Finite Group Databasebeta

← #6506500

Subgroups — #6506500

Order 512.

2206 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37426C211yesmaximal, characteristic, complemented
2#35666C211yesmaximal, characteristic
3#29118C212yesmaximal, complemented
4#29114C212yesmaximal, complemented
5#27284C214yesmaximal, complemented
6#27277C214yesmaximal, complemented
7#27268C214yesmaximal, complemented
8#27265C214yesmaximal, complemented
9#25866C212yesmaximal, complemented
10#25861C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#20603C212yesmaximal
13#20532C211yesmaximal, characteristic, complemented
14C22×2−1+4C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#491C2214yes—
19C2×#489C2212yes—
20C2×#487C2212yes—
21C2×#417C2212yes—
22C2×#419C2212yes—
23C2×#416C2212yes—
24C22×#103C2212yes—
25C2×#408C2212yes—
26#2851C2212yes—
27#2804—22no—
28#2799—22no—
29#2784—22no—
30#2748—22no—
31#2628C2212yes—
32#2442C2214yes—
33#2439C2214yes—
34#2434—22no—
35#2434C2214yes—
36#2432—22no—
37#2432C2214yes—
38#2406—22no—
39#2406—22no—
40#2311C2212yes—
41#2310C2211yescharacteristic
42#2297C2211yescharacteristic
43#2297C2211yescharacteristic
44#2074C2214yes—
45#2074C2212yes—
46#2073C2212yes—
47#2043C2214yes—
48#2043C2214yes—
49#2043C2214yes—
50#2042C2214yes—
51#2026C2212yes—
52#2025C2212yes—
53#2024C2214yes—
54#1960C2214yes—
55#1960C2212yes—
56#1958C2212yes—
57#1936C2214yes—
58#1935C2214yes—
59#1935C2214yes—
60#1933C2214yes—
61#1894C2214yes—
62#1892C2214yes—
63#1873C2214yes—
64#1872C2214yes—
65#1825C2214yes—
66#1788C2214yes—
67#1788C2214yes—
68#1787C2214yes—
69#1787C2214yes—
70#1787C2214yes—
71#1753C2214yes—
72#1752C2214yes—
73#1750C2212yes—
74#1749C2212yes—
75C2×2−1+4—28no—
76C2×2−1+4—44no—
77C2×2−1+4—42no—
78C2×2−1+4—42no—
79C22×#41—22no—
80C22×#41—21no—
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C22×M16C2312yes—
89C23×C8C2312yes—
90#536—28no—
91#536—42no—
92#536—42no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2314yes—
98C2×C4⋊Q8C2312yes—
99C2×#125C2312yes—
100C2×#124—21no—
101C2×#124C2312yes—
102C2×#123—22no—
103C2×#123—21no—
104C2×#123—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2314yes—
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—22no—
111C22×C4⋊C4C2312yes—
112C22×C42C2311yescharacteristic
113#491—48no—
114#489—44no—
115#487—44no—
116#419—44no—
117#417—44no—
118#416—44no—
119C2×#103—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132#408—28no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2312yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—24no—
140#399—22no—
141#398—24no—
142#396—24no—
143#395—22no—
144#395—24no—
145#395—24no—
146#394—24no—
147#394—24no—
148#393—22no—
149#393—24no—
150#391—22no—
151#391—22no—
152#391—24no—
153#391D412yes—
154#391C2314yes—
155#391C2312yes—
156#390—22no—
157#390D412yes—
158#390C2312yes—
159#389C2314yes—
160#389C2314yes—
161#387—24no—
162#387—24no—
163#386—22no—
164#386—22no—
165#386—22no—
166#386D412yes—
167#386D412yes—
168#385—24no—
169#385—24no—
170#385—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#382C2312yes—
175#382C2312yes—
176#381—21no—
177#381—21no—
178#380—24no—
179#380C2314yes—
180#380C2314yes—
181C4×C22⋊C4—22no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2314yes—
1902−1+4—432no—
1912−1+4—432no—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—41no—
201C2×#41—41no—
202C2×#41—42no—
203C2×#41—42no—
204C2×#41—42no—
205C2×#41—48no—
206C2×#41—44no—
207C2×#41—48no—
208C2×#41—44no—
209C2×#41—48no—
210C2×#41—44no—
211C2×#41#4118yes—
212C22×Q8—24no—
213C22×Q8—21no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—41no—
226C22×Q8—41no—
227C22×Q8—42no—
228C22×Q8—41no—
229C22×Q8—41no—
230C22×Q8—44no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C22×D4—21no—
238C22×D4—22no—
239C22×D4C2×D412yes—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—42no—
253C2×M16—42no—
254C22×C8—24no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—28no—
260C4⋊Q8—24no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—42no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267#125—44no—
268#124—24no—
269#124—42no—
270#124—44no—
271#123—44no—
272#123—44no—
273#123—48no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—48no—
277#117—24no—
278#117—41no—
279#117—41no—
280#117—42no—
281#117—42no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4C2×D411yescharacteristic
308C2×C22⋊C4C2×D411yescharacteristic
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42C2×D412yes—
315C2×C42C2412yes—
316C2×C42C2412yes—
317#103—28no—
318#103—28no—
319#103—28no—
320#103—44no—
321#103—44no—
322#103—44no—
323#103—44no—
324#103—44no—
325#103—44no—
326#103—44no—
327#102—28no—
328#102—44no—
329C22⋊C8—28no—
330C22⋊C8—28no—
331#97—42no—
332#97—44no—
333C4×C8—42no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95C2×D414yes—
345#95C2×D414yes—
346C24—21no—
347C24C22×D412yes—
348C24C22×D411yescharacteristic
349#41—216no—
350#41—48no—
351#41—48no—
352#41—48no—
353#41—48no—
354#41—48no—
355#41—48no—
356#41—48no—
357#41—432no—
358#41—464no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—216no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—24no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—42no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—432no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—82no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×Q8#12414yes—
394C2×Q8H(5,𝔽2)12yes—
395C2×Q8H(5,𝔽2)14yes—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—41no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4#12418yes—
410C2×D4#12414yes—
411C2×D4H(5,𝔽2)12yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2511yescharacteristic
453M16—84no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—84no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—42no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—42no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—82no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—42no—
493C42—42no—
494C42—82no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—21no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—44no—
504C23—41no—
505C23—41no—
506C23C2×#12414yes—
507C23C2×#12412yes—
508C23C23×D411yescharacteristic
509Q8—24no—
510Q8—24no—
511Q8—24no—
512Q8—44no—
513Q8—416no—
514Q8—48no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—44no—
519Q8—44no—
520Q8—84no—
521Q8—164no—
522D4—24no—
523D4—48no—
524D4—44no—
525D4—48no—
526D4—416no—
527D4—42no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—416no—
549C2×C4—48no—
550C2×C4—416no—
551C2×C4—48no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—84no—
568C2×C4—164no—
569C2×C4C2×#12412yes—
570C2×C4C2×#12414yes—
571C2×C4C22×#4112yes—
572C2×C4C2×H(5,𝔽2)12yes—
573C2×C4C2×H(5,𝔽2)11yescharacteristic
574C2×C4C2×H(5,𝔽2)11yescharacteristic
575C8—44no—
576C8—44no—
577C22—22no—
578C22—24no—
579C22—28no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—44no—
588C22—82no—
589C22—82no—
590C22C22×#12411yescharacteristic
591C22#348411yescharacteristic
592C22#348411yescharacteristic
593C22#358614yes—
594C4—24no—
595C4—22no—
596C4—42no—
597C4—44no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4—84no—
605C4#349012yes—
606C4#349012yes—
607C2—42no—
608C2—44no—
609C2—82no—
610C2#3536711yescharacteristic
611C2#6062712yes—
612C2#6164612yes—
613C2#6167612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37426C2semidirect
3. #29118C2semidirect
4. #29114C2semidirect
5. #27284C2semidirect
6. #27277C2semidirect
7. #27268C2semidirect
8. #27265C2semidirect
9. #25866C2semidirect
10. #25861C2semidirect
11. #21771C2semidirect
13. #20532C2semidirect

Maximal subgroups

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

subgroupindex
#374262
#356662
#291182
#291142
#272842
#272772
#272682
#272652
#258662
#258612
#217712
#206032
#205322