Finite Group Databasebeta

← #6529912

Subgroups — #6529912

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60868C211yesmaximal, characteristic, complemented
2C2×#3122C211yesmaximal, characteristic
3#60490C211yesmaximal, characteristic, complemented
4#60252C211yesmaximal, characteristic, complemented
5#28299C212yesmaximal, complemented
6#28295C212yesmaximal, complemented
7#28220C212yesmaximal, complemented
8#28217C212yesmaximal, complemented
9#25493C214yesmaximal, complemented
10#25312C214yesmaximal, complemented
11#24625C214yesmaximal, complemented
12#24520C214yesmaximal, complemented
13#21995C211yesmaximal, characteristic, complemented
14#21882C211yesmaximal, characteristic, complemented
15#20729C211yesmaximal, characteristic, complemented
16#3458C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C2×#518C2211yescharacteristic
19#3241—24no—
20#3241—24no—
21#3166C2212yescomplemented
22#3138C2212yescomplemented
23#3122—24no—
24#3122—22no—
25#3122—22no—
26#3094C2214yes—
27C2×#482C2212yes—
28#3081—24no—
29#3081—24no—
30#3077C2212yes—
31#3077C2212yes—
32#3077C2214yes—
33#3077C2212yescomplemented
34#3077C2212yescomplemented
35C2×#478C2212yes—
36C2×#459C2212yes—
37C2×#447C2212yes—
38#2987—22no—
39#2986—22no—
40#2985—24no—
41#2962C2214yescomplemented
42#2961C2212yescomplemented
43#2960C2212yescomplemented
44C2×#442C2212yes—
45C2×C4×D8C2212yes—
46#2924C2214yes—
47#2921C2211yescharacteristic
48#2921C2211yescharacteristic
49C2×#433C2211yescharacteristic
50#2911C2212yes—
51#2905C2211yescharacteristic
52C2×#418C2212yes—
53#2090C2214yescomplemented
54#2089C2214yescomplemented
55#2063C2214yes—
56#2063C2214yes—
57#2063C2214yescomplemented
58#2063C2214yescomplemented
59#2045C2214yes—
60#2045C2214yes—
61#1998C2212yescomplemented
62#1978C2214yes—
63#1978C2214yes—
64#1965C2212yescomplemented
65#1952C2214yescomplemented
66#1951C2214yescomplemented
67#1950C2214yescomplemented
68#1931C2214yes—
69#1931C2214yes—
70#1931C2214yescomplemented
71#1931C2214yescomplemented
72#1900C2214yes—
73#1855C2214yes—
74#1849C2211yescharacteristic, complemented
75#1848C2211yescharacteristic, complemented
76#1838C2211yescharacteristic, complemented
77#1837C2211yescharacteristic, complemented
78#1822C2212yes—
79#1800C2212yes—
80#1800C2212yes—
81#1800C2212yescomplemented
82#1800C2212yescomplemented
83#1783C2212yes—
84C4×#100C2214yes—
85C23×D4C2311yescharacteristic
86C22×D8C2312yes—
87C22×D8C2312yes—
88C22×M16C2312yes—
89#544—22no—
90#544D414yes—
91#544D414yes—
92#543—22no—
93#540—22no—
94#540—22no—
95#540—24no—
96#533—22no—
97#533—22no—
98#533C2312yes—
99#533C2312yes—
100C2×#127C2311yescharacteristic
101C2×#125C2311yescharacteristic
102C2×#123—22no—
103C2×#121—22no—
104C2×#121C2312yes—
105#518—24no—
106#518—22no—
107#518—22no—
108C2×C4×D4—22no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C22⋊C4—22no—
116#501—24no—
117#499—22no—
118#499—22no—
119#490—22no—
120#490—22no—
121#486—24no—
122#482—24no—
123#482—24no—
124#482—24no—
125#482—24no—
126#481—24no—
127#480—24no—
128#478—24no—
129#478—24no—
130#478—24no—
131#478—24no—
132#478—24no—
133#478—24no—
134#478—24no—
135#478—24no—
136#476—22no—
137#476—22no—
138#476—24no—
139#476—22no—
140#476—22no—
141#474—22no—
142#474—22no—
143#474—24no—
144#474—22no—
145#474—22no—
146#459—24no—
147#459—24no—
148#447—24no—
149#447—24no—
150#446—24no—
151#446—24no—
152#446—24no—
153#446—24no—
154#446—24no—
155#446—22no—
156#446—22no—
157C8×Q8—24no—
158C8×Q8—22no—
159C8×Q8—22no—
160#442—24no—
161#442—24no—
162C4×D8—22no—
163C4×D8—22no—
164C4×D8—22no—
165C4×D8—22no—
166#433—22no—
167#433—24no—
168#433—22no—
169#433—24no—
170#433—24no—
171#433D414yes—
172#433D412yes—
173#433D412yes—
174#432—24no—
175#431—24no—
176C2×#108C2314yes—
177C2×#108C2314yes—
178#428—24no—
179#428C2314yes—
180#428C2314yes—
181#428C2314yes—
182#428C2314yes—
183#427—22no—
184#427—22no—
185C2×#107C2312yes—
186C2×#106C2312yes—
187#424—24no—
188#424C2314yes—
189#424C2314yes—
190#423C2311yescharacteristic
191#423C2311yescharacteristic
192#423C2312yes—
193#423C2311yescharacteristic
194#423C2311yescharacteristic
195#423C2312yes—
196C2×C4⋊C8C2312yes—
197#418—22no—
198#418—22no—
199#418C2314yes—
200#418C2314yes—
201#418C2314yes—
202#418C2314yes—
203C2×#102—24no—
204C2×#102C2312yes—
205C2×#102C2312yes—
206C2×#102C2312yes—
207C2×#102C2312yes—
208C2×#100C2314yes—
209C2×#100C2314yes—
210C2×#100C2314yes—
211C2×#100C2314yes—
212#407C2312yes—
213#407C2312yes—
214C2×C22⋊C8C2312yes—
215#405C2314yes—
216#405C2314yes—
217C4×M16C2312yes—
218C4×M16C2312yes—
219C2×#97C2311yescharacteristic
220C2×C4×C8C2311yescharacteristic
221#343C2314yes—
222#343C2314yes—
223#343C2314yes—
224#343C2314yes—
225#341C2314yes—
226#341C2314yes—
227C25—21no—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—42no—
235C22×D4C2412yes—
236C22×D4C2412yes—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4C2411yescharacteristic
241C2×D8—24no—
242C2×D8—24no—
243C2×D8—22no—
244C2×D8—22no—
245C2×D8—24no—
246C2×D8—42no—
247C2×D8—42no—
248C2×M16—22no—
249C2×M16—44no—
250C2×M16C2412yes—
251C2×M16C2412yes—
252C2×M16C2412yes—
253C2×M16C2414yes—
254C2×M16C2412yes—
255C22×C8—22no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8C2×D412yes—
261C4⋊Q8C2×D412yes—
262C4⋊Q8C2×D414yes—
263#127—22no—
264#127—22no—
265#125—24no—
266#125—24no—
267#125—22no—
268#125—22no—
269#125C2×D414yes—
270#125C2×D412yes—
271#125C2×D412yes—
272#123—44no—
273#123—44no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281#121—24no—
282#121—24no—
283#121—44no—
284#121—44no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—44no—
299C4×D4—44no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117C2×D412yes—
305#117C2×D412yes—
306#117C2×D412yes—
307#117C2×D412yes—
308#117C2×D414yes—
309#117C2414yes—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4#4112yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4#4112yes—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#108—44no—
329#108—44no—
330#107—22no—
331#107—22no—
332#107—24no—
333#107—24no—
334#107—24no—
335#107—24no—
336#106—24no—
337#106—24no—
338#106—24no—
339#106—24no—
340#106—24no—
341C4⋊C8—24no—
342C4⋊C8—24no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345C4⋊C8—24no—
346C4⋊C8—22no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—22no—
352C4⋊C8C2×D414yes—
353C4⋊C8C2×D414yes—
354#102—22no—
355#102—22no—
356#102—22no—
357#102—22no—
358#102—24no—
359#102—24no—
360#102—24no—
361#102—24no—
362#102—24no—
363#102—24no—
364#102—24no—
365#102—24no—
366#102—24no—
367#102—24no—
368#100—28no—
369#100—44no—
370#100—44no—
371#100—44no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C22⋊C8C2×D414yes—
375C22⋊C8C2×D414yes—
376#97—24no—
377#97—24no—
378#97—22no—
379#97—22no—
380#97C2×D412yes—
381#97C2×D412yes—
382C4×C8—24no—
383C4×C8—24no—
384C4×C8—22no—
385C4×C8—22no—
386C4×C8C2×D411yescharacteristic
387C4×C8C2×D412yes—
388C4×C8C2×D411yescharacteristic
389C24—22no—
390C24—22no—
391C24—44no—
392C24—42no—
393C24—162no—
394C24C2×#4111yescharacteristic
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×D4—28no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—48no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—162no—
417C2×D4—162no—
418C2×D42−1+414yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—162no—
430C22×C4—162no—
431C22×C4—162no—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2511yescharacteristic
438D8—44no—
439D8—44no—
440D8—48no—
441M16—24no—
442M16—24no—
443M16—24no—
444M16—24no—
445M16—24no—
446M16—24no—
447M16—44no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—44no—
458C2×C8C22×D412yes—
459C2×C8C22×D414yes—
460C2×C8C22×D412yes—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4#12114yes—
475C4⋊C4#12114yes—
476C4⋊C4C22×D414yes—
477C4⋊C4C22×D414yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4#12114yes—
489C22⋊C4#12114yes—
490C42—22no—
491C42—22no—
492C42—24no—
493C42—42no—
494C42C22×D412yes—
495C42C22×D412yes—
496C42C22×D411yescharacteristic
497C42C22×D411yescharacteristic
498C42C22×D412yes—
499C23—22no—
500C23—24no—
501C23—42no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—164no—
513C23—168no—
514C23C2×#12112yes—
515C23C22×#4111yescharacteristic
516Q8—82no—
517Q8—82no—
518Q8—84no—
519D4—24no—
520D4—24no—
521D4—48no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—164no—
529D4—164no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—84no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—162no—
544C2×C4—162no—
545C2×C4—162no—
546C2×C4—162no—
547C2×C4—164no—
548C2×C4C2×#12112yes—
549C2×C4C2×#12112yes—
550C2×C4C2×#12112yes—
551C2×C4C2×#12114yes—
552C2×C4C23×D411yescharacteristic
553C2×C4C23×D411yescharacteristic
554C2×C4C2×2−1+412yes—
555C2×C4#58512yes—
556C8—22no—
557C8—24no—
558C8—22no—
559C8—42no—
560C8—42no—
561C8—42no—
562C8—42no—
563C8—44no—
564C22—22no—
565C22—24no—
566C22—22no—
567C22—44no—
568C22—82no—
569C22—84no—
570C22—84no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—162no—
577C22—162no—
578C22—168no—
579C22C22×#12111yescharacteristic
580C22#349512yes—
581C4—22no—
582C4—22no—
583C4—24no—
584C4—22no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—82no—
589C4#348611yescharacteristic
590C4#348711yescharacteristic
591C4#348912yes—
592C2—22no—
593C2—42no—
594C2—84no—
595C2—162no—
596C2#3566111yescharacteristic
597C2#6047112yes—
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3122C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60868C2semidirect
3. #60490C2semidirect
4. #60252C2semidirect
5. #28299C2semidirect
6. #28295C2semidirect
7. #28220C2semidirect
8. #28217C2semidirect
9. #25493C2semidirect
10. #25312C2semidirect
11. #24625C2semidirect
12. #24520C2semidirect
13. #21995C2semidirect
14. #21882C2semidirect
15. #20729C2semidirect
21. #3166C22semidirect
22. #3138C22semidirect
33. #3077C22semidirect
34. #3077C22semidirect
41. #2962C22semidirect
42. #2961C22semidirect
43. #2960C22semidirect
53. #2090C22semidirect
54. #2089C22semidirect
57. #2063C22semidirect
58. #2063C22semidirect
61. #1998C22semidirect
64. #1965C22semidirect
65. #1952C22semidirect
66. #1951C22semidirect
67. #1950C22semidirect
70. #1931C22semidirect
71. #1931C22semidirect
74. #1849C22semidirect
75. #1848C22semidirect
76. #1838C22semidirect
77. #1837C22semidirect
81. #1800C22semidirect
82. #1800C22semidirect

Maximal subgroups

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

subgroupindex
#608682
C2×#31222
#604902
#602522
#282992
#282952
#282202
#282172
#254932
#253122
#246252
#245202
#219952
#218822
#207292