Finite Group Databasebeta

← #760185

Subgroups — #760185

Order 512.

6354 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2403C211yesmaximal, characteristic
2C22×#386C212yesmaximal, complemented
3C2×#2280C211yesmaximal, characteristic, complemented
4#16660C214yesmaximal, complemented
5#14921C214yesmaximal, complemented
6#14886C214yesmaximal, complemented
7#14725C214yesmaximal, complemented
8#14127C211yesmaximal, characteristic, complemented
9#14020C214yesmaximal, complemented
10#13995C214yesmaximal, complemented
11#13867C212yesmaximal, complemented
12C24×D4C2212yes—
13C25×C4C2211yescharacteristic
14C22×#120C2214yes—
15C22×#117C2212yescomplemented
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2212yescomplemented
18C23×C22⋊C4C2212yescomplemented
19#2403—28no—
20C2×#394C2214yescomplemented
21C2×#392C2214yes—
22C2×#390C2214yescomplemented
23C2×#386—216no—
24C2×#386C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yescomplemented
30C2×#386C2214yescomplemented
31C2×#385C2214yescomplemented
32#2280—28no—
33C2×#381C2212yescomplemented
34C2×#380C2214yescomplemented
35C2×#379C2211yescharacteristic
36C2×#379C2211yescharacteristic
37C2×#379C2214yescomplemented
38C2×C4×C22⋊C4C2214yescomplemented
39C22×#95C2212yescomplemented
40C22×#95C2212yescomplemented
41#1443C2218yescomplemented
42#1441C2214yescomplemented
43#1440C2218yescomplemented
44#1439C2214yescomplemented
45#1439C2218yescomplemented
46#1439C2218yescomplemented
47#1438C2218yescomplemented
48#1438C2218yescomplemented
49#1437C2218yescomplemented
50#1434C2214yescomplemented
51C22⋊#95C2214yescomplemented
52C22⋊#95C2214yescomplemented
53C26C2312yes—
54C26C2311yescharacteristic
55C23×D4—28no—
56C23×D4—216no—
57C23×D4—28no—
58C23×D4—28no—
59C23×D4—48no—
60C23×D4C2314yes—
61C23×D4C2314yes—
62C23×D4C2314yes—
63C24×C4—24no—
64C24×C4—24no—
65C24×C4—44no—
66C24×C4—44no—
67C24×C4D418yescomplemented
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C2×#120—216no—
76C2×#120—216no—
77C2×#120—48no—
78C2×#117—28no—
79C2×#117—44no—
80C2×#117—44no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—48no—
87C22×C22⋊C4—48no—
88C22×C22⋊C4—48no—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2318yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2318yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2314yes—
103#394—48no—
104#392—216no—
105#390—48no—
106#386—216no—
107#386—216no—
108#386—216no—
109#386—216no—
110#386—216no—
111#386—216no—
112#386—48no—
113#385—48no—
114#381—28no—
115#380—216no—
116#379—24no—
117#379—48no—
118#379D418yes—
119C4×C22⋊C4—216no—
120C2×#95—28no—
121C2×#95—28no—
122C2×#95C2314yes—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2318yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C25—28no—
133C25—28no—
134C25—24no—
135C25—28no—
136C25—24no—
137C25—28no—
138C25—24no—
139C25—28no—
140C25—42no—
141C25—42no—
142C25—44no—
143C25—44no—
144C25—44no—
145C25C2×D414yes—
146C25C2×D414yes—
147C25#4114yes—
148C25C2414yes—
149C25C2411yescharacteristic
150C25C2412yes—
151C22×D4—264no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—216no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—48no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—464no—
171C22×D4—464no—
172C22×D4—432no—
173C22×D4—432no—
174C22×D4#41116yes—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—232no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—416no—
197C23×C4—416no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—432no—
208C23×C4—432no—
209C23×C4C2×D414yes—
210C23×C4C2×D418yes—
211C23×C4C2×D418yes—
212C23×C4C2×D414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#120—464no—
222#117—432no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—44no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
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—216no—
243C2×C22⋊C4—216no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—432no—
257C2×C22⋊C4—432no—
258C2×C22⋊C4—432no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4C2×D4116yes—
265C2×C22⋊C4C2×D418yes—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—44no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—216no—
274#95—28no—
275#95—216no—
276#95—48no—
277#95—48no—
278#95—44no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—216no—
284C24—216no—
285C24—232no—
286C24—232no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—216no—
292C24—24no—
293C24—24no—
294C24—28no—
295C24—24no—
296C24—24no—
297C24—28no—
298C24—28no—
299C24—28no—
300C24—24no—
301C24—24no—
302C24—44no—
303C24—44no—
304C24—48no—
305C24—48no—
306C24—416no—
307C24—432no—
308C24—416no—
309C24—416no—
310C24—416no—
311C24—432no—
312C24—432no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—416no—
320C24—416no—
321C24—88no—
322C24—88no—
323C24—816no—
324C24—816no—
325C24#121116yes—
326C24C22×D412yes—
327C24C22×D412yes—
328C24C22×D414yes—
329C24C22×D414yes—
330C24C2×#4118yes—
331C24C2×#4112yes—
332C24C2×#4114yes—
333C24C2×#4118yes—
334C24H(5,𝔽2)18yes—
335C24C2511yescharacteristic
336C2×D4—232no—
337C2×D4—232no—
338C2×D4—232no—
339C2×D4—232no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—432no—
343C2×D4—416no—
344C2×D4—416no—
345C2×D4—4256no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—416no—
349C2×D4—416no—
350C2×D4—416no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4—88no—
354C2×D4—816no—
355C2×D4—88no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—28no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
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—24no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—4128no—
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—48no—
423C22×C4—816no—
424C22×C4—816no—
425C22×C4C22×D414yes—
426C22×C4C22×D418yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4118yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C4H(5,𝔽2)14yes—
432C4⋊C4—416no—
433C4⋊C4—88no—
434C22⋊C4—232no—
435C22⋊C4—48no—
436C22⋊C4—432no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—816no—
446C22⋊C4—816no—
447C42—416no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—28no—
454C23—216no—
455C23—28no—
456C23—28no—
457C23—216no—
458C23—216no—
459C23—216no—
460C23—264no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—216no—
473C23—216no—
474C23—416no—
475C23—432no—
476C23—416no—
477C23—464no—
478C23—416no—
479C23—464no—
480C23—432no—
481C23—432no—
482C23—432no—
483C23—44no—
484C23—48no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—864no—
505C23—864no—
506C23—816no—
507C23—816no—
508C23—816no—
509C23—816no—
510C23C2×#12114yes—
511C23C2×#12118yes—
512C23#52518yes—
513C23#53518yes—
514C23#546116yes—
515C23#551116yes—
516C23C23×D411yescharacteristic
517C23C23×D411yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4114yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)12yes—
522C23C2×H(5,𝔽2)14yes—
523D4—464no—
524D4—464no—
525D4—832no—
526C2×C4—232no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—216no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—432no—
544C2×C4—432no—
545C2×C4—416no—
546C2×C4—432no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—816no—
572C2×C4—816no—
573C2×C4#53414yes—
574C2×C4#53414yes—
575C22—28no—
576C22—28no—
577C22—28no—
578C22—232no—
579C22—28no—
580C22—28no—
581C22—28no—
582C22—232no—
583C22—24no—
584C22—28no—
585C22—464no—
586C22—44no—
587C22—42no—
588C22—48no—
589C22—42no—
590C22—48no—
591C22—48no—
592C22—416no—
593C22—48no—
594C22—48no—
595C22—416no—
596C22—864no—
597C22—84no—
598C22—84no—
599C22—88no—
600C22—84no—
601C22—84no—
602C22—88no—
603C22#2411116yes—
604C22C22×#12111yescharacteristic
605C22C2×#52512yes—
606C22C2×#53411yescharacteristic
607C22C2×#53411yescharacteristic
608C22C2×#53512yes—
609C22C2×#54614yes—
610C22C2×#55114yes—
611C22#348114yes—
612C22#348412yes—
613C22#349012yes—
614C22#351918yes—
615C22#353714yes—
616C4—24no—
617C4—24no—
618C4—44no—
619C4—48no—
620C4—48no—
621C4—84no—
622C4—84no—
623C4—84no—
624C2—216no—
625C2—44no—
626C2—88no—
627C2C2×#241112yes—
628C2#3495611yescharacteristic
629C2#3498111yescharacteristic
630C2#3507612yes—
631C2#3745011yescharacteristic
632C2#5908914yes—
633C2#5926714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2403C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#386C2semidirect
3. C2×#2280C2semidirect
4. #16660C2semidirect
5. #14921C2semidirect
6. #14886C2semidirect
7. #14725C2semidirect
8. #14127C2semidirect
9. #14020C2semidirect
10. #13995C2semidirect
11. #13867C2semidirect
15. C22×#117C22semidirect
17. C23×C22⋊C4C22semidirect
18. C23×C22⋊C4C22semidirect
20. C2×#394C22semidirect
22. C2×#390C22semidirect
29. C2×#386C22semidirect
30. C2×#386C22semidirect
31. C2×#385C22semidirect
33. C2×#381C22semidirect
34. C2×#380C22semidirect
37. C2×#379C22semidirect
38. C2×C4×C22⋊C4C22semidirect
39. C22×#95C22semidirect
40. C22×#95C22semidirect
41. #1443C22semidirect
42. #1441C22semidirect
43. #1440C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1438C22semidirect
48. #1438C22semidirect
49. #1437C22semidirect
50. #1434C22semidirect
51. C22⋊#95C22semidirect
52. C22⋊#95C22semidirect
67. C24×C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24032
C22×#3862
C2×#22802
#166602
#149212
#148862
#147252
#141272
#140202
#139952
#138672