Finite Group Databasebeta

← #7195425

Subgroups — #7195425

Order 512.

1892 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#31994C214yesmaximal, complemented
4#31318C214yesmaximal, complemented
5#29272C211yesmaximal, characteristic, complemented
6#29254C211yesmaximal, characteristic, complemented
7#28597C212yesmaximal, complemented
8#26296C212yesmaximal, complemented
9#26230C212yesmaximal, complemented
10#26126C214yesmaximal, complemented
11#24088C214yesmaximal, complemented
12#23835C211yesmaximal, characteristic, complemented
13D4×#107C212yesmaximal, complemented
14#22260C211yesmaximal, characteristic, complemented
15#22220C211yesmaximal, characteristic, complemented
16C2×#559—21no—
17C2×#550—21no—
18C2×#547C2212yes—
19C2×#486C2211yescharacteristic
20C2×#480C2214yes—
21C2×C8×D4C2212yes—
22C22×#107C2214yes—
23#2724—22no—
24#2682—22no—
25#2582—21no—
26#2547C2212yes—
27#2547C2212yes—
28#2541C2212yes—
29#2538—21no—
30#2537C2212yes—
31#2486C2214yes—
32#2447—22no—
33#2447—22no—
34#2445—22no—
35#2416C2214yes—
36#2407—22no—
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#2093C2212yes—
47#2083C2212yes—
48#2083C2212yes—
49#2075C2214yes—
50#2075C2214yes—
51#2057C2212yes—
52#2057C2212yes—
53#2057C2212yes—
54#2033C2214yes—
55#2033C2214yes—
56#2033C2214yes—
57#2033C2214yes—
58#2023C2212yes—
59#1990C2211yescharacteristic
60#1988C2212yes—
61#1982C2212yes—
62#1957C2211yescharacteristic
63#1954C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1925C2214yes—
67#1918C2214yes—
68#1893C2214yes—
69#1893C2214yes—
70#1874C2214yes—
71#1869C2214yes—
72#1869C2214yes—
73#1862C2214yes—
74#1862C2214yes—
75#1852C2214yes—
76#1852C2214yes—
77#1852C2214yes—
78#1845C2211yescharacteristic
79#1831C2211yescharacteristic
80#1823C2214yes—
81#1808C2214yes—
82C4×#107C2212yes—
83C4×#107C2212yes—
84#1744C2211yescharacteristic
85C22×#41—22no—
86C23×C8C2314yes—
87#559—48no—
88#550—48no—
89#547—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127—22no—
94C2×#127D411yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124—22no—
97C2×#124—22no—
98C2×#123—24no—
99C2×#123—24no—
100C2×C22⋊Q8—22no—
101C2×#121—22no—
102C2×#121—24no—
103C2×#121C2314yes—
104C2×#121C2314yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119#486—24no—
120#480—48no—
121C8×D4—28no—
122C2×#107—48no—
123C2×#107C2312yes—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#398—24no—
141#397—24no—
142#397—24no—
143#397—24no—
144#396—24no—
145#394—24no—
146#394—24no—
147#392—24no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—21no—
152#391—21no—
153#390—22no—
154#390—21no—
155#390—21no—
156#390D412yes—
157#390D412yes—
158#390D412yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#386—24no—
170#385—24no—
171#385—24no—
172#385—24no—
173#384D411yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#381—24no—
179#380—22no—
180#380C2314yes—
181#380C2314yes—
182#380C2314yes—
183#380C2314yes—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C43C2311yescharacteristic
191#340C2312yes—
192#340C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2312yes—
198#340C2314yes—
199#340C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203#336C2312yes—
204C2×#41—48no—
205C2×#41—48no—
206C2×#41—84no—
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—21no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C2×D412yes—
215C22×D4C2×D412yes—
216C22×D4C2×D412yes—
217C22×D4C2412yes—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4C2414yes—
222C22×C8—28no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8C2414yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C4⋊Q8—24no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—42no—
232#127—22no—
233#127—22no—
234#127—48no—
235#127D818yes—
236#124—48no—
237#124—48no—
238#123—416no—
239#123—416no—
240C22⋊Q8—48no—
241#121—28no—
242#121—44no—
243#121—48no—
244#121—416no—
245#121—44no—
246#121—44no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4—48no—
256C4×D4—48no—
257C4×D4—48no—
258C4×D4—44no—
259C4×D4—82no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312#107—24no—
313#107—28no—
314#107—24no—
315#107—44no—
316#107—44no—
317#107—42no—
318#107—42no—
319C4⋊C8—24no—
320#102—28no—
321#102—44no—
322#102—44no—
323#102—44no—
324C22⋊C8—28no—
325C22⋊C8—44no—
326C22⋊C8—44no—
327C4×C8—22no—
328C4×C8#4114yescomplemented
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339C24—22no—
340C24C22×D414yes—
341#41—832no—
342C2×Q8—24no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4C2×D818yes—
374C2×D4C2×D818yes—
375C2×D4C2×D818yes—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—48no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—44no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D414yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2511yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—416no—
415C2×C8C2×#4114yes—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—84no—
445C4⋊C4—81no—
446C4⋊C4H(5,𝔽2)14yes—
447C22⋊C4—28no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—48no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—82no—
458C22⋊C4—84no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—41no—
463C42—41no—
464C42—44no—
465C42—44no—
466C42—44no—
467C42C2×D814yes—
468C42C2×#4112yes—
469C422−1+412yes—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C22×D818yes—
484C23C23×D411yescharacteristic
485Q8—48no—
486Q8—84no—
487Q8—84no—
488D4—216no—
489D4—48no—
490D4—48no—
491D4—44no—
492D4—44no—
493D4—84no—
494D4—816no—
495D4—84no—
496D4—168no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
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—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—88no—
538C2×C4—816no—
539C2×C4—168no—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4C22×D812yes—
543C2×C4C22×D812yes—
544C2×C4C22×D812yes—
545C2×C4#57812yes—
546C2×C4C22×#4111yescharacteristic
547C2×C4C2×H(5,𝔽2)12yes—
548C2×C4C2×2−1+411yescharacteristic
549C2×C4#58512yes—
550C8—24no—
551C8—44no—
552C8—42no—
553C22—28no—
554C22—28no—
555C22—28no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22C2×#54711yescharacteristic
568C22#348411yescharacteristic
569C22#348511yescharacteristic
570C22C23×D811yescharacteristic
571C22C2×#57811yescharacteristic
572C22#358511yescharacteristic
573C22#358611yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—44no—
578C4—41no—
579C4—44no—
580C4—41no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4#333414yes—
588C4#350612yes—
589C4#355412yes—
590C2—48no—
591C2—162no—
592C2#4535411yescharacteristic
593C2C2×#333411yescharacteristic
594C2#6129111yescharacteristic
595C2#6155911yescharacteristic
596C2#6168311yescharacteristic
597C2#6169411yescharacteristic
598C2#6181011yescharacteristic
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. #59827C2semidirect
3. #31994C2semidirect
4. #31318C2semidirect
5. #29272C2semidirect
6. #29254C2semidirect
7. #28597C2semidirect
8. #26296C2semidirect
9. #26230C2semidirect
10. #26126C2semidirect
11. #24088C2semidirect
12. #23835C2semidirect
13. D4×#107C2semidirect
14. #22260C2semidirect
15. #22220C2semidirect
328. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#598272
#598152
#319942
#313182
#292722
#292542
#285972
#262962
#262302
#261262
#240882
#238352
D4×#1072
#222602
#222202