Finite Group Databasebeta

← #4387675

Subgroups — #4387675

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16415C214yesmaximal, complemented
2#16005C214yesmaximal, complemented
3#15987C214yesmaximal, complemented
4#15797C214yesmaximal, complemented
5#15481C214yesmaximal, complemented
6#15018C214yesmaximal, complemented
7#14673C211yesmaximal, characteristic
8#14341C211yesmaximal, characteristic
9#14338C211yesmaximal, characteristic
10#14127C211yesmaximal, characteristic
11#13869C211yesmaximal, characteristic
12#13869C211yesmaximal, characteristic
13#13775C211yesmaximal, characteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C2×#400C2214yescomplemented
18C2×#385C2212yes—
19C2×#385C2212yes—
20C2×#383C2214yescomplemented
21C2×#381C2212yes—
22C2×#381C2212yes—
23C2×#380C2212yes—
24C2×#379C2211yescharacteristic
25C2×#379C2211yescharacteristic
26C2×#379C2211yescharacteristic
27C2×#379C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29#1445C2214yescomplemented
30#1445C2214yescomplemented
31#1445C2214yescomplemented
32#1443C2214yescomplemented
33#1443C2214yescomplemented
34#1443C2214yescomplemented
35#1443C2214yescomplemented
36#1442C2214yescomplemented
37#1442C2214yescomplemented
38#1442C2214yescomplemented
39#1441C2214yescomplemented
40#1440C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1440C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yescomplemented
53#1436C2214yescomplemented
54#1434C2214yescomplemented
55C22⋊#95C2212yes—
56C22⋊#95C2212yes—
57C22⋊#95C2212yes—
58C22⋊#95C2212yes—
59C22⋊#95C2212yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C26C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2314yes—
103#400—48no—
104#385—44no—
105#385—44no—
106#383—48no—
107#381—44no—
108#381—44no—
109#380—44no—
110#379—24no—
111#379—24no—
112#379—24no—
113#379—24no—
114C4×C22⋊C4—44no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C25—42no—
140C25—42no—
141C25—42no—
142C25—42no—
143C25—42no—
144C25—42no—
145C25#4114yes—
146C25#4114yes—
147C25#4114yes—
148C25C2411yescharacteristic
149C25C2411yescharacteristic
150C25C2411yescharacteristic
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
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—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—416no—
235C2×C22⋊C4—416no—
236C2×C22⋊C4—416no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C42—28no—
254C2×C42—44no—
255C2×C42—44no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—44no—
299C24—44no—
300C24—48no—
301C24—48no—
302C24—48no—
303C24—44no—
304C24—44no—
305C24—44no—
306C24—44no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—88no—
311C24—88no—
312C24—88no—
313C24—88no—
314C24#126116yes—
315C24C2×#4112yes—
316C24C2×#4112yes—
317C24C2×#4112yes—
318C24C2×#4112yes—
319C24C2×#4112yes—
320C24C2×#4112yes—
321C24C2×#4112yes—
322C24C2×#4112yes—
323C24C2×#4112yes—
324C24C2511yescharacteristic
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
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—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4H(5,𝔽2)14yes—
408C22×C4H(5,𝔽2)14yes—
409C4⋊C4—88no—
410C4⋊C4—88no—
411C4⋊C4—88no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C42—416no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—416no—
468C23—416no—
469C23—416no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—88no—
483C23—88no—
484C23—832no—
485C23—832no—
486C23—88no—
487C23—88no—
488C23C2×#12614yes—
489C23C2×#12614yes—
490C23C2×#12614yes—
491C23C22×#4111yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C2×H(5,𝔽2)12yes—
495C23C2×H(5,𝔽2)12yes—
496C23#58512yes—
497C23#58512yes—
498C23#58512yes—
499C23#58512yes—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
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—88no—
559C2×C4—88no—
560C2×C4—88no—
561C2×C4—88no—
562C2×C4—88no—
563C2×C4—88no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—48no—
580C22—48no—
581C22—48no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—832no—
592C22C22×#12611yescharacteristic
593C22#345912yes—
594C22#349011yescharacteristic
595C22#349011yescharacteristic
596C22#349111yescharacteristic
597C22#349111yescharacteristic
598C22#349111yescharacteristic
599C22#349112yes—
600C22#349112yes—
601C22#349111yescharacteristic
602C22#350512yes—
603C22#350512yes—
604C22#350512yes—
605C22#353314yes—
606C22#354514yes—
607C22#354614yes—
608C22#356614yes—
609C4—44no—
610C4—44no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—84no—
616C4—84no—
617C4—84no—
618C4—84no—
619C2—44no—
620C2—44no—
621C2—44no—
622C2#3499511yescharacteristic
623C2#3537011yescharacteristic
624C2#3545611yescharacteristic
625C2#4168814yes—
626C2#4638911yescharacteristic
627C2#4683711yescharacteristic
628C2#4683811yescharacteristic
629C2#4699911yescharacteristic
630C2#4886514yes—
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
1. #16415C2semidirect
2. #16005C2semidirect
3. #15987C2semidirect
4. #15797C2semidirect
5. #15481C2semidirect
6. #15018C2semidirect
17. C2×#400C22semidirect
20. C2×#383C22semidirect
29. #1445C22semidirect
30. #1445C22semidirect
31. #1445C22semidirect
32. #1443C22semidirect
33. #1443C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1442C22semidirect
39. #1441C22semidirect
52. #1436C22semidirect
53. #1436C22semidirect
54. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#164152
#160052
#159872
#157972
#154812
#150182
#146732
#143412
#143382
#141272
#138692
#137752