Finite Group Databasebeta

← #612652

Subgroups — #612652

Order 512.

6722 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4⋊C4C211yesmaximal, characteristic
2C2×#2473C213yesmaximal, complemented
3#14843C213yesmaximal, complemented
4#14821C213yesmaximal, complemented
5#14786C213yesmaximal, complemented
6#14728C213yesmaximal, complemented
7#14693C211yesmaximal, characteristic
8#14693C211yesmaximal, characteristic
9#14693C211yesmaximal, characteristic
10#14693C211yesmaximal, characteristic
11#14680C211yesmaximal, characteristic
12#14043C213yesmaximal, complemented
13#14039C213yesmaximal, complemented
14#13882C213yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C25×C4C2211yescharacteristic
19C22×#125C2211yescharacteristic, complemented
20C23×C4⋊C4—224no—
21C23×C4⋊C4C2213yes—
22C23×C4⋊C4C2213yes—
23C23×C4⋊C4C2213yes—
24C23×C4⋊C4C2213yes—
25#2473—224no—
26C2×#400C2213yescomplemented
27C2×#398C2213yescomplemented
28C2×#398C2213yescomplemented
29C2×#398C2213yescomplemented
30C2×#389C2213yescomplemented
31C2×#387C2213yescomplemented
32C2×#387C2213yescomplemented
33C2×#385C2213yes—
34C2×#385C2213yes—
35C2×#385C2213yes—
36C2×#385C2213yes—
37C2×#385C2213yes—
38C2×#385C2213yes—
39C2×#385C2213yes—
40C2×#385C2213yes—
41C2×#380C2213yes—
42C2×#380C2213yes—
43C2×#379C2211yescharacteristic
44C2×#379C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×#379C2211yescharacteristic
47C2×C4×C22⋊C4C2213yes—
48C2×C4×C22⋊C4C2213yes—
49#1445C2216yescomplemented
50#1444C2216yescomplemented
51#1442C2216yescomplemented
52#1441C2216yescomplemented
53#1441C2216yescomplemented
54#1441C2216yescomplemented
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
69C24×C4—24no—
70C24×C4—212no—
71C24×C4—212no—
72C24×C4—212no—
73C24×C4—212no—
74C24×C4—24no—
75C24×C4—24no—
76C24×C4—46no—
77C24×C4—46no—
78C24×C4C2313yes—
79C24×C4C2313yes—
80C24×C4C2313yes—
81C24×C4C2313yes—
82C24×C4C2313yes—
83C24×C4C2313yes—
84C2×#125—212no—
85C22×C4⋊C4—2128no—
86C22×C4⋊C4—212no—
87C22×C4⋊C4—212no—
88C22×C4⋊C4—212no—
89C22×C4⋊C4—224no—
90C22×C4⋊C4—224no—
91C22×C4⋊C4—224no—
92C22×C4⋊C4—224no—
93C22×C4⋊C4D4124yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2313yes—
98C22×C4⋊C4C2313yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2316yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2313yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2313yes—
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2313yes—
107C22×C4⋊C4C2313yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2313yes—
110C22×C22⋊C4C2313yes—
111C22×C22⋊C4C2313yes—
112C22×C22⋊C4C2313yes—
113C22×C22⋊C4C2313yes—
114C22×C22⋊C4C2313yes—
115C22×C22⋊C4C2313yes—
116C22×C22⋊C4C2313yes—
117C22×C42C2311yescharacteristic
118C22×C42C2313yes—
119#400—212no—
120#398—212no—
121#398—212no—
122#398—212no—
123#389—212no—
124#387—212no—
125#387—212no—
126#385—212no—
127#385—212no—
128#385—212no—
129#385—212no—
130#385—212no—
131#385—212no—
132#385—46no—
133#385—46no—
134#380—212no—
135#380—46no—
136#379—24no—
137#379—24no—
138#379—24no—
139#379Q818yes—
140C4×C22⋊C4—212no—
141C4×C22⋊C4—46no—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2313yes—
154C2×#95C2316yes—
155C2×#95C2316yes—
156C2×#95C2316yes—
157C2×#95C2316yes—
158C2×#95C2316yes—
159C2×#95C2316yes—
160C25—26no—
161C25—26no—
162C25—26no—
163C25—26no—
164C25C2×Q814yes—
165C25#4114yes—
166C25#4114yes—
167C25C2413yes—
168C23×C4—232no—
169C23×C4—296no—
170C23×C4—296no—
171C23×C4—26no—
172C23×C4—26no—
173C23×C4—26no—
174C23×C4—26no—
175C23×C4—212no—
176C23×C4—212no—
177C23×C4—212no—
178C23×C4—212no—
179C23×C4—232no—
180C23×C4—232no—
181C23×C4—26no—
182C23×C4—26no—
183C23×C4—26no—
184C23×C4—26no—
185C23×C4—26no—
186C23×C4—26no—
187C23×C4—212no—
188C23×C4—212no—
189C23×C4—212no—
190C23×C4—212no—
191C23×C4—26no—
192C23×C4—26no—
193C23×C4—26no—
194C23×C4—26no—
195C23×C4—46no—
196C23×C4—46no—
197C23×C4—46no—
198C23×C4—46no—
199C23×C4—448no—
200C23×C4—448no—
201C23×C4—448no—
202C23×C4—448no—
203C23×C4C2×D4112yes—
204C23×C4C2×D4112yes—
205C23×C4C2×D4112yes—
206C23×C4#41112yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2413yes—
211C23×C4C2413yes—
212C23×C4C2413yes—
213C23×C4C2413yes—
214C23×C4C2413yes—
215C23×C4C2411yescharacteristic
216C23×C4C2413yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2413yes—
221#125—416no—
222C2×C4⋊C4—296no—
223C2×C4⋊C4—296no—
224C2×C4⋊C4—296no—
225C2×C4⋊C4—212no—
226C2×C4⋊C4—212no—
227C2×C4⋊C4—212no—
228C2×C4⋊C4—212no—
229C2×C4⋊C4—212no—
230C2×C4⋊C4—212no—
231C2×C4⋊C4—296no—
232C2×C4⋊C4—26no—
233C2×C4⋊C4—26no—
234C2×C4⋊C4—26no—
235C2×C4⋊C4—26no—
236C2×C4⋊C4—26no—
237C2×C4⋊C4—26no—
238C2×C4⋊C4—26no—
239C2×C4⋊C4—212no—
240C2×C4⋊C4—212no—
241C2×C4⋊C4—26no—
242C2×C4⋊C4—26no—
243C2×C4⋊C4—26no—
244C2×C4⋊C4—46no—
245C2×C4⋊C4—46no—
246C2×C4⋊C4—46no—
247C2×C4⋊C4C2×D4112yes—
248C2×C4⋊C4C2×D4112yes—
249C2×C4⋊C4C2×D4112yes—
250C2×C4⋊C4C2×D4112yes—
251C2×C22⋊C4—26no—
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⋊C4C2×Q8112yes—
274C2×C22⋊C4C2×Q8112yes—
275C2×C42—26no—
276C2×C42—26no—
277C2×C42—46no—
278#95—26no—
279#95—26no—
280#95—26no—
281#95—26no—
282#95—26no—
283#95—26no—
284#95—26no—
285#95—26no—
286#95—26no—
287#95—26no—
288#95—26no—
289#95—26no—
290#95—212no—
291#95—212no—
292#95—46no—
293#95—46no—
294#95—46no—
295#95—46no—
296C24—28no—
297C24—224no—
298C24—224no—
299C24—224no—
300C24—248no—
301C24—224no—
302C24—224no—
303C24—224no—
304C24—224no—
305C24—224no—
306C24—224no—
307C24—26no—
308C24—26no—
309C24—26no—
310C24—26no—
311C24#125116yescomplemented
312C24C22×D416yes—
313C24C22×Q816yes—
314C24C2×#4116yes—
315C24C2×#4116yes—
316C24C2×#4116yes—
317C24H(5,𝔽2)16yes—
318C242−1+416yes—
319C24C2511yescharacteristic
320C22×C4—224no—
321C22×C4—2256no—
322C22×C4—248no—
323C22×C4—248no—
324C22×C4—248no—
325C22×C4—224no—
326C22×C4—224no—
327C22×C4—224no—
328C22×C4—26no—
329C22×C4—21no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—224no—
338C22×C4—224no—
339C22×C4—248no—
340C22×C4—224no—
341C22×C4—224no—
342C22×C4—224no—
343C22×C4—224no—
344C22×C4—23no—
345C22×C4—23no—
346C22×C4—26no—
347C22×C4—26no—
348C22×C4—26no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4—26no—
352C22×C4—23no—
353C22×C4—23no—
354C22×C4—26no—
355C22×C4—26no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—224no—
359C22×C4—224no—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—23no—
367C22×C4—23no—
368C22×C4—26no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—23no—
372C22×C4—23no—
373C22×C4—21no—
374C22×C4—23no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—23no—
378C22×C4—26no—
379C22×C4—26no—
380C22×C4—23no—
381C22×C4—26no—
382C22×C4—26no—
383C22×C4—26no—
384C22×C4—412no—
385C22×C4—412no—
386C22×C4—424no—
387C22×C4—424no—
388C22×C4—424no—
389C22×C4—424no—
390C22×C4—424no—
391C22×C4—424no—
392C22×C4—424no—
393C22×C4—424no—
394C22×C4—4128no—
395C22×C4—4128no—
396C22×C4—412no—
397C22×C4—412no—
398C22×C4—412no—
399C22×C4—412no—
400C22×C4C22×D416yes—
401C22×C4C22×D416yes—
402C22×C4C22×D416yes—
403C22×C4C22×D416yes—
404C22×C4C22×D416yes—
405C22×C4C22×D416yes—
406C22×C4C22×Q812yes—
407C22×C4C22×Q816yes—
408C22×C4C2×#41112yes—
409C4⋊C4—248no—
410C4⋊C4—224no—
411C4⋊C4—224no—
412C4⋊C4—224no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—412no—
419C4⋊C4—412no—
420C4⋊C4—86no—
421C4⋊C4—86no—
422C4⋊C4—86no—
423C4⋊C4#120116yes—
424C22⋊C4—224no—
425C22⋊C4—224no—
426C22⋊C4—412no—
427C22⋊C4—412no—
428C22⋊C4—412no—
429C22⋊C4—412no—
430C22⋊C4—412no—
431C22⋊C4—412no—
432C42—44no—
433C42—86no—
434C23—232no—
435C23—232no—
436C23—296no—
437C23—296no—
438C23—232no—
439C23—296no—
440C23—26no—
441C23—26no—
442C23—212no—
443C23—212no—
444C23—212no—
445C23—212no—
446C23—212no—
447C23—212no—
448C23—212no—
449C23—26no—
450C23—212no—
451C23—26no—
452C23—212no—
453C23—212no—
454C23—212no—
455C23—212no—
456C23—296no—
457C23#525112yes—
458C23C2×#125112yes—
459C23#539112yes—
460C23#546112yes—
461C23D4×Q8112yes—
462C23#552112yes—
463C23#554112yes—
464C23C23×D413yes—
465C23C23×Q811yescharacteristic
466C23C22×#4111yescharacteristic
467C23C22×#4113yes—
468C23C22×#4111yescharacteristic
469C23C2×H(5,𝔽2)13yes—
470C23C2×2−1+413yes—
471C2×C4—296no—
472C2×C4—296no—
473C2×C4—26no—
474C2×C4—26no—
475C2×C4—26no—
476C2×C4—26no—
477C2×C4—212no—
478C2×C4—212no—
479C2×C4—26no—
480C2×C4—26no—
481C2×C4—26no—
482C2×C4—26no—
483C2×C4—26no—
484C2×C4—26no—
485C2×C4—26no—
486C2×C4—26no—
487C2×C4—26no—
488C2×C4—26no—
489C2×C4—212no—
490C2×C4—26no—
491C2×C4—26no—
492C2×C4—448no—
493C2×C4—448no—
494C2×C4—448no—
495C2×C4—448no—
496C2×C4—41no—
497C2×C4—43no—
498C2×C4—46no—
499C2×C4—46no—
500C2×C4—43no—
501C2×C4—46no—
502C2×C4—46no—
503C2×C4—43no—
504C2×C4—46no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—43no—
509C2×C4—43no—
510C2×C4—43no—
511C2×C4—43no—
512C2×C4—43no—
513C2×C4—43no—
514C2×C4—43no—
515C2×C4—46no—
516C2×C4—46no—
517C2×C4—43no—
518C2×C4—43no—
519C2×C4—46no—
520C2×C4—43no—
521C2×C4—43no—
522C2×C4—43no—
523C2×C4—43no—
524C2×C4—43no—
525C2×C4—43no—
526C2×C4—46no—
527C2×C4—46no—
528C2×C4—46no—
529C2×C4C2×#12014yes—
530C2×C4C2×#12014yes—
531C2×C4C2×#12014yes—
532C2×C4#525112yes—
533C22—2128no—
534C22—212no—
535C22—212no—
536C22—212no—
537C22—224no—
538C22—224no—
539C22—212no—
540C22—224no—
541C22—212no—
542C22—212no—
543C22—224no—
544C22#2473124yescomplemented
545C22C22×#12011yescharacteristic
546C22C2×#52513yes—
547C22C2×#52513yes—
548C22C22×#12511yescharacteristic
549C22C2×#53913yes—
550C22C2×#54613yes—
551C22C2×D4×Q813yes—
552C22C2×#55213yes—
553C22C2×#55413yes—
554C22#348413yes—
555C22#348613yes—
556C22#349216yes—
557C4—24no—
558C4—212no—
559C4—24no—
560C4—24no—
561C4—46no—
562C4—46no—
563C4—42no—
564C4—46no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—83no—
569C4—83no—
570C4—83no—
571C2—224no—
572C2C2×#247313yes—
573C2#3498913yes—
574C2#3615211yescharacteristic
575C2#3616313yes—
576C2Q8×#12011yescharacteristic
577C2#3654613yes—
578C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2473C2semidirect
3. #14843C2semidirect
4. #14821C2semidirect
5. #14786C2semidirect
6. #14728C2semidirect
12. #14043C2semidirect
13. #14039C2semidirect
14. #13882C2semidirect
19. C22×#125C22semidirect
26. C2×#400C22semidirect
27. C2×#398C22semidirect
28. C2×#398C22semidirect
29. C2×#398C22semidirect
30. C2×#389C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
49. #1445C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1436C22semidirect
311. C24#125semidirect
544. C22#2473semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#24732
#148432
#148212
#147862
#147282
#146932
#146802
#140432
#140392
#138822
C4×#3792