Finite Group Databasebeta

← #6508801

Subgroups — #6508801

Order 512.

2240 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic, complemented
2C2×#2972C211yesmaximal, characteristic
3#35010C211yesmaximal, characteristic
4C2×#2486C211yesmaximal, characteristic
5#28098C214yesmaximal, complemented
6#27658C214yesmaximal, complemented
7#25236C218yesmaximal, complemented
8#24601C214yesmaximal, complemented
9#24551C214yesmaximal, complemented
10#21938C211yesmaximal, characteristic, complemented
11#21759C211yesmaximal, characteristic, complemented
12C2×#1823C211yesmaximal, characteristic, complemented
13C22×#123—21no—
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#483—28no—
18C2×#483C2214yes—
19C2×#483C2214yescomplemented
20#2972—28no—
21C2×C8×D4C2214yes—
22C2×C8×D4C2214yes—
23C2×#431C2212yes—
24C22×#106C2212yescomplemented
25C22×#103C2212yescomplemented
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2212yes—
28C22×C22⋊C8C2211yescharacteristic
29#2855C2212yes—
30#2848C2212yes—
31#2627—21no—
32#2626—21no—
33#2579—22no—
34#2564—22no—
35#2487C2212yes—
36#2486—22no—
37#2486—22no—
38#2486C2214yes—
39#2486C2214yes—
40#2417C2214yes—
41#2409C2212yes—
42C2×#398—21no—
43C2×#397—21no—
44#2315—21no—
45#2308—21no—
46C2×#389C2212yes—
47C2×#381—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51#2088C2218yescomplemented
52#2077C2218yescomplemented
53#2066C2218yescomplemented
54#2035C2218yescomplemented
55#1941C2218yescomplemented
56#1933C2218yescomplemented
57#1892C2214yescomplemented
58#1892C2218yescomplemented
59#1872C2214yes—
60#1853C2214yes—
61#1853C2214yescomplemented
62#1824C2212yescomplemented
63#1823C2214yes—
64#1823C2214yescomplemented
65#1807C2218yescomplemented
66#1807C2214yescomplemented
67#1787C2212yescomplemented
68C2×#340C2214yescomplemented
69C23×Q8C2311yescharacteristic
70C23×D4D411yescharacteristic
71C24×C4—21no—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C23×C8C2314yes—
75C23×C8C2312yes—
76C23×C8C2312yes—
77C2×#123—24no—
78C2×#123—24no—
79C2×#123—48no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8C2312yes—
85C2×C22⋊Q8C2312yes—
86C2×C22⋊Q8C2314yes—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4D412yes—
90C2×C4×D4D412yes—
91C2×C4×D4C2312yes—
92C2×C4×D4C2312yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D411yescharacteristic
112C22×C22⋊C4D411yescharacteristic
113C22×C42C2311yescharacteristic
114#483—216no—
115#483—216no—
116C8×D4—216no—
117C8×D4—216no—
118#431—28no—
119C2×#106—28no—
120C2×#106C2318yes—
121C2×#106C2318yes—
122C2×C4⋊C8C2314yes—
123C2×#103—28no—
124C2×#103C2318yes—
125C2×#103C2318yes—
126C2×C22⋊C8—24no—
127C2×C22⋊C8—28no—
128C2×C22⋊C8D418yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C4×C8C2314yes—
135#400—22no—
136#399—22no—
137#398—24no—
138#398—24no—
139#398—24no—
140#397—24no—
141#397—24no—
142#396—24no—
143#395—22no—
144#393—22no—
145#391C2314yes—
146#389—24no—
147#389C2318yes—
148#389C2314yes—
149#387—24no—
150#386—24no—
151#386—22no—
152#386—22no—
153#385—24no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#381—22no—
159#381—24no—
160#381—24no—
161#380—21no—
162#380—21no—
163#380—24no—
164#380—21no—
165#380—21no—
166#380—24no—
167#380C2314yes—
168#380C2312yes—
169#380C2312yes—
170#380C2318yes—
171#380C2312yes—
172#380C2312yes—
173#379—22no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—22no—
177C2×#95—22no—
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182C25C2×D411yescharacteristic
183C25C2×D411yescharacteristic
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—44no—
188C22×Q8C2412yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4SD1618yes—
192C22×D4C2×D412yes—
193C22×D4C2×D412yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C22×C8—24no—
222C22×C8—28no—
223C22×C8—44no—
224C22×C8—48no—
225C22×C8—44no—
226C22×C8C2×D418yes—
227C22×C8C2414yes—
228C22×C8C2414yes—
229#123—416no—
230#123—416no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—44no—
236C22⋊Q8—44no—
237C22⋊Q8—48no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2414yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42C2×D412yes—
299C2×C42C2412yes—
300#106—216no—
301#106—48no—
302C4⋊C8—28no—
303#103—216no—
304#103—48no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307C22⋊C8—28no—
308C22⋊C8C2×D418yes—
309C22⋊C8C2×D418yes—
310C4×C8—28no—
311#95—28no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321C24—22no—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24#12114yes—
328C24C2×SD1614yes—
329C24C2×SD1614yes—
330C24#13514yes—
331C24C22×D412yes—
332C24C22×D411yescharacteristic
333C24C22×D411yescharacteristic
334C24C22×D411yescharacteristic
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—28no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—82no—
341C2×Q8—84no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—232no—
347C2×D4C2×SD1618yes—
348C2×D4C2×SD1618yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—816no—
395C22×C4#12114yes—
396C22×C4#12114yes—
397C22×C4#12114yes—
398C22×C4C2×SD1614yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2511yescharacteristic
412C2×C8—28no—
413C2×C8—24no—
414C2×C8—28no—
415C2×C8—48no—
416C2×C8—88no—
417C2×C8—84no—
418C2×C8C22×D418yes—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—416no—
435C4⋊C4—84no—
436C4⋊C4—82no—
437C4⋊C4—84no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—416no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C42—24no—
452C42—22no—
453C42—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—28no—
465C23—44no—
466C23—44no—
467C23—48no—
468C23—48no—
469C23—48no—
470C23—82no—
471C23—84no—
472C23—82no—
473C23#465116yes—
474C23C2×#12112yes—
475C23C2×#12112yes—
476C23C2×#12112yes—
477C23C22×SD1614yes—
478C23C22×SD1612yes—
479C23C22×SD1612yes—
480C23C22×SD1612yes—
481C23C2×#13512yes—
482C23C2×#13512yes—
483C23C2×#13512yes—
484C23C23×D411yescharacteristic
485Q8—44no—
486Q8—44no—
487Q8—84no—
488D4—216no—
489D4—216no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—88no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4—88no—
533C2×C4—164no—
534C2×C4C2×#12112yes—
535C2×C4C2×#12112yes—
536C2×C4C2×#12112yes—
537C2×C4C22×SD1614yes—
538C2×C4C23×D411yescharacteristic
539C2×C4C22×#4111yescharacteristic
540C2×C4#58514yes—
541C8—28no—
542C8—44no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—28no—
557C22—416no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—88no—
562C22C2×#46514yes—
563C22C2×#46514yes—
564C22C2×#46514yes—
565C22C22×#12111yescharacteristic
566C22#348512yes—
567C22C23×SD1611yescharacteristic
568C22C22×#13511yescharacteristic
569C22#358612yes—
570C4—24no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—42no—
575C4—42no—
576C4—84no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4—84no—
581C4#348512yes—
582C4#349112yes—
583C2—24no—
584C2—24no—
585C2—24no—
586C2—44no—
587C2#3544711yescharacteristic
588C2C22×#46511yescharacteristic
589C2#6045511yescharacteristic
590C2#6169512yes—
591C2#6169612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#483C2semidirect
5. #28098C2semidirect
6. #27658C2semidirect
7. #25236C2semidirect
8. #24601C2semidirect
9. #24551C2semidirect
10. #21938C2semidirect
11. #21759C2semidirect
12. C2×#1823C2semidirect
19. C2×#483C22semidirect
24. C22×#106C22semidirect
25. C22×#103C22semidirect
51. #2088C22semidirect
52. #2077C22semidirect
53. #2066C22semidirect
54. #2035C22semidirect
55. #1941C22semidirect
56. #1933C22semidirect
57. #1892C22semidirect
58. #1892C22semidirect
61. #1853C22semidirect
62. #1824C22semidirect
64. #1823C22semidirect
65. #1807C22semidirect
66. #1807C22semidirect
67. #1787C22semidirect
68. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#29722
#350102
C2×#24862
#280982
#276582
#252362
#246012
#245512
#219382
#217592
C2×#18232