Finite Group Databasebeta

← #4464172

Subgroups — #4464172

Order 512.

3458 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17272C212yesmaximal, complemented
2#17272C212yesmaximal, complemented
3#17272C212yesmaximal, complemented
4#17204C214yesmaximal, complemented
5#17136C212yesmaximal, complemented
6#17136C212yesmaximal, complemented
7#17136C212yesmaximal, complemented
8#14935C214yesmaximal, complemented
9#14734C214yesmaximal, complemented
10#14726C214yesmaximal, complemented
11#14335C211yesmaximal, characteristic
12#14335C211yesmaximal, characteristic
13C2×#1438C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2212yes—
17C2×#399C2214yescomplemented
18C2×#399C2214yescomplemented
19C2×#396C2214yescomplemented
20C2×#396C2214yescomplemented
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#381C2212yes—
24C2×#381C2212yes—
25C2×#380C2214yes—
26C2×#379C2212yes—
27C2×#379C2212yes—
28C2×#379C2212yes—
29C2×#379C2212yes—
30C22×#95C2212yes—
31#1443C2214yescomplemented
32#1443C2214yescomplemented
33#1441C2214yescomplemented
34#1441C2214yescomplemented
35#1441C2214yescomplemented
36#1441C2214yescomplemented
37#1441C2214yescomplemented
38#1441C2214yescomplemented
39#1439C2214yescomplemented
40#1439C2214yescomplemented
41#1439C2214yescomplemented
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1438C2212yes—
52#1438C2211yescharacteristic
53#1438C2212yes—
54#1438C2212yes—
55#1438C2211yescharacteristic
56#1438C2212yes—
57#1438C2212yes—
58#1438C2211yescharacteristic
59#1438C2211yescharacteristic
60#1438C2212yes—
61#1438C2212yes—
62#1438C2212yes—
63#1438C2212yes—
64#1438C2211yescharacteristic
65#1438C2211yescharacteristic
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C26C2311yescharacteristic
69C23×D4—44no—
70C23×D4—44no—
71C23×D4—44no—
72C23×D4Q818yes—
73C23×D4C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C22⋊C4—48no—
86C22×C22⋊C4—48no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4—44no—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109#399—48no—
110#399—48no—
111#396—48no—
112#396—48no—
113#385—48no—
114#385—48no—
115#381—44no—
116#381—44no—
117#380—48no—
118#379—44no—
119#379—44no—
120#379—44no—
121#379—44no—
122C2×#95—44no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—44no—
149C25—44no—
150C25—44no—
151C25—44no—
152C25C2×Q814yes—
153C25C2×Q814yes—
154C25C2411yescharacteristic
155C25C2411yescharacteristic
156C25C2411yescharacteristic
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C22×D4—416no—
160C22×D4—416no—
161C22×D4—416no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—464no—
175C22×D4C2×Q814yes—
176C22×D4C2×Q814yes—
177C22×D4C2×Q814yes—
178C22×D4C2×Q814yes—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4C2×Q814yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—432no—
237C2×C22⋊C4—432no—
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—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266#95—28no—
267#95—42no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285C24—24no—
286C24—24no—
287C24—28no—
288C24—28no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—28no—
310C24—28no—
311C24—28no—
312C24—28no—
313C24—416no—
314C24—416no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—416no—
320C24—416no—
321C24—416no—
322C24—416no—
323C24—84no—
324C24—84no—
325C24—84no—
326C24—84no—
327C24—88no—
328C24—88no—
329C24—88no—
330C24—88no—
331C24—84no—
332C24—84no—
333C24—84no—
334C24—84no—
335C24C22×Q812yes—
336C24C22×Q812yes—
337C24C22×Q812yes—
338C24C22×Q812yes—
339C24C22×Q812yes—
340C24H(5,𝔽2)14yes—
341C24H(5,𝔽2)14yes—
342C24H(5,𝔽2)14yes—
343C24C2511yescharacteristic
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—4128no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—416no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4C22×Q812yes—
420C22×C4C22×Q812yes—
421C22×C4H(5,𝔽2)18yes—
422C4⋊C4—88no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C23—28no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—28no—
454C23—28no—
455C23—28no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—432no—
475C23—432no—
476C23—416no—
477C23—416no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—816no—
519C23—816no—
520C23—816no—
521C23—816no—
522C23—832no—
523C23—832no—
524C23—832no—
525C23—832no—
526C23#54314yes—
527C23#54314yes—
528C23#54314yes—
529C23#560116yes—
530C23C23×Q811yescharacteristic
531C23C2×H(5,𝔽2)12yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)14yes—
535C23#58514yes—
536D4—416no—
537D4—416no—
538D4—416no—
539D4—416no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—416no—
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—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
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—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—832no—
592C22—28no—
593C22—216no—
594C22—216no—
595C22—28no—
596C22—28no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—832no—
614C22—832no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22#284218yes—
624C22C2×#54311yescharacteristic
625C22C2×#54311yescharacteristic
626C22C2×#54311yescharacteristic
627C22#350114yes—
628C22C2×#56014yes—
629C22#353212yes—
630C22#353212yes—
631C22#353312yes—
632C22#353312yes—
633C22#353414yes—
634C22#354214yes—
635C22#354214yes—
636C22#354312yes—
637C22#354512yes—
638C4—42no—
639C4—42no—
640C4—48no—
641C4—84no—
642C4—84no—
643C4—84no—
644C4—84no—
645C4—84no—
646C2—28no—
647C2—82no—
648C2—82no—
649C2—82no—
650C2—82no—
651C2C2×#284211yescharacteristic
652C2#3674712yes—
653C2#3674712yes—
654C2#3713412yes—
655C2#5230814yes—
656C2#5234014yes—
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. #17272C2semidirect
2. #17272C2semidirect
3. #17272C2semidirect
4. #17204C2semidirect
5. #17136C2semidirect
6. #17136C2semidirect
7. #17136C2semidirect
8. #14935C2semidirect
9. #14734C2semidirect
10. #14726C2semidirect
17. C2×#399C22semidirect
18. C2×#399C22semidirect
19. C2×#396C22semidirect
20. C2×#396C22semidirect
31. #1443C22semidirect
32. #1443C22semidirect
33. #1441C22semidirect
34. #1441C22semidirect
35. #1441C22semidirect
36. #1441C22semidirect
37. #1441C22semidirect
38. #1441C22semidirect
39. #1439C22semidirect
40. #1439C22semidirect
41. #1439C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect

Maximal subgroups

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

subgroupindex
#172722
#172042
#171362
#149352
#147342
#147262
#143352
C2×#14382