Finite Group Databasebeta

← #7623447

Subgroups — #7623447

Order 512.

1255 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60695C211yesmaximal, characteristic
2#60667C211yesmaximal, characteristic, complemented
3#33810C214yesmaximal, complemented
4#33807C214yesmaximal, complemented
5#33804C212yesmaximal, complemented
6#33801C212yesmaximal, complemented
7#33799C212yesmaximal, complemented
8#33793C212yesmaximal, complemented
9#33730C212yesmaximal, complemented
10#33702C212yesmaximal, complemented
11#33653C212yesmaximal, complemented
12#33650C212yesmaximal, complemented
13#33629C211yesmaximal, characteristic, complemented
14#33627C211yesmaximal, characteristic, complemented
15#33626C211yesmaximal, characteristic, complemented
16C2×#2226C211yesmaximal, characteristic, complemented
17#33423C211yesmaximal, characteristic, complemented
18#3503—21no—
19#3460—21no—
20#3179—22no—
21#3176C2212yes—
22#3175C2212yes—
23#3171—22no—
24#3169C2212yes—
25#3168C2212yes—
26#3165—21no—
27#3165—22no—
28#3164—21no—
29#3164—22no—
30#3164C2211yescharacteristic
31#3164C2212yes—
32#3164C2214yescomplemented
33#3160C2211yescharacteristic
34#3160C2212yescomplemented
35C2×#500C2211yescharacteristic
36C2×#500C2212yescomplemented
37#3155C2211yescharacteristic
38#3136—21no—
39#3062—21no—
40#3061C2211yescharacteristic
41#3058C2211yescharacteristic
42C4×AGL(1,ℤ/8)—21no—
43C4×#135C2211yescharacteristic
44C2×C4×D8C2211yescharacteristic
45#2913—21no—
46#2913C2212yes—
47C4×#131C2211yescharacteristic
48#2255C2214yescomplemented
49#2255C2214yescomplemented
50#2254C2214yescomplemented
51#2253C2214yescomplemented
52#2244C2214yes—
53#2243C2214yes—
54#2242C2214yes—
55#2240C2214yes—
56#2237C2214yescomplemented
57#2235C2214yescomplemented
58#2234C2214yes—
59#2234C2214yes—
60#2234C2214yescomplemented
61#2233C2214yes—
62#2233C2214yescomplemented
63#2232C2214yes—
64#2229C2212yescomplemented
65#2228C2214yes—
66#2228C2212yescomplemented
67#2227C2212yescomplemented
68#2227C2211yescharacteristic, complemented
69#2226C2212yes—
70#2226C2212yes—
71#2226C2212yes—
72#2226C2214yes—
73#2226C2211yescharacteristic, complemented
74#2226C2212yescomplemented
75#2225C2211yescharacteristic, complemented
76#2225C2212yescomplemented
77#2224C2212yes—
78#2224C2211yescharacteristic, complemented
79#2224C2212yescomplemented
80#2164C2214yescomplemented
81#2163C2212yescomplemented
82#2162C2212yescomplemented
83#2159C2212yes—
84#2159C2212yes—
85#2158C2212yes—
86C2×#367C2212yes—
87#2150C2211yescharacteristic
88C2×C4⋊C16C2211yescharacteristic
89#2140C2212yes—
90#2139C2212yes—
91#2138C2212yes—
92C2×#357C2212yes—
93#2113C2212yes—
94#2108C2212yes—
95C2×AGL(1,ℤ/8)—21no—
96C2×#135C2311yescharacteristic
97C22×D8C2311yescharacteristic
98C2×#131D411yescharacteristic
99C2×#131C2311yescharacteristic
100Q82—41no—
101#557—41no—
102#554—42no—
103#554—41no—
104D4×Q8—42no—
105D4×Q8—41no—
106#544—21no—
107#544—22no—
108#544—24no—
109#542—22no—
110#542—41no—
111#540—22no—
112#540—41no—
113#538—42no—
114#537—21no—
115#535—21no—
116#535—21no—
117#533—21no—
118#533—22no—
119#532—22no—
120#532—41no—
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8—22no—
123#529—22no—
124#529—41no—
125#525—42no—
126#520—22no—
127C4×#41—21no—
128C4×#41D411yescharacteristic
129C2×C4×D4—21no—
130C2×M32C2312yes—
131C22×C16C2312yes—
132#501—24no—
133#501—24no—
134#501—24no—
135#501—22no—
136#500—21no—
137#500—21no—
138#500—22no—
139#500—22no—
140#500C2312yes—
141#500C2312yes—
142#500C2311yescharacteristic
143#500C2314yes—
144#500C2311yescharacteristic
145#500C2312yes—
146#499—21no—
147#499—22no—
148#499C2312yes—
149#499C2314yes—
150#499C2312yes—
151#498—21no—
152#498—22no—
153#496—22no—
154#495C2312yes—
155#493C2312yes—
156#490—22no—
157#489—22no—
158#488—22no—
159#486—22no—
160#469—22no—
161#469—22no—
162#468—22no—
163#468—22no—
164#467C2312yes—
165#466—22no—
166#466C2312yes—
167#466C2312yes—
168#466C2312yes—
169#465—22no—
170#442—22no—
171#440—22no—
172C4×Q16C2311yescharacteristic
173C4×Q16C2311yescharacteristic
174C4×SD16—22no—
175C4×SD16—21no—
176C4×SD16—21no—
177C4×D8—22no—
178C4×D8—21no—
179C4×D8—21no—
180C4×D8C2312yes—
181C4×D8C2311yescharacteristic
182C4×D8C2312yes—
183C4×D8C2311yescharacteristic
184#428—22no—
185#428—24no—
186#428—22no—
187#428—21no—
188#428—24no—
189#427—21no—
190#427—22no—
191#427C2312yes—
192#427C2314yes—
193#427C2311yescharacteristic
194C2×#107—21no—
195C2×#107—22no—
196C2×#107C2312yes—
197C2×#107C2314yes—
198C2×#107C2311yescharacteristic
199#418—21no—
200#417—21no—
201#416—21no—
202C2×#102—21no—
203#405D411yescharacteristic
204#405D412yes—
205#405C2311yescharacteristic
206#405C2312yes—
207C4×M16D412yes—
208C4×M16D411yescharacteristic
209C2×C4×C8C2311yescharacteristic
210C2×C4×C8C2312yes—
211#367C2314yes—
212#367C2314yes—
213#367C2312yes—
214#367C2312yes—
215#367C2314yes—
216#367C2314yes—
217#366C2312yes—
218#366C2312yes—
219#366C2314yes—
220#366C2314yes—
221C4⋊C16C2311yescharacteristic
222C4⋊C16C2311yescharacteristic
223C4⋊C16C2312yes—
224C4⋊C16C2312yes—
225C4⋊C16C2312yes—
226#358C2312yes—
227#358C2312yes—
228#358C2314yes—
229#357C2314yes—
230#357C2314yes—
231#357C2312yes—
232#357C2312yes—
233#357C2314yes—
234#357C2314yes—
235#357C2314yes—
236C22⋊C16C2314yes—
237C22⋊C16C2314yes—
238#346C2314yes—
239C4×C16C2314yes—
240C2×#41—21no—
241C2×#41—42no—
242C2×#41C2×D411yescharacteristic
243C22×Q8—41no—
244C22×Q8—42no—
245C22×D4—21no—
246C23×C4—41no—
247AGL(1,ℤ/8)—44no—
248#135—22no—
249#135—22no—
250C2×Q16C2411yescharacteristic
251C2×SD16—21no—
252C2×SD16—22no—
253C2×D8—21no—
254C2×D8—22no—
255C2×D8—22no—
256C2×D8—22no—
257C2×D8C2412yes—
258C2×D8C2411yescharacteristic
259#131—24no—
260#131—22no—
261#131—22no—
262C2×M16C2×D412yes—
263C2×M16C2×D411yescharacteristic
264C2×M16C2×D412yes—
265C2×M16C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
267C22×C8C2×D412yes—
268C22×C8C2412yes—
269C22×C8C2411yescharacteristic
270C4⋊Q8—22no—
271C4⋊Q8—21no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8—21no—
276C4⋊Q8—41no—
277C4⋊Q8—42no—
278C4⋊Q8—44no—
279C4⋊Q8—41no—
280C4⋊Q8—42no—
281C4⋊Q8—41no—
282C4⋊Q8—42no—
283#127—22no—
284#127—41no—
285#126—42no—
286#126—42no—
287#125—22no—
288#125—22no—
289#125—24no—
290#125—41no—
291#125—42no—
292#125—44no—
293#124—22no—
294#124—41no—
295#124—42no—
296#124—42no—
297#123—42no—
298#123—42no—
299#123—42no—
300#123—42no—
301C22⋊Q8—22no—
302C22⋊Q8—41no—
303C22⋊Q8—41no—
304C22⋊Q8—42no—
305C22⋊Q8—41no—
306C22⋊Q8—42no—
307C22⋊Q8—42no—
308C22⋊Q8—42no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312#121—22no—
313#121—22no—
314#121—22no—
315#121—42no—
316#121—41no—
317#121—42no—
318#121—41no—
319#121—41no—
320#121—42no—
321#121—42no—
322#120—42no—
323#120—42no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—42no—
327C4×Q8—41no—
328C4×Q8—42no—
329C4×Q8—41no—
330C4×Q8—42no—
331C4×Q8D811yescharacteristic
332C4×Q8D811yescharacteristic
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—44no—
338C4×D4—42no—
339C4×D4—41no—
340C4×D4—41no—
341C4×D4—42no—
342C4×D4—41no—
343C4×D4—41no—
344C4×D4—42no—
345C4×D4D812yes—
346C4×D4D811yescharacteristic
347C4×D4D812yes—
348C4×D4D811yescharacteristic
349#117—21no—
350#117—24no—
351#117—22no—
352#117—41no—
353#117—42no—
354#117C2×D411yescharacteristic
355#117C2×D412yes—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—42no—
361C2×C22⋊C4—41no—
362C2×C42—41no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365M32—24no—
366C2×C16—24no—
367C2×C16C2412yes—
368C2×C16C2414yes—
369C2×C16C2412yes—
370#107—22no—
371#107—21no—
372#107—22no—
373#107—24no—
374#107—24no—
375#107—22no—
376#107—21no—
377#107—21no—
378#107—22no—
379#107C2412yes—
380#107C2414yes—
381#107C2412yes—
382#107C2411yescharacteristic
383#107C2412yes—
384#107C2411yescharacteristic
385#106—22no—
386#106—24no—
387#106—24no—
388#106—22no—
389#106—21no—
390#106—22no—
391#103—21no—
392#103—21no—
393#103—22no—
394#102—22no—
395#102—22no—
396#102—21no—
397#102—21no—
398#102—22no—
399#102—22no—
400#102—22no—
401#97C2×D412yes—
402#97C2×D412yes—
403#97C2×D412yes—
404C4×C8C2×D412yes—
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D412yes—
408C4×C8C2412yes—
409C4×C8C2411yescharacteristic
410C4×C8C2411yescharacteristic
411C24—41no—
412#41—22no—
413#41—22no—
414#41—42no—
415#41—42no—
416#41—164no—
417C2×Q8—21no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—162no—
425C2×Q8—164no—
426C2×Q8C2×D811yescharacteristic
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4C2×D812yes—
439C2×D4C2×D811yescharacteristic
440C22×C4—41no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—162no—
449C22×C4C2×D811yescharacteristic
450C22×C4C2×D812yes—
451C22×C4C2×D811yescharacteristic
452C22×C4C2×D812yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455Q16—21no—
456Q16—21no—
457SD16—42no—
458SD16—44no—
459D8—22no—
460D8—22no—
461D8—21no—
462D8—21no—
463D8—42no—
464D8—44no—
465M16—22no—
466M16—24no—
467M16—24no—
468M16#13714yes—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8C22×D411yescharacteristic
474C2×C8C22×D412yes—
475C2×C8C22×D411yescharacteristic
476C2×C8C22×D411yescharacteristic
477C2×C8C22×D412yes—
478C2×C82−1+412yes—
479C2×C82−1+412yes—
480C2×C8C2511yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—41no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—41no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4C2×D811yescharacteristic
506C4⋊C4C2×D811yescharacteristic
507C4⋊C4C2×D812yes—
508C4⋊C4C2×D812yes—
509C22⋊C4—41no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4C2×D812yes—
523C22⋊C4C2×D812yes—
524C22⋊C4C2×D812yes—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—44no—
529C42—42no—
530C42C2×D811yescharacteristic
531C42C2×D812yes—
532C42C2×D811yescharacteristic
533C42C2×D812yes—
534C42C22×D411yescharacteristic
535C42C22×D411yescharacteristic
536C42C22×D412yes—
537C16—22no—
538C16—22no—
539C16—24no—
540C23—42no—
541C23—41no—
542C23—82no—
543C23—161no—
544C23—161no—
545C23C22×D812yes—
546C23C22×D811yescharacteristic
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—84no—
551Q8—82no—
552Q8—82no—
553Q8—162no—
554Q8—164no—
555Q8—162no—
556D4—24no—
557D4—22no—
558D4—42no—
559D4—41no—
560D4—41no—
561D4—42no—
562D4—82no—
563D4—82no—
564D4—162no—
565D4—162no—
566D4—162no—
567D4—162no—
568D4—162no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—84no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—161no—
586C2×C4—162no—
587C2×C4—161no—
588C2×C4—162no—
589C2×C4—164no—
590C2×C4#53612yes—
591C2×C4C22×D812yes—
592C2×C4C22×D811yescharacteristic
593C2×C4C22×D811yescharacteristic
594C2×C4C22×D811yescharacteristic
595C2×C4C22×D812yes—
596C2×C4C22×D812yes—
597C2×C4C22×D811yescharacteristic
598C2×C4C22×D811yescharacteristic
599C2×C4C2×#13712yes—
600C2×C4C23×D411yescharacteristic
601C8—22no—
602C8—21no—
603C8—21no—
604C8#53612yes—
605C8C2×#13714yes—
606C8C2×2−1+411yescharacteristic
607C8C2×2−1+411yescharacteristic
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—81no—
614C22—82no—
615C22—82no—
616C22—81no—
617C22—81no—
618C22—82no—
619C22—164no—
620C22#322612yes—
621C22C23×D811yescharacteristic
622C4—22no—
623C4—21no—
624C4—22no—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—81no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—81no—
635C4#322612yes—
636C4#341612yes—
637C4C2×#53611yescharacteristic
638C4C22×#13711yescharacteristic
639C2—22no—
640C2—42no—
641C2—161no—
642C2—162no—
643C2C2×#322611yescharacteristic
644C2C2×#341611yescharacteristic
645C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#60695C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60667C2semidirect
3. #33810C2semidirect
4. #33807C2semidirect
5. #33804C2semidirect
6. #33801C2semidirect
7. #33799C2semidirect
8. #33793C2semidirect
9. #33730C2semidirect
10. #33702C2semidirect
11. #33653C2semidirect
12. #33650C2semidirect
13. #33629C2semidirect
14. #33627C2semidirect
15. #33626C2semidirect
16. C2×#2226C2semidirect
17. #33423C2semidirect
32. #3164C22semidirect
34. #3160C22semidirect
36. C2×#500C22semidirect
48. #2255C22semidirect
49. #2255C22semidirect
50. #2254C22semidirect
51. #2253C22semidirect
56. #2237C22semidirect
57. #2235C22semidirect
60. #2234C22semidirect
62. #2233C22semidirect
64. #2229C22semidirect
66. #2228C22semidirect
67. #2227C22semidirect
68. #2227C22semidirect
73. #2226C22semidirect
74. #2226C22semidirect
75. #2225C22semidirect
76. #2225C22semidirect
78. #2224C22semidirect
79. #2224C22semidirect
80. #2164C22semidirect
81. #2163C22semidirect
82. #2162C22semidirect

Maximal subgroups

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

subgroupindex
#606952
#606672
#338102
#338072
#338042
#338012
#337992
#337932
#337302
#337022
#336532
#336502
#336292
#336272
#336262
C2×#22262
#334232