Finite Group Databasebeta

← #6507894

Subgroups — #6507894

Order 512.

1700 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
1#35014C211yesmaximal, characteristic
2C2×#2423C211yesmaximal, characteristic
3#29105C211yesmaximal, characteristic, complemented
4#28977C214yesmaximal, complemented
5#28843C214yesmaximal, complemented
6#28304C211yesmaximal, characteristic, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27950C211yesmaximal, characteristic, complemented
9#26965C214yesmaximal, complemented
10#25035C214yesmaximal, complemented
11#24552C212yesmaximal
12#24380C212yesmaximal
13#22079C212yesmaximal, complemented
14C4×#451C212yesmaximal, complemented
15#20523C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#484C2212yes—
19C2×#467C2214yes—
20C2×#451C2212yes—
21C2×C4×Q16C2214yescomplemented
22C22×C22⋊C8C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2848C2212yes—
26#2646—21no—
27#2626—21no—
28#2589—22no—
29#2587—22no—
30#2497C2212yes—
31#2492C2212yes—
32#2429C2212yes—
33#2423—22no—
34#2423—22no—
35#2423C2214yes—
36#2423C2214yes—
37#2423C2212yes—
38C2×#399—21no—
39C2×#398—21no—
40#2329—21no—
41#2312—21no—
42C4×C22⋊Q8C2212yes—
43C2×#391C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#387—21no—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2095C2212yescomplemented
51#2083C2214yescomplemented
52#2076C2214yes—
53#2072C2214yescomplemented
54#2072C2212yescomplemented
55#2036C2214yes—
56#2036C2214yes—
57#2024C2212yescomplemented
58#2016C2212yescomplemented
59#2001C2214yes—
60#1990C2212yescomplemented
61#1969C2214yescomplemented
62#1956C2212yescomplemented
63#1942C2214yescomplemented
64#1933C2214yescomplemented
65#1933C2214yescomplemented
66#1921C2214yescomplemented
67#1892C2214yes—
68#1892C2214yes—
69#1878C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1863C2214yescomplemented
73#1863C2214yescomplemented
74#1852C2212yes—
75#1852C2212yes—
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1808C2214yes—
80#1787C2212yes—
81C4×#103C2214yescomplemented
82C8×C22⋊C4C2214yes—
83C8×C22⋊C4C2214yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2212yes—
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×Q16C2314yes—
90C23×C8C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—44no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2312yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#484—28no—
120#467—48no—
121#451—28no—
122C4×Q16—48no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×C22⋊C8—28no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2314yes—
139#400—22no—
140#400—22no—
141#399—22no—
142#399—24no—
143#399—24no—
144#398—22no—
145#398—22no—
146#398—24no—
147#398—24no—
148#396—22no—
149#396—22no—
150#393—22no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2314yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—24no—
172#388—24no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#380—21no—
189#380—21no—
190#380—21no—
191#380—22no—
192#380C2312yes—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#379—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4C2312yes—
203C4×C22⋊C4C2312yes—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C25C2×D411yescharacteristic
220C22×Q8—24no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8C2412yes—
226C22×Q8C2412yes—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—44no—
245C22×C8—28no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C22⋊Q8—28no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—44no—
256C22⋊Q8—48no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C4×Q8—28no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—48no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4Q1618yescomplemented
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—28no—
330#107—44no—
331#107—44no—
332#103—28no—
333#103—28no—
334#103—28no—
335#103—44no—
336#103—44no—
337#103—44no—
338C22⋊C8—24no—
339C22⋊C8—28no—
340C22⋊C8#4118yes—
341C4×C8—42no—
342C4×C8—42no—
343C4×C8—44no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364C24—22no—
365C24—22no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24C2×Q1614yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—416no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8H(5,𝔽2)14yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—88no—
430C22×C4C2×Q1614yes—
431C22×C4C2×Q1614yes—
432C22×C4#13514yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4114yes—
442C22×C4C2511yescharacteristic
443Q16—416no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—88no—
450C2×C8C2×#4118yes—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—84no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4C2×Q1618yes—
492C22⋊C4C2×Q1618yes—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—42no—
499C42—48no—
500C42—82no—
501C42—82no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—44no—
514C23—44no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23#53212yes—
520C23C22×Q1614yes—
521C23C22×Q1612yes—
522C23C2×#13512yes—
523C23C23×D411yescharacteristic
524Q8—28no—
525Q8—48no—
526Q8—44no—
527Q8—44no—
528Q8—44no—
529Q8—162no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—84no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#53212yes—
586C2×C4C22×Q1612yes—
587C2×C4C22×Q1612yes—
588C2×C4C22×Q1614yes—
589C2×C4C2×#13512yes—
590C2×C4C2×#13512yes—
591C2×C4C22×#4112yes—
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4#58512yes—
594C8—28no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—88no—
610C22#317014yes—
611C22C2×#53211yescharacteristic
612C22#348411yescharacteristic
613C22#348511yescharacteristic
614C22C23×Q1611yescharacteristic
615C22C22×#13511yescharacteristic
616C22#358511yescharacteristic
617C22#358611yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—24no—
624C4—42no—
625C4—44no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4#351314yes—
635C2—24no—
636C2—44no—
637C2#3539411yescharacteristic
638C2C2×#317011yescharacteristic
639C2#6070311yescharacteristic
640C2#6156311yescharacteristic
641C2#6156611yescharacteristic
642C2#6169611yescharacteristic
643C2#6169811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29105C2semidirect
4. #28977C2semidirect
5. #28843C2semidirect
6. #28304C2semidirect
7. #27972C2semidirect
8. #27950C2semidirect
9. #26965C2semidirect
10. #25035C2semidirect
13. #22079C2semidirect
14. C4×#451C2semidirect
21. C2×C4×Q16C22semidirect
50. #2095C22semidirect
51. #2083C22semidirect
53. #2072C22semidirect
54. #2072C22semidirect
57. #2024C22semidirect
58. #2016C22semidirect
60. #1990C22semidirect
61. #1969C22semidirect
62. #1956C22semidirect
63. #1942C22semidirect
64. #1933C22semidirect
65. #1933C22semidirect
66. #1921C22semidirect
69. #1878C22semidirect
72. #1863C22semidirect
73. #1863C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
81. C4×#103C22semidirect
312. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#24232
#291052
#289772
#288432
#283042
#279722
#279502
#269652
#250352
#245522
#243802
#220792
C4×#4512
#205232