Finite Group Databasebeta

← #6508804

Subgroups — #6508804

Order 512.

2164 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic, complemented
2C2×#2972C211yesmaximal, characteristic
3#35010C211yesmaximal, characteristic
4C2×#2487C211yesmaximal, characteristic
5#28095C214yesmaximal, complemented
6#27662C214yesmaximal, complemented
7#25234C218yesmaximal, complemented
8#24602C214yesmaximal, complemented
9#24552C214yesmaximal, complemented
10#21940C211yesmaximal, characteristic, complemented
11#21759C211yesmaximal, characteristic, complemented
12C2×#1824C211yesmaximal, characteristic, complemented
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#484—28no—
17C2×#484C2214yes—
18C2×#484C2214yescomplemented
19#2972—28no—
20C2×C8×D4C2214yes—
21C2×C8×D4C2214yes—
22C2×#431C2212yes—
23C22×#107C2212yescomplemented
24C22×#103C2212yescomplemented
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2212yes—
27C22×C22⋊C8C2211yescharacteristic
28#2855C2212yes—
29#2848C2212yes—
30#2627—21no—
31#2626—21no—
32#2579—22no—
33#2564—22no—
34#2487—22no—
35#2487—22no—
36#2487C2212yes—
37#2487C2214yes—
38#2487C2214yes—
39#2417C2214yes—
40#2409C2212yes—
41C2×#400—21no—
42C2×#399—21no—
43#2315—21no—
44#2308—21no—
45C2×#389C2212yes—
46C2×#386—21no—
47C2×#385—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51#2087C2218yescomplemented
52#2076C2218yescomplemented
53#2068C2218yescomplemented
54#2036C2218yescomplemented
55#1942C2218yescomplemented
56#1933C2218yescomplemented
57#1892C2214yescomplemented
58#1892C2218yescomplemented
59#1872C2214yes—
60#1852C2214yes—
61#1852C2214yescomplemented
62#1824C2214yes—
63#1824C2212yescomplemented
64#1824C2214yescomplemented
65#1808C2218yescomplemented
66#1808C2214yescomplemented
67#1787C2212yescomplemented
68C2×#340C2214yescomplemented
69C23×Q8C2311yescharacteristic
70C23×D4D411yescharacteristic
71C24×C4—21no—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C23×C8C2314yes—
75C23×C8C2312yes—
76C23×C8C2312yes—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8C2312yes—
82C2×C22⋊Q8C2312yes—
83C2×C22⋊Q8C2314yes—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4D412yes—
87C2×C4×D4D412yes—
88C2×C4×D4C2312yes—
89C2×C4×D4C2312yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4D411yescharacteristic
109C22×C22⋊C4D411yescharacteristic
110C22×C42C2311yescharacteristic
111#484—216no—
112#484—216no—
113C8×D4—216no—
114C8×D4—216no—
115#431—28no—
116C2×#107—28no—
117C2×#107C2318yes—
118C2×#107C2318yes—
119C2×C4⋊C8C2314yes—
120C2×#103—28no—
121C2×#103C2318yes—
122C2×#103C2318yes—
123C2×C22⋊C8—24no—
124C2×C22⋊C8—28no—
125C2×C22⋊C8D418yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C4×C8C2314yes—
132#400—22no—
133#400—24no—
134#400—24no—
135#399—22no—
136#399—24no—
137#399—24no—
138#398—24no—
139#396—24no—
140#395—22no—
141#393—22no—
142#391C2314yes—
143#389—24no—
144#389C2318yes—
145#389C2314yes—
146#387—24no—
147#386—24no—
148#386—22no—
149#386—22no—
150#386—24no—
151#386—24no—
152#385—24no—
153#385—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—24no—
158#385—24no—
159#381—22no—
160#380—21no—
161#380—21no—
162#380—24no—
163#380—21no—
164#380—21no—
165#380—24no—
166#380C2314yes—
167#380C2312yes—
168#380C2312yes—
169#380C2318yes—
170#380C2312yes—
171#380C2312yes—
172#379—22no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—22no—
176C2×#95—22no—
177#340C2318yes—
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181C25C2×D411yescharacteristic
182C25C2×D411yescharacteristic
183C22×Q8—21no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—44no—
187C22×Q8C2412yes—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4Q1618yes—
191C22×D4C2×D412yes—
192C22×D4C2×D412yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—84no—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D414yes—
214C23×C4#4114yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8—24no—
220C22×C8—28no—
221C22×C8—44no—
222C22×C8—48no—
223C22×C8—44no—
224C22×C8C2×D418yes—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—44no—
232C22⋊Q8—44no—
233C22⋊Q8—48no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—84no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2414yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—84no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42C2×D412yes—
295C2×C42C2412yes—
296#107—216no—
297#107—48no—
298C4⋊C8—28no—
299#103—216no—
300#103—48no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—28no—
304C22⋊C8C2×D418yes—
305C22⋊C8C2×D418yes—
306C4×C8—28no—
307#95—22no—
308#95—28no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317C24—22no—
318C24—22no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24#12114yes—
324C24C2×Q1614yes—
325C24C2×Q1614yes—
326C24#13514yes—
327C24C22×D412yes—
328C24C22×D411yescharacteristic
329C24C22×D411yescharacteristic
330C24C22×D411yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—28no—
334C2×Q8—48no—
335C2×Q8—42no—
336C2×Q8—82no—
337C2×Q8—84no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—232no—
343C2×D4C2×Q1618yes—
344C2×D4C2×Q1618yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4—82no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—816no—
389C22×C4#12114yes—
390C22×C4#12114yes—
391C22×C4#12114yes—
392C22×C4C2×Q1614yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2511yescharacteristic
406C2×C8—28no—
407C2×C8—24no—
408C2×C8—28no—
409C2×C8—48no—
410C2×C8—88no—
411C2×C8—84no—
412C2×C8C22×D418yes—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—84no—
429C4⋊C4—82no—
430C4⋊C4—88no—
431C4⋊C4—84no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—84no—
443C22⋊C4—88no—
444C22⋊C4—84no—
445C42—24no—
446C42—22no—
447C42—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—28no—
459C23—44no—
460C23—44no—
461C23—48no—
462C23—48no—
463C23—48no—
464C23—82no—
465C23—84no—
466C23—82no—
467C23#467116yes—
468C23C2×#12112yes—
469C23C2×#12112yes—
470C23C2×#12112yes—
471C23C22×Q1614yes—
472C23C22×Q1612yes—
473C23C22×Q1612yes—
474C23C22×Q1612yes—
475C23C2×#13512yes—
476C23C2×#13512yes—
477C23C2×#13512yes—
478C23C23×D411yescharacteristic
479Q8—44no—
480Q8—44no—
481Q8—84no—
482D4—216no—
483D4—216no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—88no—
522C2×C4—88no—
523C2×C4—84no—
524C2×C4—88no—
525C2×C4—88no—
526C2×C4—84no—
527C2×C4—164no—
528C2×C4C2×#12112yes—
529C2×C4C2×#12112yes—
530C2×C4C2×#12112yes—
531C2×C4C22×Q1614yes—
532C2×C4C23×D411yescharacteristic
533C2×C4C22×#4111yescharacteristic
534C2×C4#58514yes—
535C8—28no—
536C8—44no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—28no—
550C22—24no—
551C22—416no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—88no—
556C22C2×#46714yes—
557C22C2×#46714yes—
558C22C2×#46714yes—
559C22C22×#12111yescharacteristic
560C22#348512yes—
561C22C23×Q1611yescharacteristic
562C22C22×#13511yescharacteristic
563C22#358512yes—
564C4—24no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—42no—
569C4—42no—
570C4—84no—
571C4—84no—
572C4—82no—
573C4—84no—
574C4—82no—
575C4#348512yes—
576C4#349112yes—
577C2—24no—
578C2—24no—
579C2—24no—
580C2—44no—
581C2#3544711yescharacteristic
582C2C22×#46711yescharacteristic
583C2#6045411yescharacteristic
584C2#6169812yes—
585C2#6169912yes—
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×#484C2semidirect
5. #28095C2semidirect
6. #27662C2semidirect
7. #25234C2semidirect
8. #24602C2semidirect
9. #24552C2semidirect
10. #21940C2semidirect
11. #21759C2semidirect
12. C2×#1824C2semidirect
18. C2×#484C22semidirect
23. C22×#107C22semidirect
24. C22×#103C22semidirect
51. #2087C22semidirect
52. #2076C22semidirect
53. #2068C22semidirect
54. #2036C22semidirect
55. #1942C22semidirect
56. #1933C22semidirect
57. #1892C22semidirect
58. #1892C22semidirect
61. #1852C22semidirect
63. #1824C22semidirect
64. #1824C22semidirect
65. #1808C22semidirect
66. #1808C22semidirect
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×#4842
C2×#29722
#350102
C2×#24872
#280952
#276622
#252342
#246022
#245522
#219402
#217592
C2×#18242