Finite Group Databasebeta

← #6629248

Subgroups — #6629248

Order 512.

4378 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic, complemented
3C2×#3007C211yesmaximal, characteristic
4#35311C211yesmaximal, characteristic, complemented
5#33016C218yesmaximal, complemented
6#33008C214yesmaximal, complemented
7#33007C214yesmaximal, complemented
8#20931C214yesmaximal, complemented
9#20882C214yesmaximal, complemented
10#20700C211yesmaximal, characteristic, complemented
11#20685C211yesmaximal, characteristic, complemented
12#20597C211yesmaximal, characteristic
13C23×#41C2211yescharacteristic
14C22×#137C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C2×#520—28no—
17C2×#520—22no—
18C2×#520—22no—
19C2×#520—28no—
20C2×#520C2214yes—
21C2×#520C2214yes—
22C22×C4×Q8C2212yes—
23C22×C4×D4—22no—
24C22×C4×D4C2212yes—
25C22×#117—21no—
26C22×#117C2211yescharacteristic
27C23×C4⋊C4—21no—
28C23×C4⋊C4C2211yescharacteristic
29C2×#483—28no—
30C2×#483C2214yes—
31C2×#483C2214yescomplemented
32#3007—28no—
33C2×#452C2214yes—
34C2×#451C2212yes—
35C2×#450C2212yes—
36C22×#106C2212yescomplemented
37C2×#417C2212yes—
38C2×#416C2214yes—
39C22×#103C2212yescomplemented
40C2×#407C2211yescharacteristic
41C22×C22⋊C8C2211yescharacteristic
42#2855C2214yes—
43#2625C2214yescomplemented
44#2503—21no—
45#2502—21no—
46#2499—22no—
47#2484—22no—
48#2483—22no—
49#2317C2212yescomplemented
50#2310C2212yescomplemented
51#2280C2211yescharacteristic
52#1878C22116yescomplemented
53#1877C22116yescomplemented
54#1865C22116yescomplemented
55#1864C22116yescomplemented
56#1824C2214yescomplemented
57#1823C2214yes—
58#1818C2214yescomplemented
59#1817C2214yes—
60#1788C2214yes—
61#1787C2212yes—
62#1786C2212yes—
63#1736C2214yescomplemented
64#1729C2214yescomplemented
65C22×#41—216no—
66C22×#41—24no—
67C22×#41—21no—
68C22×#41—21no—
69C22×#41D418yes—
70C22×#41C2312yes—
71C22×#41C2312yes—
72C23×Q8C2311yescharacteristic
73C23×Q8C2311yescharacteristic
74C23×D4—21no—
75C23×D4C2311yescharacteristic
76C24×C4—21no—
77C24×C4—21no—
78C24×C4—22no—
79C24×C4C2311yescharacteristic
80C2×#137—28no—
81C2×#137—216no—
82C22×Q16C2314yes—
83C22×SD16C2314yes—
84C22×M16C2312yes—
85C23×C8C2312yes—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92#520—232no—
93#520—232no—
94#520—216no—
95#520—216no—
96#520—216no—
97#520—216no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—24no—
101C2×C4×Q8C2318yes—
102C2×C4×D4—232no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4—28no—
106C2×C4×D4—28no—
107C2×C4×D4—24no—
108C2×C4×D4C2318yes—
109C2×#117—216no—
110C2×#117—24no—
111C2×#117—24no—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117D412yes—
115C2×#117D412yes—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C22×C4⋊C4—216no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—24no—
126C22×C4⋊C4—24no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D418yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4D412yes—
138C22×C42—22no—
139C22×C42C2312yes—
140#483—216no—
141#483—216no—
142#452—216no—
143#451—28no—
144#450—28no—
145C2×#106—28no—
146C2×#106C2318yes—
147#417—28no—
148#416—216no—
149C2×#103—28no—
150C2×#103C2318yes—
151C2×#103C2314yes—
152C2×#103C2318yes—
153C2×#102C2314yes—
154C2×#102C2318yes—
155#407—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161#400—24no—
162#398—24no—
163#397—28no—
164#391—22no—
165#391—22no—
166#391C2318yes—
167#389C2318yes—
168#387—28no—
169#385—24no—
170#385—24no—
171#384—22no—
172#382—22no—
173#381—24no—
174#381—21no—
175#381—24no—
176#380C2312yes—
177#380C2312yes—
178#379—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181#340C23116yes—
182#336C23116yes—
183C25—21no—
184C25C2×D411yescharacteristic
185C2×#41—216no—
186C2×#41—264no—
187C2×#41—216no—
188C2×#41—264no—
189C2×#41—28no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—28no—
196C2×#41C2×D418yes—
197C2×#41C2×D418yes—
198C2×#41#41116yes—
199C22×Q8—28no—
200C22×Q8—22no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—28no—
206C22×Q8C2×D414yes—
207C22×Q8C2412yes—
208C22×Q8C2412yes—
209C22×D4—216no—
210C22×D4—28no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—28no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4C2×D414yes—
217C22×D4C2412yes—
218C23×C4—216no—
219C23×C4—21no—
220C23×C4—28no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—28no—
224C23×C4—22no—
225C23×C4—216no—
226C23×C4—28no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—28no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—88no—
233C23×C4—84no—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#137—432no—
245C2×Q16—28no—
246C2×Q16—216no—
247C2×SD16—28no—
248C2×SD16—216no—
249C2×M16—24no—
250C2×M16—28no—
251C22×C8—24no—
252C22×C8—28no—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C4×Q8—232no—
262C4×Q8—216no—
263C4×Q8—216no—
264C4×D4—264no—
265C4×D4—264no—
266C4×D4—232no—
267C4×D4—216no—
268C4×D4—216no—
269#117—232no—
270#117—232no—
271#117—28no—
272#117—28no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—24no—
277#117—28no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—232no—
280C2×C4⋊C4—232no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D418yes—
305C2×C4⋊C4C2×D418yes—
306C2×C4⋊C4C2×D418yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2414yes—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C42—216no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—22no—
326C2×C42C2×D414yes—
327C2×C42C2414yes—
328#106—216no—
329#103—28no—
330#103—216no—
331#103—216no—
332#102—28no—
333#102—216no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338#95—22no—
339#95—22no—
340#95—28no—
341#95—28no—
342#95—28no—
343C24—28no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—42no—
349C24—44no—
350C24—44no—
351C24#12014yes—
352C24C22×D411yescharacteristic
353C24C22×D411yescharacteristic
354C24C22×D411yescharacteristic
355#41—264no—
356#41—232no—
357#41—264no—
358#41—216no—
359#41—216no—
360#41—216no—
361#41—232no—
362#41#123132yes—
363C2×Q8—216no—
364C2×Q8—28no—
365C2×Q8—216no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—28no—
372C2×Q8—42no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8C22×D414yes—
377C2×Q8C2×#4118yes—
378C2×D4—264no—
379C2×D4—216no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4C22×D414yes—
390C2×D4C2×#4118yes—
391C22×C4—28no—
392C22×C4—264no—
393C22×C4—216no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—232no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—816no—
423C22×C4—82no—
424C22×C4—816no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—832no—
428C22×C4#12014yes—
429C22×C4#12014yes—
430C22×C4#12014yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4118yes—
443C22×C4C2511yescharacteristic
444Q16—416no—
445SD16—416no—
446M16—48no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—28no—
450C2×C8—28no—
451C2×C8—48no—
452C4⋊C4—232no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—28no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—88no—
468C4⋊C4C22×D418yes—
469C22⋊C4—232no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—82no—
477C42—232no—
478C42—28no—
479C42—24no—
480C42—24no—
481C42—44no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—216no—
487C23—22no—
488C23—28no—
489C23—24no—
490C23—22no—
491C23—28no—
492C23—28no—
493C23—22no—
494C23—44no—
495C23—416no—
496C23—432no—
497C23—48no—
498C23—44no—
499C23—48no—
500C23—44no—
501C23C2×#12012yes—
502C23C2×#12012yes—
503C23C2×#12012yes—
504C23#52514yes—
505C23#57714yes—
506C23C23×D411yescharacteristic
507Q8—216no—
508Q8—24no—
509Q8—28no—
510Q8—44no—
511Q8—44no—
512Q8—84no—
513Q8C2×#12318yes—
514D4—232no—
515D4—216no—
516D4—232no—
517D4—24no—
518D4—28no—
519D4C2×#12318yes—
520C2×C4—216no—
521C2×C4—232no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—232no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—416no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—88no—
551C2×C4—88no—
552C2×C4—832no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4C2×#12012yes—
556C2×C4C2×#12012yes—
557C2×C4C2×#12012yes—
558C2×C4C2×#12318yes—
559C2×C4#52514yes—
560C2×C4#52518yes—
561C2×C4C23×D411yescharacteristic
562C2×C4C23×D411yescharacteristic
563C2×C4C22×#4114yes—
564C8—44no—
565C8—44no—
566C22—22no—
567C22—28no—
568C22—28no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—28no—
574C22—22no—
575C22—22no—
576C22—416no—
577C22—432no—
578C22—44no—
579C22—416no—
580C22—44no—
581C22—44no—
582C22#284914yes—
583C22#300914yes—
584C22C22×#12011yescharacteristic
585C22C2×#52512yes—
586C22C2×#57712yes—
587C22#358612yes—
588C4—22no—
589C4—28no—
590C4—22no—
591C4—44no—
592C4—48no—
593C4—44no—
594C4—82no—
595C4—84no—
596C4—82no—
597C4#284914yes—
598C4C22×#12312yes—
599C4C2×#52512yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—48no—
604C2C2×#284911yescharacteristic
605C2C2×#300911yescharacteristic
606C2#6036811yescharacteristic
607C2#6058514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#483C2semidirect
4. #35311C2semidirect
5. #33016C2semidirect
6. #33008C2semidirect
7. #33007C2semidirect
8. #20931C2semidirect
9. #20882C2semidirect
10. #20700C2semidirect
11. #20685C2semidirect
31. C2×#483C22semidirect
36. C22×#106C22semidirect
39. C22×#103C22semidirect
43. #2625C22semidirect
49. #2317C22semidirect
50. #2310C22semidirect
52. #1878C22semidirect
53. #1877C22semidirect
54. #1865C22semidirect
55. #1864C22semidirect
56. #1824C22semidirect
58. #1818C22semidirect
63. #1736C22semidirect
64. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C22×#4832
C2×#30072
#353112
#330162
#330082
#330072
#209312
#208822
#207002
#206852
#205972