Finite Group Databasebeta

← #375649

Subgroups — #375649

Order 512.

1379 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2643C211yesmaximal, characteristic
2#13796C211yesmaximal, characteristic, complemented
3#13778C211yesmaximal, characteristic, complemented
4#11203C212yesmaximal, complemented
5#11043C212yesmaximal
6#10960C212yesmaximal
7#10960C212yesmaximal
8#10912C212yesmaximal, complemented
9#10912C212yesmaximal, complemented
10C22×#99C2214yes—
11#2643—24no—
12#2643—24no—
13C2×#401C411yescharacteristic, complemented
14C2×#400C411yescharacteristic, complemented
15C2×#400C411yescharacteristic, complemented
16C2×#400C411yescharacteristic, complemented
17C2×#397C411yescharacteristic, complemented
18C2×#397C411yescharacteristic, complemented
19C2×#397C411yescharacteristic, complemented
20C2×#396C411yescharacteristic, complemented
21C2×#386C2211yescharacteristic
22C2×#386C2211yescharacteristic
23C2×#386C2211yescharacteristic
24C2×#385C2211yescharacteristic
25C2×#385C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C22×#95C2211yescharacteristic
31C2×#342C2214yes—
32C2×#342C2214yes—
33C2×#342C2214yes—
34#1436—24no—
35#1434—24no—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C22⋊#95C2212yes—
39C23×D4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2311yescharacteristic
57C22×C22⋊C4C2311yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C22⋊C4C2314yes—
60C22×C22⋊C4C2314yes—
61C22×C42—22no—
62C22×C42—22no—
63C2×#99—28no—
64C2×#99—28no—
65C2×#99—44no—
66#401—24no—
67#400—24no—
68#400—24no—
69#400—24no—
70#397—24no—
71#397—24no—
72#397—24no—
73#396—24no—
74#386—22no—
75#386—22no—
76#386—22no—
77#386—22no—
78#386—22no—
79#386—22no—
80#385—22no—
81#385—22no—
82#385—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#380—28no—
87C4×C22⋊C4—28no—
88C4×C22⋊C4—44no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95D414yes—
115C2×#95D414yes—
116#342—28no—
117#342—28no—
118#342—44no—
119C25C22×C411yescharacteristic
120C25C2411yescharacteristic
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—22no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4—81no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—81no—
168C2×C4⋊C4—81no—
169C2×C4⋊C4—81no—
170C2×C4⋊C4—81no—
171C2×C4⋊C4—81no—
172C2×C4⋊C4—81no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—81no—
200C2×C22⋊C4—81no—
201C2×C22⋊C4—81no—
202C2×C22⋊C4—81no—
203C2×C22⋊C4—81no—
204C2×C22⋊C4—81no—
205C2×C22⋊C4#4118yes—
206C2×C22⋊C4#4118yes—
207C2×C42—28no—
208C2×C42—44no—
209C2×C42—82no—
210C2×C42—82no—
211C2×C42—82no—
212C2×C42—82no—
213#99—416no—
214#95—28no—
215#95—28no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—44no—
229#95—82no—
230#95—82no—
231#95—82no—
232#95—82no—
233#95—82no—
234#95—82no—
235#95C22⋊C414yes—
236#95C22⋊C414yes—
237#95C22⋊C414yes—
238#95C22⋊C414yes—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24C23×C411yescharacteristic
252C24C22×D411yescharacteristic
253C24C22×D411yescharacteristic
254C24C2×#4114yes—
255C24C2×#4114yes—
256C24H(5,𝔽2)12yes—
257C242−1+412yes—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—84no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4—84no—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—88no—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C42—88no—
389C42—88no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—44no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23C22×C22⋊C411yescharacteristic
425C23C4×#4112yes—
426C23C4×#4112yes—
427C23#52012yes—
428C23#52512yes—
429C23#52512yes—
430C23#53212yes—
431C23#53212yes—
432C23#53511yescharacteristic
433C23#53611yescharacteristic
434C23#53712yes—
435C23#55114yes—
436C23#55214yes—
437C23#55314yes—
438C23#55614yes—
439D4—84no—
440D4—82no—
441D4—82no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C22—21no—
518C22—21no—
519C22—22no—
520C22—22no—
521C22—21no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—164no—
549C22#229711yescharacteristic
550C22#229711yescharacteristic
551C22#231111yescharacteristic
552C22#237314yes—
553C22#260114yes—
554C22#260312yes—
555C22#260412yes—
556C22#288311yescharacteristic
557C22#288311yescharacteristic
558C22#288311yescharacteristic
559C22#288411yescharacteristic
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—88no—
567C4—88no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—22no—
576C2—24no—
577C2—24no—
578C2—22no—
579C2—161no—
580C2#1416611yescharacteristic
581C2#2142211yescharacteristic
582C2#2142211yescharacteristic
583C2#2142311yescharacteristic
584C2#2142311yescharacteristic
585C2#2187611yescharacteristic
586C2#2187711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13796C2semidirect
3. #13778C2semidirect
4. #11203C2semidirect
8. #10912C2semidirect
9. #10912C2semidirect
13. C2×#401C4semidirect
14. C2×#400C4semidirect
15. C2×#400C4semidirect
16. C2×#400C4semidirect
17. C2×#397C4semidirect
18. C2×#397C4semidirect
19. C2×#397C4semidirect
20. C2×#396C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26432
#137962
#137782
#112032
#110432
#109602
#109122