Finite Group Databasebeta

← #281591

Subgroups — #281591

Order 512.

795 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
1#25811C211yesmaximal, characteristic
2#24752C211yesmaximal, characteristic
3#24302C211yesmaximal, characteristic
4#22952C211yesmaximal, characteristic
5C2×#1672C211yesmaximal, characteristic, complemented
6C22×#333C211yesmaximal, characteristic
7#10589C211yesmaximal, characteristic, complemented
8#10544C211yesmaximal, characteristic, complemented
9#10482C211yesmaximal, characteristic, complemented
10#10273C211yesmaximal, characteristic, complemented
11#9729C211yesmaximal, characteristic, complemented
12#9568C211yesmaximal, characteristic, complemented
13#9216C211yesmaximal, characteristic, complemented
14#9129C211yesmaximal, characteristic
15#8830C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#477C2211yescharacteristic
21C2×#467—21no—
22C2×#462C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26#2857C2211yescharacteristic
27C4×C22⋊Q8C2211yescharacteristic
28#2068—21no—
29#2058—21no—
30#2058—21no—
31#2057—21no—
32#2056—21no—
33#2024—21no—
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1984C2211yescharacteristic
39#1981C2211yescharacteristic
40#1947C2211yescharacteristic
41#1939—21no—
42#1939C2211yescharacteristic
43#1916—21no—
44#1892—21no—
45#1872—21no—
46#1853—21no—
47#1853—21no—
48#1852—21no—
49#1850—21no—
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C22⋊C4—21no—
54C4×C22⋊C8C2211yescharacteristic
55#1672—24no—
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333—24no—
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C23×Q8—21no—
70C22×Q16—21no—
71C23×C8—21no—
72C23×C8—21no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—41no—
79C2×C4⋊Q8—41no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8D411yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C42C2311yescharacteristic
89#498—22no—
90#498—22no—
91#498—41no—
92#498—41no—
93#477—22no—
94#477—22no—
95#477—41no—
96#477—41no—
97#467—44no—
98#462—22no—
99#462—41no—
100C8×Q8—22no—
101C8×Q8—41no—
102C2×#107—21no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#106—21no—
106C2×#106—21no—
107C2×#106—21no—
108C2×#106—21no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—41no—
116C2×C4⋊C8—41no—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×#103—21no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8D411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#391—21no—
137#391D411yescharacteristic
138#389—21no—
139#389—21no—
140#389D411yescharacteristic
141#387—21no—
142#385—21no—
143#384C2311yescharacteristic
144#380—21no—
145#380—21no—
146#380D411yescharacteristic
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2311yescharacteristic
149C4×C22⋊C4—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340—21no—
156#340—21no—
157#340—21no—
158#340—21no—
159#340—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#336D411yescharacteristic
166#336D411yescharacteristic
167#334—22no—
168#334—41no—
169#333—22no—
170#333—22no—
171#333—22no—
172#333D414yes—
173#332—22no—
174#332—41no—
175C22×Q8—24no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—42no—
180C22×Q8—41no—
181C22×Q8—82no—
182C22×Q8—82no—
183C22×Q8C2×D411yescharacteristic
184C23×C4—41no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188C2×Q16—42no—
189C2×Q16—42no—
190C2×Q16—81no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—81no—
202C22×C8—81no—
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C4⋊Q8—24no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223C4⋊Q8C2×D412yes—
224C4⋊Q8#4112yes—
225C22⋊Q8—22no—
226C22⋊Q8—42no—
227C22⋊Q8—42no—
228C22⋊Q8SD1614yes—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—41no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268#107—42no—
269#107—42no—
270#107—42no—
271#106—42no—
272#106—42no—
273#106—42no—
274#106—42no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288C4⋊C8—81no—
289C4⋊C8C2×D412yes—
290C4⋊C8C2×D412yes—
291#103—42no—
292#103—42no—
293#103—42no—
294#103—42no—
295#103—42no—
296#103—81no—
297C22⋊C8—22no—
298C22⋊C8—42no—
299C22⋊C8Q1614yescomplemented
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—42no—
305C4×C8—81no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—41no—
312#95—41no—
313C24#12111yescharacteristic
314C2×Q8—21no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—44no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—81no—
333C2×Q8—81no—
334C2×Q8—81no—
335C2×Q8—81no—
336C2×Q8—88no—
337C2×Q8—88no—
338C2×Q8C2×SD1612yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
355C22×C4—44no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4C2×SD1612yes—
366C22×C4C2×Q1612yes—
367C22×C4#13512yes—
368C22×C4#13712yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372Q16—84no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—84no—
402C2×C8—84no—
403C2×C8C2×Q1612yes—
404C2×C8C2×Q1612yes—
405C4⋊C4—21no—
406C4⋊C4—24no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—82no—
434C4⋊C4C2×SD1612yes—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—82no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42C22×D411yescharacteristic
449C42C2×#4111yescharacteristic
450C42H(5,𝔽2)11yescharacteristic
451C42H(5,𝔽2)11yescharacteristic
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23#46212yes—
457C23#46512yes—
458C23#47112yes—
459C23C2×#12111yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—41no—
463Q8—41no—
464Q8—82no—
465Q8—82no—
466Q8—82no—
467Q8—82no—
468Q8—88no—
469Q8—81no—
470Q8—81no—
471Q8—81no—
472Q8—81no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—88no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4#46212yes—
518C2×C4#46512yes—
519C2×C4#52511yescharacteristic
520C2×C4#53811yescharacteristic
521C2×C4D4211yescharacteristic
522C2×C4D4211yescharacteristic
523C2×C4#54611yescharacteristic
524C2×C4#54611yescharacteristic
525C2×C4C22×SD1611yescharacteristic
526C2×C4C22×Q1611yescharacteristic
527C2×C4C2×#13511yescharacteristic
528C2×C4C2×#13711yescharacteristic
529C2×C4#57711yescharacteristic
530C2×C4#57711yescharacteristic
531C2×C4#57811yescharacteristic
532C2×C4#57811yescharacteristic
533C8—22no—
534C8—22no—
535C8—41no—
536C8—42no—
537C8—42no—
538C8—41no—
539C8—81no—
540C8—81no—
541C8—81no—
542C8—81no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22#167214yescomplemented
550C22#242411yescharacteristic
551C22C2×#46211yescharacteristic
552C22#304611yescharacteristic
553C22C2×#46511yescharacteristic
554C22#306311yescharacteristic
555C22C2×#47111yescharacteristic
556C22#306811yescharacteristic
557C4—22no—
558C4—21no—
559C4—21no—
560C4—22no—
561C4—42no—
562C4—41no—
563C4—42no—
564C4—81no—
565C4—82no—
566C4—82no—
567C4—81no—
568C4—161no—
569C4—161no—
570C4#326611yescharacteristic
571C4D4×SD1611yescharacteristic
572C4#328511yescharacteristic
573C4D4×Q1611yescharacteristic
574C4#329811yescharacteristic
575C4#329911yescharacteristic
576C4#330111yescharacteristic
577C4#331211yescharacteristic
578C2—24no—
579C2C2×#167211yescharacteristic
580C2#1924111yescharacteristic
581C2#1941411yescharacteristic
582C2#1958811yescharacteristic
583C2#2508211yescharacteristic
584C2#2511411yescharacteristic
585C2#2514411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#1672C2semidirect
7. #10589C2semidirect
8. #10544C2semidirect
9. #10482C2semidirect
10. #10273C2semidirect
11. #9729C2semidirect
12. #9568C2semidirect
13. #9216C2semidirect
299. C22⋊C8Q16semidirect
549. C22#1672semidirect

Maximal subgroups

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

subgroupindex
#258112
#247522
#243022
#229522
C2×#16722
C22×#3332
#105892
#105442
#104822
#102732
#97292
#95682
#92162
#91292
#88302