Finite Group Databasebeta

← #627063

Subgroups — #627063

Order 512.

5446 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic, complemented
2C2×#2626C212yesmaximal, complemented
3C22×#390C211yesmaximal, characteristic
4#15325C218yesmaximal, complemented
5#15280C218yesmaximal, complemented
6#14720C212yesmaximal, complemented
7#14717C212yesmaximal, complemented
8#14702C212yesmaximal, complemented
9#14338C211yesmaximal, characteristic, complemented
10#14127C212yesmaximal, complemented
11#13867C212yesmaximal
12C24×D4C2211yescharacteristic
13C2×#543—216no—
14C2×#543—28no—
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2214yescomplemented
18C22×C22⋊Q8C2214yescomplemented
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C42C2211yescharacteristic
26#2626—216no—
27C2×#399C2214yescomplemented
28C2×#396C2218yescomplemented
29C2×#395C2214yescomplemented
30C2×#393C2214yescomplemented
31C2×#393C2214yescomplemented
32C2×#391C2214yescomplemented
33C2×#390—28no—
34C2×#390C2218yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yescomplemented
39C2×#380C2212yes—
40C2×#379C2212yes—
41C2×#379C2214yes—
42#1444C2218yescomplemented
43#1441C2214yescomplemented
44#1439C2218yescomplemented
45#1439C2218yescomplemented
46#1439C2214yescomplemented
47#1439C2218yescomplemented
48#1438C2218yes—
49#1436C2218yescomplemented
50C4⋊#95C2214yescomplemented
51C22⋊#95C2212yes—
52C22⋊#95C2214yes—
53C22⋊#95C2214yes—
54C22⋊#95C2214yescomplemented
55C26C2311yescharacteristic
56C26C2311yescharacteristic
57C23×Q8C2312yes—
58C23×Q8C2312yes—
59C23×D4—24no—
60C23×D4—24no—
61C23×D4—44no—
62C23×D4—44no—
63C23×D4Q818yescomplemented
64C23×D4C2314yes—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72#543—2128no—
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8D4116yescomplemented
84C22×C4⋊C4—24no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2318yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—216no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4D418yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42—24no—
115C22×C42—44no—
116C22×C42C2314yes—
117#399—216no—
118#396—232no—
119#395—48no—
120#393—216no—
121#393—48no—
122#391—48no—
123#390—232no—
124#386—48no—
125#386—48no—
126#385—216no—
127#385—48no—
128#380—28no—
129#379—28no—
130#379—216no—
131C2×#95C2318yes—
132C2×#95C2314yes—
133C2×#95C2318yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2318yes—
139C25—24no—
140C25—24no—
141C25—22no—
142C25—22no—
143C25—24no—
144C25—24no—
145C25—42no—
146C25—42no—
147C25—42no—
148C25—42no—
149C25—42no—
150C25—42no—
151C25—42no—
152C25—42no—
153C25C2×D414yes—
154C25C2×Q814yes—
155C25C2×Q814yes—
156C25#4114yes—
157C25C2412yes—
158C25C2412yes—
159C25C2411yescharacteristic
160C22×Q8—28no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8C2×D418yescomplemented
169C22×D4—232no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—432no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—416no—
178C22×D4—416no—
179C22×D4—432no—
180C22×D4C2×Q8116yes—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D414yes—
207C23×C4C2×D418yes—
208C23×C4C2×D414yes—
209C23×C4C2×Q814yes—
210C23×C4C2418yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C22⋊Q8—2128no—
219C22⋊Q8—2128no—
220C22⋊Q8—264no—
221C22⋊Q8—264no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—416no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4C2×D4116yescomplemented
242C2×C4⋊C4C2×D418yescomplemented
243C2×C22⋊C4—264no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—232no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—416no—
271C2×C22⋊C4—416no—
272C2×C22⋊C4—416no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4C2×D4116yes—
280C2×C22⋊C4C2×D4116yes—
281C2×C42—28no—
282C2×C42—28no—
283C2×C42—44no—
284C2×C42—416no—
285#95—28no—
286#95—216no—
287#95—216no—
288#95—216no—
289#95—216no—
290#95—48no—
291#95—44no—
292#95—44no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—28no—
304C24—28no—
305C24—216no—
306C24—216no—
307C24—28no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—216no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—44no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—44no—
328C24—44no—
329C24—44no—
330C24—48no—
331C24—48no—
332C24—44no—
333C24—44no—
334C24—44no—
335C24—48no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24—88no—
340C24—88no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24C22⋊Q8116yes—
345C24C22×D414yes—
346C24C22×D412yes—
347C24C22×D414yes—
348C24C22×Q814yes—
349C24C22×Q814yes—
350C24C22×Q812yes—
351C24C2×#4114yes—
352C24C2×#4112yes—
353C242−1+414yes—
354C24C2511yescharacteristic
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—416no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8—88no—
361C2×Q8—84no—
362C2×Q8—84no—
363C2×Q8—816no—
364C2×D4—232no—
365C2×D4—232no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—4128no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—416no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—416no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—816no—
425C22×C4—816no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4#120116yes—
433C22×C4C22×D418yes—
434C22×C4C22×D418yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×Q814yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C4H(5,𝔽2)14yes—
443C4⋊C4—216no—
444C4⋊C4—232no—
445C4⋊C4—232no—
446C4⋊C4—216no—
447C4⋊C4—416no—
448C4⋊C4—88no—
449C4⋊C4—816no—
450C22⋊C4—232no—
451C22⋊C4—232no—
452C22⋊C4—232no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—816no—
459C22⋊C4—816no—
460C22⋊C4—88no—
461C42—416no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—232no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—416no—
492C23—416no—
493C23—416no—
494C23—416no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—416no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23—88no—
517C23—832no—
518C23—832no—
519C23—832no—
520C23—88no—
521C23—88no—
522C23—88no—
523C23—88no—
524C23—832no—
525C23C2×#12014yes—
526C23C2×C22⋊Q818yes—
527C23C2×C22⋊Q814yes—
528C23#53614yes—
529C23#54818yes—
530C23D4×Q818yes—
531C23C23×D411yescharacteristic
532C23C23×D412yes—
533C23C23×Q811yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×2−1+412yes—
539C23#58512yes—
540Q8—816no—
541Q8—816no—
542D4—464no—
543C2×C4—232no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—216no—
561C2×C4—216no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—88no—
587C2×C4—816no—
588C2×C4—816no—
589C2×C4—88no—
590C2×C4C2×#12018yes—
591C2×C4#54318yescomplemented
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—216no—
599C22—216no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—416no—
608C22—416no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—416no—
613C22—832no—
614C22—832no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—88no—
620C22#240918yes—
621C22C22×#12011yescharacteristic
622C22C22×C22⋊Q811yescharacteristic
623C22C2×#53611yescharacteristic
624C22C2×#54312yes—
625C22#345912yes—
626C22C2×#54812yes—
627C22C2×D4×Q812yes—
628C22#348411yescharacteristic
629C22#348411yescharacteristic
630C22#348411yescharacteristic
631C22#348511yescharacteristic
632C22#349012yes—
633C22#349614yes—
634C22#349814yes—
635C22#350112yes—
636C22#353614yes—
637C22#356114yes—
638C4—28no—
639C4—28no—
640C4—48no—
641C4—44no—
642C4—44no—
643C4—48no—
644C4—82no—
645C4—88no—
646C2—28no—
647C2—48no—
648C2—84no—
649C2C2×#240911yescharacteristic
650C2#3498211yescharacteristic
651C2#3498511yescharacteristic
652C2#3534711yescharacteristic
653C2#3552711yescharacteristic
654C2#3671811yescharacteristic
655C2#3696611yescharacteristic
656C2#3936714yes—
657C2#3942014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#543C2semidirect
2. C2×#2626C2semidirect
4. #15325C2semidirect
5. #15280C2semidirect
6. #14720C2semidirect
7. #14717C2semidirect
8. #14702C2semidirect
9. #14338C2semidirect
10. #14127C2semidirect
17. C22×C22⋊Q8C22semidirect
18. C22×C22⋊Q8C22semidirect
27. C2×#399C22semidirect
28. C2×#396C22semidirect
29. C2×#395C22semidirect
30. C2×#393C22semidirect
31. C2×#393C22semidirect
32. C2×#391C22semidirect
38. C2×#385C22semidirect
42. #1444C22semidirect
43. #1441C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
49. #1436C22semidirect
50. C4⋊#95C22semidirect
54. C22⋊#95C22semidirect
63. C23×D4Q8semidirect
83. C2×C22⋊Q8D4semidirect
168. C22×Q8C2×D4semidirect
241. C2×C4⋊C4C2×D4semidirect
242. C2×C4⋊C4C2×D4semidirect
591. C2×C4#543semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#26262
C22×#3902
#153252
#152802
#147202
#147172
#147022
#143382
#141272
#138672