Finite Group Databasebeta

← #617166

Subgroups — #617166

Order 512.

5194 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2482C211yesmaximal, characteristic
4C2×#2310C211yesmaximal, characteristic, complemented
5#15985C218yesmaximal, complemented
6#15466C214yesmaximal, complemented
7#15465C214yesmaximal, complemented
8#15073C212yesmaximal, complemented
9#14918C212yesmaximal
10#14681C212yesmaximal
11#14336C211yesmaximal, characteristic, complemented
12#14263C212yesmaximal, complemented
13#13870C211yesmaximal, characteristic, complemented
14#13868C211yesmaximal, characteristic
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—28no—
19C22×C22⋊Q8—28no—
20C22×C22⋊Q8C2214yes—
21C22×C22⋊Q8C2214yes—
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29#2484—28no—
30#2482—28no—
31C2×#400C2212yescomplemented
32C2×#399C2212yes—
33C2×#398C2212yescomplemented
34C2×#397C2212yes—
35#2310—28no—
36C2×#391C2212yescomplemented
37C2×#391C2212yescomplemented
38C2×#391C2214yescomplemented
39C2×#388C2214yes—
40C2×#387C2214yescomplemented
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#381C2212yes—
50C2×#381C2212yes—
51C2×#379C2212yes—
52C2×#379C2212yes—
53#1446C2214yescomplemented
54#1446C2214yescomplemented
55#1444C2218yescomplemented
56#1442C2218yescomplemented
57#1442C2218yescomplemented
58#1441C2214yescomplemented
59#1441C2218yescomplemented
60#1440C2218yes—
61#1439C2214yes—
62#1439C2218yes—
63#1437C2214yes—
64#1436C2214yescomplemented
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2218yes—
68C22⋊#95C2218yes—
69C26C2311yescharacteristic
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—28no—
73C23×Q8—44no—
74C23×Q8D418yes—
75C23×Q8C2312yes—
76C23×Q8C2312yes—
77C24×C4—24no—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—28no—
81C24×C4—44no—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8—2128no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4D418yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—24no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4D418yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2318yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#400—28no—
138#399—28no—
139#398—28no—
140#397—28no—
141#391—28no—
142#391—28no—
143#391—216no—
144#388—216no—
145#387—216no—
146#385—28no—
147#385—28no—
148#381—28no—
149#381—28no—
150#381—24no—
151#381—24no—
152#381—28no—
153#381—28no—
154#381—44no—
155#381—44no—
156#379—28no—
157#379—44no—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25—22no—
176C25—22no—
177C25—28no—
178C25—24no—
179C25—24no—
180C25—44no—
181C25C2×D414yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—232no—
185C22×Q8—232no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—28no—
191C22×Q8—28no—
192C22×Q8—416no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8—432no—
196C22×Q8—432no—
197C22×Q8C2×D418yes—
198C22×Q8C2×D418yes—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—232no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—216no—
221C23×C4—48no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—432no—
225C23×C4—432no—
226C23×C4—416no—
227C23×C4—416no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4#4118yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C22⋊Q8—2128no—
250C22⋊Q8—2128no—
251#117—416no—
252#117—416no—
253C2×C4⋊C4—232no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—232no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C22⋊C4—232no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—232no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4C2×D418yes—
309C2×C22⋊C4C2×D418yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—44no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—216no—
328#95—48no—
329#95—48no—
330#95—48no—
331#95—48no—
332C24—24no—
333C24—24no—
334C24—216no—
335C24—216no—
336C24—28no—
337C24—216no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—28no—
344C24—216no—
345C24—48no—
346C24—432no—
347C24—432no—
348C24—48no—
349C24—48no—
350C24—44no—
351C24—44no—
352C24—48no—
353C24C22×D414yes—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)14yes—
359C242−1+418yes—
360C24C2511yescharacteristic
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—216no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—4128no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—416no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—232no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—432no—
436C22×C4—432no—
437C22×C4—4128no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4#120116yes—
450C22×C4#123116yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C4C2×#4118yes—
463C4⋊C4—232no—
464C4⋊C4—232no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—88no—
472C22⋊C4—232no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—816no—
482C42—48no—
483C42—48no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—216no—
491C23—216no—
492C23—232no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—216no—
500C23—216no—
501C23—464no—
502C23—432no—
503C23—416no—
504C23—432no—
505C23—464no—
506C23—416no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23C2×#12014yes—
515C23C2×#12314yes—
516C23#53414yes—
517C23#53618yes—
518C23#54618yes—
519C23#54618yes—
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×D411yescharacteristic
523C23C22×#4112yes—
524C23C2×H(5,𝔽2)12yes—
525C23C2×2−1+414yes—
526C23#58514yes—
527Q8—432no—
528Q8—432no—
529C2×C4—232no—
530C2×C4—232no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—464no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—42no—
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—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4C2×#12014yes—
591C2×C4C2×#12014yes—
592C2×C4C2×#12318yes—
593C22—28no—
594C22—24no—
595C22—28no—
596C22—24no—
597C22—216no—
598C22—216no—
599C22—28no—
600C22—216no—
601C22—464no—
602C22—48no—
603C22—44no—
604C22—48no—
605C22—44no—
606C22—48no—
607C22—416no—
608C22#240518yes—
609C22C22×#12011yescharacteristic
610C22C22×#12311yescharacteristic
611C22C2×#53411yescharacteristic
612C22C2×#53612yes—
613C22#344912yes—
614C22C2×#54612yes—
615C22C2×#54612yes—
616C22#348512yes—
617C22#348612yes—
618C22#348612yes—
619C22#348712yes—
620C22#349518yes—
621C22#353518yes—
622C4—28no—
623C4—24no—
624C4—24no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—84no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4—84no—
634C2—28no—
635C2—48no—
636C2C2×#240511yescharacteristic
637C2#3514911yescharacteristic
638C2#3524211yescharacteristic
639C2#3567212yes—
640C2#3718612yes—
641C2#3783514yes—
642C2#3784114yes—
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×#2484C2semidirect
4. C2×#2310C2semidirect
5. #15985C2semidirect
6. #15466C2semidirect
7. #15465C2semidirect
8. #15073C2semidirect
11. #14336C2semidirect
12. #14263C2semidirect
13. #13870C2semidirect
31. C2×#400C22semidirect
33. C2×#398C22semidirect
36. C2×#391C22semidirect
37. C2×#391C22semidirect
38. C2×#391C22semidirect
40. C2×#387C22semidirect
53. #1446C22semidirect
54. #1446C22semidirect
55. #1444C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
64. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#24842
C2×#24822
C2×#23102
#159852
#154662
#154652
#150732
#149182
#146812
#143362
#142632
#138702
#138682