Finite Group Databasebeta

← #4474524

Subgroups — #4474524

Order 512.

3450 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17234C214yesmaximal, complemented
2#16903C211yesmaximal, characteristic, complemented
3#16690C214yesmaximal, complemented
4#16258C214yesmaximal, complemented
5#16252C211yesmaximal, characteristic, complemented
6#15369C214yesmaximal, complemented
7#15273C214yesmaximal, complemented
8#15257C214yesmaximal, complemented
9#14716C212yesmaximal, complemented
10#14417C212yesmaximal, complemented
11#14341C211yesmaximal, characteristic, complemented
12C22×#121C2212yes—
13C22×#121C2214yes—
14C22×#120C2214yes—
15C23×C22⋊C4C2211yescharacteristic
16C2×#392C2214yes—
17C2×#392C2212yes—
18C2×#392C2212yes—
19C2×#392C2214yes—
20C2×#392C2214yescomplemented
21C2×#386C2214yes—
22C2×#386C2214yes—
23C2×#386C2214yes—
24C2×#386C2214yescomplemented
25C2×#386C2214yescomplemented
26C2×#381C2214yescomplemented
27C2×#379C2212yes—
28#1445C2214yescomplemented
29#1443C2214yescomplemented
30#1443C2214yescomplemented
31#1442C2214yescomplemented
32#1442C2214yescomplemented
33#1440C2214yes—
34#1440C2214yes—
35#1440C2214yes—
36#1440C2214yescomplemented
37#1440C2214yescomplemented
38#1439C2214yes—
39#1439C2212yes—
40#1439C2214yes—
41#1439C2212yes—
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1438C2214yes—
46#1438C2214yes—
47#1438C2214yes—
48#1438C2214yes—
49#1438C2214yes—
50#1438C2214yes—
51#1438C2214yescomplemented
52C22⋊#95C2211yescharacteristic
53C22⋊#95C2212yes—
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2214yescomplemented
57C26C2311yescharacteristic
58C23×D4C2314yes—
59C23×D4C2312yes—
60C23×D4C2314yes—
61C23×D4C2314yes—
62C23×D4C2314yes—
63C24×C4C2314yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C2×#121—28no—
67C2×#121—48no—
68C2×#121—48no—
69C2×#121—48no—
70C2×#121—48no—
71C2×#120—48no—
72C2×#120—48no—
73C2×#120—48no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—44no—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95#392—48no—
96#392—44no—
97#392—44no—
98#392—48no—
99#392—48no—
100#386—48no—
101#386—48no—
102#386—48no—
103#386—48no—
104#386—48no—
105#381—48no—
106#379—44no—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2312yes—
111C2×#95C2312yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C25—22no—
125C25—24no—
126C25—24no—
127C25—41no—
128C25—41no—
129C25—44no—
130C25—44no—
131C25C2412yes—
132C25C2414yes—
133C25C2411yescharacteristic
134C22×D4—28no—
135C22×D4—28no—
136C22×D4—28no—
137C22×D4—28no—
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—28no—
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—28no—
145C22×D4—24no—
146C22×D4—28no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—42no—
162C22×D4—44no—
163C22×D4—42no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C23×C4—28no—
168C23×C4—22no—
169C23×C4—28no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4#4114yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188#121—432no—
189#121—464no—
190#120—464no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—416no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—44no—
264#95—44no—
265C24—24no—
266C24—22no—
267C24—24no—
268C24—21no—
269C24—21no—
270C24—28no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—48no—
296C24—48no—
297C24—44no—
298C24—416no—
299C24—44no—
300C24—84no—
301C24—84no—
302C24—88no—
303C24—88no—
304C24—88no—
305C24—88no—
306C24—88no—
307C24C2×#4112yes—
308C24H(5,𝔽2)14yes—
309C24H(5,𝔽2)18yes—
310C24H(5,𝔽2)14yes—
311C24C2511yescharacteristic
312C2×D4—216no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—416no—
316C2×D4—416no—
317C2×D4—416no—
318C2×D4—416no—
319C2×D4—416no—
320C2×D4—416no—
321C2×D4—416no—
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—88no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4—88no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C2×D4—88no—
343C2×D4—88no—
344C2×D4—88no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—416no—
392C22×C4—44no—
393C22×C4—416no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—88no—
397C22×C4—84no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4C2×#4114yes—
403C22×C4H(5,𝔽2)18yes—
404C4⋊C4—48no—
405C4⋊C4—416no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—416no—
409C22⋊C4—416no—
410C22⋊C4—416no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C23—22no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—28no—
428C23—24no—
429C23—24no—
430C23—28no—
431C23—24no—
432C23—28no—
433C23—28no—
434C23—28no—
435C23—28no—
436C23—28no—
437C23—28no—
438C23—24no—
439C23—24no—
440C23—28no—
441C23—28no—
442C23—28no—
443C23—416no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—816no—
495C23—88no—
496C23—88no—
497C23—832no—
498C23—832no—
499C23—816no—
500C23#54014yes—
501C23C22×#4111yescharacteristic
502C23C2×H(5,𝔽2)12yes—
503C23C2×H(5,𝔽2)14yes—
504C23C2×H(5,𝔽2)12yes—
505C23C2×H(5,𝔽2)14yes—
506C23#58511yescharacteristic
507C23#58511yescharacteristic
508D4—816no—
509D4—832no—
510D4—832no—
511D4—832no—
512D4—832no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—88no—
560C2×C4—832no—
561C2×C4—88no—
562C22—28no—
563C22—24no—
564C22—48no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—48no—
569C22—48no—
570C22—48no—
571C22—48no—
572C22—48no—
573C22—44no—
574C22—44no—
575C22—48no—
576C22—44no—
577C22—48no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—832no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22C2×#54011yescharacteristic
597C22#349012yes—
598C22#349011yescharacteristic
599C22#349311yescharacteristic
600C22#350812yes—
601C22#353214yes—
602C22#353214yes—
603C22#353412yes—
604C22#353414yes—
605C22#353412yes—
606C22#353714yes—
607C22#353714yes—
608C22#354214yes—
609C4—48no—
610C4—44no—
611C4—84no—
612C4—84no—
613C4—81no—
614C4—81no—
615C4—84no—
616C4—84no—
617C2—44no—
618C2—88no—
619C2—84no—
620C2#3673111yescharacteristic
621C2#4054012yes—
622C2#4645411yescharacteristic
623C2#4699611yescharacteristic
624C2#4959512yes—
625C2#5503414yes—
626C2#5638414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17234C2semidirect
2. #16903C2semidirect
3. #16690C2semidirect
4. #16258C2semidirect
5. #16252C2semidirect
6. #15369C2semidirect
7. #15273C2semidirect
8. #15257C2semidirect
9. #14716C2semidirect
10. #14417C2semidirect
11. #14341C2semidirect
20. C2×#392C22semidirect
24. C2×#386C22semidirect
25. C2×#386C22semidirect
26. C2×#381C22semidirect
28. #1445C22semidirect
29. #1443C22semidirect
30. #1443C22semidirect
31. #1442C22semidirect
32. #1442C22semidirect
36. #1440C22semidirect
37. #1440C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
51. #1438C22semidirect
56. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#172342
#169032
#166902
#162582
#162522
#153692
#152732
#152572
#147162
#144172
#143412