Finite Group Databasebeta

← #7625139

Subgroups — #7625139

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62476C211yesmaximal, characteristic, complemented
2#62417C211yesmaximal, characteristic, complemented
3#33942C212yesmaximal, complemented
4#33940C212yesmaximal, complemented
5#33886C214yesmaximal, complemented
6#33878C214yesmaximal, complemented
7#33872C212yesmaximal, complemented
8#33871C212yesmaximal, complemented
9#33803C211yesmaximal, characteristic, complemented
10#33801C212yesmaximal, complemented
11#33785C212yesmaximal, complemented
12#33783C211yesmaximal, characteristic, complemented
13#33746C212yesmaximal, complemented
14#33738C211yesmaximal, characteristic, complemented
15#33704C211yesmaximal, characteristic, complemented
16#33701C212yesmaximal, complemented
17#33484C211yesmaximal, characteristic, complemented
18#3518—21no—
19#3508—21no—
20#3396—21no—
21#3396C2212yes—
22Q8×Q16C2211yescharacteristic, complemented
23Q8×SD16—21no—
24Q8×D8C2211yescharacteristic, complemented
25#3342C2212yes—
26#3342C2211yescharacteristic, complemented
27#3341C2212yes—
28#3341C2211yescharacteristic, complemented
29#3340—22no—
30#3340—21no—
31#3287—21no—
32#3287C2212yes—
33#3280—22no—
34#3280C2214yes—
35#3233—21no—
36#3233—22no—
37#3208—21no—
38#3207—22no—
39#3170C2211yescharacteristic
40#3169C2212yes—
41#3168C2211yescharacteristic
42#3168C2212yescomplemented
43#3160C2211yescharacteristic
44#3160C2212yescomplemented
45#3147C2212yes—
46C2×#493C2211yescharacteristic
47C8×#41C2211yescharacteristic
48#2255C2211yescharacteristic, complemented
49#2255C2212yescomplemented
50#2251C2211yescharacteristic, complemented
51#2250C2212yes—
52#2245C2212yescomplemented
53#2245C2211yescharacteristic, complemented
54#2244C2212yes—
55#2244C2212yes—
56#2244C2212yes—
57#2243C2214yescomplemented
58#2243C2214yescomplemented
59#2243C2211yescharacteristic, complemented
60#2240C2212yes—
61#2240C2212yes—
62#2237C2214yes—
63#2234C2214yescomplemented
64#2234C2214yescomplemented
65#2227C2212yescomplemented
66#2226C2214yescomplemented
67#2225C2212yescomplemented
68#2224C2214yescomplemented
69#2220C2212yes—
70#2219C2212yes—
71#2217C2214yescomplemented
72#2215C2214yescomplemented
73#2211C2214yescomplemented
74#2210C2214yescomplemented
75#2207C2212yes—
76#2206C2212yes—
77#2189C2214yescomplemented
78#2187C2212yes—
79#2186C2214yescomplemented
80#2184C2212yes—
81#2183C2212yescomplemented
82#2177C2214yescomplemented
83#2175C2214yescomplemented
84C4×Q32C2212yescomplemented
85C4×SD32C2212yescomplemented
86#2168C2214yes—
87D4×C16C2212yescomplemented
88#2163C2211yescharacteristic
89#2163C2212yescomplemented
90#2140C2214yes—
91#2138C2214yescomplemented
92#2138C2214yescomplemented
93C4×M32C2211yescharacteristic
94C2×#346C2212yescomplemented
95C2×#135—22no—
96C2×#135C2314yes—
97C22×D8—21no—
98C22×D8C2312yes—
99#559—22no—
100#559—41no—
101Q82—21no—
102#556—22no—
103#556—21no—
104#556—41no—
105#556—42no—
106#555—22no—
107#555—21no—
108#550—22no—
109#550—41no—
110D4×Q8—21no—
111#548—41no—
112#546—24no—
113#546—42no—
114#542—42no—
115#542—41no—
116#540—42no—
117#535—41no—
118#533—21no—
119#533—22no—
120#533—21no—
121#532—22no—
122#532—22no—
123C2×#127—21no—
124#529—21no—
125#529—22no—
126#519—41no—
127#518—41no—
128C4×#41D411yescharacteristic
129C2×Q32C2312yes—
130C2×SD32C2312yes—
131C2×SD32C2314yes—
132C2×D16C2314yes—
133C2×M32C2312yes—
134C22×C16C2314yes—
135#500C2312yes—
136#500C2311yescharacteristic
137#500C2312yes—
138#500C2311yescharacteristic
139#500C2311yescharacteristic
140#499—21no—
141#499—22no—
142#499C2312yes—
143#499C2311yescharacteristic
144#498—21no—
145#498—22no—
146#495—22no—
147#495C2314yes—
148#495C2311yescharacteristic
149#494C2311yescharacteristic
150#493—21no—
151#493—21no—
152#493C2312yes—
153#493C2312yes—
154#492—22no—
155#488—21no—
156#488—22no—
157#487—21no—
158#486—22no—
159#485—22no—
160#485—22no—
161#485—22no—
162#482—22no—
163#482—22no—
164#482—22no—
165#477—21no—
166#477—22no—
167#476—22no—
168#476—21no—
169#475—22no—
170#475—21no—
171#474—21no—
172#474—22no—
173#467—21no—
174#467C2311yescharacteristic
175#467C2312yes—
176#467C2312yes—
177#466—22no—
178#466C2311yescharacteristic
179#466C2312yes—
180#466C2314yes—
181#465—21no—
182#465—22no—
183#463—24no—
184#463—24no—
185#460—22no—
186#460—22no—
187#450—24no—
188#449—24no—
189#449—24no—
190C8×Q8D411yescharacteristic
191C8×Q8C2311yescharacteristic
192C4×Q16—21no—
193C4×Q16C2312yes—
194C4×Q16C2311yescharacteristic
195C4×Q16C2312yes—
196C4×SD16—21no—
197C4×SD16—22no—
198C4×D8—22no—
199C4×D8C2312yes—
200C4×D8C2311yescharacteristic
201C4×D8C2314yes—
202C8×D4D412yes—
203C8×D4D411yescharacteristic
204C8×D4C2312yes—
205C8×D4C2311yescharacteristic
206#431D412yes—
207#431D411yescharacteristic
208#427—21no—
209#427—22no—
210#427C2312yes—
211#427C2312yes—
212#427C2312yes—
213C2×C4×C8C2312yes—
214C2×C4×C8C2311yescharacteristic
215#367C2312yes—
216#367C2314yes—
217#366C2312yes—
218#366C2314yes—
219C4⋊C16C2312yes—
220C4⋊C16C2314yes—
221#358C2312yes—
222#358C2312yes—
223#358C2314yes—
224#358C2314yes—
225#358C2312yes—
226#357C2314yes—
227#357C2314yes—
228#357C2314yes—
229#357C2314yes—
230#357C2314yes—
231#357C2314yes—
232#357C2312yes—
233C22⋊C16C2312yes—
234C22⋊C16C2314yes—
235#346C2311yescharacteristic
236#346C2312yes—
237#346C2314yes—
238#346C2311yescharacteristic
239C4×C16C2312yes—
240C2×#41—24no—
241C2×#41—42no—
242C2×#41D811yescharacteristic
243C22×Q8—41no—
244C22×D4—22no—
245C22×D4—41no—
246C22×D4—41no—
247#135—44no—
248#135—44no—
249#135—84no—
250C2×Q16—21no—
251C2×Q16C2412yes—
252C2×Q16C2411yescharacteristic
253C2×SD16—21no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×D8—22no—
257C2×D8—22no—
258C2×D8—22no—
259C2×D8—44no—
260C2×D8—82no—
261C2×D8C2414yes—
262C2×D8C2411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D412yes—
265C22×C8C2411yescharacteristic
266C22×C8C2412yes—
267C4⋊Q8—21no—
268C4⋊Q8—22no—
269C4⋊Q8—21no—
270C4⋊Q8—22no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—41no—
274C4⋊Q8—42no—
275#127—22no—
276#127—22no—
277#127—42no—
278#127—41no—
279#127—41no—
280#126—42no—
281#126—42no—
282#126—42no—
283#126—42no—
284#126—41no—
285#125—21no—
286#125—22no—
287#125—41no—
288#124—21no—
289#124—24no—
290#124—42no—
291#124—41no—
292#124—42no—
293#124—42no—
294#124—42no—
295#124—42no—
296#124—42no—
297#123—42no—
298#123—41no—
299#123—44no—
300#123—44no—
301C22⋊Q8—22no—
302C22⋊Q8—21no—
303C22⋊Q8—22no—
304C22⋊Q8—42no—
305C22⋊Q8—41no—
306C22⋊Q8—41no—
307C22⋊Q8—41no—
308C22⋊Q8—42no—
309C22⋊Q8—41no—
310C22⋊Q8—42no—
311#121—22no—
312#121—21no—
313#121—24no—
314#121—41no—
315#121—42no—
316#121—42no—
317#121—42no—
318#121—42no—
319#121—44no—
320#121—44no—
321#120—44no—
322C4×Q8—22no—
323C4×Q8—21no—
324C4×Q8—22no—
325C4×Q8—41no—
326C4×Q8—41no—
327C4×Q8—42no—
328C4×Q8—42no—
329C4×Q8—41no—
330C4×Q8—41no—
331C4×Q8—42no—
332C4×Q8—41no—
333C4×Q8D811yescharacteristic
334C4×Q8C2×D411yescharacteristic
335C4×D4—22no—
336C4×D4—21no—
337C4×D4—24no—
338C4×D4—41no—
339C4×D4—42no—
340C4×D4—42no—
341C4×D4—42no—
342C4×D4—42no—
343C4×D4—42no—
344C4×D4—42no—
345C4×D4D812yes—
346C4×D4D811yescharacteristic
347C4×D4C2×D412yes—
348C4×D4C2×D411yescharacteristic
349#117—22no—
350#117—22no—
351#117—22no—
352#117—41no—
353#117—41no—
354#117—42no—
355#117—42no—
356#117—41no—
357#117D811yescharacteristic
358#117D812yes—
359C2×C22⋊C4—44no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362Q32—42no—
363SD32—44no—
364SD32—42no—
365D16—44no—
366M32—42no—
367C2×C16—44no—
368C2×C16C2412yes—
369C2×C16C2412yes—
370C2×C16C2414yes—
371#107—22no—
372#107—21no—
373#107—22no—
374#107—21no—
375#107—21no—
376#107C2412yes—
377#107C2411yescharacteristic
378#107C2412yes—
379#107C2411yescharacteristic
380#107C2412yes—
381#106—22no—
382#106—21no—
383#106—22no—
384#106—22no—
385#106—22no—
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D412yes—
388C4⋊C8C2×D411yescharacteristic
389C4⋊C8C2×D412yes—
390#103—22no—
391#103—21no—
392#103—22no—
393#103—22no—
394#103—22no—
395#103—21no—
396#102—24no—
397#102—24no—
398#102—22no—
399#102—21no—
400#102—22no—
401#102—21no—
402C22⋊C8C2×D412yes—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C4×C8C2×D412yes—
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D412yes—
408C4×C8C2×D411yescharacteristic
409C4×C8C2411yescharacteristic
410C4×C8C2411yescharacteristic
411C4×C8C2412yes—
412C24—42no—
413#41—42no—
414#41—84no—
415#41—84no—
416#41—164no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—42no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—42no—
423C2×Q8—41no—
424C2×Q8—84no—
425C2×Q8C2×D811yescharacteristic
426C2×D4—21no—
427C2×D4—24no—
428C2×D4—41no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—84no—
440C2×D4—162no—
441C2×D4C2×D811yescharacteristic
442C2×D4C2×D812yes—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4C2×D811yescharacteristic
448C22×C4C2×D812yes—
449C22×C4C2×D812yes—
450C22×C4C2×D811yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453Q16—22no—
454Q16—22no—
455Q16—22no—
456Q16—42no—
457SD16—42no—
458SD16—84no—
459SD16—84no—
460D8—22no—
461D8—24no—
462D8—24no—
463D8—44no—
464D8—82no—
465D8—82no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8C22×D412yes—
471C2×C8C22×D411yescharacteristic
472C2×C8C22×D412yes—
473C2×C8C22×D411yescharacteristic
474C2×C8C22×D411yescharacteristic
475C2×C8C2511yescharacteristic
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4C2×D811yescharacteristic
499C4⋊C4C2×D812yes—
500C4⋊C4C2×D811yescharacteristic
501C4⋊C4C2×D812yes—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—44no—
518C22⋊C4C2×D812yes—
519C22⋊C4C2×D812yes—
520C22⋊C4C2×D812yes—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42C2×D812yes—
528C42C2×D811yescharacteristic
529C42C2×D812yes—
530C42C2×D811yescharacteristic
531C42C22×D411yescharacteristic
532C42C22×D412yes—
533C42C22×D411yescharacteristic
534C16—22no—
535C16—24no—
536C16—22no—
537C23—41no—
538C23—44no—
539C23—84no—
540C23—162no—
541C23—162no—
542C23C22×D811yescharacteristic
543C23C22×D812yes—
544Q8—22no—
545Q8—42no—
546Q8—41no—
547Q8—41no—
548Q8—42no—
549Q8—84no—
550Q8—82no—
551Q8—82no—
552Q8—162no—
553Q8—161no—
554D4—22no—
555D4—42no—
556D4—44no—
557D4—41no—
558D4—44no—
559D4—41no—
560D4—84no—
561D4—84no—
562D4—84no—
563D4—162no—
564D4—162no—
565D4—162no—
566D4—161no—
567D4—164no—
568D4—164no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—84no—
585C2×C4—82no—
586C2×C4—164no—
587C2×C4—164no—
588C2×C4C22×D812yes—
589C2×C4C22×D811yescharacteristic
590C2×C4C22×D812yes—
591C2×C4C22×D811yescharacteristic
592C2×C4C22×D811yescharacteristic
593C2×C4C22×D812yes—
594C2×C4C22×D811yescharacteristic
595C2×C4C22×D811yescharacteristic
596C2×C4C23×D411yescharacteristic
597C8—21no—
598C8—22no—
599C8—21no—
600C8—22no—
601C8#57712yes—
602C8#58512yes—
603C22—22no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—21no—
609C22—84no—
610C22—81no—
611C22—81no—
612C22—84no—
613C22—84no—
614C22—81no—
615C22—324no—
616C22C23×D811yescharacteristic
617C4—22no—
618C4—21no—
619C4—21no—
620C4—22no—
621C4—21no—
622C4—22no—
623C4—21no—
624C4—82no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4#341412yes—
632C4#350011yescharacteristic
633C4C2×#57711yescharacteristic
634C2—42no—
635C2—41no—
636C2—161no—
637C2—164no—
638C2#6252911yescharacteristic
639C2C2×#341411yescharacteristic
640C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
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. #62476C2semidirect
2. #62417C2semidirect
3. #33942C2semidirect
4. #33940C2semidirect
5. #33886C2semidirect
6. #33878C2semidirect
7. #33872C2semidirect
8. #33871C2semidirect
9. #33803C2semidirect
10. #33801C2semidirect
11. #33785C2semidirect
12. #33783C2semidirect
13. #33746C2semidirect
14. #33738C2semidirect
15. #33704C2semidirect
16. #33701C2semidirect
17. #33484C2semidirect
22. Q8×Q16C22semidirect
24. Q8×D8C22semidirect
26. #3342C22semidirect
28. #3341C22semidirect
42. #3168C22semidirect
44. #3160C22semidirect
48. #2255C22semidirect
49. #2255C22semidirect
50. #2251C22semidirect
52. #2245C22semidirect
53. #2245C22semidirect
57. #2243C22semidirect
58. #2243C22semidirect
59. #2243C22semidirect
63. #2234C22semidirect
64. #2234C22semidirect
65. #2227C22semidirect
66. #2226C22semidirect
67. #2225C22semidirect
68. #2224C22semidirect
71. #2217C22semidirect
72. #2215C22semidirect
73. #2211C22semidirect
74. #2210C22semidirect
77. #2189C22semidirect
79. #2186C22semidirect
81. #2183C22semidirect
82. #2177C22semidirect
83. #2175C22semidirect
84. C4×Q32C22semidirect
85. C4×SD32C22semidirect
87. D4×C16C22semidirect
89. #2163C22semidirect
91. #2138C22semidirect
92. #2138C22semidirect
94. C2×#346C22semidirect

Maximal subgroups

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

subgroupindex
#624762
#624172
#339422
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