Finite Group Databasebeta

← #6566091

Subgroups — #6566091

Order 512.

1773 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3342C212yesmaximal
2#36188C211yesmaximal, characteristic
3#34324C211yesmaximal, characteristic
4#29866C211yesmaximal, characteristic, complemented
5#29759C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28575C212yesmaximal, complemented
8#28205C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25456C212yesmaximal, complemented
11#25215C214yesmaximal, complemented
12#24818C212yesmaximal, complemented
13#24586C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23656C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic
17#22490C212yesmaximal
18C2×#556C2212yes—
19C2×#556C2211yescharacteristic
20#3342—216no—
21C2×#500C2212yes—
22C2×#485C2214yes—
23C2×#485C2214yes—
24C2×#475C2212yes—
25C2×#475C2212yes—
26C2×#467C2214yes—
27C2×C4×Q16C2212yes—
28C2×C8×D4C2212yes—
29C2×#427C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2568C2212yes—
33#2547—21no—
34#2547—21no—
35#2499C2214yes—
36#2496—22no—
37#2496—22no—
38#2373—22no—
39#2367—21no—
40#2365—21no—
41#2365—21no—
42#2365C2211yescharacteristic
43#2362C2212yes—
44#2355—21no—
45#2354C2211yescharacteristic
46Q8×C4⋊C4C2211yescharacteristic
47#2346C2212yes—
48#2334—22no—
49#2326—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2296C2212yes—
54#2295C2211yescharacteristic
55#2285C2212yes—
56#2276—21no—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2211yescharacteristic
66#2072C2211yescharacteristic
67#2072C2212yes—
68#2072C2212yes—
69#2068C2212yes—
70#2068C2214yes—
71#2066C2214yes—
72#2056C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2016C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1949C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1928C2214yes—
82#1927C2214yes—
83#1892C2214yes—
84#1878C2214yes—
85#1873C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1805C2214yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1735C2214yes—
95C2×#336C2212yes—
96C22×Q16C2312yes—
97C23×C8C2314yes—
98#556—28no—
99#556—24no—
100#556—28no—
101#556D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—22no—
108C2×#126—24no—
109C2×#124—21no—
110C2×#124—22no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117C2314yes—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C42C2312yes—
127#500—28no—
128#485—216no—
129#485—216no—
130#475—28no—
131#475—28no—
132#467—216no—
133C4×Q16—28no—
134C8×D4—28no—
135#427—216no—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#397—24no—
155#397—24no—
156#391—22no—
157#391—22no—
158#391—21no—
159#391—21no—
160#391C2312yes—
161#391C2311yescharacteristic
162#391C2314yes—
163#391C2311yescharacteristic
164#390—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2311yescharacteristic
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—21no—
182#387—22no—
183#387—21no—
184#387—22no—
185#387—22no—
186#387—24no—
187#387—24no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#382C2314yes—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—21no—
228#376—22no—
229C43C2311yescharacteristic
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×Q16—24no—
251C2×Q16—24no—
252C2×Q16—42no—
253C22×C8—28no—
254C22×C8—28no—
255C22×C8—44no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8C2×D414yes—
266#126—216no—
267#126—416no—
268#124—28no—
269#124—48no—
270C22⋊Q8—24no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8C2×D418yes—
274C4×Q8—28no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×Q8C2×D414yes—
283C4×D4—24no—
284C4×D4—42no—
285C4×D4C2×D414yes—
286C4×D4#4114yes—
287#117—24no—
288#117—28no—
289#117—28no—
290#117—44no—
291#117C2×D418yes—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—28no—
355#107—28no—
356#106—28no—
357#106—28no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#103—24no—
362#103—28no—
363#103—24no—
364#103—24no—
365#103—28no—
366#103—42no—
367#103—44no—
368C22⋊C8—28no—
369C4×C8—24no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24C22×D412yes—
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8C22×D412yes—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4#12114yes—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D414yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4114yes—
440C22×C4C2511yescharacteristic
441Q16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—416no—
451C2×C82−1+414yes—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4#12114yes—
486C4⋊C4C22×D412yes—
487C4⋊C4C22×D414yes—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4#12118yes—
497C42—28no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—42no—
505C42—42no—
506C42—41no—
507C42—44no—
508C42—48no—
509C42—44no—
510C42—44no—
511C42—42no—
512C42—82no—
513C42—82no—
514C42#13514yes—
515C42C22×D412yes—
516C42C2×#4112yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23C2×#12114yes—
524C23C23×D411yescharacteristic
525Q8—24no—
526Q8—42no—
527Q8—48no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—88no—
533D4—28no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4C2×#12114yes—
585C2×C4C2×#12112yes—
586C2×C4C2×#12112yes—
587C2×C4C2×#13512yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4#57512yes—
591C2×C4#57812yes—
592C2×C4C23×D411yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×2−1+412yes—
595C2×C4#58512yes—
596C8—24no—
597C8—42no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22C22×#12111yescharacteristic
607C22#348511yescharacteristic
608C22#349911yescharacteristic
609C22C22×#13511yescharacteristic
610C22C2×#57511yescharacteristic
611C22C2×#57811yescharacteristic
612C22#358611yescharacteristic
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—24no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—44no—
624C4—82no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4—82no—
629C4—84no—
630C4#303514yes—
631C4#348912yes—
632C4#351512yes—
633C2—44no—
634C2#3602011yescharacteristic
635C2C2×#303511yescharacteristic
636C2#6043211yescharacteristic
637C2#6170111yescharacteristic
638C2#6173011yescharacteristic
639C2#6247311yescharacteristic
640C2#6250711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #29866C2semidirect
5. #29759C2semidirect
6. #28689C2semidirect
7. #28575C2semidirect
8. #28205C2semidirect
9. #27652C2semidirect
10. #25456C2semidirect
11. #25215C2semidirect
12. #24818C2semidirect
13. #24586C2semidirect
14. #23679C2semidirect
15. #23656C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33422
#361882
#343242
#298662
#297592
#286892
#285752
#282052
#276522
#254562
#252152
#248182
#245862
#236792
#236562
#226962
#224902