Finite Group Databasebeta

← #4452700

Subgroups — #4452700

Order 512.

3361 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2622C211yesmaximal, characteristic
2C22×#393C211yesmaximal, characteristic, complemented
3#16878C214yesmaximal, complemented
4#16837C214yesmaximal, complemented
5#16788C214yesmaximal, complemented
6#16787C214yesmaximal, complemented
7#15026C211yesmaximal, characteristic, complemented
8#14946C211yesmaximal, characteristic, complemented
9#14874C211yesmaximal, characteristic, complemented
10#14873C211yesmaximal, characteristic, complemented
11#14727C212yesmaximal, complemented
12#14718C212yesmaximal
13#14044C212yesmaximal, complemented
14#14020C212yesmaximal, complemented
15#13775C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20#2622—28no—
21C2×#397C2212yes—
22C2×#397C2212yes—
23C2×#397C2214yes—
24C2×#395C2214yescomplemented
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#393—28no—
29C2×#393C2214yes—
30C2×#393C2214yes—
31C2×#393C2214yescomplemented
32C2×#386C2211yescharacteristic
33C2×#386C2212yes—
34C2×#386C2211yescharacteristic
35C2×#386C2212yes—
36C2×#385C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#379C2212yes—
41C2×#379C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2212yes—
45#1446C2214yescomplemented
46#1446C2214yescomplemented
47#1445C2214yescomplemented
48#1443C2214yescomplemented
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yescomplemented
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yescomplemented
60#1438C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yescomplemented
64#1436C2214yescomplemented
65C4⋊#95C2214yescomplemented
66#1434C2214yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C26C2311yescharacteristic
74C23×Q8—24no—
75C23×Q8—44no—
76C23×Q8—48no—
77C23×Q8C2314yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#120—24no—
86C2×#120—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—48no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4D418yes—
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2314yes—
115#397—28no—
116#397—28no—
117#397—48no—
118#395—48no—
119#394—28no—
120#394—28no—
121#394—48no—
122#393—216no—
123#393—216no—
124#393—48no—
125#386—24no—
126#386—24no—
127#386—28no—
128#386—44no—
129#385—24no—
130#385—24no—
131#385—28no—
132#385—44no—
133#379—24no—
134#379—28no—
135C4×C22⋊C4—44no—
136C2×#95—28no—
137C2×#95—24no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25—22no—
161C25—22no—
162C25—24no—
163C25—24no—
164C25—44no—
165C25—44no—
166C25C2×D414yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×Q8—28no—
172C22×Q8—28no—
173C22×Q8—28no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—432no—
179C22×Q8—416no—
180C22×Q8—464no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—44no—
191C22×D4—44no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219#120—416no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—416no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—432no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D418yes—
279C2×C22⋊C4C2×D418yes—
280C2×C22⋊C4#4118yes—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
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—
304#95—44no—
305#95—44no—
306C24—24no—
307C24—24no—
308C24—28no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—24no—
320C24—24no—
321C24—28no—
322C24—28no—
323C24—21no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—24no—
330C24—48no—
331C24—48no—
332C24—416no—
333C24—416no—
334C24—48no—
335C24—48no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24—88no—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C2×#4114yes—
344C24H(5,𝔽2)14yes—
345C242−1+412yes—
346C242−1+412yes—
347C24C2511yescharacteristic
348C2×Q8—416no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—416no—
352C2×Q8—416no—
353C2×Q8—416no—
354C2×Q8—4128no—
355C2×Q8—416no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—88no—
366C2×D4—88no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C2×#4118yes—
441C22×C4H(5,𝔽2)18yes—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C22⋊C4—216no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C42—88no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—432no—
485C23—416no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—88no—
516C23—816no—
517C23—88no—
518C23—832no—
519C23—832no—
520C23#53214yes—
521C23#53414yes—
522C23#53614yes—
523C23#53918yes—
524C23#54218yes—
525C23C23×D411yescharacteristic
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)14yes—
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23#58514yes—
532Q8—464no—
533D4—88no—
534D4—88no—
535D4—88no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—432no—
549C2×C4—416no—
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—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
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—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—88no—
598C2×C4—816no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—216no—
603C22—216no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—832no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—81no—
621C22—81no—
622C22—82no—
623C22—82no—
624C22—84no—
625C22—84no—
626C22—81no—
627C22#266318yes—
628C22C2×#53211yescharacteristic
629C22C2×#53411yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#53912yes—
632C22C2×#54212yes—
633C22#348412yes—
634C22#348512yes—
635C22#351414yes—
636C22#351414yes—
637C22#353214yes—
638C22#354114yes—
639C22#354114yes—
640C22#354514yes—
641C4—44no—
642C4—48no—
643C4—48no—
644C4—44no—
645C4—81no—
646C4—81no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—84no—
651C4—81no—
652C2—28no—
653C2—82no—
654C2—82no—
655C2—161no—
656C2C2×#266311yescharacteristic
657C2#3546612yes—
658C2#3671411yescharacteristic
659C2#3671612yes—
660C2#3711911yescharacteristic
661C2#4144814yes—
662C2#4144914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#393C2semidirect
3. #16878C2semidirect
4. #16837C2semidirect
5. #16788C2semidirect
6. #16787C2semidirect
7. #15026C2semidirect
8. #14946C2semidirect
9. #14874C2semidirect
10. #14873C2semidirect
11. #14727C2semidirect
13. #14044C2semidirect
14. #14020C2semidirect
15. #13775C2semidirect
24. C2×#395C22semidirect
31. C2×#393C22semidirect
45. #1446C22semidirect
46. #1446C22semidirect
47. #1445C22semidirect
48. #1443C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
56. #1440C22semidirect
59. #1439C22semidirect
63. #1437C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26222
C22×#3932
#168782
#168372
#167882
#167872
#150262
#149462
#148742
#148732
#147272
#147182
#140442
#140202
#137752