Finite Group Databasebeta

← #4386444

Subgroups — #4386444

Order 512.

3702 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic, complemented
2C2×#2308C211yesmaximal, characteristic
3#16298C214yesmaximal, complemented
4#15962C214yesmaximal, complemented
5#15358C214yesmaximal, complemented
6#15335C214yesmaximal, complemented
7#15309C214yesmaximal, complemented
8#14726C214yesmaximal, complemented
9#14674C211yesmaximal, characteristic, complemented
10#14297C212yesmaximal, complemented
11#13902C212yesmaximal
12C24×D4C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14#2608—28no—
15C2×#399C2214yescomplemented
16C2×#396C2214yes—
17C2×#396C2214yescomplemented
18C2×#396C2214yescomplemented
19#2308—28no—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#389C2214yescomplemented
25C2×#387C2214yescomplemented
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#385C2214yescomplemented
29C2×#385C2214yescomplemented
30C2×#381C2211yescharacteristic
31C2×#381C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2214yescomplemented
34C2×#379C2212yes—
35C2×#379C2214yes—
36C2×#379C2214yes—
37C2×C4×C22⋊C4C2212yes—
38#1444C2214yescomplemented
39#1442C2214yescomplemented
40#1441C2214yescomplemented
41#1441C2214yescomplemented
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58C22⋊#95C2214yescomplemented
59C26C2311yescharacteristic
60C26C2311yescharacteristic
61C23×D4—24no—
62C23×D4—24no—
63C23×D4—44no—
64C23×D4—44no—
65C23×D4Q818yes—
66C23×D4C2312yes—
67C23×D4C2312yes—
68C24×C4C2314yes—
69C24×C4C2311yescharacteristic
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C22⋊C4—28no—
79C22×C22⋊C4—44no—
80C22×C22⋊C4—44no—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C42C2312yes—
97C22×C42C2312yes—
98#399—48no—
99#396—216no—
100#396—48no—
101#396—48no—
102#390—44no—
103#390—44no—
104#390—44no—
105#390—44no—
106#389—216no—
107#387—216no—
108#386—216no—
109#386—216no—
110#385—48no—
111#385—48no—
112#381—24no—
113#381D418yes—
114#380—28no—
115#380—48no—
116#379—28no—
117#379—48no—
118#379—48no—
119C4×C22⋊C4—28no—
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—
134C25—22no—
135C25—22no—
136C25—24no—
137C25—24no—
138C25—22no—
139C25—22no—
140C25—42no—
141C25—42no—
142C25—44no—
143C25—44no—
144C25—44no—
145C25—44no—
146C25C2×Q814yes—
147C25C2×Q814yes—
148C25C2411yescharacteristic
149C25C2411yescharacteristic
150C25C2411yescharacteristic
151C25C2412yes—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—232no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—432no—
166C22×D4—432no—
167C22×D4—416no—
168C22×D4—416no—
169C22×D4C2×Q818yes—
170C22×D4C2×Q818yes—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4C2×D414yes—
181C23×C4C2×Q814yes—
182C23×C4#4114yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
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×C22⋊C4—24no—
215C2×C22⋊C4—28no—
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—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—432no—
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—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C2×D418yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
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#95C2×D4116yes—
280C24—216no—
281C24—28no—
282C24—28no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—28no—
296C24—216no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—24no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—28no—
310C24—48no—
311C24—416no—
312C24—416no—
313C24—48no—
314C24—44no—
315C24—44no—
316C24—88no—
317C24—88no—
318C24—84no—
319C24—84no—
320C24—84no—
321C24—84no—
322C24—88no—
323C24—88no—
324C24—88no—
325C24—88no—
326C24—88no—
327C24—88no—
328C24—88no—
329C24—88no—
330C24C22×D412yes—
331C24C22×Q812yes—
332C24C22×Q812yes—
333C24C22×Q812yes—
334C24C22×Q814yes—
335C24C2×#4112yes—
336C24H(5,𝔽2)14yes—
337C24H(5,𝔽2)14yes—
338C24C2511yescharacteristic
339C2×D4—216no—
340C2×D4—216no—
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—4128no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
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—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—48no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4C22⋊Q8116yes—
410C22×C4C22×D418yes—
411C22×C4C22×D414yes—
412C22×C4C22×Q814yes—
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)18yes—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—48no—
424C22⋊C4—416no—
425C22⋊C4—416no—
426C22⋊C4—416no—
427C22⋊C4—416no—
428C22⋊C4—48no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C42—84no—
439C42—84no—
440C23—232no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—416no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—832no—
503C23—816no—
504C23—816no—
505C23—88no—
506C23—88no—
507C23—816no—
508C23—832no—
509C23—832no—
510C23—88no—
511C23—88no—
512C23—816no—
513C23—832no—
514C23C2×C22⋊Q814yes—
515C23#54318yes—
516C23C23×D411yescharacteristic
517C23C23×Q811yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)14yes—
522C23#58514yes—
523D4—432no—
524D4—432no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—416no—
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—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—832no—
575C2×C4C2×C22⋊Q818yes—
576C22—216no—
577C22—24no—
578C22—28no—
579C22—28no—
580C22—44no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22—48no—
585C22—48no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—832no—
601C22—832no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—82no—
608C22—84no—
609C22—84no—
610C22C22×C22⋊Q811yescharacteristic
611C22C2×#54312yes—
612C22#348412yes—
613C22#348814yes—
614C22#349011yescharacteristic
615C22#349011yescharacteristic
616C22#350114yes—
617C22#350812yes—
618C22#351012yes—
619C22#353414yes—
620C22#353714yes—
621C22#354214yes—
622C22#356014yes—
623C4—28no—
624C4—48no—
625C4—48no—
626C4—84no—
627C4—84no—
628C4—84no—
629C4—84no—
630C2—48no—
631C2—44no—
632C2—82no—
633C2—82no—
634C2#3498211yescharacteristic
635C2#3551512yes—
636C2#4060914yes—
637C2#4645411yescharacteristic
638C2#4645711yescharacteristic
639C2#4703312yes—
640C2#5063414yes—
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. C2×#2608C2semidirect
3. #16298C2semidirect
4. #15962C2semidirect
5. #15358C2semidirect
6. #15335C2semidirect
7. #15309C2semidirect
8. #14726C2semidirect
9. #14674C2semidirect
10. #14297C2semidirect
15. C2×#399C22semidirect
17. C2×#396C22semidirect
18. C2×#396C22semidirect
24. C2×#389C22semidirect
25. C2×#387C22semidirect
28. C2×#385C22semidirect
29. C2×#385C22semidirect
33. C2×#380C22semidirect
38. #1444C22semidirect
39. #1442C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
C2×#23082
#162982
#159622
#153582
#153352
#153092
#147262
#146742
#142972
#139022