Finite Group Databasebeta

← #4928982

Subgroups — #4928982

Order 512.

5706 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic, complemented
2C2×#2482C211yesmaximal, characteristic, complemented
3C2×#2404C211yesmaximal, characteristic
4C22×#399C211yesmaximal, characteristic, complemented
5#14905C214yesmaximal, complemented
6#14875C214yesmaximal, complemented
7#14796C214yesmaximal, complemented
8#14777C214yesmaximal, complemented
9#14342C211yesmaximal, characteristic, complemented
10#14015C214yesmaximal, complemented
11#13967C214yesmaximal, complemented
12#13958C211yesmaximal, characteristic, complemented
13#13956C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#120C2212yes—
16C2×#520—28no—
17C2×#520—28no—
18C2×#520—28no—
19C22×C4×Q8C2212yescomplemented
20C22×C4×D4C2214yes—
21C22×C4×D4C2212yescomplemented
22C22×#117C2211yescharacteristic
23C22×#117C2212yescomplemented
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2212yescomplemented
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2212yes—
29#2482—28no—
30#2404—28no—
31C2×#399—28no—
32C2×#399C2214yes—
33C2×#399C2214yes—
34C2×#399C2212yes—
35C2×#397C2212yes—
36C2×#393C2212yes—
37C2×#392C2212yes—
38C2×#388C2214yes—
39C2×#386C2214yes—
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#386C2218yes—
43C2×#385C2212yes—
44C2×#385C2218yescomplemented
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C2×#379C2212yes—
48C22×#95C2211yescharacteristic
49#1446C2212yescomplemented
50#1445C2214yescomplemented
51#1445C2214yescomplemented
52#1445C2212yescomplemented
53#1445C2212yescomplemented
54#1444C2218yescomplemented
55#1439C2218yes—
56#1439C2218yescomplemented
57#1438C2218yes—
58#1438C2212yescomplemented
59#1437C2218yes—
60#1436C2218yescomplemented
61#1434C2214yescomplemented
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yescomplemented
65C26C2311yescharacteristic
66C22×#41—216no—
67C22×#41—24no—
68C22×#41—24no—
69C22×#41D418yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C23×D4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C2×#120—28no—
81C2×#120—216no—
82#520—2128no—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—28no—
86C2×C4×D4—216no—
87C2×C4×D4—28no—
88C2×C4×D4—216no—
89C2×C4×D4—28no—
90C2×C4×D4—216no—
91C2×C4×D4—28no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117D418yescomplemented
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—48no—
105C22×C4⋊C4D418yescomplemented
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2318yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—48no—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132#399—216no—
133#399—216no—
134#399—28no—
135#397—28no—
136#393—28no—
137#392—28no—
138#388—216no—
139#386—28no—
140#386—216no—
141#386—216no—
142#386—416no—
143#385—28no—
144#385—416no—
145#381—24no—
146#381—24no—
147#379—28no—
148C2×#95—24no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C25—22no—
163C25—24no—
164C25—22no—
165C25—22no—
166C25—28no—
167C25—44no—
168C25—42no—
169C25C2412yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C2×#41—264no—
173C2×#41—264no—
174C2×#41—264no—
175C2×#41—232no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—28no—
179C22×Q8C2×D414yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—216no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—44no—
190C22×D4—48no—
191C22×D4C2×D414yes—
192C22×D4C2×D418yes—
193C22×D4#41116yes—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—22no—
207C23×C4—28no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—48no—
215C23×C4—48no—
216C23×C4C2×D418yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D418yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231#120—432no—
232C4×Q8—264no—
233C4×D4—2128no—
234C4×D4—264no—
235#117—264no—
236#117—232no—
237C2×C4⋊C4—264no—
238C2×C4⋊C4—232no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4—416no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4C2×D418yes—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42C2×D418yes—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—216no—
308#95—216no—
309#95—48no—
310#95—48no—
311#95—48no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—28no—
319C24—24no—
320C24—24no—
321C24—28no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—28no—
326C24—28no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—48no—
331C24—48no—
332C24—432no—
333C24—48no—
334C24—44no—
335C24—44no—
336C24—48no—
337C24—48no—
338C24—48no—
339C24—84no—
340C24—84no—
341C24—816no—
342C24C22×D414yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C2×#4118yes—
347C24C2×#4118yes—
348C242−1+414yes—
349C24C2511yescharacteristic
350#41—2256no—
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—28no—
354C2×Q8—216no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4—232no—
361C2×D4—232no—
362C2×D4—28no—
363C2×D4—416no—
364C2×D4—432no—
365C2×D4—416no—
366C2×D4—88no—
367C2×D4—816no—
368C2×D4#123132yes—
369C22×C4—232no—
370C22×C4—232no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—232no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—432no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—816no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—816no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4#120116yes—
430C22×C4C22×D414yes—
431C22×C4C22×D418yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C2×#4118yes—
441C22×C4H(5,𝔽2)18yes—
442C4⋊C4—232no—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—28no—
447C4⋊C4—432no—
448C4⋊C4—88no—
449C22⋊C4—232no—
450C22⋊C4—216no—
451C22⋊C4—432no—
452C22⋊C4—416no—
453C22⋊C4—432no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C42—232no—
462C42—216no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—216no—
471C23—216no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—216no—
479C23—216no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—216no—
484C23—432no—
485C23—416no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—832no—
507C23—832no—
508C23—816no—
509C23—88no—
510C23—88no—
511C23C2×#12014yes—
512C23C2×#12318yes—
513C23C2×#12318yes—
514C23#52518yes—
515C23#52518yes—
516C23C23×D411yescharacteristic
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C22×#4114yes—
520C23C2×H(5,𝔽2)14yes—
521C23C2×2−1+412yes—
522C23#58512yes—
523Q8—232no—
524D4—232no—
525D4—264no—
526D4—832no—
527C2×C4—264no—
528C2×C4—232no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—216no—
548C2×C4—432no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—816no—
574C2×C4—832no—
575C2×C4—88no—
576C2×C4C2×#12014yes—
577C2×C4C2×#12014yes—
578C2×C4C2×#12318yes—
579C22—28no—
580C22—28no—
581C22—24no—
582C22—216no—
583C22—216no—
584C22—24no—
585C22—28no—
586C22—216no—
587C22—48no—
588C22—48no—
589C22—44no—
590C22—48no—
591C22—416no—
592C22—44no—
593C22—48no—
594C22—44no—
595C22—48no—
596C22—48no—
597C22—832no—
598C22—82no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—82no—
603C22#284918yes—
604C22C22×#12011yescharacteristic
605C22C22×#12312yes—
606C22C2×#52512yes—
607C22C2×#52512yes—
608C22#348412yes—
609C22#348611yescharacteristic
610C22#348711yescharacteristic
611C22#348912yes—
612C22#348912yes—
613C22#349812yes—
614C22#349812yes—
615C22#351318yes—
616C22#352018yes—
617C4—24no—
618C4—24no—
619C4—28no—
620C4—48no—
621C4—44no—
622C4—48no—
623C4—44no—
624C4—84no—
625C4—84no—
626C4—82no—
627C2—28no—
628C2—48no—
629C2—84no—
630C2#3535612yes—
631C2#3535712yes—
632C2#3552711yescharacteristic
633C2#3565911yescharacteristic
634C2#3598714yes—
635C2#3599014yes—
636C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#520C2semidirect
2. C2×#2482C2semidirect
4. C22×#399C2semidirect
5. #14905C2semidirect
6. #14875C2semidirect
7. #14796C2semidirect
8. #14777C2semidirect
9. #14342C2semidirect
10. #14015C2semidirect
11. #13967C2semidirect
12. #13958C2semidirect
13. #13956C2semidirect
19. C22×C4×Q8C22semidirect
21. C22×C4×D4C22semidirect
23. C22×#117C22semidirect
26. C23×C4⋊C4C22semidirect
44. C2×#385C22semidirect
49. #1446C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
56. #1439C22semidirect
58. #1438C22semidirect
60. #1436C22semidirect
61. #1434C22semidirect
64. C22⋊#95C22semidirect
97. C2×#117D4semidirect
105. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#24822
C2×#24042
C22×#3992
#149052
#148752
#147962
#147772
#143422
#140152
#139672
#139582
#139562