Finite Group Databasebeta

← #627804

Subgroups — #627804

Order 512.

2942 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3#15710C214yesmaximal, complemented
4#15697C214yesmaximal, complemented
5#15648C214yesmaximal, complemented
6#15641C214yesmaximal, complemented
7#14820C214yesmaximal
8#14702C211yesmaximal, characteristic, complemented
9#14686C211yesmaximal, characteristic, complemented
10#14347C211yesmaximal, characteristic, complemented
11#14281C211yesmaximal, characteristic, complemented
12#14047C214yesmaximal, complemented
13C2×#1437C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C23×C42C2211yescharacteristic
18#2626—28no—
19C2×#399C2212yes—
20C2×#397C2214yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#393C2212yes—
24#2312—28no—
25C2×#391C2212yescomplemented
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yescomplemented
29C2×#385C2212yes—
30C2×#384C2212yescomplemented
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2214yes—
35C2×#379C2212yes—
36C2×#379C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2212yes—
40#1446C2212yescomplemented
41#1445C2212yescomplemented
42#1445C2214yescomplemented
43#1443C2214yescomplemented
44#1442C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1437C2214yes—
56#1437C2212yes—
57#1437C2212yes—
58#1436C2214yescomplemented
59#1436C2212yescomplemented
60#1436C2214yescomplemented
61#1436C2212yescomplemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2214yescomplemented
64C4⋊#95C2212yescomplemented
65#1434C2214yescomplemented
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—44no—
90C22×C22⋊C4—44no—
91C22×C22⋊C4D418yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42—24no—
105C22×C42—44no—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2314yes—
110#399—28no—
111#397—48no—
112#397—48no—
113#396—216no—
114#393—28no—
115#391—44no—
116#388—216no—
117#388—216no—
118#387—48no—
119#385—28no—
120#384—44no—
121#380—24no—
122#380—24no—
123#380—24no—
124#380—216no—
125#379—28no—
126#379—28no—
127C4×C22⋊C4—28no—
128C2×#95—24no—
129C2×#95—44no—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25—24no—
154C25—22no—
155C25—22no—
156C25—24no—
157C25—42no—
158C25—42no—
159C25—42no—
160C25—42no—
161C25C2×D414yes—
162C25C2412yes—
163C25C2411yescharacteristic
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—42no—
168C22×Q8—42no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4#4118yes—
191C23×C4#4114yes—
192C23×C4#4114yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C22⋊Q8—416no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C22⋊C4—232no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
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—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—432no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C2×D418yes—
263C2×C22⋊C4C2×D418yes—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—416no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42—44no—
275#95—28no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—28no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—28no—
312C24—28no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—82no—
322C24—82no—
323C24—84no—
324C24—84no—
325C24—82no—
326C24—82no—
327C24—84no—
328C24—84no—
329C24—84no—
330C24—84no—
331C24C22×D414yes—
332C24C22×D412yes—
333C24C2×#4112yes—
334C24C2×#4112yes—
335C24C2×#4114yes—
336C242−1+414yes—
337C24C2511yescharacteristic
338C2×Q8—48no—
339C2×Q8—44no—
340C2×Q8—82no—
341C2×Q8—82no—
342C2×Q8—84no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—216no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
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—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—84no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4#123116yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4114yes—
432C22×C4H(5,𝔽2)14yes—
433C22×C4H(5,𝔽2)14yes—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—416no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C22⋊C4—216no—
443C22⋊C4—216no—
444C22⋊C4—416no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—416no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C42—28no—
455C42—48no—
456C42—44no—
457C42—88no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—432no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—416no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—88no—
498C23—816no—
499C23—88no—
500C23—816no—
501C23—816no—
502C23—816no—
503C23—816no—
504C23C2×#12314yes—
505C23#53214yes—
506C23#53614yes—
507C23#55218yes—
508C23#55618yes—
509C23C23×D411yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4112yes—
513C23C2×H(5,𝔽2)12yes—
514C23C2×H(5,𝔽2)12yes—
515C23C2×2−1+412yes—
516C23#58512yes—
517C23#58512yes—
518Q8—88no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—416no—
539C2×C4—432no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—88no—
586C2×C4—816no—
587C2×C4C2×#12314yes—
588C2×C4C2×#12314yes—
589C2×C4#53818yes—
590C22—24no—
591C22—28no—
592C22—28no—
593C22—28no—
594C22—28no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—44no—
599C22—48no—
600C22—48no—
601C22—44no—
602C22—44no—
603C22—832no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22#249918yes—
609C22C22×#12311yescharacteristic
610C22C2×#53211yescharacteristic
611C22C2×#53611yescharacteristic
612C22C2×#53812yes—
613C22#345512yes—
614C22C2×#55212yes—
615C22C2×#55612yes—
616C22#349011yescharacteristic
617C22#349111yescharacteristic
618C22#349412yes—
619C22#349612yes—
620C22#349612yes—
621C22#350512yes—
622C22#350512yes—
623C22#350612yes—
624C22#350811yescharacteristic
625C22#351011yescharacteristic
626C22#353512yes—
627C22#353612yes—
628C22#356112yes—
629C22#356412yes—
630C4—24no—
631C4—24no—
632C4—28no—
633C4—44no—
634C4—44no—
635C4—82no—
636C4—84no—
637C4—84no—
638C4—84no—
639C4—84no—
640C4—82no—
641C2—28no—
642C2—84no—
643C2C2×#249911yescharacteristic
644C2#3500512yes—
645C2#3536011yescharacteristic
646C2#3554111yescharacteristic
647C2#3674012yes—
648C2#3788112yes—
649C2#3788512yes—
650C2#3950812yes—
651C2#3951112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15710C2semidirect
4. #15697C2semidirect
5. #15648C2semidirect
6. #15641C2semidirect
8. #14702C2semidirect
9. #14686C2semidirect
10. #14347C2semidirect
11. #14281C2semidirect
12. #14047C2semidirect
25. C2×#391C22semidirect
28. C2×#387C22semidirect
30. C2×#384C22semidirect
40. #1446C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C2×#23122
#157102
#156972
#156482
#156412
#148202
#147022
#146862
#143472
#142812
#140472
C2×#14372