Finite Group Databasebeta

← #1409004

Subgroups — #1409004

Order 512.

4117 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C213yesmaximal, complemented
2C2×#2472C213yesmaximal
3C2×#2658C211yesmaximal, characteristic, complemented
4C22×#399C211yesmaximal, characteristic, complemented
5#17062C213yesmaximal, complemented
6#16811C213yesmaximal, complemented
7#16511C213yesmaximal, complemented
8#16510C213yesmaximal, complemented
9#15155C211yesmaximal, characteristic, complemented
10#14910C211yesmaximal, characteristic, complemented
11#14836C211yesmaximal, characteristic, complemented
12#14777C213yesmaximal, complemented
13#14734C213yesmaximal, complemented
14#14322C211yesmaximal, characteristic, complemented
15#13809C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2213yescomplemented
17C22×#124C2213yes—
18C22×C22⋊Q8C2213yes—
19C22×#120C2211yescharacteristic
20C23×C4⋊C4C2213yes—
21C23×C22⋊C4C2213yes—
22#2658—28no—
23#2472—224no—
24#2440—224no—
25C2×#399—28no—
26C2×#399C2211yescharacteristic
27C2×#399C2213yes—
28C2×#399C2213yes—
29C2×#399C2213yes—
30C2×#399C2213yes—
31C2×#399C2211yescharacteristic
32C2×#399C2211yescharacteristic, complemented
33C2×#399C2213yescomplemented
34C2×#399C2213yescomplemented
35C2×#398C2216yescomplemented
36C2×#397C2213yes—
37C2×#395C2216yescomplemented
38C2×#394C2213yes—
39C2×#394C2213yes—
40C2×#393C2213yes—
41C2×#390C2213yes—
42C2×#389C2213yescomplemented
43C2×#389C2213yescomplemented
44C2×#388C2213yes—
45C2×#388C2213yes—
46C2×#386C2213yes—
47C2×#386C2213yes—
48C2×#385C2216yes—
49C2×#384C2213yescomplemented
50C2×#380C2213yes—
51C2×#379C2213yes—
52C2×C4×C22⋊C4C2213yes—
53C2×#376C2211yescharacteristic, complemented
54C22×#95C2211yescharacteristic
55#1445C2213yescomplemented
56#1445C2213yescomplemented
57#1445C2213yescomplemented
58#1444C2211yescharacteristic, complemented
59#1444C2216yescomplemented
60#1439C2213yes—
61#1439C2216yes—
62#1439C2213yes—
63#1439C2216yes—
64#1439C2216yescomplemented
65#1439C2213yescomplemented
66#1434C2213yescomplemented
67#1434C2213yescomplemented
68C22⋊#95C2213yes—
69C4×#95C2211yescharacteristic, complemented
70C26C2311yescharacteristic
71C23×Q8C2313yes—
72C23×D4C2311yescharacteristic
73C23×D4C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2313yes—
76C24×C4C2313yes—
77C2×C4⋊Q8—212no—
78C2×C4⋊Q8—212no—
79C2×C4⋊Q8—46no—
80C2×#124—212no—
81C2×#124—212no—
82C2×#124—46no—
83C2×C22⋊Q8—224no—
84C2×C22⋊Q8—212no—
85C2×#120—212no—
86C22×C4⋊C4—212no—
87C22×C4⋊C4—212no—
88C22×C4⋊C4—46no—
89C22×C4⋊C4C2313yes—
90C22×C4⋊C4C2313yes—
91C22×C4⋊C4C2313yes—
92C22×C4⋊C4C2313yes—
93C22×C4⋊C4C2313yes—
94C22×C4⋊C4C2316yes—
95C22×C4⋊C4C2316yes—
96C22×C4⋊C4C2313yes—
97C22×C4⋊C4C2316yes—
98C22×C4⋊C4C2313yes—
99C22×C22⋊C4—212no—
100C22×C22⋊C4—212no—
101C22×C22⋊C4—46no—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
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⋊C4C2313yes—
114C22×C22⋊C4C2313yes—
115C22×C22⋊C4C2313yes—
116C22×C22⋊C4C2313yes—
117C22×C22⋊C4C2313yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2313yes—
120C22×C42C2313yes—
121#399—212no—
122#399—212no—
123#399—212no—
124#399—212no—
125#399—24no—
126#399—24no—
127#399—46no—
128#399—46no—
129#399D418yescomplemented
130#398—224no—
131#397—212no—
132#395—224no—
133#394—212no—
134#394—212no—
135#393—212no—
136#390—212no—
137#389—212no—
138#389—212no—
139#388—212no—
140#388—212no—
141#386—212no—
142#386—212no—
143#385—224no—
144#384—212no—
145#380—212no—
146#379—212no—
147C4×C22⋊C4D4124yes—
148#376—24no—
149C2×#95—24no—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2316yes—
153C2×#95C2316yes—
154C2×#95C2313yes—
155C2×#95C2316yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2313yes—
160C2×#95C2316yes—
161C2×#95C2313yes—
162C2×#95C2313yes—
163C2×#95C2311yescharacteristic
164C25—22no—
165C25—22no—
166C25—22no—
167C25—26no—
168C25—26no—
169C25—43no—
170C25—43no—
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×Q8—26no—
176C22×Q8—26no—
177C22×Q8—212no—
178C22×Q8—43no—
179C22×Q8—46no—
180C22×D4—26no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—26no—
185C22×D4—26no—
186C22×D4—26no—
187C22×D4—43no—
188C22×D4—43no—
189C23×C4—212no—
190C23×C4—26no—
191C23×C4—26no—
192C23×C4—26no—
193C23×C4—212no—
194C23×C4—26no—
195C23×C4—46no—
196C23×C4—43no—
197C23×C4—46no—
198C23×C4C2×D4112yes—
199C23×C4C2413yes—
200C23×C4C2413yes—
201C23×C4C2413yes—
202C23×C4C2411yescharacteristic
203C23×C4C2413yes—
204C23×C4C2413yes—
205C23×C4C2413yes—
206C23×C4C2413yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2413yes—
210C4⋊Q8—448no—
211#124—448no—
212C22⋊Q8—448no—
213#120—416no—
214C2×C4⋊C4—26no—
215C2×C4⋊C4—26no—
216C2×C4⋊C4—26no—
217C2×C4⋊C4—212no—
218C2×C4⋊C4—212no—
219C2×C4⋊C4—26no—
220C2×C4⋊C4—212no—
221C2×C4⋊C4—26no—
222C2×C4⋊C4—26no—
223C2×C4⋊C4—212no—
224C2×C4⋊C4—212no—
225C2×C4⋊C4—26no—
226C2×C4⋊C4—26no—
227C2×C4⋊C4—212no—
228C2×C4⋊C4—212no—
229C2×C4⋊C4—26no—
230C2×C4⋊C4—26no—
231C2×C4⋊C4—26no—
232C2×C4⋊C4—26no—
233C2×C4⋊C4—212no—
234C2×C4⋊C4—448no—
235C2×C4⋊C4—46no—
236C2×C4⋊C4—46no—
237C2×C4⋊C4—46no—
238C2×C4⋊C4—46no—
239C2×C4⋊C4—46no—
240C2×C4⋊C4C2×D4112yes—
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—212no—
248C2×C22⋊C4—26no—
249C2×C22⋊C4—26no—
250C2×C22⋊C4—26no—
251C2×C22⋊C4—26no—
252C2×C22⋊C4—26no—
253C2×C22⋊C4—26no—
254C2×C22⋊C4—26no—
255C2×C22⋊C4—26no—
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—448no—
270C2×C22⋊C4—43no—
271C2×C22⋊C4—43no—
272C2×C22⋊C4—43no—
273C2×C22⋊C4—43no—
274C2×C22⋊C4—43no—
275C2×C22⋊C4—46no—
276C2×C22⋊C4—46no—
277C2×C22⋊C4—46no—
278C2×C22⋊C4—46no—
279C2×C22⋊C4C2×D4112yes—
280C2×C22⋊C4C2×D4112yes—
281C2×C22⋊C4C2×D4112yes—
282C2×C42—26no—
283C2×C42—26no—
284C2×C42—46no—
285C2×C42C2×D4112yes—
286C2×C42C2×D4112yes—
287#95—26no—
288#95—26no—
289#95—212no—
290#95—212no—
291#95—212no—
292#95—26no—
293#95—26no—
294#95—26no—
295#95—26no—
296#95—26no—
297#95—46no—
298#95C2×D414yes—
299#95C2×D4112yes—
300#95C2×D4112yes—
301C24—21no—
302C24—23no—
303C24—21no—
304C24—23no—
305C24—28no—
306C24—21no—
307C24—23no—
308C24—23no—
309C24—23no—
310C24—23no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—23no—
315C24—23no—
316C24—23no—
317C24—23no—
318C24—21no—
319C24—23no—
320C24—23no—
321C24—21no—
322C24—23no—
323C24—21no—
324C24—23no—
325C24—412no—
326C24—412no—
327C24—412no—
328C24—412no—
329C24—412no—
330C24—412no—
331C24—44no—
332C24—412no—
333C24—86no—
334C24—86no—
335C24—812no—
336C24C22×D412yes—
337C24C22×D416yes—
338C242−1+412yes—
339C242−1+412yes—
340C242−1+416yes—
341C24C2511yescharacteristic
342C2×Q8—412no—
343C2×Q8—424no—
344C2×Q8—412no—
345C2×Q8—86no—
346C2×Q8—812no—
347C2×D4—412no—
348C2×D4—412no—
349C2×D4—44no—
350C2×D4—412no—
351C2×D4—412no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—86no—
355C2×D4—86no—
356C22×C4—21no—
357C22×C4—23no—
358C22×C4—26no—
359C22×C4—21no—
360C22×C4—23no—
361C22×C4—26no—
362C22×C4—224no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—26no—
367C22×C4—26no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—26no—
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—26no—
378C22×C4—23no—
379C22×C4—26no—
380C22×C4—26no—
381C22×C4—224no—
382C22×C4—23no—
383C22×C4—26no—
384C22×C4—23no—
385C22×C4—23no—
386C22×C4—23no—
387C22×C4—26no—
388C22×C4—26no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—26no—
394C22×C4—26no—
395C22×C4—23no—
396C22×C4—21no—
397C22×C4—23no—
398C22×C4—26no—
399C22×C4—23no—
400C22×C4—412no—
401C22×C4—424no—
402C22×C4—424no—
403C22×C4—412no—
404C22×C4—424no—
405C22×C4—412no—
406C22×C4—86no—
407C22×C4—812no—
408C22×C4C22×D416yes—
409C22×C4C22×D416yes—
410C22×C4C22×D416yes—
411C22×C4C22×D416yes—
412C22×C4C22×D416yes—
413C22×C4C22×D416yes—
414C22×C4C22×D416yes—
415C22×C4C22×D416yes—
416C22×C4H(5,𝔽2)16yes—
417C4⋊C4—224no—
418C4⋊C4—424no—
419C4⋊C4—424no—
420C4⋊C4—412no—
421C4⋊C4—412no—
422C4⋊C4—412no—
423C4⋊C4—412no—
424C4⋊C4—412no—
425C4⋊C4—412no—
426C4⋊C4—812no—
427C22⋊C4—224no—
428C22⋊C4—224no—
429C22⋊C4—412no—
430C22⋊C4—412no—
431C22⋊C4—412no—
432C22⋊C4—412no—
433C22⋊C4—412no—
434C22⋊C4—412no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—412no—
438C22⋊C4—412no—
439C22⋊C4—86no—
440C22⋊C4—86no—
441C22⋊C4—86no—
442C22⋊C4—86no—
443C22⋊C4#127116yes—
444C42—224no—
445C42—412no—
446C23—26no—
447C23—26no—
448C23—22no—
449C23—26no—
450C23—26no—
451C23—26no—
452C23—212no—
453C23—26no—
454C23—26no—
455C23—26no—
456C23—22no—
457C23—22no—
458C23—26no—
459C23—26no—
460C23—26no—
461C23—448no—
462C23—41no—
463C23—41no—
464C23—43no—
465C23—43no—
466C23—43no—
467C23—43no—
468C23—43no—
469C23—43no—
470C23—43no—
471C23—43no—
472C23—43no—
473C23—43no—
474C23—43no—
475C23—46no—
476C23—46no—
477C23—43no—
478C23—43no—
479C23—43no—
480C23—41no—
481C23—43no—
482C23—43no—
483C23—43no—
484C23—43no—
485C23—41no—
486C23—43no—
487C23—43no—
488C23—43no—
489C23—46no—
490C23—43no—
491C23—43no—
492C23—824no—
493C23—88no—
494C23—88no—
495C23—824no—
496C23—824no—
497C23—88no—
498C23C2×#12714yes—
499C23#536112yes—
500C23#536112yes—
501C23C23×D413yes—
502C23C23×D411yescharacteristic
503C23C2×H(5,𝔽2)13yes—
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+413yes—
506C23C2×2−1+411yescharacteristic
507C23#58513yes—
508Q8—824no—
509D4—88no—
510D4—88no—
511D4—88no—
512C2×C4—26no—
513C2×C4—26no—
514C2×C4—26no—
515C2×C4—26no—
516C2×C4—212no—
517C2×C4—26no—
518C2×C4—26no—
519C2×C4—212no—
520C2×C4—26no—
521C2×C4—26no—
522C2×C4—26no—
523C2×C4—26no—
524C2×C4—26no—
525C2×C4—26no—
526C2×C4—212no—
527C2×C4—26no—
528C2×C4—26no—
529C2×C4—26no—
530C2×C4—26no—
531C2×C4—448no—
532C2×C4—43no—
533C2×C4—43no—
534C2×C4—43no—
535C2×C4—46no—
536C2×C4—43no—
537C2×C4—46no—
538C2×C4—46no—
539C2×C4—46no—
540C2×C4—43no—
541C2×C4—43no—
542C2×C4—46no—
543C2×C4—41no—
544C2×C4—43no—
545C2×C4—43no—
546C2×C4—43no—
547C2×C4—43no—
548C2×C4—43no—
549C2×C4—43no—
550C2×C4—43no—
551C2×C4—46no—
552C2×C4—46no—
553C2×C4—43no—
554C2×C4—46no—
555C2×C4—43no—
556C2×C4—46no—
557C2×C4—43no—
558C2×C4—43no—
559C2×C4—46no—
560C2×C4—46no—
561C2×C4—824no—
562C2×C4C2×#12714yes—
563C2×C4C2×#12714yes—
564C2×C4D42112yes—
565C22—212no—
566C22—212no—
567C22—24no—
568C22—212no—
569C22—24no—
570C22—212no—
571C22—46no—
572C22—46no—
573C22—46no—
574C22—42no—
575C22—42no—
576C22—46no—
577C22—832no—
578C22—83no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—83no—
583C22—83no—
584C22—81no—
585C22—83no—
586C22—83no—
587C22—83no—
588C22—81no—
589C22—81no—
590C22—83no—
591C22—81no—
592C22—83no—
593C22#265818yescomplemented
594C22C22×#12711yescharacteristic
595C22C2×#53613yes—
596C22C2×#53613yes—
597C22C2×D4213yes—
598C22#348513yes—
599C22#348611yescharacteristic
600C22#348913yes—
601C22#348913yes—
602C22#349813yes—
603C22#353516yes—
604C22#353813yes—
605C22#354113yes—
606C4—212no—
607C4—24no—
608C4—24no—
609C4—46no—
610C4—46no—
611C4—46no—
612C4—46no—
613C4—83no—
614C4—83no—
615C4—81no—
616C4—83no—
617C2—28no—
618C2—82no—
619C2—82no—
620C2—161no—
621C2C2×#265811yescharacteristic
622C2#3606011yescharacteristic
623C2D4×#12711yescharacteristic
624C2#3764713yes—
625C2#3765113yes—
626C2#4137013yes—
627C2#4195313yes—
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×#2440C2semidirect
3. C2×#2658C2semidirect
4. C22×#399C2semidirect
5. #17062C2semidirect
6. #16811C2semidirect
7. #16511C2semidirect
8. #16510C2semidirect
9. #15155C2semidirect
10. #14910C2semidirect
11. #14836C2semidirect
12. #14777C2semidirect
13. #14734C2semidirect
14. #14322C2semidirect
15. #13809C2semidirect
16. C22×C4⋊Q8C22semidirect
32. C2×#399C22semidirect
33. C2×#399C22semidirect
34. C2×#399C22semidirect
35. C2×#398C22semidirect
37. C2×#395C22semidirect
42. C2×#389C22semidirect
43. C2×#389C22semidirect
49. C2×#384C22semidirect
53. C2×#376C22semidirect
55. #1445C22semidirect
56. #1445C22semidirect
57. #1445C22semidirect
58. #1444C22semidirect
59. #1444C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
69. C4×#95C22semidirect
129. #399D4semidirect
593. C22#2658semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#24722
C2×#26582
C22×#3992
#170622
#168112
#165112
#165102
#151552
#149102
#148362
#147772
#147342
#143222
#138092