Finite Group Databasebeta

← #4314283

Subgroups — #4314283

Order 512.

2878 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2709C212yesmaximal
3C2×#2328C211yesmaximal, characteristic
4#17011C212yesmaximal, complemented
5#16688C212yesmaximal, complemented
6#16465C214yesmaximal, complemented
7#15494C214yesmaximal, complemented
8#15383C214yesmaximal, complemented
9#15122C212yesmaximal, complemented
10#14824C212yesmaximal, complemented
11#14737C214yesmaximal, complemented
12#14486C211yesmaximal, characteristic, complemented
13#13968C212yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×#123C2212yes—
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2724—28no—
19#2709—216no—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#397C2212yes—
23C2×#394C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26#2328—28no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2212yes—
36C2×#385C2214yes—
37C2×#382C2211yescharacteristic
38C2×#381C2212yes—
39C2×#380C2212yes—
40C2×#376C2211yescharacteristic
41C22×#95C2212yes—
42#1445C2212yes—
43#1445C2212yes—
44#1444C2212yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2212yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1436C2214yes—
63#1434C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C2×#127—24no—
78C2×#127—24no—
79C2×#127D418yes—
80C2×#123—28no—
81C2×#123—28no—
82C2×#123—28no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#398—28no—
113#397—216no—
114#397—44no—
115#394—216no—
116#394—216no—
117#394—216no—
118#390—24no—
119#390D418yes—
120#389—28no—
121#387—216no—
122#386—216no—
123#386—216no—
124#386—28no—
125#386—216no—
126#386—216no—
127#385—216no—
128#382—24no—
129#381—44no—
130#380—44no—
131#376—24no—
132C2×#95—44no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C25C2412yes—
157C25C2412yes—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4C2×D4116yes—
175C22×D4C2×D414yes—
176C22×D4C2×D414yes—
177C22×D4C2×D414yes—
178C22×D4#4114yes—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#127—232no—
213#123—432no—
214C4×D4—416no—
215C4×D4—416no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
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⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C2×D4116yes—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42C2×D414yes—
282C2×C42C2×D414yes—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—44no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—42no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—22no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24C22×D414yes—
325C24C22×D414yes—
326C24C22×D414yes—
327C24C2×#4114yes—
328C24C2511yescharacteristic
329C2×D4—232no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—416no—
344C2×D4—416no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4#121116yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
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—24no—
403C22×C4—24no—
404C22×C4—24no—
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—88no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—88no—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)12yes—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—416no—
438C4⋊C4—44no—
439C4⋊C4—416no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C22⋊C4—416no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—84no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—28no—
456C42—28no—
457C42—44no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—24no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C2×#12118yes—
504C23C23×D411yescharacteristic
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23#58511yescharacteristic
510C23#58511yescharacteristic
511C23#58514yes—
512C23#58514yes—
513D4—416no—
514D4—416no—
515D4—832no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
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—44no—
559C2×C4—44no—
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—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—816no—
579C2×C4—816no—
580C2×C4—816no—
581C2×C4—816no—
582C2×C4C2×#12114yes—
583C22—28no—
584C22—216no—
585C22—44no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—48no—
595C22—44no—
596C22—48no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—84no—
604C22—84no—
605C22C22×#12111yescharacteristic
606C22#345911yescharacteristic
607C22#345911yescharacteristic
608C22#348411yescharacteristic
609C22#348512yes—
610C22#348514yes—
611C22#348511yescharacteristic
612C22#349611yescharacteristic
613C22#349611yescharacteristic
614C22#350012yes—
615C22#352114yes—
616C22#353312yes—
617C22#353312yes—
618C22#354012yes—
619C22#355112yes—
620C22#356614yes—
621C22#356714yes—
622C4—24no—
623C4—42no—
624C4—44no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—82no—
629C4—84no—
630C4—84no—
631C4—84no—
632C4—84no—
633C2—48no—
634C2—84no—
635C2#3498411yescharacteristic
636C2#3600912yes—
637C2#3708012yes—
638C2#4225212yes—
639C2#4681712yes—
640C2#4780012yes—
641C2#5197811yescharacteristic
642C2#5205411yescharacteristic
643C2#5206412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2724C2semidirect
4. #17011C2semidirect
5. #16688C2semidirect
6. #16465C2semidirect
7. #15494C2semidirect
8. #15383C2semidirect
9. #15122C2semidirect
10. #14824C2semidirect
11. #14737C2semidirect
12. #14486C2semidirect
13. #13968C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#27092
C2×#23282
#170112
#166882
#164652
#154942
#153832
#151222
#148242
#147372
#144862
#139682