Finite Group Databasebeta

← #6479667

Subgroups — #6479667

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35324C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#32964C212yesmaximal
4#32915C212yesmaximal
5#29915C212yesmaximal
6#29627C212yesmaximal
7#29509C214yesmaximal, complemented
8#29371C212yesmaximal, complemented
9#29370C212yesmaximal, complemented
10#29129C211yesmaximal, characteristic, complemented
11#29126C211yesmaximal, characteristic, complemented
12#28385C211yesmaximal, characteristic, complemented
13#28373C211yesmaximal, characteristic, complemented
14#28213C212yesmaximal, complemented
15#28170C212yesmaximal, complemented
16#22054C214yesmaximal
17#20781C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519C2212yes—
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#451C2214yes—
25#2864C2211yescharacteristic
26#2862C2211yescharacteristic
27#2855C2212yes—
28#2645C2212yes—
29#2628C2211yescharacteristic
30#2621C2211yescharacteristic
31#2588—22no—
32#2575—22no—
33#2528—22no—
34#2527—22no—
35#2365—22no—
36#2365—22no—
37#2344—22no—
38#2344—22no—
39#2331C2211yescharacteristic
40#2327C2211yescharacteristic
41#2327C2212yes—
42#2327C2212yes—
43#2323C2212yes—
44#2310C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49#2295C2212yes—
50#2285C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2212yes—
53#2099C2212yes—
54#2096C2212yes—
55#2095C2212yes—
56#2085C2214yes—
57#2074C2214yes—
58#2043C2214yes—
59#2043C2212yes—
60#2043C2212yes—
61#2016C2214yes—
62#2016C2212yes—
63#2016C2212yes—
64#1971C2214yes—
65#1969C2214yes—
66#1935C2214yes—
67#1933C2214yes—
68#1928C2214yes—
69#1927C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1878C2214yes—
74#1873C2214yes—
75#1865C2214yes—
76#1863C2214yes—
77#1854C2214yes—
78#1839C2212yes—
79#1832C2212yes—
80#1817C2212yes—
81#1805C2212yes—
82#1805C2212yes—
83#1805C2212yes—
84#1805C2212yes—
85#1798C2212yes—
86#1787C2214yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2211yescharacteristic
89#1750C2212yes—
90#1746C2212yes—
91C23×Q8C2312yes—
92C22×Q16C2314yes—
93C22×Q16C2314yes—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C22⋊Q8C2312yes—
102#519—44no—
103#519—44no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×#117—22no—
112C2×#117—22no—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C42C2311yescharacteristic
118#477—44no—
119#475—44no—
120#462—44no—
121#462—44no—
122#451—216no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#401—22no—
139#400—21no—
140#400—22no—
141#400—22no—
142#400—22no—
143#399—21no—
144#398—21no—
145#398—21no—
146#397—21no—
147#397—21no—
148#395—21no—
149#393—21no—
150#391—24no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#385—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—22no—
190#382C2311yescharacteristic
191#382C2311yescharacteristic
192#382C2312yes—
193#381—22no—
194#381—24no—
195#380—22no—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4D412yes—
208#376—22no—
209#376C2311yescharacteristic
210#376C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222C22×Q8—22no—
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8#4118yes—
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C22×Q8C2412yes—
233C23×C4—22no—
234C23×C4C2×D412yes—
235C23×C4C2411yescharacteristic
236C2×Q16—28no—
237C2×Q16—28no—
238C2×Q16—44no—
239C2×Q16—44no—
240C2×Q16—44no—
241C2×Q16—44no—
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—216no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C22⋊Q8—28no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—42no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—42no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×Q8—48no—
263C4×Q8—48no—
264#117—24no—
265#117—41no—
266#117—41no—
267#117—48no—
268#117—48no—
269#117—42no—
270#117—42no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4C2×D411yescharacteristic
302C2×C22⋊C4C2×D411yescharacteristic
303C2×C22⋊C4C2×D412yes—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#106—44no—
325C4⋊C8—24no—
326C4⋊C8—24no—
327#103—28no—
328#103—28no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334#103—44no—
335C22⋊C8—28no—
336#97—42no—
337C4×C8—42no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365C24C22×D411yescharacteristic
366C2×Q8—28no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—82no—
389C2×Q8C2×#4114yes—
390C2×Q8C2×#4114yes—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C2×#4114yes—
418C22×C4C2511yescharacteristic
419Q16—88no—
420Q16—88no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—216no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—82no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—42no—
473C42—42no—
474C42—44no—
475C42—44no—
476C42—44no—
477C42—44no—
478C42—44no—
479C42—44no—
480C42—82no—
481C42H(5,𝔽2)14yes—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23#52512yes—
487C23C23×D411yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—24no—
491Q8—24no—
492Q8—416no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—44no—
498Q8—44no—
499Q8—162no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—48no—
524C2×C4—416no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4#52512yes—
546C2×C4#53512yes—
547C2×C4#53512yes—
548C2×C4#57814yes—
549C2×C4C22×#4112yes—
550C2×C4C2×H(5,𝔽2)12yes—
551C2×C4#58511yescharacteristic
552C2×C4#58511yescharacteristic
553C8—44no—
554C8—44no—
555C22—22no—
556C22—24no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22C2×#52511yescharacteristic
561C22C2×#53511yescharacteristic
562C22#344911yescharacteristic
563C22C2×#57812yes—
564C22#358612yes—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—24no—
569C4—44no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#350512yes—
583C4#350512yes—
584C2—42no—
585C2#3510211yescharacteristic
586C2#6058312yes—
587C2#6082111yescharacteristic
588C2#6082811yescharacteristic
589C2#6099512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #29509C2semidirect
8. #29371C2semidirect
9. #29370C2semidirect
10. #29129C2semidirect
11. #29126C2semidirect
12. #28385C2semidirect
13. #28373C2semidirect
14. #28213C2semidirect
15. #28170C2semidirect

Maximal subgroups

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

subgroupindex
#353242
#342092
#329642
#329152
#299152
#296272
#295092
#293712
#293702
#291292
#291262
#283852
#283732
#282132
#281702
#220542
#207812