Finite Group Databasebeta

← #612660

Subgroups — #612660

Order 512.

6857 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2481C213yesmaximal, complemented
3#14843C213yesmaximal, complemented
4#14822C213yesmaximal, complemented
5#14776C213yesmaximal, complemented
6#14729C213yesmaximal, complemented
7#14694C211yesmaximal, characteristic
8#14694C211yesmaximal, characteristic
9#14694C211yesmaximal, characteristic
10#14693C211yesmaximal, characteristic, complemented
11#14680C211yesmaximal, characteristic, complemented
12#14039C213yesmaximal, complemented
13#14019C213yesmaximal, complemented
14#13882C213yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C25×C4C2211yescharacteristic
19C22×#126C2211yescharacteristic, complemented
20C22×#120C2211yescharacteristic
21C23×C4⋊C4—224no—
22C23×C4⋊C4C2213yes—
23C23×C4⋊C4C2213yes—
24C23×C4⋊C4C2213yes—
25C23×C4⋊C4C2213yes—
26#2481—224no—
27C2×#400C2213yescomplemented
28C2×#400C2213yescomplemented
29C2×#397C2213yescomplemented
30C2×#396C2213yescomplemented
31C2×#388C2213yescomplemented
32C2×#387C2213yescomplemented
33C2×#387C2213yescomplemented
34C2×#386C2213yes—
35C2×#386C2213yes—
36C2×#386C2213yes—
37C2×#386C2213yes—
38C2×#386C2213yes—
39C2×#386C2213yes—
40C2×#385C2213yes—
41C2×#385C2213yes—
42C2×#380C2213yes—
43C2×#380C2213yes—
44C2×#379C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×#379C2211yescharacteristic
47C2×C4×C22⋊C4C2213yes—
48C2×C4×C22⋊C4C2213yes—
49#1445C2216yescomplemented
50#1442C2216yescomplemented
51#1442C2216yescomplemented
52#1441C2216yescomplemented
53#1441C2216yescomplemented
54#1439C2216yescomplemented
55#1436C2216yescomplemented
56C22⋊#95C2213yes—
57C22⋊#95C2213yes—
58C22⋊#95C2213yes—
59C22⋊#95C2213yes—
60C22⋊#95C2213yes—
61C22⋊#95C2213yes—
62C22⋊#95C2213yes—
63C22⋊#95C2213yes—
64C22⋊#95C2213yes—
65C22⋊#95C2213yes—
66C22⋊#95C2213yes—
67C22⋊#95C2213yes—
68C26C2311yescharacteristic
69C23×D4C2313yes—
70C24×C4—24no—
71C24×C4—212no—
72C24×C4—212no—
73C24×C4—212no—
74C24×C4—212no—
75C24×C4—24no—
76C24×C4—24no—
77C24×C4—46no—
78C24×C4—46no—
79C24×C4C2313yes—
80C24×C4C2313yes—
81C24×C4C2313yes—
82C24×C4C2313yes—
83C24×C4C2313yes—
84C24×C4C2313yes—
85C2×#126—212no—
86C2×#120—24no—
87C2×#120—24no—
88C2×#120—24no—
89C22×C4⋊C4—2128no—
90C22×C4⋊C4—212no—
91C22×C4⋊C4—212no—
92C22×C4⋊C4—212no—
93C22×C4⋊C4—224no—
94C22×C4⋊C4—224no—
95C22×C4⋊C4—224no—
96C22×C4⋊C4—224no—
97C22×C4⋊C4D4124yescomplemented
98C22×C4⋊C4C2313yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2316yes—
102C22×C4⋊C4C2313yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2313yes—
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2313yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2313yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2313yes—
113C22×C22⋊C4C2313yes—
114C22×C22⋊C4C2313yes—
115C22×C22⋊C4C2313yes—
116C22×C22⋊C4C2313yes—
117C22×C22⋊C4C2313yes—
118C22×C22⋊C4C2313yes—
119C22×C22⋊C4C2313yes—
120C22×C42C2311yescharacteristic
121C22×C42C2313yes—
122#400—212no—
123#400—212no—
124#397—212no—
125#396—212no—
126#388—212no—
127#387—212no—
128#387—212no—
129#386—212no—
130#386—212no—
131#386—212no—
132#386—212no—
133#386—46no—
134#386—46no—
135#385—212no—
136#385—212no—
137#380—212no—
138#380—46no—
139#379—24no—
140#379—24no—
141#379—24no—
142C4×C22⋊C4—212no—
143C4×C22⋊C4—46no—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2313yes—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2316yes—
157C2×#95C2316yes—
158C2×#95C2316yes—
159C2×#95C2316yes—
160C2×#95C2316yes—
161C2×#95C2316yes—
162C25—26no—
163C25—26no—
164C25—26no—
165C25—26no—
166C25#4114yes—
167C25#4114yes—
168C25#4114yes—
169C25C2411yescharacteristic
170C25C2413yes—
171C22×D4—26no—
172C22×D4—26no—
173C22×D4—26no—
174C22×D4—26no—
175C22×D4—26no—
176C22×D4—26no—
177C22×D4—26no—
178C23×C4—232no—
179C23×C4—296no—
180C23×C4—296no—
181C23×C4—26no—
182C23×C4—26no—
183C23×C4—26no—
184C23×C4—26no—
185C23×C4—212no—
186C23×C4—212no—
187C23×C4—212no—
188C23×C4—212no—
189C23×C4—232no—
190C23×C4—232no—
191C23×C4—26no—
192C23×C4—26no—
193C23×C4—26no—
194C23×C4—26no—
195C23×C4—26no—
196C23×C4—26no—
197C23×C4—212no—
198C23×C4—212no—
199C23×C4—212no—
200C23×C4—212no—
201C23×C4—26no—
202C23×C4—26no—
203C23×C4—26no—
204C23×C4—26no—
205C23×C4—46no—
206C23×C4—46no—
207C23×C4—46no—
208C23×C4—46no—
209C23×C4—448no—
210C23×C4—448no—
211C23×C4—448no—
212C23×C4—448no—
213C23×C4C2×D4112yes—
214C23×C4C2×D4112yes—
215C23×C4C2×D4112yes—
216C23×C4#41112yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2413yes—
220C23×C4C2411yescharacteristic
221C23×C4C2413yes—
222C23×C4C2413yes—
223C23×C4C2413yes—
224C23×C4C2413yes—
225C23×C4C2413yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2413yes—
230#126—416no—
231#120—416no—
232C2×C4⋊C4—296no—
233C2×C4⋊C4—296no—
234C2×C4⋊C4—296no—
235C2×C4⋊C4—212no—
236C2×C4⋊C4—212no—
237C2×C4⋊C4—212no—
238C2×C4⋊C4—212no—
239C2×C4⋊C4—212no—
240C2×C4⋊C4—212no—
241C2×C4⋊C4—296no—
242C2×C4⋊C4—26no—
243C2×C4⋊C4—26no—
244C2×C4⋊C4—26no—
245C2×C4⋊C4—212no—
246C2×C4⋊C4—26no—
247C2×C4⋊C4—46no—
248C2×C4⋊C4C2×D4112yes—
249C2×C4⋊C4C2×D4112yes—
250C2×C4⋊C4C2×D4112yes—
251C2×C4⋊C4C2×D4112yes—
252C2×C22⋊C4—26no—
253C2×C22⋊C4—26no—
254C2×C22⋊C4—26no—
255C2×C22⋊C4—26no—
256C2×C22⋊C4—26no—
257C2×C22⋊C4—26no—
258C2×C22⋊C4—26no—
259C2×C22⋊C4—26no—
260C2×C22⋊C4—26no—
261C2×C22⋊C4—26no—
262C2×C22⋊C4—26no—
263C2×C22⋊C4—26no—
264C2×C22⋊C4—26no—
265C2×C22⋊C4—26no—
266C2×C22⋊C4—26no—
267C2×C22⋊C4—26no—
268C2×C22⋊C4—26no—
269C2×C22⋊C4—26no—
270C2×C22⋊C4—26no—
271C2×C22⋊C4—26no—
272C2×C22⋊C4—26no—
273C2×C22⋊C4—26no—
274C2×C22⋊C4—26no—
275C2×C22⋊C4—26no—
276C2×C22⋊C4—26no—
277C2×C22⋊C4—212no—
278C2×C22⋊C4—26no—
279C2×C22⋊C4—26no—
280C2×C22⋊C4—46no—
281C2×C22⋊C4—46no—
282C2×C42—26no—
283C2×C42—26no—
284C2×C42—46no—
285#95—26no—
286#95—26no—
287#95—26no—
288#95—26no—
289#95—26no—
290#95—26no—
291#95—26no—
292#95—26no—
293#95—26no—
294#95—26no—
295#95—26no—
296#95—26no—
297#95—212no—
298#95—212no—
299#95—46no—
300#95—46no—
301#95—46no—
302#95—46no—
303C24—28no—
304C24—224no—
305C24—224no—
306C24—224no—
307C24—248no—
308C24—224no—
309C24—224no—
310C24—224no—
311C24—224no—
312C24—224no—
313C24—224no—
314C24—26no—
315C24—26no—
316C24—26no—
317C24—26no—
318C24—23no—
319C24—23no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—23no—
324C24—23no—
325C24#126116yescomplemented
326C24C22×D416yes—
327C24C2×#4116yes—
328C24C2×#4116yes—
329C24C2×#4116yes—
330C24C2×#4116yes—
331C24H(5,𝔽2)16yes—
332C242−1+416yes—
333C24C2511yescharacteristic
334C2×D4—412no—
335C2×D4—412no—
336C2×D4—412no—
337C2×D4—412no—
338C2×D4—412no—
339C2×D4—412no—
340C2×D4—412no—
341C22×C4—224no—
342C22×C4—2256no—
343C22×C4—248no—
344C22×C4—248no—
345C22×C4—248no—
346C22×C4—224no—
347C22×C4—224no—
348C22×C4—224no—
349C22×C4—26no—
350C22×C4—21no—
351C22×C4—23no—
352C22×C4—23no—
353C22×C4—23no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—224no—
359C22×C4—224no—
360C22×C4—248no—
361C22×C4—224no—
362C22×C4—224no—
363C22×C4—224no—
364C22×C4—224no—
365C22×C4—26no—
366C22×C4—26no—
367C22×C4—23no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—26no—
372C22×C4—26no—
373C22×C4—224no—
374C22×C4—224no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—23no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4—26no—
385C22×C4—26no—
386C22×C4—21no—
387C22×C4—23no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—26no—
398C22×C4—26no—
399C22×C4—26no—
400C22×C4—26no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—412no—
407C22×C4—412no—
408C22×C4—424no—
409C22×C4—424no—
410C22×C4—424no—
411C22×C4—424no—
412C22×C4—424no—
413C22×C4—424no—
414C22×C4—424no—
415C22×C4—424no—
416C22×C4—4128no—
417C22×C4—4128no—
418C22×C4—412no—
419C22×C4—412no—
420C22×C4—412no—
421C22×C4—412no—
422C22×C4C22×D416yes—
423C22×C4C22×D416yes—
424C22×C4C22×D416yes—
425C22×C4C22×D416yes—
426C22×C4C22×D416yes—
427C22×C4C22×D416yes—
428C22×C4C2×#41112yes—
429C4⋊C4—248no—
430C4⋊C4—224no—
431C4⋊C4—224no—
432C4⋊C4—224no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—412no—
436C4⋊C4—86no—
437C4⋊C4#120116yescomplemented
438C22⋊C4—412no—
439C22⋊C4—412no—
440C22⋊C4—412no—
441C22⋊C4—412no—
442C22⋊C4—412no—
443C22⋊C4—412no—
444C22⋊C4—412no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—412no—
449C22⋊C4—86no—
450C22⋊C4—86no—
451C42—44no—
452C42—86no—
453C23—232no—
454C23—232no—
455C23—296no—
456C23—296no—
457C23—232no—
458C23—296no—
459C23—26no—
460C23—26no—
461C23—212no—
462C23—212no—
463C23—212no—
464C23—212no—
465C23—212no—
466C23—212no—
467C23—212no—
468C23—26no—
469C23—212no—
470C23—26no—
471C23—212no—
472C23—212no—
473C23—212no—
474C23—212no—
475C23—296no—
476C23—41no—
477C23—43no—
478C23—43no—
479C23—41no—
480C23—43no—
481C23—43no—
482C23—43no—
483C23—43no—
484C23—43no—
485C23—43no—
486C23—43no—
487C23—46no—
488C23#525112yes—
489C23C2×#126112yes—
490C23#539112yes—
491C23#546112yes—
492C23#548112yes—
493C23#552112yes—
494C23#553112yes—
495C23C23×D413yes—
496C23C22×#4111yescharacteristic
497C23C22×#4113yes—
498C23C22×#4111yescharacteristic
499C23C22×#4111yescharacteristic
500C23C2×H(5,𝔽2)13yes—
501C23C2×2−1+413yes—
502D4—824no—
503C2×C4—296no—
504C2×C4—296no—
505C2×C4—26no—
506C2×C4—26no—
507C2×C4—26no—
508C2×C4—26no—
509C2×C4—212no—
510C2×C4—212no—
511C2×C4—26no—
512C2×C4—26no—
513C2×C4—26no—
514C2×C4—26no—
515C2×C4—26no—
516C2×C4—26no—
517C2×C4—26no—
518C2×C4—26no—
519C2×C4—26no—
520C2×C4—26no—
521C2×C4—448no—
522C2×C4—448no—
523C2×C4—448no—
524C2×C4—448no—
525C2×C4—46no—
526C2×C4—46no—
527C2×C4—46no—
528C2×C4—46no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—43no—
533C2×C4—43no—
534C2×C4—43no—
535C2×C4—43no—
536C2×C4—43no—
537C2×C4—43no—
538C2×C4—43no—
539C2×C4—46no—
540C2×C4—43no—
541C2×C4—43no—
542C2×C4—43no—
543C2×C4—43no—
544C2×C4—43no—
545C2×C4—43no—
546C2×C4—43no—
547C2×C4—43no—
548C2×C4—46no—
549C2×C4—46no—
550C2×C4—43no—
551C2×C4—43no—
552C2×C4—43no—
553C2×C4—43no—
554C2×C4—43no—
555C2×C4—43no—
556C2×C4—43no—
557C2×C4—46no—
558C2×C4—46no—
559C2×C4—46no—
560C2×C4C2×#12014yes—
561C2×C4C2×#12014yes—
562C2×C4C2×#12014yes—
563C2×C4#525112yes—
564C22—2128no—
565C22—212no—
566C22—212no—
567C22—212no—
568C22—224no—
569C22—224no—
570C22—212no—
571C22—224no—
572C22—212no—
573C22—212no—
574C22—224no—
575C22—81no—
576C22—81no—
577C22—83no—
578C22—83no—
579C22—83no—
580C22—81no—
581C22—83no—
582C22#2481124yescomplemented
583C22C22×#12011yescharacteristic
584C22C2×#52513yes—
585C22C2×#52513yes—
586C22C22×#12611yescharacteristic
587C22C2×#53913yes—
588C22C2×#54613yes—
589C22C2×#54813yes—
590C22C2×#55213yes—
591C22C2×#55313yes—
592C22#348413yes—
593C22#348613yes—
594C22#349216yes—
595C4—24no—
596C4—212no—
597C4—24no—
598C4—24no—
599C4—46no—
600C4—46no—
601C4—81no—
602C4—83no—
603C4—83no—
604C4—81no—
605C4—81no—
606C4—83no—
607C4—83no—
608C2—224no—
609C2—161no—
610C2C2×#248113yes—
611C2#3498913yes—
612C2#3615211yescharacteristic
613C2#3616313yes—
614C2#3657511yescharacteristic
615C2#3658113yes—
616C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×C4⋊C4C2semidirect
2. C2×#2481C2semidirect
3. #14843C2semidirect
4. #14822C2semidirect
5. #14776C2semidirect
6. #14729C2semidirect
10. #14693C2semidirect
11. #14680C2semidirect
12. #14039C2semidirect
13. #14019C2semidirect
14. #13882C2semidirect
15. C4×#379C2semidirect
19. C22×#126C22semidirect
27. C2×#400C22semidirect
28. C2×#400C22semidirect
29. C2×#397C22semidirect
30. C2×#396C22semidirect
31. C2×#388C22semidirect
32. C2×#387C22semidirect
33. C2×#387C22semidirect
49. #1445C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1439C22semidirect
55. #1436C22semidirect
97. C22×C4⋊C4D4semidirect
325. C24#126semidirect
437. C4⋊C4#120semidirect
582. C22#2481semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#24812
#148432
#148222
#147762
#147292
#146942
#146932
#146802
#140392
#140192
#138822
C4×#3792