Finite Group Databasebeta

← #947583

Subgroups — #947583

Order 512.

4026 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic, complemented
2C2×#2408C211yesmaximal, characteristic
3#17141C214yesmaximal, complemented
4#16298C214yesmaximal, complemented
5#16288C214yesmaximal, complemented
6#15266C214yesmaximal, complemented
7#15195C212yesmaximal, complemented
8#15134C212yesmaximal, complemented
9#15130C212yesmaximal, complemented
10#14260C212yesmaximal, complemented
11#14137C212yesmaximal, complemented
12#13902C212yesmaximal, complemented
13#13847C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#121C2212yes—
16C22×#120C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic, complemented
19#2487—28no—
20#2408—28no—
21C2×#400C2212yescomplemented
22C2×#399C2212yes—
23C2×#396C2218yescomplemented
24C2×#394C2212yes—
25C2×#394C2218yescomplemented
26C2×#392C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yescomplemented
33C2×#386C2212yes—
34C2×#386C2218yescomplemented
35C2×#385C2212yescomplemented
36C2×#384C2212yescomplemented
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic, complemented
40C2×#379C2212yes—
41C2×#379C2218yes—
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yescomplemented
45C2×#376C2211yescharacteristic, complemented
46#1445C2212yescomplemented
47#1442C2218yescomplemented
48#1440C2218yescomplemented
49#1439C2212yes—
50#1439C2218yes—
51#1439C2212yescomplemented
52#1439C2218yescomplemented
53#1439C2218yescomplemented
54#1439C2218yescomplemented
55#1439C2218yescomplemented
56#1438C2212yes—
57#1438C2218yes—
58C4⋊#95C2212yescomplemented
59C4⋊#95C2212yescomplemented
60C4⋊#95C2212yescomplemented
61C26C2312yes—
62C23×D4—28no—
63C23×D4—24no—
64C23×D4—42no—
65C23×D4—42no—
66C23×D4—44no—
67C23×D4C2312yes—
68C23×D4C2314yes—
69C23×D4C2311yescharacteristic
70C23×D4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#121—28no—
75C2×#121—28no—
76C2×#121—44no—
77C2×#120—28no—
78C2×#120—48no—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C2×C4×D4D418yes—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2318yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110#400—28no—
111#399—28no—
112#396—416no—
113#394—28no—
114#394—416no—
115#392—28no—
116#390—28no—
117#390—28no—
118#390—44no—
119#390—44no—
120#390—44no—
121#389—28no—
122#386—28no—
123#386—416no—
124#385—28no—
125#384—44no—
126#380—24no—
127#380—28no—
128#380—24no—
129#379—28no—
130#379—416no—
131C4×C4⋊C4—42no—
132C4×C22⋊C4—44no—
133C4×C22⋊C4—44no—
134#376—42no—
135C2×#95C2312yes—
136C2×#95C2318yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C25—24no—
148C25—28no—
149C25—28no—
150C25—44no—
151C25—42no—
152C25—42no—
153C25—44no—
154C25—48no—
155C25C2414yes—
156C25C2411yescharacteristic
157C25C2411yescharacteristic
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—216no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—432no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—416no—
178C22×D4—44no—
179C22×D4—48no—
180C22×D4—416no—
181C22×D4—432no—
182C22×D4—416no—
183C22×D4C2×D414yes—
184C22×D4#4118yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—48no—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209#121—432no—
210#120—432no—
211C4×D4—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—416no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4C2×D414yes—
271C2×C22⋊C4C2×D414yes—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42C2×D414yes—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—216no—
286#95—42no—
287#95—42no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—48no—
292#95—48no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—28no—
308C24—28no—
309C24—28no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—28no—
315C24—416no—
316C24—48no—
317C24—432no—
318C24—48no—
319C24—416no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—88no—
324C24—816no—
325C24—88no—
326C24—88no—
327C24—816no—
328C24—88no—
329C24—816no—
330C24—88no—
331C24C22×D412yes—
332C24C22×D412yes—
333C24C2×#4118yes—
334C24H(5,𝔽2)14yes—
335C242−1+414yes—
336C24C2511yescharacteristic
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—216no—
340C2×D4—216no—
341C2×D4—216no—
342C2×D4—44no—
343C2×D4—416no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—416no—
349C2×D4—48no—
350C2×D4—4128no—
351C2×D4—416no—
352C2×D4—432no—
353C2×D4—416no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—88no—
359C2×D4—816no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4114yes—
423C22×C4H(5,𝔽2)18yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)14yes—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—88no—
434C4⋊C4—816no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—88no—
444C22⋊C4—816no—
445C22⋊C4—816no—
446C22⋊C4—88no—
447C22⋊C4—816no—
448C22⋊C4—816no—
449C42—28no—
450C42—48no—
451C42—48no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—216no—
472C23—216no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—44no—
478C23—44no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—432no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—44no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—832no—
507C23—816no—
508C23—832no—
509C23—864no—
510C23—832no—
511C23—816no—
512C23—832no—
513C23#53514yes—
514C23#53614yes—
515C23C23×D411yescharacteristic
516C23C22×#4112yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)14yes—
522C23C2×2−1+412yes—
523D4—416no—
524D4—432no—
525D4—432no—
526D4—832no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—816no—
578C2×C4#53414yes—
579C2×C4#53818yes—
580C2×C4#54618yes—
581C2×C4#55114yes—
582C2×C4#56014yes—
583C22—24no—
584C22—24no—
585C22—28no—
586C22—28no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—48no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—416no—
600C22—416no—
601C22—48no—
602C22—48no—
603C22—48no—
604C22—864no—
605C22—84no—
606C22—88no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—88no—
611C22C2×#53411yescharacteristic
612C22C2×#53511yescharacteristic
613C22C2×#53611yescharacteristic
614C22C2×#53812yes—
615C22C2×#54612yes—
616C22C2×#55111yescharacteristic
617C22#349812yes—
618C22#351914yes—
619C22#352014yes—
620C22C2×#56011yescharacteristic
621C22#353612yes—
622C22#353612yes—
623C22#353614yes—
624C22#353712yes—
625C22#353714yes—
626C22#354912yes—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—48no—
632C4—84no—
633C4—84no—
634C4—84no—
635C4—84no—
636C4#269718yes—
637C2—44no—
638C2—44no—
639C2—88no—
640C2#3729711yescharacteristic
641C2#3745111yescharacteristic
642C2#3807812yes—
643C2#3808012yes—
644C2#4260614yes—
645C2#4537712yes—
646C2C2×#269711yescharacteristic
647C2#5298812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2408C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2487C2semidirect
3. #17141C2semidirect
4. #16298C2semidirect
5. #16288C2semidirect
6. #15266C2semidirect
7. #15195C2semidirect
8. #15134C2semidirect
9. #15130C2semidirect
10. #14260C2semidirect
11. #14137C2semidirect
12. #13902C2semidirect
13. #13847C2semidirect
18. C23×C4⋊C4C22semidirect
21. C2×#400C22semidirect
23. C2×#396C22semidirect
25. C2×#394C22semidirect
32. C2×#389C22semidirect
34. C2×#386C22semidirect
35. C2×#385C22semidirect
36. C2×#384C22semidirect
39. C2×#380C22semidirect
42. C2×C4×C4⋊C4C22semidirect
44. C2×C4×C22⋊C4C22semidirect
45. C2×#376C22semidirect
46. #1445C22semidirect
47. #1442C22semidirect
48. #1440C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#24082
#171412
#162982
#162882
#152662
#151952
#151342
#151302
#142602
#141372
#139022
#138472