Finite Group Databasebeta

← #4456462

Subgroups — #4456462

Order 512.

3028 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2539C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic, complemented
3#16434C214yesmaximal, complemented
4#16360C214yesmaximal, complemented
5#16276C214yesmaximal, complemented
6#16255C214yesmaximal, complemented
7#14947C214yesmaximal, complemented
8#14778C214yesmaximal, complemented
9#14290C211yesmaximal, characteristic, complemented
10#14165C211yesmaximal, characteristic, complemented
11#14136C211yesmaximal, characteristic, complemented
12#13868C211yesmaximal, characteristic, complemented
13#13852C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C22×#120C2212yes—
17C22×#117C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2539—28no—
20C2×#400C2214yescomplemented
21C2×#399C2212yes—
22C2×#399C2214yescomplemented
23C2×#397C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#392C2212yes—
27#2312—28no—
28C2×#390C2214yes—
29C2×#390C2212yescomplemented
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#381C2211yescharacteristic
33C2×#380C2212yes—
34C2×#379C2212yes—
35C2×#379C2212yes—
36C2×C4×C4⋊C4C2212yescomplemented
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40#1445C2212yescomplemented
41#1441C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yescomplemented
59#1437C2212yes—
60#1437C2212yes—
61#1437C2212yescomplemented
62#1437C2212yescomplemented
63#1437C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C4⋊#95C2212yescomplemented
67C22⋊#95C2212yes—
68C4×#95C2212yescomplemented
69C26C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×#123—28no—
76C2×#123—48no—
77C2×#121—24no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#120—28no—
81C2×#120—48no—
82C2×#117—24no—
83C2×#117—44no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4D418yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116#400—48no—
117#399—28no—
118#399—48no—
119#397—48no—
120#394—48no—
121#394—48no—
122#392—28no—
123#390—216no—
124#390—44no—
125#388—216no—
126#388—216no—
127#381—24no—
128#380—28no—
129#379—28no—
130#379—44no—
131C4×C4⋊C4—44no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25—22no—
154C25—22no—
155C25—44no—
156C25—44no—
157C25—42no—
158C25—42no—
159C25C2×D414yes—
160C25C2412yes—
161C25C2412yes—
162C25C2411yescharacteristic
163C22×D4—24no—
164C22×D4—28no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—28no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—42no—
178C22×D4—44no—
179C22×D4—44no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4C2×D414yes—
186C23×C4C2×D414yes—
187C23×C4#4114yes—
188C23×C4#4114yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199#123—432no—
200#121—416no—
201#120—432no—
202#117—416no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—232no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
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—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D418yes—
272C2×C22⋊C4C2×D418yes—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—28no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—82no—
332C24—82no—
333C24—84no—
334C24—84no—
335C24—84no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24—88no—
340C24C22×D414yes—
341C24C22×D412yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24C2511yescharacteristic
345C2×D4—416no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—416no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—82no—
353C2×D4—82no—
354C2×D4—84no—
355C2×D4—88no—
356C2×D4—88no—
357C2×D4—88no—
358C2×D4—88no—
359C2×D4—88no—
360C2×D4—88no—
361C2×D4—88no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4#123116yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)18yes—
428C22×C4H(5,𝔽2)14yes—
429C4⋊C4—416no—
430C4⋊C4—48no—
431C4⋊C4—416no—
432C4⋊C4—44no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C22⋊C4—216no—
436C22⋊C4—216no—
437C22⋊C4—216no—
438C22⋊C4—416no—
439C22⋊C4—48no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—84no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C42—28no—
454C42—44no—
455C42—84no—
456C42—84no—
457C42—84no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—416no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—88no—
516C23—88no—
517C23—832no—
518C23—816no—
519C23—816no—
520C23—832no—
521C23C2×#12314yes—
522C23C23×D411yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4111yescharacteristic
525C23C2×H(5,𝔽2)14yes—
526C23C2×H(5,𝔽2)12yes—
527C23#58514yes—
528C23#58512yes—
529D4—88no—
530D4—832no—
531D4—816no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—416no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—88no—
590C2×C4C2×#12314yes—
591C2×C4C2×#12314yes—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—48no—
596C22—44no—
597C22—44no—
598C22—48no—
599C22—44no—
600C22—44no—
601C22—48no—
602C22—48no—
603C22—44no—
604C22—832no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—82no—
616C22—84no—
617C22—84no—
618C22C22×#12311yescharacteristic
619C22#344912yes—
620C22#345512yes—
621C22#345512yes—
622C22#348412yes—
623C22#349011yescharacteristic
624C22#349111yescharacteristic
625C22#349612yes—
626C22#350512yes—
627C22#350812yes—
628C22#350812yes—
629C22#353414yes—
630C22#354514yes—
631C22#354614yes—
632C22#356214yes—
633C4—24no—
634C4—24no—
635C4—48no—
636C4—44no—
637C4—44no—
638C4—84no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4—84no—
643C2—44no—
644C2—84no—
645C2—162no—
646C2#3463511yescharacteristic
647C2#3536011yescharacteristic
648C2#3555411yescharacteristic
649C2#3681212yes—
650C2#3701012yes—
651C2#4052414yes—
652C2#4068614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2539C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2312C2semidirect
3. #16434C2semidirect
4. #16360C2semidirect
5. #16276C2semidirect
6. #16255C2semidirect
7. #14947C2semidirect
8. #14778C2semidirect
9. #14290C2semidirect
10. #14165C2semidirect
11. #14136C2semidirect
12. #13868C2semidirect
13. #13852C2semidirect
20. C2×#400C22semidirect
22. C2×#399C22semidirect
29. C2×#390C22semidirect
36. C2×C4×C4⋊C4C22semidirect
40. #1445C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
53. #1439C22semidirect
58. #1438C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
68. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25392
C2×#23122
#164342
#163602
#162762
#162552
#149472
#147782
#142902
#141652
#141362
#138682
#138522