Finite Group Databasebeta

← #6640638

Subgroups — #6640638

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic
3#32983C212yesmaximal
4#32284C214yesmaximal, complemented
5#32052C214yesmaximal, complemented
6#31936C214yesmaximal, complemented
7#31840C214yesmaximal, complemented
8#29597C212yesmaximal
9#29105C211yesmaximal, characteristic, complemented
10#28304C211yesmaximal, characteristic, complemented
11#28121C212yesmaximal, complemented
12#27961C211yesmaximal, characteristic, complemented
13#27910C211yesmaximal, characteristic, complemented
14#27648C212yesmaximal, complemented
15#21902C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#484C2212yes—
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#451C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2848C2212yes—
25#2646—21no—
26#2626—21no—
27#2589—22no—
28#2587—22no—
29#2502—22no—
30#2502—22no—
31#2502C2214yes—
32#2502C2214yes—
33#2497C2212yes—
34#2492C2212yes—
35#2429C2212yes—
36#2423C2212yes—
37C2×#398—21no—
38C2×#398—21no—
39#2329—21no—
40#2312—21no—
41C4×C22⋊Q8C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#385—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2100C2214yescomplemented
50#2095C2214yescomplemented
51#2095C2212yescomplemented
52#2083C2214yescomplemented
53#2072C2214yescomplemented
54#2072C2212yescomplemented
55#2068C2214yescomplemented
56#2056C2214yescomplemented
57#2036C2214yes—
58#2036C2214yes—
59#2024C2212yescomplemented
60#2016C2212yescomplemented
61#2001C2214yes—
62#2001C2214yes—
63#1990C2212yescomplemented
64#1969C2214yescomplemented
65#1956C2212yescomplemented
66#1947C2214yescomplemented
67#1933C2214yescomplemented
68#1928C2214yescomplemented
69#1923C2214yes—
70#1923C2214yes—
71#1878C2214yescomplemented
72#1872C2214yes—
73#1872C2214yes—
74#1852C2214yes—
75#1852C2214yes—
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1803C2214yescomplemented
82#1797C2212yes—
83#1787C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×Q16C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8C2312yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#484—44no—
119#484—44no—
120#484—44no—
121#451—44no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#399—22no—
141#398—22no—
142#398—22no—
143#398—24no—
144#398—24no—
145#398—24no—
146#398—24no—
147#396—22no—
148#396—22no—
149#393—22no—
150#391—21no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391C2312yes—
156#391C2314yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—24no—
178#385—24no—
179#385—24no—
180#385—24no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384C2314yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#380—21no—
190#380—21no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#379—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4C2312yes—
202C4×C22⋊C4C2312yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C25C2×D411yescharacteristic
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8C2412yes—
225C22×Q8C2412yes—
226C23×C4—24no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—82no—
236C23×C4—82no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C2×Q16—44no—
246C2×Q16—44no—
247C2×Q16—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C4×Q8—44no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
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—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4Q1618yescomplemented
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—44no—
326#107—44no—
327#107—44no—
328C4⋊C8—44no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334#103—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337C22⋊C8—44no—
338C22⋊C8—44no—
339C4×C8—42no—
340C4×C8—42no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—22no—
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#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C24—22no—
362C24—22no—
363C24—41no—
364C24—41no—
365C24—41no—
366C24—41no—
367C24C2×Q1614yes—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×Q8—88no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
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—84no—
428C22×C4—84no—
429C22×C4—816no—
430C22×C4C2×Q1614yes—
431C22×C4C2×Q1614yes—
432C22×C4#13714yes—
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×C4H(5,𝔽2)14yes—
442C22×C4C2511yescharacteristic
443Q16—88no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—82no—
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—82no—
489C22⋊C4—82no—
490C22⋊C4C2×Q1618yes—
491C22⋊C4C2×Q1618yes—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—42no—
496C42—41no—
497C42—41no—
498C42—82no—
499C42—82no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23#53512yes—
518C23C22×Q1614yes—
519C23C22×Q1612yes—
520C23C2×#13712yes—
521C23C23×D411yescharacteristic
522Q8—44no—
523Q8—44no—
524Q8—44no—
525Q8—44no—
526Q8—84no—
527Q8—162no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
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—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—816no—
583C2×C4#53512yes—
584C2×C4C22×Q1614yes—
585C2×C4C22×Q1612yes—
586C2×C4C22×Q1612yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4#58512yes—
591C2×C4#58512yes—
592C8—44no—
593C8—44no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—416no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22#320914yes—
609C22C2×#53511yescharacteristic
610C22#348511yescharacteristic
611C22#348511yescharacteristic
612C22C23×Q1611yescharacteristic
613C22C22×#13711yescharacteristic
614C22#358511yescharacteristic
615C22#358511yescharacteristic
616C4—24no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4#353314yes—
633C2—24no—
634C2—44no—
635C2#3695611yescharacteristic
636C2C2×#320911yescharacteristic
637C2#6080611yescharacteristic
638C2#6169811yescharacteristic
639C2#6169811yescharacteristic
640C2#6170911yescharacteristic
641C2#6171911yescharacteristic
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
4. #32284C2semidirect
5. #32052C2semidirect
6. #31936C2semidirect
7. #31840C2semidirect
9. #29105C2semidirect
10. #28304C2semidirect
11. #28121C2semidirect
12. #27961C2semidirect
13. #27910C2semidirect
14. #27648C2semidirect
49. #2100C22semidirect
50. #2095C22semidirect
51. #2095C22semidirect
52. #2083C22semidirect
53. #2072C22semidirect
54. #2072C22semidirect
55. #2068C22semidirect
56. #2056C22semidirect
59. #2024C22semidirect
60. #2016C22semidirect
63. #1990C22semidirect
64. #1969C22semidirect
65. #1956C22semidirect
66. #1947C22semidirect
67. #1933C22semidirect
68. #1928C22semidirect
71. #1878C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
81. #1803C22semidirect
310. 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×#25022
#329832
#322842
#320522
#319362
#318402
#295972
#291052
#283042
#281212
#279612
#279102
#276482
#219022