Finite Group Databasebeta

← #5021233

Subgroups — #5021233

Order 512.

3560 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic, complemented
2C2×#2375C211yesmaximal, characteristic
3#16039C213yesmaximal, complemented
4#15961C213yesmaximal, complemented
5#15951C213yesmaximal, complemented
6#15672C213yesmaximal, complemented
7#15357C213yesmaximal, complemented
8#15356C213yesmaximal, complemented
9#15329C213yesmaximal, complemented
10#15326C213yesmaximal, complemented
11#14694C211yesmaximal, characteristic
12#14693C211yesmaximal, characteristic, complemented
13#14673C211yesmaximal, characteristic, complemented
14#14673C211yesmaximal, characteristic, complemented
15#14657C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2213yes—
18C22×C22⋊Q8C2211yescharacteristic, complemented
19C22×#120C2211yescharacteristic
20#2848—28no—
21C2×#400C2213yescomplemented
22C2×#398C2213yescomplemented
23#2375—28no—
24C2×#390C2213yes—
25C2×#388C2213yes—
26C2×#388C2213yes—
27C2×#386C2213yes—
28C2×#386C2213yes—
29C2×#386C2213yes—
30C2×#385C2213yes—
31C2×#385C2213yes—
32C2×#385C2213yes—
33C2×#385C2213yes—
34C2×#385C2213yescomplemented
35C2×#380C2213yes—
36C2×#379C2211yescharacteristic
37C2×#379C2211yescharacteristic
38C2×#379C2211yescharacteristic
39C2×#379C2211yescharacteristic
40C2×#379C2211yescharacteristic
41#1446C2213yescomplemented
42#1445C2213yescomplemented
43#1444C2213yescomplemented
44#1442C2213yescomplemented
45#1442C2213yescomplemented
46#1442C2213yescomplemented
47#1441C2213yescomplemented
48#1441C2213yescomplemented
49#1441C2213yescomplemented
50#1441C2213yescomplemented
51#1440C2213yes—
52#1440C2213yes—
53#1439C2213yes—
54#1439C2213yes—
55#1439C2213yes—
56#1439C2213yes—
57#1439C2213yes—
58#1439C2213yes—
59#1439C2213yes—
60#1439C2213yes—
61#1439C2213yescomplemented
62#1439C2213yescomplemented
63#1439C2213yescomplemented
64#1439C2213yescomplemented
65#1438C2213yes—
66#1438C2213yes—
67#1436C2213yescomplemented
68#1436C2213yescomplemented
69C4⋊#95C2213yescomplemented
70#1434C2213yescomplemented
71C22⋊#95C2213yes—
72C22⋊#95C2213yes—
73C22⋊#95C2213yes—
74C22⋊#95C2213yes—
75C26C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C23×D4C2313yes—
78C24×C4—24no—
79C24×C4—46no—
80C24×C4—46no—
81C24×C4C2313yes—
82C24×C4C2313yes—
83C24×C4C2311yescharacteristic
84C2×C22⋊Q8—212no—
85C2×C22⋊Q8—46no—
86C2×C22⋊Q8—46no—
87C2×C22⋊Q8—46no—
88C2×#120—24no—
89C2×#120—24no—
90C2×#120Q818yescomplemented
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2313yes—
93C22×C4⋊C4C2313yes—
94C22×C4⋊C4C2313yes—
95C22×C4⋊C4C2313yes—
96C22×C4⋊C4C2313yes—
97C22×C4⋊C4C2313yes—
98C22×C4⋊C4C2313yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2313yes—
103C22×C22⋊C4C2313yes—
104C22×C22⋊C4C2313yes—
105C22×C22⋊C4C2313yes—
106C22×C22⋊C4C2313yes—
107C22×C22⋊C4C2313yes—
108C22×C22⋊C4C2313yes—
109C22×C22⋊C4C2313yes—
110C22×C22⋊C4C2313yes—
111C22×C22⋊C4C2313yes—
112C22×C22⋊C4C2313yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2313yes—
115C22×C22⋊C4C2313yes—
116C22×C22⋊C4C2313yes—
117C22×C42C2313yes—
118#400—46no—
119#398—46no—
120#390—212no—
121#388—212no—
122#388—212no—
123#386—212no—
124#386—46no—
125#386—46no—
126#385—212no—
127#385—212no—
128#385—46no—
129#385—46no—
130#385—46no—
131#380—212no—
132#379—24no—
133#379—24no—
134#379—24no—
135#379—24no—
136#379D418yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2313yes—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2313yes—
160C2×#95C2313yes—
161C25—43no—
162C25—43no—
163C25—43no—
164C25—43no—
165C25C2×D414yes—
166C25C2×Q814yes—
167C25#4114yes—
168C25C2411yescharacteristic
169C25C2413yes—
170C22×Q8—22no—
171C22×Q8—26no—
172C22×Q8—43no—
173C22×D4—26no—
174C22×D4—26no—
175C22×D4—43no—
176C22×D4—43no—
177C22×D4—43no—
178C22×D4—43no—
179C22×D4C2×Q8112yes—
180C23×C4—26no—
181C23×C4—26no—
182C23×C4—26no—
183C23×C4—26no—
184C23×C4—26no—
185C23×C4—26no—
186C23×C4—26no—
187C23×C4—26no—
188C23×C4—22no—
189C23×C4—43no—
190C23×C4—43no—
191C23×C4—43no—
192C23×C4—43no—
193C23×C4—43no—
194C23×C4—43no—
195C23×C4—43no—
196C23×C4—43no—
197C23×C4—448no—
198C23×C4—448no—
199C23×C4—416no—
200C23×C4C2411yescharacteristic
201C23×C4C2413yes—
202C23×C4C2413yes—
203C23×C4C2413yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2413yes—
207C23×C4C2411yescharacteristic
208C23×C4C2413yes—
209C23×C4C2411yescharacteristic
210C23×C4C2413yes—
211C23×C4C2413yes—
212C23×C4C2411yescharacteristic
213C22⋊Q8—448no—
214C22⋊Q8—416no—
215#120—232no—
216C2×C4⋊C4—26no—
217C2×C4⋊C4—26no—
218C2×C4⋊C4—26no—
219C2×C4⋊C4—26no—
220C2×C4⋊C4—26no—
221C2×C4⋊C4—26no—
222C2×C4⋊C4—26no—
223C2×C4⋊C4—26no—
224C2×C4⋊C4—26no—
225C2×C4⋊C4—43no—
226C2×C4⋊C4—43no—
227C2×C4⋊C4—43no—
228C2×C4⋊C4—43no—
229C2×C4⋊C4—43no—
230C2×C4⋊C4—43no—
231C2×C4⋊C4—43no—
232C2×C4⋊C4—43no—
233C2×C4⋊C4—43no—
234C2×C4⋊C4—43no—
235C2×C4⋊C4—43no—
236C2×C4⋊C4—43no—
237C2×C4⋊C4—43no—
238C2×C4⋊C4—43no—
239C2×C4⋊C4—43no—
240C2×C22⋊C4—26no—
241C2×C22⋊C4—26no—
242C2×C22⋊C4—26no—
243C2×C22⋊C4—26no—
244C2×C22⋊C4—26no—
245C2×C22⋊C4—26no—
246C2×C22⋊C4—26no—
247C2×C22⋊C4—26no—
248C2×C22⋊C4—26no—
249C2×C22⋊C4—26no—
250C2×C22⋊C4—26no—
251C2×C22⋊C4—26no—
252C2×C22⋊C4—26no—
253C2×C22⋊C4—43no—
254C2×C22⋊C4—43no—
255C2×C22⋊C4—43no—
256C2×C22⋊C4—43no—
257C2×C22⋊C4—43no—
258C2×C22⋊C4—43no—
259C2×C22⋊C4—43no—
260C2×C22⋊C4—43no—
261C2×C22⋊C4—43no—
262C2×C22⋊C4—43no—
263C2×C22⋊C4—43no—
264C2×C22⋊C4—43no—
265C2×C22⋊C4—43no—
266C2×C22⋊C4—43no—
267C2×C22⋊C4—43no—
268C2×C22⋊C4—43no—
269C2×C22⋊C4—43no—
270C2×C22⋊C4—43no—
271C2×C22⋊C4—43no—
272C2×C22⋊C4—43no—
273C2×C22⋊C4—43no—
274C2×C22⋊C4—43no—
275C2×C22⋊C4—43no—
276C2×C22⋊C4—43no—
277C2×C22⋊C4—43no—
278C2×C22⋊C4—43no—
279C2×C22⋊C4—43no—
280C2×C22⋊C4—43no—
281C2×C22⋊C4C2×D4112yes—
282C2×C22⋊C4C2×D4112yes—
283C2×C22⋊C4C2×Q8112yes—
284C2×C42—26no—
285C2×C42—43no—
286C2×C42—43no—
287#95—26no—
288#95—26no—
289#95—26no—
290#95—26no—
291#95—26no—
292#95—26no—
293#95—26no—
294#95—26no—
295#95—43no—
296#95—43no—
297#95—43no—
298#95—43no—
299#95—43no—
300#95—43no—
301#95—43no—
302#95—43no—
303#95—43no—
304#95—43no—
305#95—43no—
306#95—43no—
307#95—43no—
308#95—43no—
309#95—43no—
310#95—43no—
311C24—23no—
312C24—23no—
313C24—23no—
314C24—23no—
315C24—23no—
316C24—23no—
317C24—23no—
318C24—23no—
319C24—23no—
320C24—23no—
321C24—23no—
322C24—23no—
323C24—21no—
324C24—21no—
325C24—23no—
326C24—23no—
327C24—23no—
328C24—23no—
329C24—412no—
330C24—412no—
331C24—412no—
332C24—412no—
333C24—412no—
334C24—412no—
335C24—44no—
336C24—412no—
337C24—412no—
338C24—412no—
339C24—412no—
340C24—412no—
341C24C22⋊Q8116yescomplemented
342C24C22×D416yes—
343C24C22×Q812yes—
344C24C22×Q816yes—
345C24C2×#4116yes—
346C24C2511yescharacteristic
347C2×Q8—412no—
348C2×Q8—44no—
349C2×Q8—86no—
350C2×D4—224no—
351C2×D4—412no—
352C2×D4—412no—
353C2×D4—86no—
354C2×D4—86no—
355C2×D4—86no—
356C2×D4—86no—
357C22×C4—21no—
358C22×C4—23no—
359C22×C4—224no—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—23no—
367C22×C4—23no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—23no—
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—23no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—21no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—21no—
384C22×C4—23no—
385C22×C4—21no—
386C22×C4—23no—
387C22×C4—21no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4—23no—
409C22×C4—23no—
410C22×C4—23no—
411C22×C4—23no—
412C22×C4—23no—
413C22×C4—23no—
414C22×C4—23no—
415C22×C4—23no—
416C22×C4—23no—
417C22×C4—23no—
418C22×C4—23no—
419C22×C4—23no—
420C22×C4—412no—
421C22×C4—412no—
422C22×C4—412no—
423C22×C4—412no—
424C22×C4—412no—
425C22×C4—412no—
426C22×C4—412no—
427C22×C4—412no—
428C22×C4—412no—
429C22×C4—412no—
430C22×C4—412no—
431C22×C4—4128no—
432C22×C4—412no—
433C22×C4—412no—
434C22×C4—44no—
435C22×C4—86no—
436C22×C4—86no—
437C22×C4C22×D416yes—
438C22×C4C22×D412yes—
439C22×C4C22×Q816yes—
440C22×C4H(5,𝔽2)16yes—
441C22×C4H(5,𝔽2)16yes—
442C4⋊C4—412no—
443C4⋊C4—412no—
444C4⋊C4—412no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—86no—
449C4⋊C4—86no—
450C4⋊C4—86no—
451C4⋊C4—86no—
452C22⋊C4—224no—
453C22⋊C4—412no—
454C22⋊C4—412no—
455C22⋊C4—412no—
456C22⋊C4—412no—
457C22⋊C4—412no—
458C22⋊C4—412no—
459C22⋊C4—412no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—86no—
463C22⋊C4—86no—
464C22⋊C4—86no—
465C22⋊C4—86no—
466C22⋊C4—86no—
467C22⋊C4—86no—
468C42—86no—
469C23—26no—
470C23—26no—
471C23—26no—
472C23—26no—
473C23—26no—
474C23—26no—
475C23—26no—
476C23—26no—
477C23—26no—
478C23—26no—
479C23—26no—
480C23—26no—
481C23—26no—
482C23—26no—
483C23—26no—
484C23—26no—
485C23—26no—
486C23—26no—
487C23—22no—
488C23—26no—
489C23—26no—
490C23—448no—
491C23—416no—
492C23—448no—
493C23—448no—
494C23—416no—
495C23—448no—
496C23—416no—
497C23—43no—
498C23—43no—
499C23—43no—
500C23—43no—
501C23—43no—
502C23—43no—
503C23—43no—
504C23—43no—
505C23—43no—
506C23—43no—
507C23—43no—
508C23—43no—
509C23—43no—
510C23—43no—
511C23—43no—
512C23—43no—
513C23—43no—
514C23—43no—
515C23—43no—
516C23—43no—
517C23—41no—
518C23—43no—
519C23—43no—
520C23—43no—
521C23—43no—
522C23—43no—
523C23C2×C22⋊Q8112yes—
524C23C23×D411yescharacteristic
525C23C23×Q811yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)13yes—
528C23C2×H(5,𝔽2)13yes—
529C23#58513yes—
530C23#58513yes—
531Q8—88no—
532D4—824no—
533C2×C4—26no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—26no—
537C2×C4—26no—
538C2×C4—26no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—26no—
545C2×C4—448no—
546C2×C4—448no—
547C2×C4—43no—
548C2×C4—43no—
549C2×C4—43no—
550C2×C4—43no—
551C2×C4—43no—
552C2×C4—43no—
553C2×C4—41no—
554C2×C4—43no—
555C2×C4—43no—
556C2×C4—43no—
557C2×C4—43no—
558C2×C4—43no—
559C2×C4—43no—
560C2×C4—43no—
561C2×C4—43no—
562C2×C4—43no—
563C2×C4—43no—
564C2×C4—43no—
565C2×C4—43no—
566C2×C4—43no—
567C2×C4—43no—
568C2×C4—43no—
569C2×C4—43no—
570C2×C4—43no—
571C2×C4—43no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—43no—
575C2×C4—43no—
576C2×C4—43no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—43no—
582C2×C4—43no—
583C2×C4—43no—
584C2×C4—43no—
585C2×C4—43no—
586C2×C4—43no—
587C2×C4—43no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—43no—
591C2×C4—43no—
592C2×C4—43no—
593C2×C4—43no—
594C2×C4—43no—
595C2×C4—43no—
596C2×C4—43no—
597C2×C4—43no—
598C2×C4—43no—
599C2×C4—43no—
600C2×C4—43no—
601C2×C4—88no—
602C2×C4H(3,𝔽4)14yes—
603C22—212no—
604C22—212no—
605C22—212no—
606C22—464no—
607C22—46no—
608C22—46no—
609C22—46no—
610C22—46no—
611C22—46no—
612C22—46no—
613C22—46no—
614C22—46no—
615C22—46no—
616C22—46no—
617C22—46no—
618C22—46no—
619C22—46no—
620C22—46no—
621C22—42no—
622C22—81no—
623C22—83no—
624C22—83no—
625C22—81no—
626C22C22×C22⋊Q811yescharacteristic
627C22#348413yes—
628C22#348513yes—
629C22#350113yes—
630C22#350113yes—
631C22#350513yes—
632C22#350513yes—
633C22C2×H(3,𝔽4)11yescharacteristic
634C22#352913yes—
635C22#353213yes—
636C22#353913yes—
637C22#353913yes—
638C22#354013yes—
639C4—24no—
640C4—46no—
641C4—46no—
642C4—42no—
643C4—46no—
644C4—81no—
645C4—81no—
646C4—83no—
647C4—83no—
648C4—83no—
649C4—83no—
650C4—81no—
651C4—81no—
652C2—412no—
653C2—82no—
654C2#3534913yes—
655C2#4543911yescharacteristic
656C2#4544013yes—
657C2#4549213yes—
658C2#4549411yescharacteristic
659C2#4550713yes—
660C2#4550911yescharacteristic
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
1. C2×#2848C2semidirect
3. #16039C2semidirect
4. #15961C2semidirect
5. #15951C2semidirect
6. #15672C2semidirect
7. #15357C2semidirect
8. #15356C2semidirect
9. #15329C2semidirect
10. #15326C2semidirect
12. #14693C2semidirect
13. #14673C2semidirect
14. #14673C2semidirect
18. C22×C22⋊Q8C22semidirect
21. C2×#400C22semidirect
22. C2×#398C22semidirect
34. C2×#385C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. C4⋊#95C22semidirect
70. #1434C22semidirect
90. C2×#120Q8semidirect
341. C24C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#23752
#160392
#159612
#159512
#156722
#153572
#153562
#153292
#153262
#146942
#146932
#146732
#146572