Finite Group Databasebeta

← #1393181

Subgroups — #1393181

Order 512.

5522 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#2418C213yesmaximal
3C2×#2614C211yesmaximal, characteristic, complemented
4C22×#393C211yesmaximal, characteristic, complemented
5#16511C213yesmaximal, complemented
6#16506C213yesmaximal, complemented
7#16263C213yesmaximal, complemented
8#16251C213yesmaximal, complemented
9#15176C211yesmaximal, characteristic, complemented
10#14874C211yesmaximal, characteristic, complemented
11#14873C211yesmaximal, characteristic, complemented
12#14758C213yesmaximal, complemented
13#14717C213yesmaximal, complemented
14#14301C211yesmaximal, characteristic, complemented
15#13894C213yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C24×Q8C2211yescharacteristic, complemented
18C2×D4×Q8—224no—
19C22×C4⋊Q8C2213yescomplemented
20C22×#123C2213yes—
21C22×C22⋊Q8C2213yes—
22C22×C22⋊Q8C2213yescomplemented
23C22×#120C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic, complemented
25C22×C4×D4C2213yes—
26C23×C22⋊C4C2213yes—
27#2614—28no—
28#2418—224no—
29C2×#399C2213yes—
30C2×#397C2213yes—
31C2×#395C2211yescharacteristic, complemented
32C2×#395C2213yescomplemented
33C2×#395C2213yescomplemented
34C2×#394C2213yes—
35C2×#394C2213yes—
36C2×#393—28no—
37C2×#393C2211yescharacteristic
38C2×#393C2213yes—
39C2×#393C2213yes—
40C2×#393C2213yes—
41C2×#393C2213yes—
42C2×#393C2211yescharacteristic
43C2×#388C2213yes—
44C2×#388C2213yes—
45C2×#386C2213yes—
46C2×#386C2213yes—
47C2×#385C2213yes—
48C2×#381C2213yescomplemented
49C2×#380C2213yes—
50C2×#379C2213yes—
51C22×#95C2213yes—
52#1446C2211yescharacteristic, complemented
53#1446C2213yescomplemented
54#1446C2213yescomplemented
55#1445C2216yescomplemented
56#1444C2216yescomplemented
57#1444C2213yescomplemented
58#1441C2213yescomplemented
59#1441C2216yescomplemented
60#1440C2216yescomplemented
61#1439C2213yes—
62#1439C2216yes—
63#1439C2213yes—
64#1439C2216yes—
65#1434C2213yescomplemented
66#1434C2213yescomplemented
67C22⋊#95C2213yes—
68C22⋊#95C2216yes—
69C22⋊#95C2213yescomplemented
70C26C2311yescharacteristic
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—212no—
74C23×Q8—212no—
75C23×Q8—46no—
76C23×Q8—46no—
77C23×Q8C2311yescharacteristic
78C23×Q8C2313yes—
79C23×Q8C2313yes—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4C2313yes—
84C24×C4C2313yes—
85C24×C4C2313yes—
86D4×Q8—2128no—
87C2×C4⋊Q8—224no—
88C2×C4⋊Q8—212no—
89C2×#123—212no—
90C2×#123—212no—
91C2×#123—46no—
92C2×C22⋊Q8—212no—
93C2×C22⋊Q8—212no—
94C2×C22⋊Q8—224no—
95C2×C22⋊Q8—224no—
96C2×#120—212no—
97C2×C4×Q8—212no—
98C2×C4×D4—224no—
99C2×C4×D4D4124yes—
100C22×C4⋊C4C2313yes—
101C22×C4⋊C4C2313yes—
102C22×C4⋊C4C2316yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2313yes—
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2313yes—
107C22×C22⋊C4—212no—
108C22×C22⋊C4—212no—
109C22×C22⋊C4—46no—
110C22×C22⋊C4C2313yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2313yes—
115C22×C22⋊C4C2313yes—
116C22×C22⋊C4C2313yes—
117C22×C22⋊C4C2313yes—
118C22×C22⋊C4C2313yes—
119C22×C22⋊C4C2313yes—
120C22×C22⋊C4C2313yes—
121C22×C22⋊C4C2313yes—
122C22×C22⋊C4C2313yes—
123C22×C22⋊C4C2313yes—
124C22×C22⋊C4C2313yes—
125C22×C22⋊C4C2313yes—
126C22×C42C2313yes—
127#399—212no—
128#397—212no—
129#395—24no—
130#395—212no—
131#395—46no—
132#394—212no—
133#394—212no—
134#393—24no—
135#393—212no—
136#393—212no—
137#393—212no—
138#393—46no—
139#393D418yescomplemented
140#388—212no—
141#388—212no—
142#386—212no—
143#386—212no—
144#385—212no—
145#381—212no—
146#380—212no—
147#379—212no—
148C2×#95—212no—
149C2×#95C2313yes—
150C2×#95C2316yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2316yes—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2313yes—
160C2×#95C2316yes—
161C2×#95C2316yes—
162C2×#95C2316yes—
163C2×#95C2316yes—
164C2×#95C2313yes—
165C25—22no—
166C25—22no—
167C25—22no—
168C25—26no—
169C25—26no—
170C25—43no—
171C25—43no—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C22×Q8—232no—
177C22×Q8—232no—
178C22×Q8—26no—
179C22×Q8—212no—
180C22×Q8—212no—
181C22×Q8—26no—
182C22×Q8—26no—
183C22×Q8—26no—
184C22×Q8—26no—
185C22×Q8—26no—
186C22×Q8—26no—
187C22×Q8—26no—
188C22×Q8—46no—
189C22×Q8—46no—
190C22×Q8—448no—
191C22×Q8—448no—
192C22×Q8—448no—
193C22×Q8—448no—
194C22×Q8C2×D414yes—
195C22×D4—22no—
196C22×D4—26no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—26no—
200C22×D4—26no—
201C22×D4—43no—
202C22×D4—43no—
203C22×D4C2×D4112yes—
204C23×C4—212no—
205C23×C4—212no—
206C23×C4—26no—
207C23×C4—26no—
208C23×C4—26no—
209C23×C4—26no—
210C23×C4—26no—
211C23×C4—212no—
212C23×C4—26no—
213C23×C4—46no—
214C23×C4—46no—
215C23×C4—43no—
216C23×C4—46no—
217C23×C4C2×D4112yes—
218C23×C4C2×D4112yes—
219C23×C4C2413yes—
220C23×C4C2413yes—
221C23×C4C2413yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2413yes—
225C23×C4C2413yes—
226C23×C4C2413yes—
227C23×C4C2413yes—
228C23×C4C2411yescharacteristic
229C23×C4C2413yes—
230C4⋊Q8—296no—
231#123—448no—
232C22⋊Q8—296no—
233C22⋊Q8—296no—
234#120—416no—
235C4×Q8—232no—
236C4×D4—296no—
237C2×C4⋊C4—212no—
238C2×C4⋊C4—212no—
239C2×C4⋊C4—26no—
240C2×C4⋊C4—212no—
241C2×C4⋊C4—212no—
242C2×C4⋊C4—212no—
243C2×C4⋊C4—26no—
244C2×C4⋊C4—26no—
245C2×C4⋊C4—26no—
246C2×C4⋊C4—26no—
247C2×C4⋊C4—26no—
248C2×C4⋊C4—26no—
249C2×C4⋊C4—26no—
250C2×C4⋊C4—43no—
251C2×C4⋊C4—46no—
252C2×C4⋊C4—46no—
253C2×C4⋊C4C2×D4112yes—
254C2×C22⋊C4—212no—
255C2×C22⋊C4—212no—
256C2×C22⋊C4—26no—
257C2×C22⋊C4—26no—
258C2×C22⋊C4—26no—
259C2×C22⋊C4—26no—
260C2×C22⋊C4—26no—
261C2×C22⋊C4—26no—
262C2×C22⋊C4—26no—
263C2×C22⋊C4—26no—
264C2×C22⋊C4—26no—
265C2×C22⋊C4—26no—
266C2×C22⋊C4—26no—
267C2×C22⋊C4—26no—
268C2×C22⋊C4—26no—
269C2×C22⋊C4—26no—
270C2×C22⋊C4—26no—
271C2×C22⋊C4—26no—
272C2×C22⋊C4—26no—
273C2×C22⋊C4—26no—
274C2×C22⋊C4—26no—
275C2×C22⋊C4—26no—
276C2×C22⋊C4—26no—
277C2×C22⋊C4—26no—
278C2×C22⋊C4—26no—
279C2×C22⋊C4—26no—
280C2×C22⋊C4—448no—
281C2×C22⋊C4—43no—
282C2×C22⋊C4—43no—
283C2×C22⋊C4—43no—
284C2×C22⋊C4—43no—
285C2×C22⋊C4—43no—
286C2×C22⋊C4—46no—
287C2×C22⋊C4—46no—
288C2×C22⋊C4—46no—
289C2×C22⋊C4C2×D4112yes—
290C2×C22⋊C4C2×D4112yes—
291C2×C22⋊C4C2×D4112yes—
292C2×C42—212no—
293C2×C42C2×D4112yes—
294#95—212no—
295#95—26no—
296#95—26no—
297#95—26no—
298#95—212no—
299#95—26no—
300#95—26no—
301#95—26no—
302#95—26no—
303#95—26no—
304#95—26no—
305#95—46no—
306#95—46no—
307#95—46no—
308#95—46no—
309#95C2×D4112yes—
310C24—21no—
311C24—23no—
312C24—21no—
313C24—23no—
314C24—21no—
315C24—23no—
316C24—21no—
317C24—23no—
318C24—21no—
319C24—23no—
320C24—28no—
321C24—21no—
322C24—23no—
323C24—23no—
324C24—23no—
325C24—23no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—23no—
330C24—23no—
331C24—23no—
332C24—23no—
333C24—412no—
334C24—412no—
335C24—412no—
336C24—412no—
337C24—412no—
338C24—412no—
339C24—44no—
340C24—412no—
341C24—86no—
342C24—86no—
343C24—812no—
344C24C22×D412yes—
345C24C22×D416yes—
346C24C22×D416yes—
347C242−1+412yes—
348C242−1+412yes—
349C242−1+416yes—
350C24C2511yescharacteristic
351C2×Q8—28no—
352C2×Q8—224no—
353C2×Q8—224no—
354C2×Q8—224no—
355C2×Q8—224no—
356C2×Q8—424no—
357C2×Q8—424no—
358C2×Q8—424no—
359C2×Q8—424no—
360C2×Q8—412no—
361C2×Q8—412no—
362C2×Q8—4128no—
363C2×Q8—4128no—
364C2×Q8—412no—
365C2×Q8—412no—
366C2×D4—224no—
367C2×D4—224no—
368C2×D4—412no—
369C2×D4—412no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—86no—
373C2×D4—86no—
374C2×D4#120116yes—
375C22×C4—26no—
376C22×C4—26no—
377C22×C4—21no—
378C22×C4—23no—
379C22×C4—224no—
380C22×C4—224no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4—26no—
385C22×C4—26no—
386C22×C4—26no—
387C22×C4—26no—
388C22×C4—26no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—26no—
394C22×C4—23no—
395C22×C4—26no—
396C22×C4—26no—
397C22×C4—224no—
398C22×C4—224no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—21no—
407C22×C4—26no—
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—26no—
417C22×C4—21no—
418C22×C4—23no—
419C22×C4—26no—
420C22×C4—26no—
421C22×C4—23no—
422C22×C4—412no—
423C22×C4—412no—
424C22×C4—424no—
425C22×C4—424no—
426C22×C4—424no—
427C22×C4—424no—
428C22×C4—412no—
429C22×C4—424no—
430C22×C4—412no—
431C22×C4—86no—
432C22×C4—812no—
433C22×C4C22×D416yes—
434C22×C4C22×D416yes—
435C22×C4C22×D416yes—
436C22×C4C22×D416yes—
437C22×C4C22×D416yes—
438C22×C4C22×D416yes—
439C22×C4C22×D416yes—
440C22×C4H(5,𝔽2)16yes—
441C4⋊C4—224no—
442C4⋊C4—248no—
443C4⋊C4—224no—
444C4⋊C4—412no—
445C4⋊C4—412no—
446C4⋊C4—86no—
447C4⋊C4—86no—
448C22⋊C4—224no—
449C22⋊C4—224no—
450C22⋊C4—224no—
451C22⋊C4—412no—
452C22⋊C4—412no—
453C22⋊C4—412no—
454C22⋊C4—412no—
455C22⋊C4—412no—
456C22⋊C4—44no—
457C22⋊C4—412no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—412no—
461C22⋊C4—86no—
462C22⋊C4—86no—
463C22⋊C4—86no—
464C42—224no—
465C23—26no—
466C23—26no—
467C23—22no—
468C23—26no—
469C23—26no—
470C23—26no—
471C23—26no—
472C23—26no—
473C23—212no—
474C23—26no—
475C23—26no—
476C23—26no—
477C23—26no—
478C23—26no—
479C23—26no—
480C23—26no—
481C23—22no—
482C23—22no—
483C23—26no—
484C23—26no—
485C23—26no—
486C23—448no—
487C23—41no—
488C23—43no—
489C23—43no—
490C23—43no—
491C23—46no—
492C23—41no—
493C23—41no—
494C23—43no—
495C23—43no—
496C23—43no—
497C23—43no—
498C23—43no—
499C23—43no—
500C23—43no—
501C23—43no—
502C23—43no—
503C23—43no—
504C23—43no—
505C23—43no—
506C23—43no—
507C23—43no—
508C23—46no—
509C23—46no—
510C23—824no—
511C23—824no—
512C23—88no—
513C23—88no—
514C23—824no—
515C23—88no—
516C23C2×#12014yes—
517C23C2×#12014yes—
518C23#536112yes—
519C23#536112yes—
520C23C23×D413yes—
521C23C23×D411yescharacteristic
522C23C2×H(5,𝔽2)13yes—
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+413yes—
525C23C2×2−1+411yescharacteristic
526C23#58513yes—
527Q8—232no—
528Q8—448no—
529Q8—448no—
530D4—232no—
531D4—88no—
532D4—88no—
533C2×C4—26no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—212no—
537C2×C4—26no—
538C2×C4—212no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—212no—
549C2×C4—26no—
550C2×C4—448no—
551C2×C4—448no—
552C2×C4—43no—
553C2×C4—46no—
554C2×C4—46no—
555C2×C4—46no—
556C2×C4—46no—
557C2×C4—46no—
558C2×C4—43no—
559C2×C4—46no—
560C2×C4—46no—
561C2×C4—41no—
562C2×C4—41no—
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—46no—
575C2×C4—43no—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—43no—
579C2×C4—43no—
580C2×C4—43no—
581C2×C4—46no—
582C2×C4—46no—
583C2×C4—46no—
584C2×C4—824no—
585C2×C4C2×#12014yes—
586C2×C4D42112yes—
587C22—212no—
588C22—212no—
589C22—212no—
590C22—212no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—46no—
597C22—46no—
598C22—46no—
599C22—46no—
600C22—46no—
601C22—46no—
602C22—46no—
603C22—46no—
604C22—42no—
605C22—42no—
606C22—832no—
607C22—81no—
608C22—81no—
609C22—83no—
610C22—83no—
611C22—83no—
612C22—83no—
613C22—81no—
614C22—81no—
615C22#261418yescomplemented
616C22C22×#12011yescharacteristic
617C22C2×#53613yes—
618C22C2×#53613yes—
619C22C2×D4213yes—
620C22#348413yes—
621C22#348611yescharacteristic
622C22#348913yes—
623C22#348913yes—
624C22#350013yes—
625C22#353813yes—
626C22#354113yes—
627C22#356116yes—
628C4—24no—
629C4—212no—
630C4—46no—
631C4—46no—
632C4—46no—
633C4—46no—
634C4—83no—
635C4—83no—
636C4—83no—
637C4—81no—
638C4—81no—
639C2—28no—
640C2—44no—
641C2—82no—
642C2—82no—
643C2C2×#261411yescharacteristic
644C2#3607811yescharacteristic
645C2D4×#12011yescharacteristic
646C2#3765113yes—
647C2#3767613yes—
648C2#4049113yes—
649C2#4059513yes—
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. C22×D4×Q8C2semidirect
3. C2×#2614C2semidirect
4. C22×#393C2semidirect
5. #16511C2semidirect
6. #16506C2semidirect
7. #16263C2semidirect
8. #16251C2semidirect
9. #15176C2semidirect
10. #14874C2semidirect
11. #14873C2semidirect
12. #14758C2semidirect
13. #14717C2semidirect
14. #14301C2semidirect
15. #13894C2semidirect
17. C24×Q8C22semidirect
19. C22×C4⋊Q8C22semidirect
22. C22×C22⋊Q8C22semidirect
24. C22×C4×Q8C22semidirect
31. C2×#395C22semidirect
32. C2×#395C22semidirect
33. C2×#395C22semidirect
48. C2×#381C22semidirect
52. #1446C22semidirect
53. #1446C22semidirect
54. #1446C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1444C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1440C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
69. C22⋊#95C22semidirect
139. #393D4semidirect
615. C22#2614semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#24182
C2×#26142
C22×#3932
#165112
#165062
#162632
#162512
#151762
#148742
#148732
#147582
#147172
#143012
#138942