Finite Group Databasebeta

← #7624512

Subgroups — #7624512

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62417C211yesmaximal, characteristic, complemented
2#61430C211yesmaximal, characteristic, complemented
3#33963C212yesmaximal, complemented
4#33932C212yesmaximal, complemented
5#33871C212yesmaximal, complemented
6#33864C214yesmaximal, complemented
7#33853C212yesmaximal, complemented
8#33848C214yesmaximal, complemented
9#33813C212yesmaximal, complemented
10#33812C211yesmaximal, characteristic, complemented
11#33778C212yesmaximal, complemented
12#33771C211yesmaximal, characteristic, complemented
13#33666C212yesmaximal, complemented
14#33664C211yesmaximal, characteristic, complemented
15#33579C212yesmaximal, complemented
16C2×#2207C211yesmaximal, characteristic, complemented
17#33502C211yesmaximal, characteristic, complemented
18#3508—21no—
19#3480—21no—
20C2×#508C2214yescomplemented
21C22×SD32C2212yes—
22#3396—21no—
23#3396C2212yes—
24#3396C2211yescharacteristic, complemented
25#3383C2211yescharacteristic, complemented
26#3365—21no—
27#3342C2211yescharacteristic, complemented
28#3339C2212yes—
29#3304—21no—
30#3301—22no—
31#3287—21no—
32#3287C2212yes—
33#3287C2211yescharacteristic, complemented
34#3285C2212yes—
35#3280—22no—
36#3280C2214yes—
37#3208—21no—
38#3207—22no—
39#3170C2211yescharacteristic
40#3169C2211yescharacteristic
41#3169C2212yes—
42#3168C2212yescomplemented
43#3147C2212yes—
44C2×#493C2211yescharacteristic
45#3103—21no—
46#3102—22no—
47C8×#41C2211yescharacteristic
48C4×#135C2212yescomplemented
49C2×C4×Q16C2211yescharacteristic
50#2249C2214yescomplemented
51#2246C2212yes—
52#2242C2214yescomplemented
53#2240C2212yes—
54#2236C2212yes—
55#2234C2214yescomplemented
56#2228C2214yescomplemented
57#2227C2211yescharacteristic, complemented
58#2227C2211yescharacteristic, complemented
59#2226C2212yescomplemented
60#2225C2212yescomplemented
61#2217C2214yes—
62#2217C2214yescomplemented
63#2215C2214yescomplemented
64#2211C2214yescomplemented
65#2210C2214yescomplemented
66#2210C2214yescomplemented
67#2209C2211yescharacteristic, complemented
68#2209C2211yescharacteristic, complemented
69#2208C2212yescomplemented
70#2207C2212yes—
71#2207C2212yes—
72#2207C2212yes—
73#2207C2212yes—
74#2207C2212yescomplemented
75#2206C2212yes—
76#2206C2212yes—
77#2189C2214yescomplemented
78#2188C2212yes—
79#2186C2214yescomplemented
80#2186C2214yescomplemented
81#2184C2214yes—
82Q8×C16C2212yescomplemented
83C4×Q32C2212yescomplemented
84C4×Q32C2212yescomplemented
85C4×SD32C2214yescomplemented
86C4×D16C2214yescomplemented
87D4×C16C2214yes—
88D4×C16C2212yescomplemented
89#2162C2211yescharacteristic
90#2162C2212yescomplemented
91C2×C4⋊C16C2211yescharacteristic
92C2×C4⋊C16C2212yescomplemented
93#2138C2214yescomplemented
94C2×#357C2212yes—
95C2×#135—22no—
96C2×#135C2312yes—
97C2×#135C2314yes—
98C22×Q16C2311yescharacteristic
99C22×D8—21no—
100C22×D8C2312yes—
101#559—21no—
102#559—22no—
103#559—41no—
104#559—41no—
105#557—21no—
106#556—21no—
107#556—41no—
108#553—22no—
109#553—42no—
110#550—21no—
111#550—22no—
112#550—41no—
113#548—22no—
114#546—24no—
115#546—42no—
116#540—42no—
117#535—41no—
118#533—21no—
119#532—21no—
120#532—22no—
121#532—22no—
122#532—41no—
123C2×#127—21no—
124#529—22no—
125#529—42no—
126#518—41no—
127C4×#41—22no—
128C4×#41—41no—
129C4×#41D411yescharacteristic
130C2×C4×Q8—21no—
131#508—48no—
132C2×Q32C2312yes—
133C2×Q32C2312yes—
134C2×SD32—44no—
135C2×SD32C2314yes—
136C2×SD32C2314yes—
137C2×SD32C2314yes—
138C2×D16C2314yes—
139C2×D16C2314yes—
140C22×C16C2312yes—
141C22×C16C2314yes—
142#500C2311yescharacteristic
143#500C2311yescharacteristic
144#499C2312yes—
145#495—22no—
146#495C2314yes—
147#495C2312yes—
148#494C2311yescharacteristic
149#493—21no—
150#493—21no—
151#493C2312yes—
152#493C2311yescharacteristic
153#493C2312yes—
154#492—22no—
155#488—22no—
156#487—21no—
157#486—21no—
158#486—22no—
159#485—22no—
160#482—22no—
161#481—22no—
162#479—22no—
163#478—22no—
164#475—21no—
165#475—21no—
166#467—21no—
167#467C2312yes—
168#467C2312yes—
169#466—22no—
170#466—21no—
171#466C2312yes—
172#466C2312yes—
173#466C2314yes—
174#465—22no—
175#464—22no—
176#463—22no—
177#463—24no—
178#463—24no—
179#460—21no—
180#460—21no—
181#460—22no—
182#460—22no—
183#452—22no—
184#450—24no—
185#449—22no—
186#449—24no—
187#449—24no—
188#448—22no—
189C8×Q8D411yescharacteristic
190C8×Q8C2311yescharacteristic
191C4×Q16—21no—
192C4×Q16—21no—
193C4×Q16—21no—
194C4×Q16C2311yescharacteristic
195C4×Q16C2311yescharacteristic
196C4×Q16C2312yes—
197C4×Q16C2312yes—
198C4×Q16C2311yescharacteristic
199C4×Q16C2311yescharacteristic
200C4×SD16—22no—
201C4×SD16—22no—
202C4×SD16—22no—
203C4×D8—21no—
204C4×D8—22no—
205C4×D8C2312yes—
206C4×D8C2314yes—
207C4×D8C2312yes—
208C8×D4D412yes—
209C8×D4D411yescharacteristic
210C8×D4C2312yes—
211C8×D4C2311yescharacteristic
212#431D412yes—
213#431D411yescharacteristic
214#427C2312yes—
215C2×#107C2311yescharacteristic
216#416—22no—
217C2×#103—21no—
218C2×C4×C8C2311yescharacteristic
219C2×C4×C8C2312yes—
220#367C2314yes—
221#366C2312yes—
222#366C2312yes—
223C4⋊C16C2312yes—
224C4⋊C16C2312yes—
225C4⋊C16C2311yescharacteristic
226C4⋊C16C2314yes—
227C4⋊C16C2311yescharacteristic
228#358C2312yes—
229#358C2314yes—
230#358C2312yes—
231#358C2312yes—
232#358C2312yes—
233#357C2314yes—
234#357C2314yes—
235#357C2314yes—
236#357C2314yes—
237#357C2314yes—
238C22⋊C16C2312yes—
239C22⋊C16C2314yes—
240C4×C16C2312yes—
241C4×C16C2314yes—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—41no—
245C2×#41—42no—
246C2×#41D811yescharacteristic
247C22×Q8—21no—
248C22×D4—22no—
249C22×D4—41no—
250C22×D4—41no—
251#135—24no—
252#135—42no—
253#135—44no—
254#135—44no—
255#135—84no—
256C2×Q16—22no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—41no—
260C2×Q16C2411yescharacteristic
261C2×Q16C2412yes—
262C2×Q16C2411yescharacteristic
263C2×SD16—22no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—22no—
267C2×D8—22no—
268C2×D8—22no—
269C2×D8—22no—
270C2×D8—21no—
271C2×D8—42no—
272C2×D8—42no—
273C2×D8—82no—
274C2×D8C2414yes—
275C2×D8C2412yes—
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2412yes—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—41no—
284#127—22no—
285#127—22no—
286#127—21no—
287#127—41no—
288#127—42no—
289#127—41no—
290#127—41no—
291#126—42no—
292#126—42no—
293#126—42no—
294#125—22no—
295#125—42no—
296#125—41no—
297#124—22no—
298#124—24no—
299#124—41no—
300#124—42no—
301#124—41no—
302#124—42no—
303#124—42no—
304#124—41no—
305#123—42no—
306#123—42no—
307#123—44no—
308#123—44no—
309#123—42no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—41no—
313C22⋊Q8—42no—
314#121—22no—
315#121—24no—
316#121—22no—
317#121—42no—
318#121—42no—
319#121—42no—
320#121—42no—
321#121—42no—
322#121—41no—
323#121—42no—
324#121—44no—
325#121—44no—
326#120—44no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—41no—
334C4×Q8—41no—
335C4×Q8—41no—
336C4×Q8—41no—
337C4×Q8—41no—
338C4×Q8D811yescharacteristic
339C4×Q8C2×D411yescharacteristic
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—24no—
343C4×D4—42no—
344C4×D4—42no—
345C4×D4—42no—
346C4×D4—41no—
347C4×D4—41no—
348C4×D4—42no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—42no—
352C4×D4—42no—
353C4×D4—42no—
354C4×D4—41no—
355C4×D4—42no—
356C4×D4D812yes—
357C4×D4D811yescharacteristic
358C4×D4C2×D412yes—
359C4×D4C2×D411yescharacteristic
360#117—22no—
361#117—41no—
362#117—42no—
363#117—42no—
364#117D811yescharacteristic
365#117D812yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—41no—
368C2×C22⋊C4—44no—
369C2×C42—41no—
370C2×C42—42no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373Q32—24no—
374Q32—42no—
375SD32—28no—
376SD32—44no—
377SD32—44no—
378D16—44no—
379D16—44no—
380C2×C16—42no—
381C2×C16—44no—
382C2×C16C2412yes—
383C2×C16C2414yes—
384C2×C16C2412yes—
385#107—21no—
386#107—21no—
387#107—21no—
388#107C2411yescharacteristic
389#107C2411yescharacteristic
390#107C2412yes—
391#107C2412yes—
392#106—22no—
393#106—22no—
394C4⋊C8C2×D411yescharacteristic
395C4⋊C8C2×D412yes—
396C4⋊C8C2×D411yescharacteristic
397C4⋊C8C2×D412yes—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—22no—
403#103—22no—
404#103—22no—
405#103—22no—
406#102—22no—
407#102—22no—
408#102—24no—
409#102—22no—
410#102—24no—
411#102—22no—
412C22⋊C8C2×D412yes—
413C22⋊C8C2×D412yes—
414C22⋊C8C2×D412yes—
415C4×C8C2×D411yescharacteristic
416C4×C8C2×D412yes—
417C4×C8C2×D412yes—
418C4×C8C2×D411yescharacteristic
419C4×C8C2412yes—
420C4×C8C2411yescharacteristic
421C4×C8C2411yescharacteristic
422C24—42no—
423#41—24no—
424#41—44no—
425#41—82no—
426#41—84no—
427#41—84no—
428#41—84no—
429#41—164no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—42no—
434C2×Q8—41no—
435C2×Q8—41no—
436C2×Q8—41no—
437C2×Q8—81no—
438C2×Q8C2×D811yescharacteristic
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—41no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4—84no—
456C2×D4—162no—
457C2×D4C2×D812yes—
458C2×D4C2×D811yescharacteristic
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4C2×D812yes—
467C22×C4C2×D811yescharacteristic
468C22×C4C2×D812yes—
469C22×C4C2×D811yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472Q16—21no—
473Q16—22no—
474Q16—21no—
475Q16—21no—
476Q16—22no—
477Q16—42no—
478Q16—42no—
479Q16H(5,𝔽2)12yes—
480SD16—44no—
481SD16—82no—
482SD16—84no—
483SD16—84no—
484D8—22no—
485D8—24no—
486D8—22no—
487D8—24no—
488D8—44no—
489D8—81no—
490D8—82no—
491D8—82no—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—22no—
496C2×C8—41no—
497C2×C8—42no—
498C2×C8C22×D411yescharacteristic
499C2×C8C22×D412yes—
500C2×C8C22×D411yescharacteristic
501C2×C8C22×D412yes—
502C2×C8C22×D411yescharacteristic
503C2×C8C2511yescharacteristic
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—41no—
519C4⋊C4—42no—
520C4⋊C4—41no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4C2×D811yescharacteristic
524C4⋊C4C2×D811yescharacteristic
525C4⋊C4C2×D812yes—
526C4⋊C4C2×D812yes—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4C2×D812yes—
541C22⋊C4C2×D812yes—
542C22⋊C4C2×D812yes—
543C42—41no—
544C42—42no—
545C42—41no—
546C42—42no—
547C42—41no—
548C42—41no—
549C42—42no—
550C42C2×D812yes—
551C42C2×D812yes—
552C42C2×D811yescharacteristic
553C42C2×D811yescharacteristic
554C42C22×D411yescharacteristic
555C42C22×D412yes—
556C42C22×D411yescharacteristic
557C16—22no—
558C16—24no—
559C16—22no—
560C23—42no—
561C23—44no—
562C23—84no—
563C23—162no—
564C23—162no—
565C23C22×D812yes—
566C23C22×D811yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—42no—
571Q8—41no—
572Q8—42no—
573Q8—41no—
574Q8—82no—
575Q8—82no—
576Q8—82no—
577D4—24no—
578D4—22no—
579D4—44no—
580D4—42no—
581D4—44no—
582D4—42no—
583D4—84no—
584D4—84no—
585D4—84no—
586D4—84no—
587D4—84no—
588D4—82no—
589D4—161no—
590D4—162no—
591D4—162no—
592D4—162no—
593D4—164no—
594D4—164no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—82no—
611C2×C4—84no—
612C2×C4—162no—
613C2×C4—164no—
614C2×C4—164no—
615C2×C4—161no—
616C2×C4#50814yescomplemented
617C2×C4C22×D812yes—
618C2×C4C22×D812yes—
619C2×C4C22×D811yescharacteristic
620C2×C4C22×D811yescharacteristic
621C2×C4C22×D812yes—
622C2×C4C22×D811yescharacteristic
623C2×C4C22×D811yescharacteristic
624C2×C4C22×D811yescharacteristic
625C2×C4C23×D411yescharacteristic
626C8—21no—
627C8—21no—
628C8—22no—
629C8—21no—
630C8—22no—
631C8C2×H(5,𝔽2)11yescharacteristic
632C8#58511yescharacteristic
633C22—21no—
634C22—21no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—82no—
639C22—84no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—84no—
644C22—324no—
645C22C2×#50812yes—
646C22C23×D811yescharacteristic
647C4—22no—
648C4—21no—
649C4—22no—
650C4—21no—
651C4—22no—
652C4—21no—
653C4—42no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4C2×#50812yes—
661C4C2×#50812yes—
662C4#348811yescharacteristic
663C4#358511yescharacteristic
664C2—22no—
665C2—42no—
666C2—162no—
667C2—164no—
668C2#6194711yescharacteristic
669C2C22×#50811yescharacteristic
670C2#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. #62417C2semidirect
2. #61430C2semidirect
3. #33963C2semidirect
4. #33932C2semidirect
5. #33871C2semidirect
6. #33864C2semidirect
7. #33853C2semidirect
8. #33848C2semidirect
9. #33813C2semidirect
10. #33812C2semidirect
11. #33778C2semidirect
12. #33771C2semidirect
13. #33666C2semidirect
14. #33664C2semidirect
15. #33579C2semidirect
16. C2×#2207C2semidirect
17. #33502C2semidirect
20. C2×#508C22semidirect
24. #3396C22semidirect
25. #3383C22semidirect
27. #3342C22semidirect
33. #3287C22semidirect
42. #3168C22semidirect
48. C4×#135C22semidirect
50. #2249C22semidirect
52. #2242C22semidirect
55. #2234C22semidirect
56. #2228C22semidirect
57. #2227C22semidirect
58. #2227C22semidirect
59. #2226C22semidirect
60. #2225C22semidirect
62. #2217C22semidirect
63. #2215C22semidirect
64. #2211C22semidirect
65. #2210C22semidirect
66. #2210C22semidirect
67. #2209C22semidirect
68. #2209C22semidirect
69. #2208C22semidirect
74. #2207C22semidirect
77. #2189C22semidirect
79. #2186C22semidirect
80. #2186C22semidirect
82. Q8×C16C22semidirect
83. C4×Q32C22semidirect
84. C4×Q32C22semidirect
85. C4×SD32C22semidirect
86. C4×D16C22semidirect
88. D4×C16C22semidirect
90. #2162C22semidirect
92. C2×C4⋊C16C22semidirect
93. #2138C22semidirect
616. C2×C4#508semidirect

Maximal subgroups

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

subgroupindex
#624172
#614302
#339632
#339322
#338712
#338642
#338532
#338482
#338132
#338122
#337782
#337712
#336662
#336642
#335792
C2×#22072
#335022