Finite Group Databasebeta

← #636596

Subgroups — #636596

Order 512.

5540 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2579C212yesmaximal, complemented
3C2×C4×#120C211yesmaximal, characteristic
4#16265C214yesmaximal, complemented
5#16264C214yesmaximal, complemented
6#15092C211yesmaximal, characteristic, complemented
7#15091C211yesmaximal, characteristic, complemented
8#15001C214yesmaximal, complemented
9#14874C211yesmaximal, characteristic, complemented
10#14873C211yesmaximal, characteristic, complemented
11#14723C214yesmaximal, complemented
12#14680C211yesmaximal, characteristic, complemented
13#14262C212yesmaximal, complemented
14#14255C212yesmaximal
15#13923C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×C22⋊Q8—28no—
19C22×C22⋊Q8—216no—
20C22×C22⋊Q8C2214yes—
21C22×C22⋊Q8C2214yes—
22C22×#120C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×C4×D4C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C22⋊C4C2212yes—
28#2579—216no—
29C2×#398C2214yescomplemented
30C2×#396C2214yescomplemented
31C2×#395C2214yescomplemented
32C2×#393C2214yescomplemented
33C4×#120—28no—
34C2×#387C2214yescomplemented
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2211yescharacteristic
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2211yescharacteristic
42C2×#385C2212yes—
43C2×#385C2214yescomplemented
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×#379C2212yes—
47C2×#379C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1446C2214yescomplemented
51#1446C2214yescomplemented
52#1444C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2218yescomplemented
55#1441C2214yescomplemented
56#1440C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1439C2214yescomplemented
60#1438C2218yes—
61#1436C2218yescomplemented
62C22⋊#95C2211yescharacteristic
63C22⋊#95C2212yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×Q8—28no—
71C23×Q8—24no—
72C23×Q8—48no—
73C23×Q8C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C24×C4—28no—
77C24×C4—48no—
78C24×C4D418yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×C22⋊Q8—2128no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—232no—
90C2×C22⋊Q8—232no—
91C2×#120—24no—
92C2×#120—24no—
93C2×#120Q818yescomplemented
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—24no—
97C2×C4×D4—48no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4D4116yescomplemented
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#398—216no—
131#396—216no—
132#395—48no—
133#393—48no—
134#387—216no—
135#386—28no—
136#386—28no—
137#386—24no—
138#386—28no—
139#385—28no—
140#385—28no—
141#385—24no—
142#385—28no—
143#385—216no—
144#380—28no—
145#380—48no—
146#379—24no—
147#379—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—24no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C25—24no—
165C25—24no—
166C25—44no—
167C25—44no—
168C25C2×D414yes—
169C25C2×Q814yes—
170C25#4114yes—
171C25C2411yescharacteristic
172C25C2412yes—
173C25C2411yescharacteristic
174C22×Q8—232no—
175C22×Q8—216no—
176C22×Q8—28no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—48no—
180C22×Q8—464no—
181C22×Q8—432no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4C2×Q814yes—
191C22×D4C2×Q818yescomplemented
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—232no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—432no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—48no—
210C23×C4—48no—
211C23×C4—44no—
212C23×C4—464no—
213C23×C4C2×D414yes—
214C23×C4C2×D418yes—
215C23×C4C2×D418yes—
216C23×C4C2×D418yes—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2×D418yes—
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C22⋊Q8—2128no—
233C22⋊Q8—2128no—
234#120—232no—
235C4×D4—432no—
236C4×D4—416no—
237C2×C4⋊C4—232no—
238C2×C4⋊C4—264no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4C2×D4116yes—
256C2×C4⋊C4C2×D4116yes—
257C2×C22⋊C4—264no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4C2×Q814yes—
291C2×C22⋊C4C2×Q818yescomplemented
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—44no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—48no—
307#95—48no—
308#95—48no—
309#95—48no—
310C24—24no—
311C24—24no—
312C24—28no—
313C24—28no—
314C24—216no—
315C24—216no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—48no—
329C24—48no—
330C24—416no—
331C24—416no—
332C24—416no—
333C24—416no—
334C24—432no—
335C24C22⋊Q8116yes—
336C24C22×D414yes—
337C24C22×D412yes—
338C24C22×D414yes—
339C24C22×Q812yes—
340C24C22×Q814yes—
341C24C22×Q812yes—
342C24C2×#4114yes—
343C24C2×#4112yes—
344C242−1+414yes—
345C24C2511yescharacteristic
346C2×Q8—232no—
347C2×Q8—432no—
348C2×Q8—432no—
349C2×Q8—4128no—
350C2×Q8—416no—
351C2×Q8—416no—
352C2×D4—28no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—416no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—416no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—48no—
362C22×C4—232no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—432no—
418C22×C4—432no—
419C22×C4—4128no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—44no—
426C22×C4—88no—
427C22×C4#120116yes—
428C22×C4C22×D418yes—
429C22×C4C22×D418yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q812yes—
437C22×C4H(5,𝔽2)18yes—
438C4⋊C4—232no—
439C4⋊C4—232no—
440C4⋊C4—232no—
441C4⋊C4—416no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C22⋊C4—264no—
447C22⋊C4—28no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—88no—
459C22⋊C4—816no—
460C42—48no—
461C42—44no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—216no—
469C23—24no—
470C23—232no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—22no—
483C23—24no—
484C23—28no—
485C23—432no—
486C23—416no—
487C23—464no—
488C23—416no—
489C23—432no—
490C23—464no—
491C23—48no—
492C23—48no—
493C23—48no—
494C23—48no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—48no—
504C23C2×#12014yes—
505C23C2×C22⋊Q818yes—
506C23C2×C22⋊Q814yes—
507C23#53614yes—
508C23#54818yes—
509C23D4×Q818yes—
510C23C23×D411yescharacteristic
511C23C23×D412yes—
512C23C23×Q811yescharacteristic
513C23C22×#4111yescharacteristic
514C23C2×H(5,𝔽2)14yes—
515C23C2×2−1+412yes—
516C23#58514yes—
517Q8—464no—
518D4—432no—
519D4—416no—
520C2×C4—232no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—216no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—432no—
540C2×C4—432no—
541C2×C4—432no—
542C2×C4—432no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—88no—
574C2×C4C2×#12014yes—
575C2×C4C2×#12014yes—
576C2×C4#53418yes—
577C22—216no—
578C22—24no—
579C22—28no—
580C22—24no—
581C22—216no—
582C22—28no—
583C22—28no—
584C22—28no—
585C22—28no—
586C22—464no—
587C22—416no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—416no—
593C22—42no—
594C22—48no—
595C22—44no—
596C22—84no—
597C22—81no—
598C22—82no—
599C22—81no—
600C22#240918yes—
601C22C22×#12011yescharacteristic
602C22C22×C22⋊Q811yescharacteristic
603C22C2×#53412yes—
604C22C2×#53611yescharacteristic
605C22#344914yes—
606C22#344912yes—
607C22C2×#54812yes—
608C22C2×D4×Q812yes—
609C22#348414yes—
610C22#350114yes—
611C22#350514yes—
612C22#354118yes—
613C4—28no—
614C4—24no—
615C4—24no—
616C4—48no—
617C4—42no—
618C4—48no—
619C4—44no—
620C4—84no—
621C4—84no—
622C4—81no—
623C4—82no—
624C4—81no—
625C2—28no—
626C2—48no—
627C2—82no—
628C2#3463211yescharacteristic
629C2C2×#240911yescharacteristic
630C2#3498011yescharacteristic
631C2#3501012yes—
632C2#3744912yes—
633C2#3958814yes—
634C2#3971614yes—
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. C23×C22⋊Q8C2semidirect
2. C2×#2579C2semidirect
4. #16265C2semidirect
5. #16264C2semidirect
6. #15092C2semidirect
7. #15091C2semidirect
8. #15001C2semidirect
9. #14874C2semidirect
10. #14873C2semidirect
11. #14723C2semidirect
12. #14680C2semidirect
13. #14262C2semidirect
15. #13923C2semidirect
29. C2×#398C22semidirect
30. C2×#396C22semidirect
31. C2×#395C22semidirect
32. C2×#393C22semidirect
34. C2×#387C22semidirect
43. C2×#385C22semidirect
50. #1446C22semidirect
51. #1446C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
59. #1439C22semidirect
61. #1436C22semidirect
93. C2×#120Q8semidirect
102. C22×C4⋊C4D4semidirect
191. C22×D4C2×Q8semidirect
291. C2×C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#25792
C2×C4×#1202
#162652
#162642
#150922
#150912
#150012
#148742
#148732
#147232
#146802
#142622
#142552
#139232