Finite Group Databasebeta

← #7623477

Subgroups — #7623477

Order 512.

1375 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60714C211yesmaximal, characteristic, complemented
2#60695C211yesmaximal, characteristic, complemented
3#33825C214yesmaximal, complemented
4#33823C214yesmaximal, complemented
5#33819C212yesmaximal, complemented
6#33817C212yesmaximal, complemented
7#33813C212yesmaximal, complemented
8#33811C212yesmaximal, complemented
9#33631C212yesmaximal, complemented
10#33627C211yesmaximal, characteristic, complemented
11#33626C211yesmaximal, characteristic, complemented
12#33612C212yesmaximal, complemented
13#33607C212yesmaximal, complemented
14#33585C212yesmaximal, complemented
15#33582C211yesmaximal, characteristic, complemented
16C2×#2206C211yesmaximal, characteristic, complemented
17#33398C211yesmaximal, characteristic, complemented
18#3508—21no—
19#3460—21no—
20C2×#509C2212yes—
21C22×D16C2212yes—
22C2×#504C2212yes—
23#3184—22no—
24#3179—22no—
25#3176—22no—
26#3176C2212yescomplemented
27#3176C2214yescomplemented
28#3175C2212yescomplemented
29#3171—22no—
30#3169C2212yescomplemented
31#3169C2212yescomplemented
32#3169C2212yescomplemented
33#3168C2212yescomplemented
34#3164C2211yescharacteristic
35#3160C2211yescharacteristic
36#3155C2211yescharacteristic
37C2×#493C2211yescharacteristic
38#3138—21no—
39#3136—21no—
40#3062—21no—
41#3061C2211yescharacteristic
42#3058—21no—
43#3058C2211yescharacteristic
44#3058C2212yes—
45#2948—21no—
46#2948C2212yes—
47C4×AGL(1,ℤ/8)—21no—
48C4×#135C2211yescharacteristic
49C2×C4×D8C2211yescharacteristic
50C4×#131C2211yescharacteristic
51#2229C2212yescomplemented
52#2228C2214yescomplemented
53#2228C2214yescomplemented
54#2228C2214yescomplemented
55#2228C2212yescomplemented
56#2227C2211yescharacteristic, complemented
57#2226C2212yescomplemented
58#2226C2211yescharacteristic, complemented
59#2226C2214yescomplemented
60#2225C2211yescharacteristic, complemented
61#2224C2211yescharacteristic, complemented
62#2224C2212yescomplemented
63#2224C2214yescomplemented
64#2220C2214yescomplemented
65#2218C2214yescomplemented
66#2218C2214yescomplemented
67#2218C2214yescomplemented
68#2217C2214yescomplemented
69#2215C2214yescomplemented
70#2215C2214yescomplemented
71#2215C2214yescomplemented
72#2210C2214yescomplemented
73#2210C2214yescomplemented
74#2207C2212yescomplemented
75#2207C2212yescomplemented
76#2206C2212yes—
77#2206C2212yes—
78#2206C2214yes—
79#2206C2212yescomplemented
80#2206C2214yescomplemented
81#2206C2214yescomplemented
82#2206C2212yescomplemented
83#2159C2212yes—
84#2158C2212yes—
85#2151C2212yes—
86#2151C2214yescomplemented
87#2150C2211yescharacteristic
88#2150C2212yescomplemented
89C2×C4⋊C16C2211yescharacteristic
90C2×C4⋊C16C2212yescomplemented
91#2140C2212yes—
92#2140C2212yes—
93#2138C2212yes—
94C2×#357C2212yes—
95C2×AGL(1,ℤ/8)—21no—
96C2×#135C2311yescharacteristic
97C22×D8—21no—
98C22×D8C2312yes—
99C22×D8C2311yescharacteristic
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#540—24no—
112#540—22no—
113#540—41no—
114#538—42no—
115#537—21no—
116#535—21no—
117#535—22no—
118#535—21no—
119#533—21no—
120#532—22no—
121#532—22no—
122#532—22no—
123#532—41no—
124C2×#127—21no—
125#529—22no—
126#529—41no—
127#525—42no—
128#518—22no—
129C4×#41—21no—
130C4×#41D411yescharacteristic
131C2×C4×D4—21no—
132#509—44no—
133C2×SD32C2314yes—
134C2×D16—44no—
135C2×D16C2314yes—
136C2×D16C2314yes—
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#496—22no—
152#495C2312yes—
153#493—21no—
154#493—21no—
155#493C2312yes—
156#493C2312yes—
157#493C2312yes—
158#490—22no—
159#490—24no—
160#489—22no—
161#488—22no—
162#486—22no—
163#486—22no—
164#486—22no—
165#469—22no—
166#469—22no—
167#469—24no—
168#469—24no—
169#468—22no—
170#468—22no—
171#467C2312yes—
172#466—22no—
173#466—22no—
174#466—22no—
175#466C2312yes—
176#466C2312yes—
177#466C2312yes—
178#466C2314yes—
179#466C2314yes—
180#465—22no—
181#442—24no—
182#442—22no—
183#442—24no—
184#440—22no—
185C4×Q16C2311yescharacteristic
186C4×Q16C2311yescharacteristic
187C4×SD16—22no—
188C4×SD16—21no—
189C4×SD16—21no—
190C4×D8—22no—
191C4×D8—22no—
192C4×D8—22no—
193C4×D8—21no—
194C4×D8—21no—
195C4×D8C2312yes—
196C4×D8C2311yescharacteristic
197C4×D8C2314yes—
198C4×D8C2314yes—
199C4×D8C2312yes—
200C4×D8C2311yescharacteristic
201#428—21no—
202#427—21no—
203#427C2312yes—
204#427C2311yescharacteristic
205C2×#107C2311yescharacteristic
206#418—21no—
207#418—22no—
208#418—22no—
209#417—21no—
210#416—21no—
211C2×#102—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#357C2314yes—
236#357C2314yes—
237#357C2312yes—
238#357C2312yes—
239#357C2314yes—
240#357C2314yes—
241#357C2314yes—
242#357C2314yes—
243C2×#41—21no—
244C2×#41—42no—
245C2×#41—42no—
246C2×#41—42no—
247C2×#41C2×D411yescharacteristic
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—41no—
251C22×D4—41no—
252C23×C4—41no—
253AGL(1,ℤ/8)—44no—
254#135—22no—
255#135—22no—
256C2×Q16C2411yescharacteristic
257C2×SD16—21no—
258C2×SD16—22no—
259C2×D8—22no—
260C2×D8—22no—
261C2×D8—21no—
262C2×D8—22no—
263C2×D8—22no—
264C2×D8—22no—
265C2×D8—24no—
266C2×D8—24no—
267C2×D8—22no—
268C2×D8—44no—
269C2×D8C2414yes—
270C2×D8C2412yes—
271C2×D8C2411yescharacteristic
272#131—22no—
273#131—22no—
274#131—42no—
275C2×M16C2×D411yescharacteristic
276C2×M16C2×D411yescharacteristic
277C2×M16C2×D412yes—
278C2×M16C2×D412yes—
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D412yes—
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—22no—
286C4⋊Q8—41no—
287#127—22no—
288#127—22no—
289#127—22no—
290#127—41no—
291#127—42no—
292#127—41no—
293#127—41no—
294#126—42no—
295#126—42no—
296#125—22no—
297#125—22no—
298#125—41no—
299#124—22no—
300#124—41no—
301#124—42no—
302#124—42no—
303#124—42no—
304#124—42no—
305#124—42no—
306#123—42no—
307#123—42no—
308#123—42no—
309#123—44no—
310#123—44no—
311#123—42no—
312C22⋊Q8—22no—
313C22⋊Q8—41no—
314C22⋊Q8—41no—
315C22⋊Q8—41no—
316C22⋊Q8—42no—
317C22⋊Q8—42no—
318#121—22no—
319#121—24no—
320#121—24no—
321#121—22no—
322#121—22no—
323#121—42no—
324#121—41no—
325#121—42no—
326#121—42no—
327#121—42no—
328#121—41no—
329#121—41no—
330#121—42no—
331#121—42no—
332#121—44no—
333#121—44no—
334#120—42no—
335#120—44no—
336#120—42no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—41no—
340C4×Q8—41no—
341C4×Q8D811yescharacteristic
342C4×Q8D811yescharacteristic
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—24no—
348C4×D4—24no—
349C4×D4—41no—
350C4×D4—41no—
351C4×D4—42no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4—41no—
355C4×D4—41no—
356C4×D4—42no—
357C4×D4—42no—
358C4×D4—42no—
359C4×D4D812yes—
360C4×D4D811yescharacteristic
361C4×D4D812yes—
362C4×D4D811yescharacteristic
363#117—21no—
364#117—22no—
365#117—41no—
366#117—42no—
367#117C2×D411yescharacteristic
368#117C2×D412yes—
369C2×C4⋊C4—21no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—44no—
372C2×C42—41no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D412yes—
375SD32—44no—
376D16—44no—
377D16—44no—
378D16—44no—
379M32—24no—
380M32—44no—
381C2×C16—24no—
382C2×C16—44no—
383C2×C16C2414yes—
384C2×C16C2412yes—
385C2×C16C2412yes—
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—21no—
400#103—21no—
401#103—22no—
402#102—22no—
403#102—22no—
404#102—21no—
405#102—21no—
406#102—22no—
407#102—22no—
408#102—24no—
409#102—24no—
410#102—24no—
411#102—24no—
412#102—22no—
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—
424C24—41no—
425#41—22no—
426#41—22no—
427#41—42no—
428#41—42no—
429#41—164no—
430#41—164no—
431#41—164no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—42no—
436C2×Q8C2×D811yescharacteristic
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—41no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—162no—
459C2×D4—162no—
460C2×D4C2×D812yes—
461C2×D4C2×D811yescharacteristic
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—162no—
471C22×C4C2×D812yes—
472C22×C4C2×D811yescharacteristic
473C22×C4C2×D811yescharacteristic
474C22×C4C2×D812yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477Q16—21no—
478Q16—21no—
479SD16—44no—
480SD16—42no—
481D8—22no—
482D8—21no—
483D8—24no—
484D8—24no—
485D8—22no—
486D8—21no—
487D8—44no—
488D8—44no—
489D8—44no—
490D8—42no—
491D8—44no—
492M16—22no—
493M16—22no—
494M16—24no—
495M16—42no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—42no—
500C2×C8C22×D411yescharacteristic
501C2×C8C22×D412yes—
502C2×C8C22×D412yes—
503C2×C8C22×D411yescharacteristic
504C2×C8C22×D411yescharacteristic
505C2×C8H(5,𝔽2)12yes—
506C2×C82−1+412yes—
507C2×C8C2511yescharacteristic
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4C2×D811yescharacteristic
525C4⋊C4C2×D811yescharacteristic
526C4⋊C4C2×D812yes—
527C4⋊C4C2×D812yes—
528C22⋊C4—42no—
529C22⋊C4—41no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4C2×D812yes—
545C22⋊C4C2×D812yes—
546C22⋊C4C2×D812yes—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—42no—
551C42C2×D811yescharacteristic
552C42C2×D812yes—
553C42C2×D811yescharacteristic
554C42C2×D812yes—
555C42C22×D411yescharacteristic
556C42C22×D412yes—
557C42C22×D411yescharacteristic
558C16—22no—
559C16—22no—
560C16—24no—
561C23—42no—
562C23—41no—
563C23—44no—
564C23—82no—
565C23—84no—
566C23—161no—
567C23—161no—
568C23—162no—
569C23—162no—
570C23C22×D812yes—
571C23C22×D811yescharacteristic
572Q8—22no—
573Q8—41no—
574Q8—41no—
575Q8—82no—
576Q8—82no—
577Q8—162no—
578D4—24no—
579D4—22no—
580D4—42no—
581D4—44no—
582D4—41no—
583D4—44no—
584D4—41no—
585D4—42no—
586D4—84no—
587D4—82no—
588D4—84no—
589D4—82no—
590D4—84no—
591D4—162no—
592D4—162no—
593D4—162no—
594D4—162no—
595D4—162no—
596D4—162no—
597D4—162no—
598D4—164no—
599D4—164no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—161no—
616C2×C4—162no—
617C2×C4—164no—
618C2×C4—164no—
619C2×C4—161no—
620C2×C4#53712yes—
621C2×C4C22×D811yescharacteristic
622C2×C4C22×D811yescharacteristic
623C2×C4C22×D812yes—
624C2×C4C22×D812yes—
625C2×C4C22×D812yes—
626C2×C4C22×D811yescharacteristic
627C2×C4C22×D811yescharacteristic
628C2×C4C22×D811yescharacteristic
629C2×C4#57512yes—
630C2×C4C23×D411yescharacteristic
631C8—22no—
632C8—21no—
633C8—21no—
634C8—22no—
635C8#53712yes—
636C8C2×H(5,𝔽2)11yescharacteristic
637C8C2×2−1+411yescharacteristic
638C22—21no—
639C22—21no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—82no—
644C22—81no—
645C22—84no—
646C22—84no—
647C22—81no—
648C22—82no—
649C22—81no—
650C22—82no—
651C22—84no—
652C22—164no—
653C22—324no—
654C22#324512yes—
655C22C23×D811yescharacteristic
656C4—21no—
657C4—22no—
658C4—21no—
659C4—22no—
660C4—21no—
661C4—22no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—81no—
667C4—81no—
668C4#324512yes—
669C4#341512yes—
670C4C2×#53711yescharacteristic
671C4C2×#57511yescharacteristic
672C2—22no—
673C2—42no—
674C2—161no—
675C2—162no—
676C2—164no—
677C2C2×#324511yescharacteristic
678C2C2×#341511yescharacteristic
679C2#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. #60695C2semidirect
3. #33825C2semidirect
4. #33823C2semidirect
5. #33819C2semidirect
6. #33817C2semidirect
7. #33813C2semidirect
8. #33811C2semidirect
9. #33631C2semidirect
10. #33627C2semidirect
11. #33626C2semidirect
12. #33612C2semidirect
13. #33607C2semidirect
14. #33585C2semidirect
15. #33582C2semidirect
16. C2×#2206C2semidirect
17. #33398C2semidirect
26. #3176C22semidirect
27. #3176C22semidirect
28. #3175C22semidirect
30. #3169C22semidirect
31. #3169C22semidirect
32. #3169C22semidirect
33. #3168C22semidirect
51. #2229C22semidirect
52. #2228C22semidirect
53. #2228C22semidirect
54. #2228C22semidirect
55. #2228C22semidirect
56. #2227C22semidirect
57. #2226C22semidirect
58. #2226C22semidirect
59. #2226C22semidirect
60. #2225C22semidirect
61. #2224C22semidirect
62. #2224C22semidirect
63. #2224C22semidirect
64. #2220C22semidirect
65. #2218C22semidirect
66. #2218C22semidirect
67. #2218C22semidirect
68. #2217C22semidirect
69. #2215C22semidirect
70. #2215C22semidirect
71. #2215C22semidirect
72. #2210C22semidirect
73. #2210C22semidirect
74. #2207C22semidirect
75. #2207C22semidirect
79. #2206C22semidirect
80. #2206C22semidirect
81. #2206C22semidirect
82. #2206C22semidirect
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
#606952
#338252
#338232
#338192
#338172
#338132
#338112
#336312
#336272
#336262
#336122
#336072
#335852
#335822
C2×#22062
#333982