Finite Group Databasebeta

← #6458768

Subgroups — #6458768

Order 512.

1942 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35273C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#27904C218yesmaximal, complemented
4#27892C212yesmaximal, complemented
5#27891C212yesmaximal, complemented
6#27890C212yesmaximal, complemented
7#27889C212yesmaximal, complemented
8#23485C214yesmaximal, complemented
9#21200C211yesmaximal, characteristic, complemented
10#21195C211yesmaximal, characteristic, complemented
11#21062C214yesmaximal, complemented
12#20800C211yesmaximal, characteristic, complemented
13C2×#1834C211yesmaximal, characteristic, complemented
14C4×#408C211yesmaximal, characteristic, complemented
15C2×#537C2212yes—
16C2×#532C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#428C2214yes—
19C2×#417C2214yes—
20C2×#416C2214yes—
21C2×#408C2212yes—
22C2×C4×M16C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2857C2212yes—
25#2856C2212yes—
26#2620C2214yes—
27#2538—21no—
28#2538—21no—
29#2537—21no—
30#2537—21no—
31#2416—22no—
32#2416—22no—
33#2355—21no—
34#2355—21no—
35#2354—21no—
36#2354—21no—
37#2352—22no—
38#2352—22no—
39#2314C2214yes—
40#2311C2212yes—
41#2311C2212yes—
42#2285C2212yes—
43#2285C2212yes—
44#2285C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47#2274C2212yes—
48#2036C2218yes—
49#2035C2218yes—
50#2034C2218yes—
51#2033C2218yes—
52#1959C2214yes—
53#1958C2214yes—
54#1957C2211yescharacteristic
55#1957C2211yescharacteristic
56#1957C2212yes—
57#1956C2211yescharacteristic
58#1956C2211yescharacteristic
59#1956C2212yes—
60#1928C2218yes—
61#1927C2218yes—
62#1926C2218yes—
63#1925C2218yes—
64#1847C2214yes—
65#1845C2212yes—
66#1844C2212yes—
67#1834C2214yes—
68#1834C2212yes—
69#1834C2212yes—
70#1750C2214yes—
71C4×C22⋊C8C2212yes—
72C4×C22⋊C8C2212yes—
73C2×#340C2214yes—
74#1729C2212yes—
75C22×#41C2312yes—
76C22×#41C2311yescharacteristic
77C22×M16C2312yes—
78C23×C8C2312yes—
79#537—48no—
80#532—24no—
81#532—24no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#124C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8C2314yes—
87C2×#121—24no—
88C2×#121—22no—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2314yes—
94C4×#41—24no—
95C4×#41—24no—
96C2×C4×Q8D411yescharacteristic
97C2×C4×Q8D411yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4D412yes—
101C2×C4×D4D412yes—
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×#117—22no—
105C2×#117—22no—
106C2×#117C2312yes—
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#428—48no—
118C2×#107C2318yes—
119C2×#106C2318yes—
120#417—216no—
121#416—48no—
122C2×#103C2318yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2318yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128#408—28no—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131C4×M16—24no—
132C2×#97C2312yes—
133C2×C4×C8—24no—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138#396—24no—
139#396—24no—
140#394—24no—
141#394—24no—
142#391—22no—
143#391C2314yes—
144#390—22no—
145#390—22no—
146#390—22no—
147#390C2314yes—
148#389—22no—
149#389—22no—
150#388—24no—
151#388—24no—
152#387—24no—
153#387—24no—
154#386—24no—
155#386—24no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384—21no—
160#384C2312yes—
161#384C2312yes—
162#384C2312yes—
163#384C2312yes—
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#382—22no—
167#382—22no—
168#382C2312yes—
169#380—22no—
170#380—22no—
171#380—21no—
172#380—21no—
173#380—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380C2314yes—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4—24no—
188#340C2318yes—
189#340C2318yes—
190#336C2314yes—
191#336C2314yes—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—28no—
199C2×#41—28no—
200C22×Q8C2×D411yescharacteristic
201C22×Q8C2412yes—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4C2×D412yes—
207C22×D4C2×D411yescharacteristic
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C2×M16—24no—
219C2×M16—24no—
220C2×M16#4118yes—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2414yes—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229#127—22no—
230#127—22no—
231#124—24no—
232#124—24no—
233C22⋊Q8—48no—
234C22⋊Q8—48no—
235#121—48no—
236#121—416no—
237#121—48no—
238#121—44no—
239#121—44no—
240#121—84no—
241#121—84no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4—44no—
250#117—22no—
251#117—22no—
252#117—24no—
253#117—24no—
254#117—28no—
255#117—84no—
256#117—84no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42#4114yes—
310C2×C42#4114yes—
311C2×C42C2412yes—
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—48no—
315#106—48no—
316#103—216no—
317#103—44no—
318#103—44no—
319#102—216no—
320#102—44no—
321#102—44no—
322C22⋊C8—28no—
323C22⋊C8—28no—
324#97—24no—
325C4×C8—22no—
326C4×C8—22no—
327C4×C8—24no—
328C4×C8—24no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337C24—22no—
338C24C22×D412yes—
339C24C22×D411yescharacteristic
340#41—48no—
341#41—48no—
342#41—416no—
343#41—416no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—84no—
372C2×D4—84no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4#12314yes—
402C22×C4#12314yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4114yes—
414C22×C4H(5,𝔽2)14yes—
415C22×C4C2511yescharacteristic
416M16—216no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—48no—
426C2×C8C2×#4118yes—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—28no—
435C4⋊C4—48no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C42—24no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—44no—
469C42—84no—
470C42—84no—
471C42#12314yes—
472C42#12314yes—
473C42C2×#4112yes—
474C42C2×#4112yes—
475C42C2×#4112yes—
476C42C2×#4112yes—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C2×#12312yes—
496C23#53212yes—
497C23#53512yes—
498C23C23×D411yescharacteristic
499Q8—44no—
500Q8—44no—
501Q8—44no—
502D4—48no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—88no—
507D4—84no—
508D4—84no—
509D4—168no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—88no—
545C2×C4—164no—
546C2×C4—164no—
547C2×C4C2×#12312yes—
548C2×C4C2×#12312yes—
549C2×C4C2×#12312yes—
550C2×C4C2×#12312yes—
551C2×C4C2×#12312yes—
552C2×C4#53212yes—
553C2×C4#53512yes—
554C2×C4#55318yes—
555C2×C4#55518yes—
556C2×C4#57612yes—
557C2×C4#57712yes—
558C2×C4#57712yes—
559C2×C4#57812yes—
560C2×C4C22×#4111yescharacteristic
561C2×C4C22×#4112yes—
562C2×C4C22×#4111yescharacteristic
563C2×C4C2×H(5,𝔽2)12yes—
564C8—28no—
565C8—42no—
566C8—42no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22#250014yes—
588C22C22×#12311yescharacteristic
589C22C2×#53211yescharacteristic
590C22C2×#53511yescharacteristic
591C22C2×#57611yescharacteristic
592C22C2×#57711yescharacteristic
593C22C2×#57711yescharacteristic
594C22C2×#57811yescharacteristic
595C4—22no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—84no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4#250014yes—
606C4#309814yes—
607C4#309914yes—
608C4C2×#55312yes—
609C4C2×#55512yes—
610C2—24no—
611C2—82no—
612C2—162no—
613C2C2×#250011yescharacteristic
614C2C2×#309811yescharacteristic
615C2C2×#309911yescharacteristic
616C2#6072211yescharacteristic
617C2#6072311yescharacteristic
618C2#6082211yescharacteristic
619C2#6082411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #27904C2semidirect
4. #27892C2semidirect
5. #27891C2semidirect
6. #27890C2semidirect
7. #27889C2semidirect
8. #23485C2semidirect
9. #21200C2semidirect
10. #21195C2semidirect
11. #21062C2semidirect
12. #20800C2semidirect
13. C2×#1834C2semidirect
14. C4×#408C2semidirect

Maximal subgroups

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

subgroupindex
#352732
#341882
#279042
#278922
#278912
#278902
#278892
#234852
#212002
#211952
#210622
#208002
C2×#18342
C4×#4082