yescharacteristic
42#3168C2212yes—
43#3160C2212yescomplemented
44C2×#500C2211yescharacteristic, complemented
45#3147C2211yescharacteristic
46C2×#495C2212yes—
47C8×#41C2211yescharacteristic
48#2255C2212yescomplemented
49#2255C2211yescharacteristic, complemented
50#2251C2211yescharacteristic
51#2251C2212yescomplemented
52#2245C2211yescharacteristic, complemented
53#2243C2212yes—
54#2243C2211yescharacteristic
55#2243C2212yescomplemented
56#2243C2212yescomplemented
57#2243C2212yescomplemented
58#2242C2214yes—
59#2242C2214yescomplemented
60#2241C2212yescomplemented
61#2240C2212yes—
62#2237C2214yescomplemented
63#2236C2212yescomplemented
64#2234C2214yes—
65#2233C2212yes—
66#2229C2214yescomplemented
67#2228C2214yescomplemented
68#2225C2212yescomplemented
69#2224C2212yes—
70#2220C2212yes—
71#2220C2214yes—
72#2219C2214yescomplemented
73#2219C2212yescomplemented
74#2211C2214yes—
75#2211C2214yescomplemented
76#2209C2212yescomplemented
77#2207C2212yes—
78#2189C2214yescomplemented
79#2188C2214yescomplemented
80#2187C2214yes—
81Q8×C16C2212yescomplemented
82C4×Q32C2214yescomplemented
83C4×Q32C2212yescomplemented
84C4×SD32C2212yes—
85C4×SD32C2214yescomplemented
86#2169C2212yes—
87#2169C2214yescomplemented
88#2163C2211yescharacteristic, complemented
89#2163C2212yescomplemented
90#2141C2214yescomplemented
91#2140C2214yes—
92#2139C2214yescomplemented
93C4×M32C2212yescomplemented
94C4×M32C2211yescharacteristic, complemented
95C2×#135—21no—
96C2×#135C2312yes—
97C22×Q16C2312yes—
98C22×SD16—22no—
99C22×D8C2312yes—
100#559—21no—
101#557—22no—
102#557—41no—
103#556—22no—
104#556—21no—
105#556—41no—
106#555—41no—
107#552—24no—
108#552—42no—
109#551—42no—
110D4×Q8—21no—
111#548—22no—
112#546—22no—
113#546—41no—
114D42—21no—
115#542—41no—
116#541—42no—
117#538—42no—
118#536—41no—
119#533—22no—
120#533—21no—
121#533—22no—
122#532—21no—
123C2×C4⋊Q8—21no—
124#529—22no—
125#529—21no—
126C2×#124—22no—
127#519—41no—
128#518—41no—
129C4×#41D411yescharacteristic
130C2×Q32C2312yes—
131C2×Q32C2314yes—
132C2×SD32C2312yes—
133C2×SD32C2314yes—
134C2×M32C2312yes—
135C2×M32C2314yes—
136#500—21no—
137#500—21no—
138#500C2312yes—
139#500C2312yes—
140#500C2311yescharacteristic
141#499—22no—
142#499C2314yes—
143#499C2311yescharacteristic
144#498—22no—
145#495—22no—
146#495—22no—
147#495—21no—
148#495C2312yes—
149#495C2312yes—
150#495C2311yescharacteristic
151#495C2312yes—
152#494C2311yescharacteristic
153#494C2311yescharacteristic
154#493C2311yescharacteristic
155#492—21no—
156#488—21no—
157#488—22no—
158#487—22no—
159#487—21no—
160#485—24no—
161#485—24no—
162#483—24no—
163#479—24no—
164#479—24no—
165#475—22no—
166#475—22no—
167#467—21no—
168#467—22no—
169#467C2312yes—
170#467C2314yes—
171#467C2311yescharacteristic
172#467C2312yes—
173#466C2311yescharacteristic
174#466C2312yes—
175#465—22no—
176#465—21no—
177#464—22no—
178#464—22no—
179#463—22no—
180#463—22no—
181#462—21no—
182#461—21no—
183#460—21no—
184#459—21no—
185#452—22no—
186#451—22no—
187#450—22no—
188#449—22no—
189#448—22no—
190#447—22no—
191C8×Q8D411yescharacteristic
192C8×Q8C2311yescharacteristic
193C4×Q16—22no—
194C4×Q16—21no—
195C4×Q16C2314yes—
196C4×Q16C2312yes—
197C4×Q16C2312yes—
198C4×Q16C2311yescharacteristic
199C4×SD16—21no—
200C4×SD16—22no—
201C4×D8C2311yescharacteristic
202C4×D8C2312yes—
203C8×D4D412yes—
204C8×D4D411yescharacteristic
205C8×D4C2312yes—
206C8×D4C2311yescharacteristic
207#431D412yes—
208#431D411yescharacteristic
209#427—22no—
210#427C2314yes—
211C2×#107—21no—
212C2×#107C2312yes—
213C2×C4×C8C2312yes—
214C2×C4×C8C2311yescharacteristic
215#367C2312yes—
216#367C2314yes—
217#366C2314yes—
218#366C2312yes—
219C4⋊C16C2312yes—
220C4⋊C16C2314yes—
221#358C2314yes—
222#358C2314yes—
223#358C2312yes—
224#358C2314yes—
225#358C2314yes—
226#358C2312yes—
227#358C2312yes—
228#357C2314yes—
229#357C2314yes—
230#357C2312yes—
231#357C2314yes—
232#357C2314yes—
233C22⋊C16C2314yes—
234C22⋊C16C2312yes—
235#346C2311yescharacteristic
236#346C2312yes—
237#346C2311yescharacteristic
238C4×C16C2312yes—
239C4×C16C2314yes—
240C2×#41—22no—
241C2×#41D811yescharacteristic
242C22×Q8—22no—
243C22×Q8—41no—
244C22×D4—22no—
245C22×D4—41no—
246#135—44no—
247#135—82no—
248C2×Q16—21no—
249C2×Q16—22no—
250C2×Q16—21no—
251C2×Q16—44no—
252C2×Q16C2411yescharacteristic
253C2×Q16C2412yes—
254C2×Q16C2412yes—
255C2×SD16—21no—
256C2×SD16—22no—
257C2×SD16—22no—
258C2×SD16—22no—
259C2×SD16—22no—
260C2×SD16—84no—
261C2×D8—22no—
262C2×D8—21no—
263C2×D8—44no—
264C2×D8C2411yescharacteristic
265C2×D8C2412yes—
266C22×C8C2×D412yes—
267C22×C8C2×D411yescharacteristic
268C22×C8C2412yes—
269C22×C8C2411yescharacteristic
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—42no—
276C4⋊Q8—41no—
277C4⋊Q8—41no—
278C4⋊Q8—41no—
279#127—21no—
280#126—42no—
281#126—44no—
282#126—44no—
283#126—42no—
284#126—41no—
285#125—21no—
286#125—24no—
287#125—44no—
288#125—44no—
289#125—41no—
290#125—42no—
291#124—21no—
292#124—22no—
293#124—22no—
294#124—22no—
295#124—41no—
296#124—42no—
297#124—42no—
298#124—42no—
299#124—42no—
300#123—44no—
301#123—41no—
302#123—42no—
303#123—42no—
304#123—42no—
305C22⋊Q8—22no—
306C22⋊Q8—21no—
307C22⋊Q8—22no—
308C22⋊Q8—24no—
309C22⋊Q8—41no—
310C22⋊Q8—42no—
311C22⋊Q8—42no—
312C22⋊Q8—41no—
313C22⋊Q8—42no—
314C22⋊Q8—42no—
315C22⋊Q8—42no—
316#121—22no—
317#121—21no—
318#121—42no—
319#121—41no—
320#121—41no—
321#121—41no—
322#121—42no—
323#120—42no—
324#120—42no—
325C4×Q8—22no—
326C4×Q8—21no—
327C4×Q8—24no—
328C4×Q8—22no—
329C4×Q8—42no—
330C4×Q8—42no—
331C4×Q8—42no—
332C4×Q8—42no—
333C4×Q8—42no—
334C4×Q8—41no—
335C4×Q8D811yescharacteristic
336C4×Q8C2×D411yescharacteristic
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—41no—
340C4×D4—42no—
341C4×D4—41no—
342C4×D4—41no—
343C4×D4—42no—
344C4×D4—41no—
345C4×D4—41no—
346C4×D4D811yescharacteristic
347C4×D4D812yes—
348C4×D4C2×D411yescharacteristic
349C4×D4C2×D412yes—
350#117—24no—
351#117—41no—
352#117—42no—
353#117—41no—
354#117—44no—
355#117—41no—
356#117—42no—
357#117D811yescharacteristic
358#117D812yes—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—41no—
361C2×C22⋊C4—42no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366Q32—42no—
367Q32—44no—
368SD32—42no—
369SD32—44no—
370M32—44no—
371M32—42no—
372C2×C16C2412yes—
373C2×C16C2412yes—
374C2×C16C2414yes—
375#107—22no—
376#107—22no—
377#107—22no—
378#107—21no—
379#107C2414yes—
380#107C2412yes—
381#107C2411yescharacteristic
382#107C2411yescharacteristic
383#106—24no—
384#106—24no—
385#106—21no—
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D412yes—
388C4⋊C8C2×D411yescharacteristic
389C4⋊C8C2×D412yes—
390#103—21no—
391#103—22no—
392#103—24no—
393#103—22no—
394#103—22no—
395#103—21no—
396#103—24no—
397#103—22no—
398#102—21no—
399#102—22no—
400#102—22no—
401#102—21no—
402C22⋊C8C2×D412yes—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C4×C8C2×D412yes—
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D412yes—
409C4×C8C2411yescharacteristic
410C4×C8C2412yes—
411C4×C8C2411yescharacteristic
412C24—42no—
413#41—42no—
414#41—82no—
415#41—82no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—41no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—84no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8C2×D811yescharacteristic
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—41no—
431C2×D4—42no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—42no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—84no—
438C2×D4C2×D811yescharacteristic
439C2×D4C2×D812yes—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C2×D811yescharacteristic
446C22×C4C2×D812yes—
447C22×C4C2×D812yes—
448C22×C4C2×D811yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451Q16—22no—
452Q16—22no—
453Q16—22no—
454Q16—24no—
455Q16—42no—
456Q16—82no—
457Q16—81no—
458SD16—42no—
459SD16—44no—
460SD16—82no—
461SD16—82no—
462SD16—82no—
463D8—22no—
464D8—24no—
465D8—82no—
466D8—81no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8C22×D412yes—
472C2×C8C22×D411yescharacteristic
473C2×C8C22×D412yes—
474C2×C8C22×D411yescharacteristic
475C2×C8C22×D411yescharacteristic
476C2×C8C2511yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—44no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—41no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4C2×D811yescharacteristic
502C4⋊C4C2×D812yes—
503C4⋊C4C2×D811yescharacteristic
504C4⋊C4C2×D812yes—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—41no—
521C22⋊C4—42no—
522C22⋊C4C2×D812yes—
523C22⋊C4C2×D812yes—
524C22⋊C4C2×D812yes—
525C42—44no—
526C42—42no—
527C42—44no—
528C42—41no—
529C42—42no—
530C42C2×D812yes—
531C42C2×D811yescharacteristic
532C42C2×D811yescharacteristic
533C42C2×D812yes—
534C42C22×D411yescharacteristic
535C42C22×D412yes—
536C42C22×D411yescharacteristic
537C16—22no—
538C16—22no—
539C16—24no—
540C23—41no—
541C23—42no—
542C23—84no—
543C23—162no—
544C23—162no—
545C23C22×D812yes—
546C23C22×D811yescharacteristic
547Q8—22no—
548Q8—42no—
549Q8—41no—
550Q8—42no—
551Q8—41no—
552Q8—42no—
553Q8—42no—
554Q8—82no—
555Q8—84no—
556Q8—84no—
557Q8—162no—
558D4—22no—
559D4—42no—
560D4—41no—
561D4—42no—
562D4—42no—
563D4—41no—
564D4—84no—
565D4—82no—
566D4—161no—
567D4—162no—
568D4—161no—
569D4—162no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—84no—
586C2×C4—82no—
587C2×C4—162no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4C22×D812yes—
591C2×C4C22×D812yes—
592C2×C4C22×D811yescharacteristic
593C2×C4C22×D811yescharacteristic
594C2×C4C22×D811yescharacteristic
595C2×C4C22×D812yes—
596C2×C4C22×D811yescharacteristic
597C2×C4C22×D811yescharacteristic
598C2×C4C23×D411yescharacteristic
599C8—22no—
600C8—21no—
601C8—21no—
602C8—22no—
603C8#57712yes—
604C8#58512yes—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—21no—
610C22—22no—
611C22—82no—
612C22—81no—
613C22—81no—
614C22—82no—
615C22—82no—
616C22—81no—
617C22—324no—
618C22C23×D811yescharacteristic
619C4—21no—
620C4—22no—
621C4—21no—
622C4—22no—
623C4—21no—
624C4—22no—
625C4—21no—
626C4—82no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4—84no—
631C4—81no—
632C4—82no—
633C4#341412yes—
634C4#350011yescharacteristic
635C4C2×#57711yescharacteristic
636C2—41no—
637C2—42no—
638C2—162no—
639C2—161no—
640C2#6252911yescharacteristic
641C2C2×#341411yescharacteristic
642C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62419C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62473C2semidirect
3. #33932C2semidirect
4. #33931C2semidirect
5. #33895C2semidirect
6. #33894C2semidirect
7. #33891C2semidirect
8. #33890C2semidirect
9. #33806C2semidirect
10. #33805C2semidirect
11. #33782C2semidirect
12. #33781C2semidirect
13. #33738C2semidirect
14. #33738C2semidirect
15. #33707C2semidirect
16. #33706C2semidirect
17. #33485C2semidirect
24. #3383C22semidirect
28. #3336C22semidirect
39. #3170C22semidirect
40. #3170C22semidirect
43. #3160C22semidirect
44. C2×#500C22semidirect
48. #2255C22semidirect
49. #2255C22semidirect
51. #2251C22semidirect
52. #2245C22semidirect
55. #2243C22semidirect
56. #2243C22semidirect
57. #2243C22semidirect
59. #2242C22semidirect
60. #2241C22semidirect
62. #2237C22semidirect
63. #2236C22semidirect
66. #2229C22semidirect
67. #2228C22semidirect
68. #2225C22semidirect
72. #2219C22semidirect
73. #2219C22semidirect
75. #2211C22semidirect
76. #2209C22semidirect
78. #2189C22semidirect
79. #2188C22semidirect
81. Q8×C16C22semidirect
82. C4×Q32C22semidirect
83. C4×Q32C22semidirect
85. C4×SD32C22semidirect
87. #2169C22semidirect
88. #2163C22semidirect
89. #2163C22semidirect
90. #2141C22semidirect
92. #2139C22semidirect
93. C4×M32C22semidirect
94. C4×M32C22semidirect

Maximal subgroups

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

subgroupindex
#624732
#624192
#339322
#339312
#338952
#338942
#338912
#338902
#338062
#338052
#337822
#337812
#337382
#337072
#337062
#334852