Finite Group Databasebeta

← #7623479

Subgroups — #7623479

Order 512.

1343 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60714C211yesmaximal, characteristic, complemented
2#60696C211yesmaximal, characteristic, complemented
3#33826C214yesmaximal, complemented
4#33823C214yesmaximal, complemented
5#33820C212yesmaximal, complemented
6#33817C212yesmaximal, complemented
7#33813C212yesmaximal, complemented
8#33812C212yesmaximal, complemented
9#33631C212yesmaximal, complemented
10#33628C211yesmaximal, characteristic, complemented
11#33626C211yesmaximal, characteristic, complemented
12#33612C212yesmaximal, complemented
13#33609C212yesmaximal, complemented
14#33586C212yesmaximal, complemented
15#33582C211yesmaximal, characteristic, complemented
16C2×#2207C211yesmaximal, characteristic, complemented
17#33398C211yesmaximal, characteristic, complemented
18#3508—21no—
19#3461—21no—
20C2×#509C2212yes—
21C22×SD32C2212yes—
22C2×#504C2212yes—
23#3184—22no—
24#3178—22no—
25#3177C2212yescomplemented
26#3176—22no—
27#3176C2214yes—
28#3175C2212yescomplemented
29#3172—22no—
30#3170C2212yescomplemented
31#3169C2212yes—
32#3169C2212yes—
33#3168C2212yescomplemented
34#3164C2211yescharacteristic
35#3160C2211yescharacteristic
36#3156C2211yescharacteristic
37C2×#493C2211yescharacteristic
38#3138—21no—
39#3137—21no—
40#3063—21no—
41#3061C2211yescharacteristic
42#3059C2211yescharacteristic
43#3058—21no—
44#3058C2212yes—
45#2948—21no—
46#2948C2212yes—
47C4×#137—21no—
48C4×#135C2211yescharacteristic
49C2×C4×Q16C2211yescharacteristic
50C4×#131C2211yescharacteristic
51#2229C2214yescomplemented
52#2229C2212yescomplemented
53#2228C2214yescomplemented
54#2228C2214yescomplemented
55#2228C2212yescomplemented
56#2227C2212yescomplemented
57#2227C2211yescharacteristic, complemented
58#2226C2214yescomplemented
59#2226C2211yescharacteristic, complemented
60#2225C2211yescharacteristic, complemented
61#2225C2212yescomplemented
62#2224C2211yescharacteristic, complemented
63#2224C2214yescomplemented
64#2220C2214yescomplemented
65#2220C2214yescomplemented
66#2220C2214yescomplemented
67#2218C2214yescomplemented
68#2217C2214yescomplemented
69#2217C2214yescomplemented
70#2217C2214yescomplemented
71#2215C2214yescomplemented
72#2210C2214yescomplemented
73#2210C2214yescomplemented
74#2208C2214yescomplemented
75#2208C2214yescomplemented
76#2207C2212yes—
77#2207C2212yes—
78#2207C2212yes—
79#2207C2214yes—
80#2207C2212yes—
81#2206C2212yes—
82#2206C2212yes—
83#2159C2212yes—
84#2158C2212yes—
85#2151C2212yes—
86#2151C2214yescomplemented
87#2150C2211yescharacteristic
88#2150C2212yescomplemented
89C2×C4⋊C16C2211yescharacteristic
90C2×C4⋊C16C2212yescomplemented
91#2141C2212yes—
92#2140C2212yes—
93#2138C2212yes—
94C2×#357C2212yes—
95C2×#137—21no—
96C2×#135C2311yescharacteristic
97C22×Q16C2311yescharacteristic
98C22×D8—21no—
99C22×D8C2312yes—
100C2×#131D411yescharacteristic
101C2×#131C2311yescharacteristic
102#559—41no—
103#559—41no—
104#550—41no—
105#550—41no—
106#546—42no—
107#546—42no—
108#544—21no—
109#542—22no—
110#542—41no—
111#541—22no—
112#541—41no—
113#540—24no—
114#539—42no—
115#537—21no—
116#536—21no—
117#536—21no—
118#535—22no—
119#533—21no—
120#533—22no—
121#533—41no—
122#532—22no—
123#532—22no—
124C2×#127—21no—
125#529—22no—
126#529—42no—
127#529—41no—
128#518—22no—
129C4×#41—21no—
130C4×#41D411yescharacteristic
131C2×C4×Q8—21no—
132#509—44no—
133C2×SD32—44no—
134C2×SD32C2314yes—
135C2×SD32C2314yes—
136C2×SD32C2314yes—
137C2×D16C2314yes—
138#504—44no—
139C2×M32C2312yes—
140C2×M32C2314yes—
141C22×C16C2312yes—
142C22×C16C2314yes—
143#501—22no—
144#500C2311yescharacteristic
145#500C2311yescharacteristic
146#500C2312yes—
147#499—21no—
148#499C2312yes—
149#499C2312yes—
150#498—21no—
151#497—22no—
152#495C2312yes—
153#494C2312yes—
154#493—21no—
155#493—21no—
156#493C2312yes—
157#493C2312yes—
158#491—22no—
159#490—24no—
160#489—22no—
161#488—22no—
162#487—22no—
163#486—22no—
164#486—22no—
165#470—22no—
166#470—22no—
167#469—24no—
168#469—24no—
169#468—22no—
170#468—22no—
171#467—22no—
172#467C2312yes—
173#467C2312yes—
174#467C2312yes—
175#466—22no—
176#466—22no—
177#466C2312yes—
178#466C2314yes—
179#466C2314yes—
180#465—22no—
181#442—24no—
182#442—24no—
183#441—22no—
184#440—22no—
185C4×Q16—22no—
186C4×Q16—21no—
187C4×Q16—21no—
188C4×Q16C2312yes—
189C4×Q16C2311yescharacteristic
190C4×Q16C2312yes—
191C4×Q16C2311yescharacteristic
192C4×SD16—22no—
193C4×SD16—21no—
194C4×SD16—21no—
195C4×D8—22no—
196C4×D8—22no—
197C4×D8C2311yescharacteristic
198C4×D8C2314yes—
199C4×D8C2314yes—
200C4×D8C2311yescharacteristic
201#428—21no—
202#427—21no—
203#427C2312yes—
204#427C2311yescharacteristic
205C2×#107C2311yescharacteristic
206#419—21no—
207#418—22no—
208#418—22no—
209#417—21no—
210#416—21no—
211C2×#103—21no—
212#405D411yescharacteristic
213#405D412yes—
214#405C2311yescharacteristic
215#405C2312yes—
216C4×M16D411yescharacteristic
217C4×M16D412yes—
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2312yes—
220#367C2312yes—
221#367C2312yes—
222#367C2314yes—
223#366C2312yes—
224#366C2312yes—
225#366C2314yes—
226C4⋊C16C2311yescharacteristic
227C4⋊C16C2311yescharacteristic
228C4⋊C16C2312yes—
229C4⋊C16C2312yes—
230C4⋊C16C2314yes—
231C4⋊C16C2312yes—
232C4⋊C16C2314yes—
233#358C2312yes—
234#358C2312yes—
235#358C2314yes—
236#358C2314yes—
237#357C2314yes—
238#357C2314yes—
239#357C2312yes—
240#357C2312yes—
241#357C2314yes—
242#357C2314yes—
243C2×#41—21no—
244C2×#41—42no—
245C2×#41—42no—
246C2×#41C2×D411yescharacteristic
247C22×Q8—21no—
248C22×D4—22no—
249C22×D4—41no—
250C22×D4—41no—
251#137—44no—
252#135—22no—
253#135—22no—
254C2×Q16—21no—
255C2×Q16—22no—
256C2×Q16—22no—
257C2×Q16—22no—
258C2×Q16C2412yes—
259C2×Q16C2411yescharacteristic
260C2×SD16—21no—
261C2×SD16—22no—
262C2×D8—22no—
263C2×D8—22no—
264C2×D8—22no—
265C2×D8—24no—
266C2×D8—24no—
267C2×D8—44no—
268C2×D8C2414yes—
269C2×D8C2411yescharacteristic
270#131—22no—
271#131—22no—
272#131—42no—
273C2×M16C2×D411yescharacteristic
274C2×M16C2×D411yescharacteristic
275C2×M16C2×D412yes—
276C2×M16C2×D412yes—
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D412yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—41no—
286C4⋊Q8—41no—
287#127—22no—
288#127—22no—
289#127—42no—
290#127—41no—
291#127—41no—
292#126—42no—
293#126—42no—
294#126—42no—
295#126—42no—
296#125—22no—
297#125—22no—
298#125—42no—
299#125—41no—
300#125—42no—
301#125—42no—
302#124—22no—
303#124—42no—
304#124—41no—
305#124—42no—
306#124—42no—
307#124—42no—
308#123—42no—
309#123—42no—
310#123—44no—
311#123—44no—
312#123—42no—
313#123—42no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—41no—
318C22⋊Q8—42no—
319C22⋊Q8—41no—
320C22⋊Q8—41no—
321C22⋊Q8—41no—
322C22⋊Q8—41no—
323C22⋊Q8—42no—
324C22⋊Q8—42no—
325#121—22no—
326#121—24no—
327#121—24no—
328#121—42no—
329#121—41no—
330#121—42no—
331#121—42no—
332#121—44no—
333#121—44no—
334#120—44no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—41no—
340C4×Q8—41no—
341C4×Q8—42no—
342C4×Q8—41no—
343C4×Q8—41no—
344C4×Q8D811yescharacteristic
345C4×Q8D811yescharacteristic
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—24no—
349C4×D4—24no—
350C4×D4—41no—
351C4×D4—41no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4—42no—
355C4×D4—42no—
356C4×D4—42no—
357C4×D4D812yes—
358C4×D4D811yescharacteristic
359C4×D4D812yes—
360C4×D4D811yescharacteristic
361#117—21no—
362#117—22no—
363#117—42no—
364#117—41no—
365#117—42no—
366#117C2×D411yescharacteristic
367#117C2×D412yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—41no—
370C2×C22⋊C4—44no—
371C2×C42—41no—
372C2×C42—41no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D412yes—
375SD32—44no—
376SD32—44no—
377SD32—44no—
378D16—44no—
379M32—24no—
380M32—44no—
381C2×C16—24no—
382C2×C16—44no—
383C2×C16C2412yes—
384C2×C16C2412yes—
385C2×C16C2414yes—
386#107—21no—
387#107—21no—
388#107—21no—
389#107—22no—
390#107C2412yes—
391#107C2411yescharacteristic
392#107C2411yescharacteristic
393#107C2412yes—
394#107C2412yes—
395#106—22no—
396#106—21no—
397#106—22no—
398#106—22no—
399#103—22no—
400#103—22no—
401#103—21no—
402#103—21no—
403#103—22no—
404#103—22no—
405#103—22no—
406#102—21no—
407#102—21no—
408#102—22no—
409#102—24no—
410#102—24no—
411#102—24no—
412#102—24no—
413#97C2×D412yes—
414#97C2×D412yes—
415#97C2×D412yes—
416C4×C8C2×D412yes—
417C4×C8C2×D411yescharacteristic
418C4×C8C2×D412yes—
419C4×C8C2×D411yescharacteristic
420C4×C8C2411yescharacteristic
421C4×C8C2412yes—
422C4×C8C2411yescharacteristic
423C24—42no—
424#41—22no—
425#41—22no—
426#41—42no—
427#41—42no—
428#41—164no—
429#41—164no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8C2×D811yescharacteristic
439C2×D4—21no—
440C2×D4—24no—
441C2×D4—41no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—162no—
453C2×D4—162no—
454C2×D4C2×D812yes—
455C2×D4C2×D811yescharacteristic
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4C2×D811yescharacteristic
464C22×C4C2×D812yes—
465C22×C4C2×D811yescharacteristic
466C22×C4C2×D812yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469Q16—22no—
470Q16—21no—
471Q16—22no—
472Q16—21no—
473Q16—44no—
474Q16—42no—
475SD16—44no—
476SD16—42no—
477D8—21no—
478D8—24no—
479D8—24no—
480D8—21no—
481D8—44no—
482D8—44no—
483D8—44no—
484M16—22no—
485M16—22no—
486M16—24no—
487M16—42no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—42no—
492C2×C8C22×D411yescharacteristic
493C2×C8C22×D412yes—
494C2×C8C22×D412yes—
495C2×C8C22×D411yescharacteristic
496C2×C8C22×D411yescharacteristic
497C2×C8H(5,𝔽2)12yes—
498C2×C82−1+412yes—
499C2×C8C2511yescharacteristic
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4C2×D811yescharacteristic
523C4⋊C4C2×D811yescharacteristic
524C4⋊C4C2×D812yes—
525C4⋊C4C2×D812yes—
526C22⋊C4—42no—
527C22⋊C4—41no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4C2×D812yes—
541C22⋊C4C2×D812yes—
542C22⋊C4C2×D812yes—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—41no—
547C42—42no—
548C42—42no—
549C42C2×D811yescharacteristic
550C42C2×D812yes—
551C42C2×D811yescharacteristic
552C42C2×D812yes—
553C42C22×D411yescharacteristic
554C42C22×D412yes—
555C42C22×D411yescharacteristic
556C16—22no—
557C16—22no—
558C16—24no—
559C23—41no—
560C23—44no—
561C23—84no—
562C23—162no—
563C23—162no—
564C23C22×D812yes—
565C23C22×D811yescharacteristic
566Q8—22no—
567Q8—41no—
568Q8—42no—
569Q8—42no—
570Q8—41no—
571Q8—82no—
572Q8—82no—
573Q8—82no—
574Q8—162no—
575D4—24no—
576D4—22no—
577D4—44no—
578D4—41no—
579D4—44no—
580D4—41no—
581D4—84no—
582D4—82no—
583D4—84no—
584D4—84no—
585D4—162no—
586D4—162no—
587D4—162no—
588D4—164no—
589D4—164no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—161no—
607C2×C4—164no—
608C2×C4—164no—
609C2×C4—161no—
610C2×C4#53712yes—
611C2×C4C22×D811yescharacteristic
612C2×C4C22×D811yescharacteristic
613C2×C4C22×D811yescharacteristic
614C2×C4C22×D812yes—
615C2×C4C22×D812yes—
616C2×C4C22×D812yes—
617C2×C4C22×D811yescharacteristic
618C2×C4C22×D811yescharacteristic
619C2×C4#57512yes—
620C2×C4C23×D411yescharacteristic
621C8—22no—
622C8—21no—
623C8—21no—
624C8—22no—
625C8#53712yes—
626C8C2×H(5,𝔽2)11yescharacteristic
627C8C2×2−1+411yescharacteristic
628C22—21no—
629C22—21no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—81no—
634C22—84no—
635C22—84no—
636C22—81no—
637C22—81no—
638C22—84no—
639C22—324no—
640C22#324512yes—
641C22C23×D811yescharacteristic
642C4—22no—
643C4—21no—
644C4—21no—
645C4—22no—
646C4—21no—
647C4—22no—
648C4—81no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4—82no—
655C4#324512yes—
656C4#341612yes—
657C4C2×#53711yescharacteristic
658C4C2×#57511yescharacteristic
659C2—22no—
660C2—42no—
661C2—161no—
662C2—164no—
663C2C2×#324511yescharacteristic
664C2C2×#341611yescharacteristic
665C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
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
1. #60714C2semidirect
2. #60696C2semidirect
3. #33826C2semidirect
4. #33823C2semidirect
5. #33820C2semidirect
6. #33817C2semidirect
7. #33813C2semidirect
8. #33812C2semidirect
9. #33631C2semidirect
10. #33628C2semidirect
11. #33626C2semidirect
12. #33612C2semidirect
13. #33609C2semidirect
14. #33586C2semidirect
15. #33582C2semidirect
16. C2×#2207C2semidirect
17. #33398C2semidirect
25. #3177C22semidirect
28. #3175C22semidirect
30. #3170C22semidirect
33. #3168C22semidirect
51. #2229C22semidirect
52. #2229C22semidirect
53. #2228C22semidirect
54. #2228C22semidirect
55. #2228C22semidirect
56. #2227C22semidirect
57. #2227C22semidirect
58. #2226C22semidirect
59. #2226C22semidirect
60. #2225C22semidirect
61. #2225C22semidirect
62. #2224C22semidirect
63. #2224C22semidirect
64. #2220C22semidirect
65. #2220C22semidirect
66. #2220C22semidirect
67. #2218C22semidirect
68. #2217C22semidirect
69. #2217C22semidirect
70. #2217C22semidirect
71. #2215C22semidirect
72. #2210C22semidirect
73. #2210C22semidirect
74. #2208C22semidirect
75. #2208C22semidirect
86. #2151C22semidirect
88. #2150C22semidirect
90. C2×C4⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#607142
#606962
#338262
#338232
#338202
#338172
#338132
#338122
#336312
#336282
#336262
#336122
#336092
#335862
#335822
C2×#22072
#333982