Finite Group Databasebeta

← #6629155

Subgroups — #6629155

Order 512.

4524 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic, complemented
2C2×#3094C211yesmaximal, characteristic, complemented
3C22×#480C211yesmaximal, characteristic, complemented
4#35250C211yesmaximal, characteristic
5#32986C218yesmaximal, complemented
6#32980C214yesmaximal, complemented
7#32978C214yesmaximal, complemented
8#20914C214yesmaximal, complemented
9#20847C214yesmaximal, complemented
10#20700C211yesmaximal, characteristic, complemented
11#20699C211yesmaximal, characteristic, complemented
12#20581C211yesmaximal, characteristic, complemented
13C23×#41C2211yescharacteristic
14C2×#535C2212yes—
15C22×#121C2211yescharacteristic
16C2×#520—22no—
17C2×#520—22no—
18C2×#520—28no—
19C2×#520—28no—
20C2×#520C2214yes—
21C2×#520C2214yescomplemented
22C22×C4×Q8—21no—
23C22×C4×D4—22no—
24C22×C4×D4—21no—
25C22×#117C2211yescharacteristic
26C22×#117C2212yescomplemented
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2212yescomplemented
29#3094—28no—
30C2×#481C2212yes—
31C2×#482C2214yescomplemented
32C2×#480—28no—
33C2×#480C2214yes—
34C2×#480C2212yes—
35C2×#480C2214yescomplemented
36C2×#428C2212yescomplemented
37C22×#107C2212yescomplemented
38C2×#418C2212yescomplemented
39C22×#102C2212yescomplemented
40C2×#408C2214yes—
41C2×#407C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43#2847C2214yes—
44#2615C2214yes—
45#2436—21no—
46#2433—21no—
47#2431—22no—
48#2404—22no—
49#2403—22no—
50#2311C2212yes—
51#2280C2211yescharacteristic
52#1875C22116yescomplemented
53#1874C22116yescomplemented
54#1854C22116yescomplemented
55#1852C22116yescomplemented
56#1825C2214yescomplemented
57#1824C2212yescomplemented
58#1824C2214yescomplemented
59#1823C2212yescomplemented
60#1823C2214yescomplemented
61#1780C2214yescomplemented
62#1779C2214yescomplemented
63#1736C2214yescomplemented
64#1735C2214yescomplemented
65C22×#41—216no—
66C22×#41—24no—
67C22×#41—24no—
68C22×#41D412yes—
69C22×#41D412yes—
70C22×#41C2312yes—
71C22×#41C2312yes—
72C23×Q8D411yescharacteristic
73C23×D4—21no—
74C23×D4D412yes—
75C23×D4D411yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4—21no—
78C24×C4—22no—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C22×M16C2312yes—
82C23×C8C2312yes—
83#535—216no—
84C2×#127—22no—
85C2×#124—22no—
86C2×#121—21no—
87C2×#121—21no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#121C2312yes—
91C2×#121C2312yes—
92C2×#121C2318yes—
93C2×#120—24no—
94C2×#120—24no—
95#520—232no—
96#520—232no—
97#520—216no—
98#520—216no—
99#520—216no—
100#520—216no—
101C2×C4×Q8—216no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×D4—232no—
105C2×C4×D4—216no—
106C2×C4×D4—28no—
107C2×C4×D4—24no—
108C2×C4×D4—28no—
109C2×C4×D4—24no—
110C2×#117—24no—
111C2×#117—21no—
112C2×#117—21no—
113C2×#117—24no—
114C2×#117—28no—
115C2×#117—28no—
116C2×#117D418yescomplemented
117C2×#117C2312yes—
118C2×#117C2318yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—24no—
127C22×C4⋊C4—28no—
128C22×C4⋊C4—28no—
129C22×C4⋊C4D418yescomplemented
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2318yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2312yes—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—21no—
140C22×C42—21no—
141C22×C42—22no—
142#482—216no—
143#481—28no—
144#480—216no—
145#480—216no—
146#480—28no—
147#428—28no—
148C2×#107—28no—
149C2×#107C2318yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152#418—28no—
153C2×#102—28no—
154C2×#102C2318yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157#408—216no—
158#407—24no—
159C2×C22⋊C8—24no—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C23116yes—
165#393—24no—
166#392—28no—
167#392—24no—
168#391—22no—
169#390—22no—
170#386—24no—
171#386—24no—
172#386—28no—
173#381—21no—
174#380C2318yes—
175#380C2312yes—
176#380C2312yes—
177#379—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180#340C23116yes—
181#340C2318yes—
182#340C2318yes—
183C25—21no—
184C25C2×D412yes—
185C25C2×D411yescharacteristic
186C2×#41—264no—
187C2×#41—264no—
188C2×#41—28no—
189C2×#41—216no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—216no—
198C2×#41—28no—
199C2×#41#41116yes—
200C22×Q8—28no—
201C22×Q8—24no—
202C22×Q8C2×D412yes—
203C22×Q8C2×D412yes—
204C22×D4—28no—
205C22×D4—216no—
206C22×D4—28no—
207C22×D4—22no—
208C22×D4—28no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—28no—
215C22×D4—44no—
216C22×D4—84no—
217C22×D4C2×D412yes—
218C22×D4C2×D414yes—
219C22×D4C2×D414yes—
220C22×D4C2×D412yes—
221C22×D4C2412yes—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—216no—
225C23×C4—28no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—216no—
230C23×C4—28no—
231C23×C4—28no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—84no—
236C23×C4—88no—
237C23×C4C2×D414yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C2×M16—24no—
249C2×M16—28no—
250C22×C8—24no—
251C22×C8—28no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254#127—48no—
255#124—48no—
256#121—24no—
257#121—24no—
258#121—24no—
259#121—24no—
260#121—232no—
261#121—44no—
262#121—44no—
263#120—416no—
264#120—88no—
265C4×Q8—232no—
266C4×Q8—232no—
267C4×D4—264no—
268C4×D4—264no—
269C4×D4—232no—
270C4×D4—232no—
271#117—232no—
272#117—28no—
273#117—28no—
274#117—24no—
275#117—24no—
276#117—216no—
277#117—216no—
278#117—28no—
279#117C2×D418yes—
280#117C2×D418yes—
281C2×C4⋊C4—216no—
282C2×C4⋊C4—232no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—216no—
298C2×C4⋊C4—216no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D418yes—
304C2×C4⋊C4C2×D418yes—
305C2×C4⋊C4C2×D418yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—216no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—44no—
325C2×C42—216no—
326C2×C42—28no—
327C2×C42—28no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331#107—216no—
332#107—28no—
333#106—28no—
334#102—216no—
335#102—28no—
336#102—28no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—232no—
342#95—22no—
343#95—22no—
344#95—28no—
345C24—24no—
346C24—28no—
347C24—24no—
348C24—24no—
349C24—28no—
350C24—22no—
351C24—42no—
352C24—44no—
353C24—42no—
354C24—44no—
355C24—82no—
356C24—82no—
357C24#12014yes—
358C24C22×D411yescharacteristic
359C24C22×D414yes—
360C24C22×D411yescharacteristic
361C24C22×D411yescharacteristic
362#41—264no—
363#41—264no—
364#41—232no—
365#41—232no—
366#41—232no—
367#41—232no—
368C2×Q8—232no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—28no—
372C2×Q8—24no—
373C2×Q8#12318yes—
374C2×D4—264no—
375C2×D4—232no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—28no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—416no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—84no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—816no—
400C2×D4#123116yes—
401C2×D4#12318yes—
402C22×C4—232no—
403C22×C4—232no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—232no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—816no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—816no—
438C22×C4—82no—
439C22×C4—832no—
440C22×C4#12014yes—
441C22×C4#12014yes—
442C22×C4#12014yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#41116yes—
454C22×C4C2×#4118yes—
455C22×C4C2511yescharacteristic
456M16—48no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—28no—
460C2×C8—28no—
461C2×C8—48no—
462C4⋊C4—232no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—28no—
474C4⋊C4—42no—
475C4⋊C4C22×D414yes—
476C4⋊C4C22×D414yes—
477C4⋊C4C22×D418yes—
478C22⋊C4—232no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—416no—
486C22⋊C4—82no—
487C22⋊C4—88no—
488C42—232no—
489C42—28no—
490C42—28no—
491C42—44no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—216no—
497C23—22no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—22no—
506C23—44no—
507C23—416no—
508C23—432no—
509C23—48no—
510C23—48no—
511C23—44no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23—168no—
519C23C2×#12012yes—
520C23C2×#12012yes—
521C23C2×#12012yes—
522C23C2×#12318yes—
523C23C2×#12314yes—
524C23#57614yes—
525C23C23×D411yescharacteristic
526Q8—216no—
527Q8—216no—
528D4—216no—
529D4—232no—
530D4—232no—
531D4—216no—
532D4—44no—
533D4—44no—
534D4—88no—
535D4—88no—
536D4—84no—
537D4—1616no—
538C2×C4—216no—
539C2×C4—232no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—216no—
552C2×C4—216no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—416no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—88no—
568C2×C4—88no—
569C2×C4—832no—
570C2×C4—84no—
571C2×C4C2×#12012yes—
572C2×C4C2×#12012yes—
573C2×C4C2×#12012yes—
574C2×C4C2×#12318yes—
575C2×C4C2×#12314yes—
576C2×C4#52518yes—
577C2×C4#52518yes—
578C2×C4C23×D411yescharacteristic
579C2×C4C23×D411yescharacteristic
580C2×C4C22×#4114yes—
581C8—44no—
582C8—44no—
583C22—22no—
584C22—28no—
585C22—28no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—28no—
591C22—216no—
592C22—22no—
593C22—22no—
594C22—416no—
595C22—432no—
596C22—44no—
597C22—48no—
598C22—44no—
599C22—44no—
600C22—82no—
601C22—88no—
602C22—82no—
603C22—82no—
604C22—168no—
605C22#284914yes—
606C22#300814yes—
607C22C22×#12011yescharacteristic
608C22C22×#12312yes—
609C22C2×#57612yes—
610C22#358612yes—
611C4—22no—
612C4—22no—
613C4—28no—
614C4—48no—
615C4—44no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4#284914yes—
620C4C2×#52512yes—
621C4C2×#52512yes—
622C2—24no—
623C2—24no—
624C2—42no—
625C2—48no—
626C2—162no—
627C2C2×#284911yescharacteristic
628C2C2×#300811yescharacteristic
629C2#6036811yescharacteristic
630C2#6052114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35250C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#520C2semidirect
2. C2×#3094C2semidirect
3. C22×#480C2semidirect
5. #32986C2semidirect
6. #32980C2semidirect
7. #32978C2semidirect
8. #20914C2semidirect
9. #20847C2semidirect
10. #20700C2semidirect
11. #20699C2semidirect
12. #20581C2semidirect
21. C2×#520C22semidirect
26. C22×#117C22semidirect
28. C23×C4⋊C4C22semidirect
31. C2×#482C22semidirect
35. C2×#480C22semidirect
36. C2×#428C22semidirect
37. C22×#107C22semidirect
38. C2×#418C22semidirect
39. C22×#102C22semidirect
52. #1875C22semidirect
53. #1874C22semidirect
54. #1854C22semidirect
55. #1852C22semidirect
56. #1825C22semidirect
57. #1824C22semidirect
58. #1824C22semidirect
59. #1823C22semidirect
60. #1823C22semidirect
61. #1780C22semidirect
62. #1779C22semidirect
63. #1736C22semidirect
64. #1735C22semidirect
116. C2×#117D4semidirect
129. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#30942
C22×#4802
#352502
#329862
#329802
#329782
#209142
#208472
#207002
#206992
#205812