Finite Group Databasebeta

← #7195421

Subgroups — #7195421

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#31996C214yesmaximal, complemented
4#31322C214yesmaximal, complemented
5#29273C211yesmaximal, characteristic, complemented
6#29258C211yesmaximal, characteristic, complemented
7#28599C212yesmaximal, complemented
8#26300C212yesmaximal, complemented
9#26232C212yesmaximal, complemented
10#26130C214yesmaximal, complemented
11#24089C214yesmaximal, complemented
12#23837C211yesmaximal, characteristic, complemented
13D4×#106C212yesmaximal, complemented
14#22260C211yesmaximal, characteristic, complemented
15#22219C211yesmaximal, characteristic, complemented
16C2×#550—21no—
17C2×#547C2212yes—
18C2×#486C2211yescharacteristic
19C2×#481C2214yes—
20C2×C8×D4C2212yes—
21C22×#106C2214yes—
22#2804—22no—
23#2682—22no—
24#2580—22no—
25#2580—22no—
26#2571—21no—
27#2563C2212yes—
28#2563C2212yes—
29#2541C2212yes—
30#2538—21no—
31#2537C2212yes—
32#2487C2214yes—
33#2445—22no—
34#2416C2214yes—
35#2407—22no—
36#2366—21no—
37D4×C4⋊C4C2212yes—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2099C2212yes—
46#2094C2212yes—
47#2084C2212yes—
48#2084C2212yes—
49#2077C2214yes—
50#2077C2214yes—
51#2058C2212yes—
52#2058C2212yes—
53#2058C2212yes—
54#2034C2214yes—
55#2034C2214yes—
56#2034C2214yes—
57#2034C2214yes—
58#2023C2212yes—
59#1990C2211yescharacteristic
60#1987C2212yes—
61#1982C2212yes—
62#1957C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1926C2214yes—
67#1919C2214yes—
68#1893C2214yes—
69#1893C2214yes—
70#1874C2214yes—
71#1869C2214yes—
72#1869C2214yes—
73#1864C2214yes—
74#1864C2214yes—
75#1853C2214yes—
76#1853C2214yes—
77#1853C2214yes—
78#1844C2211yescharacteristic
79#1831C2211yescharacteristic
80#1823C2214yes—
81#1807C2214yes—
82C4×#106C2212yes—
83C4×#106C2212yes—
84#1744C2211yescharacteristic
85C22×#41—22no—
86C23×C8C2314yes—
87#550—48no—
88#547—48no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127—22no—
93C2×#127D411yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#123—24no—
96C2×C22⋊Q8—22no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121C2314yes—
100C2×#121C2314yes—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115#486—24no—
116#481—48no—
117C8×D4—28no—
118C2×#106—48no—
119C2×#106C2312yes—
120C2×#106C2314yes—
121C2×#106C2312yes—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#400—24no—
137#399—24no—
138#396—24no—
139#394—24no—
140#394—24no—
141#393—24no—
142#393—24no—
143#392—24no—
144#391—21no—
145#391—21no—
146#391—21no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—21no—
151#390—21no—
152#390D412yes—
153#390D412yes—
154#390D412yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#386—24no—
166#386—24no—
167#385—24no—
168#385—24no—
169#385—24no—
170#385—24no—
171#384D411yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#382—22no—
177#382—22no—
178#380—22no—
179#380C2314yes—
180#380C2314yes—
181#380C2314yes—
182#380C2314yes—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2312yes—
191#340C2312yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2312yes—
197#340C2314yes—
198#340C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203C2×#41—48no—
204C2×#41—48no—
205C2×#41—84no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—21no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C2×D412yes—
214C22×D4C2×D412yes—
215C22×D4C2×D412yes—
216C22×D4C2412yes—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4C2414yes—
221C22×C8—28no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8C2414yes—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C4⋊Q8—24no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—42no—
231#127—22no—
232#127—22no—
233#127—48no—
234#127SD1618yes—
235#123—416no—
236C22⋊Q8—48no—
237#121—28no—
238#121—44no—
239#121—48no—
240#121—416no—
241#121—44no—
242#121—44no—
243C4×Q8—44no—
244C4×Q8—82no—
245C4×Q8—82no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—48no—
252C4×D4—44no—
253C4×D4—84no—
254C4×D4—84no—
255C4×D4—82no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#106—24no—
309#106—28no—
310#106—24no—
311#106—44no—
312#106—44no—
313#106—42no—
314#106—42no—
315C4⋊C8—24no—
316#102—28no—
317#102—44no—
318#102—44no—
319#102—44no—
320C22⋊C8—28no—
321C22⋊C8—44no—
322C22⋊C8—44no—
323C4×C8—22no—
324C4×C8#4114yescomplemented
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335C24—22no—
336C24C22×D414yes—
337#41—832no—
338C2×Q8—24no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8—81no—
346C2×Q8—81no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4C2×SD1618yes—
370C2×D4C2×SD1618yes—
371C2×D4C2×SD1618yes—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D414yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2511yescharacteristic
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—416no—
411C2×C8C2×#4114yes—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—81no—
436C4⋊C4—84no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—81no—
442C4⋊C4H(5,𝔽2)14yes—
443C22⋊C4—28no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—41no—
459C42—41no—
460C42—44no—
461C42—44no—
462C42—44no—
463C42C2×SD1614yes—
464C42C2×#4112yes—
465C422−1+412yes—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C22×SD1618yes—
480C23C23×D411yescharacteristic
481Q8—48no—
482Q8—84no—
483Q8—84no—
484D4—216no—
485D4—48no—
486D4—48no—
487D4—44no—
488D4—44no—
489D4—84no—
490D4—816no—
491D4—84no—
492D4—168no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—88no—
534C2×C4—816no—
535C2×C4—168no—
536C2×C4#54712yes—
537C2×C4#54712yes—
538C2×C4C22×SD1612yes—
539C2×C4C22×SD1612yes—
540C2×C4C22×SD1612yes—
541C2×C4#57712yes—
542C2×C4C22×#4111yescharacteristic
543C2×C4C2×H(5,𝔽2)12yes—
544C2×C4C2×2−1+411yescharacteristic
545C2×C4#58512yes—
546C8—24no—
547C8—44no—
548C8—42no—
549C22—28no—
550C22—28no—
551C22—28no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22C2×#54711yescharacteristic
564C22#348411yescharacteristic
565C22#348511yescharacteristic
566C22C23×SD1611yescharacteristic
567C22C2×#57711yescharacteristic
568C22#358511yescharacteristic
569C22#358511yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—44no—
574C4—41no—
575C4—44no—
576C4—41no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4#333514yes—
584C4#350612yes—
585C4#355412yes—
586C2—48no—
587C2—162no—
588C2#4535411yescharacteristic
589C2C2×#333511yescharacteristic
590C2#6129311yescharacteristic
591C2#6156011yescharacteristic
592C2#6168211yescharacteristic
593C2#6169711yescharacteristic
594C2#6180811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59864C2semidirect
3. #31996C2semidirect
4. #31322C2semidirect
5. #29273C2semidirect
6. #29258C2semidirect
7. #28599C2semidirect
8. #26300C2semidirect
9. #26232C2semidirect
10. #26130C2semidirect
11. #24089C2semidirect
12. #23837C2semidirect
13. D4×#106C2semidirect
14. #22260C2semidirect
15. #22219C2semidirect
324. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#598642
#598152
#319962
#313222
#292732
#292582
#285992
#263002
#262322
#261302
#240892
#238372
D4×#1062
#222602
#222192