Finite Group Databasebeta

← #4308775

Subgroups — #4308775

Order 512.

4436 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic, complemented
2C2×#2847C211yesmaximal, characteristic
3C2×#2639C211yesmaximal, characteristic, complemented
4C22×#380C211yesmaximal, characteristic, complemented
5#16401C214yesmaximal, complemented
6#16400C214yesmaximal, complemented
7#16265C214yesmaximal, complemented
8#16261C214yesmaximal, complemented
9#14729C214yesmaximal, complemented
10#14725C214yesmaximal, complemented
11#14353C211yesmaximal, characteristic, complemented
12#14281C211yesmaximal, characteristic, complemented
13#14271C211yesmaximal, characteristic, complemented
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×#121C2211yescharacteristic
18C22×#121C2212yes—
19C22×#120C2212yes—
20C23×C4⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22#2848—28no—
23#2847—28no—
24#2639—28no—
25C2×#400C2212yescomplemented
26C2×#397C2212yescomplemented
27C2×#396C2218yescomplemented
28C2×#394C2218yes—
29C2×#394C2212yescomplemented
30C2×#393C2212yescomplemented
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2212yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#380—28no—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×#380C2214yes—
44C2×#380C2214yes—
45C2×#379C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1446C2212yescomplemented
49#1445C2212yescomplemented
50#1445C2212yescomplemented
51#1444C2214yescomplemented
52#1444C2214yescomplemented
53#1441C2218yescomplemented
54#1441C2214yescomplemented
55#1441C2218yescomplemented
56#1440C2218yes—
57#1439C2218yes—
58#1439C2218yescomplemented
59#1439C2214yescomplemented
60#1438C2218yes—
61#1438C2212yescomplemented
62C22⋊#95C2214yes—
63C26C2311yescharacteristic
64C23×Q8C2311yescharacteristic
65C23×D4C2312yes—
66C23×D4C2314yes—
67C23×D4C2311yescharacteristic
68C24×C4—24no—
69C24×C4—24no—
70C24×C4—48no—
71C24×C4D418yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—48no—
83C2×#121—28no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—24no—
87C2×#121—48no—
88C2×#120—28no—
89C2×#120—48no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—48no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2318yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2318yes—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2318yes—
114#400—28no—
115#397—28no—
116#396—416no—
117#394—28no—
118#394—416no—
119#393—28no—
120#386—28no—
121#386—216no—
122#386—216no—
123#386—48no—
124#385—28no—
125#385—216no—
126#380—28no—
127#380—24no—
128#380—28no—
129#380—24no—
130#380—216no—
131#380—216no—
132#380—48no—
133#379—28no—
134C2×#95—24no—
135C2×#95D418yescomplemented
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C2×#95C2318yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C25—22no—
151C25—22no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25—24no—
156C25—44no—
157C25C2×D414yes—
158C25C2412yes—
159C25C2412yes—
160C25C2411yescharacteristic
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—44no—
165C22×D4—28no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—48no—
177C22×D4—44no—
178C22×D4—48no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—232no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—432no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—432no—
203C23×C4—432no—
204C23×C4C2×D414yes—
205C23×C4C2×D418yes—
206C23×C4C2×D418yes—
207C23×C4C2×D418yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4#4114yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C22⋊Q8—432no—
222C22⋊Q8—416no—
223#121—432no—
224#121—416no—
225#120—432no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—432no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—432no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—48no—
287#95—48no—
288#95—48no—
289#95—48no—
290#95C2×D418yes—
291#95C2×D418yes—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—28no—
296C24—28no—
297C24—216no—
298C24—28no—
299C24—216no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—28no—
309C24—28no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—28no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—48no—
319C24—416no—
320C24—416no—
321C24—416no—
322C24—416no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24C22×D414yes—
328C24C22×D412yes—
329C24C22×D412yes—
330C24C2×#4112yes—
331C24H(5,𝔽2)14yes—
332C24H(5,𝔽2)14yes—
333C242−1+414yes—
334C242−1+414yes—
335C24C2511yescharacteristic
336C2×Q8—48no—
337C2×Q8—44no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×D4—48no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—416no—
349C2×D4—44no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—816no—
353C2×D4—816no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—216no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—4128no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4#121116yes—
421C22×C4C22×D418yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)18yes—
430C4⋊C4—432no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—416no—
437C4⋊C4—816no—
438C22⋊C4—432no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—816no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—232no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—432no—
477C23—432no—
478C23—416no—
479C23—432no—
480C23—432no—
481C23—464no—
482C23—416no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23C2×#12114yes—
508C23#53514yes—
509C23#53614yes—
510C23#53718yes—
511C23#53718yes—
512C23#54218yes—
513C23C23×D411yescharacteristic
514C23C23×D411yescharacteristic
515C23C22×#4111yescharacteristic
516C23C2×H(5,𝔽2)14yes—
517C23C2×H(5,𝔽2)12yes—
518C23C2×H(5,𝔽2)12yes—
519C23C2×2−1+412yes—
520C23C2×2−1+412yes—
521Q8—88no—
522D4—88no—
523D4—832no—
524D4—816no—
525C2×C4—232no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—416no—
540C2×C4—432no—
541C2×C4—432no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—816no—
574C2×C4—816no—
575C2×C4C2×#12114yes—
576C2×C4C2×#12114yes—
577C2×C4#53418yes—
578C22—28no—
579C22—28no—
580C22—28no—
581C22—28no—
582C22—216no—
583C22—24no—
584C22—24no—
585C22—28no—
586C22—28no—
587C22—28no—
588C22—464no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—48no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—44no—
597C22—82no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—88no—
603C22—82no—
604C22—82no—
605C22#262018yes—
606C22C22×#12111yescharacteristic
607C22C2×#53412yes—
608C22C2×#53511yescharacteristic
609C22C2×#53611yescharacteristic
610C22C2×#53712yes—
611C22C2×#53712yes—
612C22C2×#54212yes—
613C22#348411yescharacteristic
614C22#348412yes—
615C22#348611yescharacteristic
616C22#349012yes—
617C22#349212yes—
618C22#353614yes—
619C22#353614yes—
620C22#353714yes—
621C22#353814yes—
622C4—28no—
623C4—24no—
624C4—24no—
625C4—48no—
626C4—82no—
627C4—84no—
628C4—84no—
629C4—84no—
630C4—82no—
631C4—82no—
632C2—28no—
633C2—48no—
634C2—162no—
635C2C2×#262011yescharacteristic
636C2#3498111yescharacteristic
637C2#3522311yescharacteristic
638C2#3670612yes—
639C2#3671912yes—
640C2#3672914yes—
641C2#3743912yes—
642C2#3744212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
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×#2848C2semidirect
3. C2×#2639C2semidirect
4. C22×#380C2semidirect
5. #16401C2semidirect
6. #16400C2semidirect
7. #16265C2semidirect
8. #16261C2semidirect
9. #14729C2semidirect
10. #14725C2semidirect
11. #14353C2semidirect
12. #14281C2semidirect
13. #14271C2semidirect
25. C2×#400C22semidirect
26. C2×#397C22semidirect
27. C2×#396C22semidirect
29. C2×#394C22semidirect
30. C2×#393C22semidirect
48. #1446C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
61. #1438C22semidirect
135. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#28472
C2×#26392
C22×#3802
#164012
#164002
#162652
#162612
#147292
#147252
#143532
#142812
#142712