Finite Group Databasebeta

← #496804

Subgroups — #496804

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33433C211yesmaximal, characteristic, complemented
2#33333C211yesmaximal, characteristic, complemented
3#21466C211yesmaximal, characteristic
4#13196C211yesmaximal, characteristic, complemented
5#13195C211yesmaximal, characteristic, complemented
6#13125C211yesmaximal, characteristic, complemented
7#13124C211yesmaximal, characteristic, complemented
8#13122C211yesmaximal, characteristic, complemented
9#13122C211yesmaximal, characteristic, complemented
10#12980C211yesmaximal, characteristic, complemented
11#12980C211yesmaximal, characteristic, complemented
12#12978C211yesmaximal, characteristic, complemented
13#12914C211yesmaximal, characteristic, complemented
14#12683C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#504C2211yescharacteristic, complemented
18C22×M32C2211yescharacteristic
19C22×M32C2211yescharacteristic
20#3002C411yescharacteristic
21C2×#443C2211yescharacteristic
22C2×#443C2211yescharacteristic
23#2929C2211yescharacteristic
24#2169—21no—
25#2169—21no—
26#2168—21no—
27#2168—21no—
28C2×#364C2211yescharacteristic, complemented
29C2×#364C2211yescharacteristic, complemented
30#2151C2211yescharacteristic, complemented
31C2×#350C2211yescharacteristic
32C2×#350C2211yescharacteristic
33C2×#350C2211yescharacteristic
34C2×#350C2211yescharacteristic
35C2×#350C2211yescharacteristic, complemented
36C2×#350C2211yescharacteristic, complemented
37#2113—21no—
38#2113—21no—
39#2113C2211yescharacteristic
40#2113C2211yescharacteristic
41#2113C2211yescharacteristic
42#2113C2211yescharacteristic
43#2113C2211yescharacteristic, complemented
44#2112—21no—
45#2112—21no—
46#2112C2211yescharacteristic
47C2×C22⋊C16C2211yescharacteristic
48#2108C2211yescharacteristic, complemented
49C2×#346C2211yescharacteristic, complemented
50C2×C4×C16C2211yescharacteristic, complemented
51#1876C411yescharacteristic
52#1876C411yescharacteristic
53#1871C411yescharacteristic
54#1866C411yescharacteristic
55#1866C411yescharacteristic
56#1855C411yescharacteristic
57#1789C411yescharacteristic
58#1764C2211yescharacteristic
59#1764C2211yescharacteristic
60C8×C22⋊C4C2211yescharacteristic
61#1379C2211yescharacteristic, complemented
62#1379C2211yescharacteristic, complemented
63#1376C2211yescharacteristic, complemented
64#1376C2211yescharacteristic, complemented
65#1374C2211yescharacteristic, complemented
66#1374C2211yescharacteristic, complemented
67C22×#41C2×C411yescharacteristic
68C2×#135C2×C411yescharacteristic
69C2×#135C2×C411yescharacteristic
70C2×#131—21no—
71C2×#131—21no—
72C2×#131—41no—
73C2×#131—41no—
74C2×#131—41no—
75C2×#131—41no—
76C2×#131C2×C411yescharacteristic
77C2×#131C2×C411yescharacteristic
78C2×#131C2311yescharacteristic
79C2×#131C2311yescharacteristic
80C2×#131C2311yescharacteristic
81C22×M16—21no—
82C22×M16C2×C411yescharacteristic
83C23×C8—21no—
84C23×C8C2311yescharacteristic
85#525C2×C411yescharacteristic
86#504—41no—
87#504—41no—
88C2×M32—21no—
89C2×M32—21no—
90C2×M32—41no—
91C2×M32—41no—
92C2×M32—41no—
93C2×M32—41no—
94C2×M32D411yescharacteristic
95C2×M32D411yescharacteristic
96C2×M32D411yescharacteristic
97C2×M32D411yescharacteristic
98C2×M32C2311yescharacteristic
99C2×M32C2311yescharacteristic
100C2×M32C2311yescharacteristic
101C2×M32C2311yescharacteristic
102C22×C16—21no—
103C22×C16—21no—
104C22×C16C2311yescharacteristic
105C22×C16C2311yescharacteristic
106C22×C16C2311yescharacteristic
107C22×C16C2311yescharacteristic
108#452C811yescharacteristic
109#452C811yescharacteristic
110#451C811yescharacteristic
111#450C811yescharacteristic
112#449C811yescharacteristic
113#449C811yescharacteristic
114#448C811yescharacteristic
115#447C811yescharacteristic
116#443—41no—
117#443—41no—
118#443—41no—
119#443—41no—
120#435—21no—
121#435—21no—
122C8×D4—21no—
123C8×D4—21no—
124C2×#108C2×C411yescharacteristic
125C2×#108C2×C411yescharacteristic
126#424C2×C411yescharacteristic
127C2×#104C2×C411yescharacteristic
128C2×#104C2×C411yescharacteristic
129C2×#104C2×C411yescharacteristic
130C2×#104C2×C411yescharacteristic
131C2×#104C2×C411yescharacteristic
132C2×#104C2×C411yescharacteristic
133#416C2×C411yescharacteristic
134#416C2×C411yescharacteristic
135#408C2×C411yescharacteristic
136#407C2×C411yescharacteristic
137C2×C22⋊C8C2×C411yescharacteristic
138#405C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C4×C22⋊C4C2×C411yescharacteristic
142#364—41no—
143#364—41no—
144C4⋊C16—21no—
145C4⋊C16—21no—
146#350—41no—
147#350—41no—
148#350—41no—
149#350—41no—
150#350—41no—
151#350—41no—
152C22⋊C16—21no—
153C22⋊C16—21no—
154C22⋊C16—21no—
155C22⋊C16—21no—
156C22⋊C16—21no—
157C22⋊C16—21no—
158C22⋊C16—21no—
159C22⋊C16—21no—
160C22⋊C16—21no—
161C22⋊C16—21no—
162C22⋊C16—21no—
163C22⋊C16—21no—
164C22⋊C16—21no—
165C22⋊C16—21no—
166#346—21no—
167#346—41no—
168C4×C16—21no—
169C4×C16—41no—
170#341C2×C411yescharacteristic
171#341C2×C411yescharacteristic
172#339C2×C411yescharacteristic
173#339C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—41no—
179C2×#41—41no—
180C2×#41—41no—
181C2×#41—41no—
182C2×#41C2×C811yescharacteristic
183C2×#41C2×C811yescharacteristic
184C2×#41C22×C411yescharacteristic
185C2×#41C22×C411yescharacteristic
186C2×#41C22×C411yescharacteristic
187C22×Q8C2×C811yescharacteristic
188C22×D4—21no—
189C22×D4C2×C811yescharacteristic
190C23×C4—21no—
191C23×C4C22×C411yescharacteristic
192#135—41no—
193#135—41no—
194#135—41no—
195#135—41no—
196C2×Q16C2×C811yescharacteristic
197C2×Q16C2×C811yescharacteristic
198C2×SD16C2×C811yescharacteristic
199C2×SD16C2×C811yescharacteristic
200C2×SD16C2×C811yescharacteristic
201C2×SD16C2×C811yescharacteristic
202C2×D8C2×C811yescharacteristic
203C2×D8C2×C811yescharacteristic
204#131—21no—
205#131—21no—
206#131—21no—
207#131—21no—
208#131—21no—
209#131—21no—
210#131—21no—
211#131—21no—
212#131—21no—
213#131—21no—
214#131—21no—
215#131—21no—
216#131—41no—
217#131—41no—
218#131—41no—
219#131—41no—
220#131—42no—
221#131—42no—
222#131—42no—
223#131—42no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—42no—
231C2×M16—42no—
232C2×M16—41no—
233C2×M16—41no—
234C2×M16C22×C411yescharacteristic
235C2×M16C22×C411yescharacteristic
236C2×M16C22×C411yescharacteristic
237C2×M16C22×C411yescharacteristic
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—42no—
249C22×C8—42no—
250C22×C8C22×C411yescharacteristic
251C22×C8C22×C411yescharacteristic
252C22×C8C22×C411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8#4111yescharacteristic
256C22×C8#4111yescharacteristic
257C22×C8C2411yescharacteristic
258#123—21no—
259C22⋊Q8C2×C811yescharacteristic
260C22⋊Q8C2×C811yescharacteristic
261#121C2×C811yescharacteristic
262#121C2×C811yescharacteristic
263#120—21no—
264C4×D4—21no—
265C4×D4—21no—
266#117C22×C411yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270M32—21no—
271M32—21no—
272M32—21no—
273M32—21no—
274M32—21no—
275M32—21no—
276M32—21no—
277M32—21no—
278M32—41no—
279M32—41no—
280M32—41no—
281M32—41no—
282M32—42no—
283M32—42no—
284C2×C16—21no—
285C2×C16—21no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16—21no—
292C2×C16—21no—
293C2×C16—21no—
294C2×C16—21no—
295C2×C16—21no—
296C2×C16—21no—
297C2×C16—21no—
298C2×C16—42no—
299C2×C16—42no—
300C2×C16C2×D411yescharacteristic
301C2×C16C2×D411yescharacteristic
302C2×C16C2×D411yescharacteristic
303C2×C16C2×D411yescharacteristic
304#108—41no—
305#108—41no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308#104—41no—
309#104—41no—
310#104—41no—
311#104—41no—
312#104—41no—
313#104—41no—
314#103C2×C811yescharacteristic
315#103C2×C811yescharacteristic
316#103C2×C811yescharacteristic
317#103C2×C811yescharacteristic
318#102C2×C811yescharacteristic
319#102C2×C811yescharacteristic
320#102C2×C811yescharacteristic
321#102C2×C811yescharacteristic
322C22⋊C8—21no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8C2×C811yescharacteristic
327C22⋊C8C2×C811yescharacteristic
328C22⋊C8C2×C811yescharacteristic
329C22⋊C8C2×C811yescharacteristic
330#97—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336#95—21no—
337C24—21no—
338C24#13111yescharacteristic
339#41—21no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—21no—
344#41—21no—
345#41—21no—
346#41—21no—
347#41—21no—
348#41—21no—
349#41—21no—
350#41—21no—
351#41—41no—
352#41—41no—
353#41—41no—
354#41—41no—
355#41—42no—
356#41—42no—
357#41—42no—
358#41—42no—
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8C22×C811yescharacteristic
363C2×Q8C22×C811yescharacteristic
364C2×Q8C22×C811yescharacteristic
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4C22×C811yescharacteristic
378C2×D4C22×C811yescharacteristic
379C2×D4C22×C811yescharacteristic
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C22×C811yescharacteristic
397C22×C4C22×C811yescharacteristic
398C22×C4#13111yescharacteristic
399C22×C4C23×C411yescharacteristic
400Q16—41no—
401Q16—41no—
402SD16—41no—
403SD16—41no—
404SD16—41no—
405SD16—41no—
406D8—41no—
407D8—41no—
408M16—21no—
409M16—21no—
410M16—21no—
411M16—21no—
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—41no—
424M16—41no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8C4×D411yescharacteristic
453C2×C8C4×D411yescharacteristic
454C2×C8C22×C811yescharacteristic
455C2×C8C22×C811yescharacteristic
456C2×C8C22×C811yescharacteristic
457C2×C8C22×C811yescharacteristic
458C2×C8C22×D411yescharacteristic
459C2×C8C2×#4111yescharacteristic
460C2×C8H(5,𝔽2)11yescharacteristic
461C2×C8H(5,𝔽2)11yescharacteristic
462C2×C8H(5,𝔽2)11yescharacteristic
463C2×C8H(5,𝔽2)11yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4C22×C811yescharacteristic
467C4⋊C4C22×C811yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—42no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C16—21no—
480C16—21no—
481C16—21no—
482C16—21no—
483C16—41no—
484C16—41no—
485C16—41no—
486C16—41no—
487C16—41no—
488C16—41no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—81no—
498C23—81no—
499C23C8×D411yescharacteristic
500C23C8×D411yescharacteristic
501C23C2×#13111yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8—21no—
508Q8—42no—
509D4—21no—
510D4—21no—
511D4—21no—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—81no—
525D4—81no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4C8×D411yescharacteristic
557C2×C4C8×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4#51811yescharacteristic
560C2×C4#51811yescharacteristic
561C2×C4C23×C811yescharacteristic
562C2×C4#56811yescharacteristic
563C2×C4#56811yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8#53411yescharacteristic
575C8#53511yescharacteristic
576C8#53811yescharacteristic
577C8#53811yescharacteristic
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—84no—
598C22C2×C8×D411yescharacteristic
599C22#297611yescharacteristic
600C22#297611yescharacteristic
601C4—21no—
602C4—21no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4#232111yescharacteristic
611C4#293011yescharacteristic
612C4#297211yescharacteristic
613C4#297211yescharacteristic
614C2—21no—
615C2—21no—
616C2—41no—
617C2—41no—
618C2—81no—
619C2—81no—
620C2—81no—
621C2#2202811yescharacteristic
622C2#3344211yescharacteristic
623C2#3344311yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radical#3002C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×M32 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33433C2semidirect
2. #33333C2semidirect
4. #13196C2semidirect
5. #13195C2semidirect
6. #13125C2semidirect
7. #13124C2semidirect
8. #13122C2semidirect
9. #13122C2semidirect
10. #12980C2semidirect
11. #12980C2semidirect
12. #12978C2semidirect
13. #12914C2semidirect
14. #12683C2semidirect
15. #12612C2semidirect
17. C2×#504C22semidirect
28. C2×#364C22semidirect
29. C2×#364C22semidirect
30. #2151C22semidirect
35. C2×#350C22semidirect
36. C2×#350C22semidirect
43. #2113C22semidirect
48. #2108C22semidirect
49. C2×#346C22semidirect
50. C2×C4×C16C22semidirect
61. #1379C22semidirect
62. #1379C22semidirect
63. #1376C22semidirect
64. #1376C22semidirect
65. #1374C22semidirect
66. #1374C22semidirect

Maximal subgroups

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

subgroupindex
#334332
#333332
#214662
#131962
#131952
#131252
#131242
#131222
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