Finite Group Databasebeta

← #4453288

Subgroups — #4453288

Order 512.

5998 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#534C211yesmaximal, characteristic
2C2×#2405C212yesmaximal, complemented
3C22×#397C211yesmaximal, characteristic, complemented
4#16657C218yesmaximal, complemented
5#16255C218yesmaximal, complemented
6#14926C212yesmaximal, complemented
7#14926C212yesmaximal, complemented
8#14913C212yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14341C211yesmaximal, characteristic, complemented
11#13867C212yesmaximal, complemented
12C24×D4C2212yes—
13C2×#534—216no—
14C2×#534—28no—
15C22×#123C2214yes—
16C22×#123C2214yescomplemented
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#120C2214yes—
20C23×C4⋊C4C2211yescharacteristic, complemented
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2405—216no—
25C2×#397—28no—
26C2×#397C2218yescomplemented
27C2×#394C2218yes—
28C2×#394C2214yescomplemented
29C2×#386C2214yescomplemented
30C2×#386C2214yescomplemented
31C2×#386C2214yescomplemented
32C2×#385C2214yescomplemented
33C2×#379C2212yes—
34C2×#379C2214yescomplemented
35C2×#379C2214yescomplemented
36C22×#95C2211yescharacteristic, complemented
37C22×#95C2211yescharacteristic, complemented
38C22×#95C2211yescharacteristic, complemented
39#1444C2214yescomplemented
40#1444C2214yescomplemented
41#1442C2214yescomplemented
42#1441C2214yescomplemented
43#1440C2218yescomplemented
44#1440C2218yescomplemented
45#1439C2218yescomplemented
46#1439C2218yescomplemented
47#1439C2218yescomplemented
48#1438C2218yes—
49#1438C2214yescomplemented
50#1438C2214yescomplemented
51#1438C2214yescomplemented
52C22⋊#95C2212yescomplemented
53C22⋊#95C2212yescomplemented
54C22⋊#95C2214yescomplemented
55C26C2311yescharacteristic
56C26C2312yes—
57C23×D4—28no—
58C23×D4—28no—
59C23×D4—48no—
60C23×D4—48no—
61C23×D4D4116yes—
62C23×D4C2314yes—
63C23×D4C2312yes—
64C23×D4C2312yes—
65C23×D4C2314yes—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73#534—2128no—
74C2×#123—216no—
75C2×#123—216no—
76C2×#123—216no—
77C2×#123—216no—
78C2×#123—48no—
79C2×#123—48no—
80C2×#121—216no—
81C2×#121—216no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#120—216no—
85C2×#120—216no—
86C2×#120—216no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2318yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4D418yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2318yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2318yes—
111#397—232no—
112#394—232no—
113#394—48no—
114#386—216no—
115#386—48no—
116#386—48no—
117#385—216no—
118#379—28no—
119#379—216no—
120#379—48no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95C2318yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C25—24no—
135C25—24no—
136C25—24no—
137C25—24no—
138C25—42no—
139C25—44no—
140C25—44no—
141C25—42no—
142C25—42no—
143C25—42no—
144C25—44no—
145C25—44no—
146C25—44no—
147C25—44no—
148C25C2×D414yes—
149C25C2×D418yes—
150C25C2×D418yes—
151C25C2414yes—
152C25C2412yes—
153C25C2411yescharacteristic
154C22×D4—264no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—464no—
176C22×D4—464no—
177C22×D4—432no—
178C22×D4—432no—
179C22×D4C2×D4116yes—
180C22×D4C2×D4116yes—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D418yes—
210C23×C4C2418yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218#123—2128no—
219#123—464no—
220#121—264no—
221#121—264no—
222#120—2128no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—416no—
234C2×C22⋊C4—232no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—416no—
262C2×C22⋊C4—416no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4C2×D4116yes—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—216no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—48no—
281#95—48no—
282#95—44no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—232no—
294C24—28no—
295C24—28no—
296C24—24no—
297C24—24no—
298C24—28no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—28no—
304C24—28no—
305C24—28no—
306C24—216no—
307C24—44no—
308C24—416no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24—416no—
313C24—416no—
314C24—416no—
315C24—416no—
316C24—416no—
317C24—48no—
318C24—48no—
319C24—88no—
320C24—88no—
321C24—88no—
322C24—88no—
323C24—84no—
324C24—84no—
325C24—88no—
326C24—84no—
327C24—84no—
328C24—88no—
329C24—816no—
330C24—816no—
331C24—816no—
332C24—816no—
333C24#120116yes—
334C24C22×D418yes—
335C24C22×D414yes—
336C24C22×D414yes—
337C24C22×D414yes—
338C24C22×D414yes—
339C24C22×D412yes—
340C24H(5,𝔽2)14yes—
341C24H(5,𝔽2)14yes—
342C24C2511yescharacteristic
343C2×D4—232no—
344C2×D4—232no—
345C2×D4—232no—
346C2×D4—232no—
347C2×D4—216no—
348C2×D4—216no—
349C2×D4—416no—
350C2×D4—416no—
351C2×D4—4256no—
352C2×D4—416no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—84no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—816no—
365C2×D4—816no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—22no—
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—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—416no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—416no—
411C22×C4—44no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—416no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D418yes—
433C22×C4H(5,𝔽2)14yes—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—416no—
437C4⋊C4—432no—
438C22⋊C4—232no—
439C22⋊C4—232no—
440C22⋊C4—232no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—432no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—816no—
447C22⋊C4—88no—
448C22⋊C4—816no—
449C22⋊C4—816no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—24no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—232no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—216no—
477C23—216no—
478C23—416no—
479C23—416no—
480C23—432no—
481C23—48no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—44no—
493C23—44no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—864no—
504C23—88no—
505C23—88no—
506C23—88no—
507C23—88no—
508C23—816no—
509C23—816no—
510C23—864no—
511C23—864no—
512C23—816no—
513C23—816no—
514C23—816no—
515C23—816no—
516C23C2×#12018yes—
517C23C2×#12014yes—
518C23#534116yes—
519C23#53514yes—
520C23#53514yes—
521C23C23×D412yes—
522C23C23×D411yescharacteristic
523C23C2×H(5,𝔽2)12yes—
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)12yes—
526C23#58512yes—
527C23#58512yes—
528C23#58512yes—
529D4—464no—
530D4—464no—
531D4—816no—
532D4—816no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—416no—
545C2×C4—432no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4—88no—
579C22—24no—
580C22—232no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—28no—
587C22—28no—
588C22—28no—
589C22—416no—
590C22—416no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—48no—
596C22—416no—
597C22—416no—
598C22—48no—
599C22—48no—
600C22—832no—
601C22—864no—
602C22—88no—
603C22—88no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22#261318yes—
609C22C22×#12011yescharacteristic
610C22C2×#53414yes—
611C22C2×#53511yescharacteristic
612C22C2×#53511yescharacteristic
613C22#348411yescharacteristic
614C22#348511yescharacteristic
615C22#348514yes—
616C22#348511yescharacteristic
617C22#348511yescharacteristic
618C22#349614yes—
619C22#353414yes—
620C22#355814yes—
621C22#356014yes—
622C22#356014yes—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—48no—
627C4—82no—
628C4—88no—
629C4—82no—
630C4—82no—
631C2—28no—
632C2—48no—
633C2—88no—
634C2C2×#261311yescharacteristic
635C2#3534711yescharacteristic
636C2#3544611yescharacteristic
637C2#3672111yescharacteristic
638C2#3696711yescharacteristic
639C2#3696911yescharacteristic
640C2#3696911yescharacteristic
641C2#4051018yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#534C2
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×#2405C2semidirect
3. C22×#397C2semidirect
4. #16657C2semidirect
5. #16255C2semidirect
6. #14926C2semidirect
7. #14926C2semidirect
8. #14913C2semidirect
9. #14719C2semidirect
10. #14341C2semidirect
11. #13867C2semidirect
16. C22×#123C22semidirect
20. C23×C4⋊C4C22semidirect
26. C2×#397C22semidirect
28. C2×#394C22semidirect
29. C2×#386C22semidirect
30. C2×#386C22semidirect
31. C2×#386C22semidirect
32. C2×#385C22semidirect
34. C2×#379C22semidirect
35. C2×#379C22semidirect
36. C22×#95C22semidirect
37. C22×#95C22semidirect
38. C22×#95C22semidirect
39. #1444C22semidirect
40. #1444C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1440C22semidirect
44. #1440C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
49. #1438C22semidirect
50. #1438C22semidirect
51. #1438C22semidirect
52. C22⋊#95C22semidirect
53. C22⋊#95C22semidirect
54. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5342
C2×#24052
C22×#3972
#166572
#162552
#149262
#149132
#147192
#143412
#138672