Finite Group Databasebeta

← #6475926

Subgroups — #6475926

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35310C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic
3#32984C212yesmaximal
4#32882C212yesmaximal
5#29963C212yesmaximal, complemented
6#29724C212yesmaximal, complemented
7#29112C211yesmaximal, characteristic, complemented
8#29106C211yesmaximal, characteristic, complemented
9#28313C211yesmaximal, characteristic, complemented
10#28304C211yesmaximal, characteristic, complemented
11#28124C212yesmaximal, complemented
12#28121C212yesmaximal, complemented
13#27596C212yesmaximal, complemented
14#24514C214yesmaximal, complemented
15#23943C212yesmaximal, complemented
16#22054C214yesmaximal, complemented
17#20716C211yesmaximal, characteristic
18C24×Q8C2211yescharacteristic
19C2×#541C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C2×#484C2212yes—
23C2×#483C2212yes—
24C2×#451C2214yes—
25C2×#451C2214yes—
26C2×#451C2212yes—
27C2×#451C2212yes—
28C2×#433C2212yes—
29#2857C2211yescharacteristic
30#2851C2211yescharacteristic
31#2848C2212yes—
32#2629C2211yescharacteristic
33#2625C2212yes—
34#2621C2211yescharacteristic
35#2480—22no—
36#2478—22no—
37#2475—22no—
38#2473—22no—
39#2323C2212yes—
40#2314C2211yescharacteristic
41#2310—22no—
42#2310—22no—
43#2310C2212yes—
44#2310C2212yes—
45#2310C2212yes—
46#2310C2212yes—
47C4×C22⋊Q8C2212yes—
48C2×#391—22no—
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#381—22no—
54#2096C2212yescomplemented
55#2095C2212yescomplemented
56#2074C2212yescomplemented
57#2074C2212yescomplemented
58#2068C2214yescomplemented
59#2066C2214yescomplemented
60#2056C2214yescomplemented
61#2056C2214yescomplemented
62#2026C2212yescomplemented
63#2026C2212yescomplemented
64#2016C2212yescomplemented
65#2016C2212yescomplemented
66#1971C2214yescomplemented
67#1969C2214yescomplemented
68#1935C2214yescomplemented
69#1933C2214yescomplemented
70#1928C2214yescomplemented
71#1927C2214yescomplemented
72#1878C2214yescomplemented
73#1878C2214yescomplemented
74#1873C2214yes—
75#1873C2214yes—
76#1854C2214yes—
77#1854C2214yes—
78#1824C2212yes—
79#1823C2212yes—
80#1818C2212yes—
81#1798C2212yes—
82#1797C2212yes—
83#1787C2214yes—
84#1787C2212yes—
85#1787C2212yes—
86#1787C2214yes—
87#1779C2212yes—
88#1750C2212yes—
89#1749C2211yescharacteristic
90C4×C22⋊C8C2211yescharacteristic
91C23×Q8—28no—
92C23×Q8—21no—
93C23×Q8—21no—
94C23×Q8—21no—
95C23×Q8—44no—
96C23×Q8—44no—
97C23×Q8C2312yes—
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C23×Q8C2312yes—
101C24×C4—21no—
102C24×C4C2311yescharacteristic
103C22×Q16C2314yes—
104C22×Q16C2314yes—
105#541—44no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—22no—
108C2×#125—22no—
109C2×#125C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×#117—21no—
115C2×#117—21no—
116C2×#117—44no—
117C2×#117—44no—
118C2×#117D412yes—
119C2×#117D412yes—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4D412yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4D412yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#484—44no—
135#483—44no—
136#451—216no—
137#451—48no—
138#451—44no—
139#451—44no—
140#433—44no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×#97C2312yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#397—22no—
163#397—22no—
164#397—22no—
165#396—22no—
166#391—24no—
167#391—24no—
168#391—24no—
169#391—24no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384C2311yescharacteristic
201#382C2311yescharacteristic
202#381—24no—
203#381—24no—
204#381—24no—
205#381—24no—
206#380—22no—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210#379—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—22no—
214C2×#95—22no—
215C2×#95—22no—
216C2×#95—22no—
217C2×#95—22no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2314yes—
223#336C2312yes—
224#336C2312yes—
225C25C2×D411yescharacteristic
226C22×Q8—28no—
227C22×Q8—28no—
228C22×Q8—28no—
229C22×Q8—24no—
230C22×Q8—24no—
231C22×Q8—24no—
232C22×Q8—24no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—24no—
238C22×Q8—28no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—464no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8—44no—
253C22×Q8#4118yes—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×Q8C2412yes—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—28no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—44no—
268C23×C4—44no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D412yes—
273C23×C4C2×D412yes—
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C2×Q16—28no—
278C2×Q16—28no—
279C2×Q16—44no—
280C2×Q16—44no—
281C2×Q16—44no—
282C2×Q16—44no—
283C22×C8C2414yes—
284C22×C8C2414yes—
285C4⋊Q8—42no—
286#126—48no—
287#125—42no—
288#125—48no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C4×Q8—44no—
294#117—24no—
295#117—24no—
296#117—42no—
297#117—42no—
298#117—44no—
299#117—44no—
300#117—44no—
301#117—44no—
302#117—44no—
303#117—44no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—44no—
355#106—44no—
356C4⋊C8—44no—
357#103—28no—
358#103—28no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365C22⋊C8—28no—
366C22⋊C8—28no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#97—42no—
370C4×C8—42no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—24no—
394C24—42no—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C2×Q8—28no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—416no—
413C2×Q8—416no—
414C2×Q8—416no—
415C2×Q8—416no—
416C2×Q8—416no—
417C2×Q8—432no—
418C2×Q8—432no—
419C2×Q8—432no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—416no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8C2×#4114yes—
430C2×Q8C2×#4114yes—
431C22×C4—232no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—432no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4114yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4C2511yescharacteristic
489Q16—88no—
490Q16—88no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42—82no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—22no—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—48no—
553C23—48no—
554C23—44no—
555C23—44no—
556C23#52512yes—
557C23#53412yes—
558C23#53512yes—
559C23#57814yes—
560C23C23×D411yescharacteristic
561Q8—28no—
562Q8—28no—
563Q8—416no—
564Q8—416no—
565Q8—48no—
566Q8—48no—
567Q8—44no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4#52512yes—
626C2×C4#53412yes—
627C2×C4#53512yes—
628C2×C4C22×#4112yes—
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4C2×H(5,𝔽2)12yes—
631C8—44no—
632C8—44no—
633C22—24no—
634C22—22no—
635C22—24no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—416no—
642C22—42no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—42no—
648C22#322314yes—
649C22C2×#52511yescharacteristic
650C22C2×#53411yescharacteristic
651C22C2×#53511yescharacteristic
652C22C2×#57812yes—
653C22#358612yes—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#349014yes—
670C2—24no—
671C2—42no—
672C2—42no—
673C2#3505011yescharacteristic
674C2#6058312yes—
675C2#6075912yes—
676C2C2×#322311yescharacteristic
677C2#6080711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29963C2semidirect
6. #29724C2semidirect
7. #29112C2semidirect
8. #29106C2semidirect
9. #28313C2semidirect
10. #28304C2semidirect
11. #28124C2semidirect
12. #28121C2semidirect
13. #27596C2semidirect
14. #24514C2semidirect
15. #23943C2semidirect
16. #22054C2semidirect
54. #2096C22semidirect
55. #2095C22semidirect
56. #2074C22semidirect
57. #2074C22semidirect
58. #2068C22semidirect
59. #2066C22semidirect
60. #2056C22semidirect
61. #2056C22semidirect
62. #2026C22semidirect
63. #2026C22semidirect
64. #2016C22semidirect
65. #2016C22semidirect
66. #1971C22semidirect
67. #1969C22semidirect
68. #1935C22semidirect
69. #1933C22semidirect
70. #1928C22semidirect
71. #1927C22semidirect
72. #1878C22semidirect
73. #1878C22semidirect

Maximal subgroups

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

subgroupindex
#353102
C2×#23102
#329842
#328822
#299632
#297242
#291122
#291062
#283132
#283042
#281242
#281212
#275962
#245142
#239432
#220542
#207162