Finite Group Databasebeta

← #4310263

Subgroups — #4310263

Order 512.

3694 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#17059C212yesmaximal, complemented
4#17058C212yesmaximal, complemented
5#16400C214yesmaximal, complemented
6#16399C214yesmaximal, complemented
7#16366C212yesmaximal, complemented
8#16365C212yesmaximal, complemented
9#14843C212yesmaximal, complemented
10#14786C212yesmaximal, complemented
11#14727C214yesmaximal
12#14680C211yesmaximal, characteristic
13#14281C212yesmaximal, complemented
14#14275C212yesmaximal, complemented
15C25×C4C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22#2848—28no—
23C2×#400C2214yescomplemented
24C2×#398C2214yescomplemented
25C2×#397C2218yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#385C2214yes—
30C2×#380—28no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#379C2212yes—
42C2×#379C2212yes—
43C22×#95C2212yes—
44#1446C2214yescomplemented
45#1445C2214yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1444C2214yescomplemented
49#1444C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2218yescomplemented
53#1441C2218yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2218yes—
57#1440C2218yes—
58#1439C2218yes—
59C22⋊#95C2212yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C26C2311yescharacteristic
66C23×Q8C2311yescharacteristic
67C24×C4—28no—
68C24×C4—44no—
69C24×C4—44no—
70C24×C4D418yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2318yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—48no—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114#400—48no—
115#398—48no—
116#397—416no—
117#385—28no—
118#385—28no—
119#385—28no—
120#385—48no—
121#380—24no—
122#380—24no—
123#380—24no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380—28no—
129#380—44no—
130#380—44no—
131#379—28no—
132#379—44no—
133C2×#95—28no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—22no—
153C25—24no—
154C25—22no—
155C25—42no—
156C25—42no—
157C25C2×D414yes—
158C25C2411yescharacteristic
159C25C2411yescharacteristic
160C25C2411yescharacteristic
161C22×Q8—22no—
162C22×Q8—24no—
163C22×Q8—42no—
164C22×Q8—42no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—232no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—416no—
183C23×C4—416no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—464no—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4#4118yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2418yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C22⋊Q8—416no—
214C22⋊Q8—416no—
215C22⋊Q8—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—432no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—48no—
280#95—48no—
281#95—48no—
282#95—48no—
283#95—48no—
284#95—48no—
285#95—48no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—232no—
291C24—28no—
292C24—28no—
293C24—28no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—21no—
308C24—28no—
309C24—28no—
310C24—44no—
311C24—44no—
312C24—48no—
313C24—416no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—416no—
318C24—416no—
319C24—48no—
320C24—48no—
321C24C22×D412yes—
322C24C22×D412yes—
323C24C22×D412yes—
324C24C2×#4114yes—
325C24H(5,𝔽2)14yes—
326C24H(5,𝔽2)12yes—
327C242−1+414yes—
328C242−1+414yes—
329C242−1+412yes—
330C24C2511yescharacteristic
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—88no—
335C22×C4—232no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—4128no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—88no—
408C22×C4—816no—
409C22×C4—88no—
410C22×C4#123116yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4H(5,𝔽2)14yes—
418C22×C4H(5,𝔽2)14yes—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—88no—
430C4⋊C4—816no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—416no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—28no—
447C23—28no—
448C23—24no—
449C23—28no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—28no—
454C23—232no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—28no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—416no—
476C23—416no—
477C23—416no—
478C23—416no—
479C23—416no—
480C23—464no—
481C23—464no—
482C23—416no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23C2×#12314yes—
496C23#53514yes—
497C23#53614yes—
498C23#53614yes—
499C23#53714yes—
500C23#54118yes—
501C23#54218yes—
502C23C23×D411yescharacteristic
503C23C22×#4112yes—
504C23C2×H(5,𝔽2)12yes—
505C23C2×H(5,𝔽2)12yes—
506C23C2×H(5,𝔽2)12yes—
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×2−1+412yes—
509C23C2×2−1+412yes—
510C23C2×2−1+411yescharacteristic
511Q8—88no—
512C2×C4—232no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—416no—
526C2×C4—416no—
527C2×C4—416no—
528C2×C4—416no—
529C2×C4—416no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—416no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4C2×#12314yes—
567C2×C4C2×#12314yes—
568C2×C4#53414yes—
569C2×C4#53414yes—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—216no—
575C22—216no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—464no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—48no—
592C22—48no—
593C22—48no—
594C22—48no—
595C22—42no—
596C22—42no—
597C22#262518yes—
598C22C22×#12311yescharacteristic
599C22C2×#53411yescharacteristic
600C22C2×#53411yescharacteristic
601C22C2×#53511yescharacteristic
602C22C2×#53611yescharacteristic
603C22C2×#53611yescharacteristic
604C22C2×#53711yescharacteristic
605C22C2×#54112yes—
606C22C2×#54212yes—
607C22#349014yes—
608C22#349214yes—
609C22#353612yes—
610C22#353612yes—
611C22#353614yes—
612C22#353712yes—
613C22#353812yes—
614C22#353814yes—
615C4—24no—
616C4—24no—
617C4—24no—
618C4—24no—
619C4—82no—
620C4—84no—
621C4—82no—
622C4—84no—
623C4—88no—
624C4—82no—
625C4—82no—
626C2—28no—
627C2—44no—
628C2—44no—
629C2C2×#262511yescharacteristic
630C2#3499211yescharacteristic
631C2#3522711yescharacteristic
632C2#3672814yes—
633C2#3672914yes—
634C2#3743911yescharacteristic
635C2#3744211yescharacteristic
636C2#3745111yescharacteristic
637C2#3745311yescharacteristic
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
3. #17059C2semidirect
4. #17058C2semidirect
5. #16400C2semidirect
6. #16399C2semidirect
7. #16366C2semidirect
8. #16365C2semidirect
9. #14843C2semidirect
10. #14786C2semidirect
13. #14281C2semidirect
14. #14275C2semidirect
23. C2×#400C22semidirect
24. C2×#398C22semidirect
44. #1446C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C22×#3802
#170592
#170582
#164002
#163992
#163662
#163652
#148432
#147862
#147272
#146802
#142812
#142752