Finite Group Databasebeta

← #6473106

Subgroups — #6473106

Order 512.

2172 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3200C212yesmaximal
2#35316C211yesmaximal, characteristic, complemented
3#34269C211yesmaximal, characteristic
4#33117C212yesmaximal, complemented
5#32914C212yesmaximal, complemented
6#29759C212yesmaximal, complemented
7#29714C212yesmaximal, complemented
8#28577C212yesmaximal, complemented
9#28571C212yesmaximal, complemented
10#28128C214yesmaximal, complemented
11#27650C212yesmaximal, complemented
12#24766C214yesmaximal, complemented
13#24209C212yesmaximal, complemented
14#21771C212yesmaximal, complemented
15#20630C211yesmaximal, characteristic, complemented
16C2×#536C2212yes—
17C2×#535C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic, complemented
19C2×#518C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3200—216no—
22C2×#485C2214yes—
23C2×#482C2214yes—
24C2×#468C2214yes—
25C2×#465C2214yes—
26C2×#461C2214yes—
27C2×#448C2214yes—
28C2×#450C2214yes—
29C2×#424C2212yescomplemented
30C2×#408C2212yes—
31#2857C2212yes—
32#2857C2212yescomplemented
33#2621C2212yes—
34#2621C2212yescomplemented
35#2547—22no—
36#2546—22no—
37#2486—22no—
38#2486—22no—
39#2367—22no—
40#2366—22no—
41#2358—22no—
42#2358—22no—
43#2324C2212yes—
44#2324C2212yes—
45#2311C2211yescharacteristic
46#2310C2212yescomplemented
47C4×#121C2212yes—
48C4×#121C2212yes—
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51#2285C2211yescharacteristic, complemented
52#2274C2211yescharacteristic
53#2084C2212yescomplemented
54#2083C2212yescomplemented
55#2071C2212yes—
56#2071C2212yes—
57#2068C2214yescomplemented
58#2066C2214yescomplemented
59#2055C2214yes—
60#2055C2214yes—
61#2043C2212yescomplemented
62#2043C2212yescomplemented
63#2013C2212yes—
64#2013C2212yes—
65#1981C2214yescomplemented
66#1949C2214yescomplemented
67#1928C2214yescomplemented
68#1927C2214yescomplemented
69#1894C2214yescomplemented
70#1892C2214yescomplemented
71#1877C2214yes—
72#1877C2214yes—
73#1873C2214yes—
74#1872C2214yes—
75#1825C2214yescomplemented
76#1804C2214yes—
77#1798C2212yes—
78#1798C2212yescomplemented
79#1797C2212yes—
80#1797C2212yescomplemented
81#1787C2214yescomplemented
82#1728C2211yescharacteristic, complemented
83C2×#336C2211yescharacteristic, complemented
84C22×#41C2311yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C22×SD16C2314yes—
88C22×SD16C2314yes—
89C22×M16C2312yes—
90C23×C8C2312yes—
91#536—28no—
92#536—28no—
93#535—24no—
94#535D418yes—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—44no—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100C2×#127—21no—
101C2×#124—21no—
102C2×#124—22no—
103C2×#123—22no—
104C2×#123—22no—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2314yes—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C2×#120—21no—
110C2×#120—21no—
111#518—44no—
112C4×#41—44no—
113C2×C4×Q8D412yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4D412yes—
118C2×C4×D4D412yes—
119C2×C4×D4D412yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#485—216no—
136#482—216no—
137#468—216no—
138#465—216no—
139#461—48no—
140#450—216no—
141#448—216no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144#424—44no—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155#408—28no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#398—24no—
159#398—24no—
160#397—24no—
161#397—24no—
162#391—21no—
163#391—24no—
164#391C2314yes—
165#391C2314yes—
166#390—21no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#386—21no—
183#386—21no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384C2312yes—
192#382C2312yes—
193#381—24no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217C25—21no—
218C2×#41—42no—
219C2×#41—42no—
220C2×#41C2×D414yes—
221C2×#41#4114yes—
222C22×Q8—22no—
223C22×Q8—24no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8C2×D411yescharacteristic
230C22×Q8C2414yes—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4C2×D414yes—
238C22×D4C2×D411yescharacteristic
239C22×D4C2×D411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×M16—24no—
259C2×M16—42no—
260C2×M16—42no—
261C22×C8—24no—
262C22×C8—42no—
263C22×C8—42no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—28no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270#127—28no—
271#124—28no—
272#124—48no—
273#123—48no—
274#123—48no—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—28no—
279#121—24no—
280#121—24no—
281#121C2×D418yes—
282#121C2×D418yes—
283#120—28no—
284#120—28no—
285C4×Q8—44no—
286C4×D4—28no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—84no—
293C4×D4—84no—
294#117—22no—
295#117—24no—
296#117—24no—
297#117—24no—
298#117—24no—
299#117—42no—
300#117—42no—
301#117C2×D414yes—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
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—22no—
336C2×C22⋊C4—48no—
337C2×C22⋊C4—84no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—28no—
357#106—28no—
358C4⋊C8—28no—
359C4⋊C8—44no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—28no—
367#102—28no—
368C22⋊C8—28no—
369C22⋊C8—28no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381C24—22no—
382C24—24no—
383C24—42no—
384C24—42no—
385C24—82no—
386C24—82no—
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390#41—48no—
391#41—48no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—84no—
404C2×Q8—84no—
405C2×Q8—84no—
406C2×Q8—84no—
407C2×Q8#12114yes—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—48no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—48no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4#12114yes—
429C2×D4#12114yes—
430C2×D4#12114yes—
431C2×D4C22×D414yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477SD16—88no—
478SD16—88no—
479M16—84no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—84no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—82no—
505C4⋊C4C22×D414yes—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—82no—
534C42—82no—
535C42H(5,𝔽2)14yes—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—48no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—88no—
557C23—162no—
558C23—164no—
559C23—162no—
560C23C2×#12112yes—
561C23C2×#12112yes—
562C23C2×#12112yes—
563C23C23×D411yescharacteristic
564Q8—44no—
565Q8—44no—
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—88no—
570D4—28no—
571D4—44no—
572D4—44no—
573D4—44no—
574D4—48no—
575D4—44no—
576D4—88no—
577D4—88no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—164no—
620C2×C4—164no—
621C2×C4—162no—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4C2×#12112yes—
625C2×C4#53414yes—
626C2×C4#57714yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4#58512yes—
631C8—44no—
632C8—44no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22C22×#12111yescharacteristic
657C22C2×#53411yescharacteristic
658C22#344911yescharacteristic
659C22C2×#57712yes—
660C22#358612yes—
661C4—24no—
662C4—24no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4#349612yes—
674C4#350512yes—
675C2—42no—
676C2—42no—
677C2—42no—
678C2—84no—
679C2#3500011yescharacteristic
680C2#6042912yes—
681C2#6075812yes—
682C2#6101211yescharacteristic
683C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35316C2semidirect
4. #33117C2semidirect
5. #32914C2semidirect
6. #29759C2semidirect
7. #29714C2semidirect
8. #28577C2semidirect
9. #28571C2semidirect
10. #28128C2semidirect
11. #27650C2semidirect
12. #24766C2semidirect
13. #24209C2semidirect
14. #21771C2semidirect
15. #20630C2semidirect
18. C22×C4⋊Q8C22semidirect
29. C2×#424C22semidirect
32. #2857C22semidirect
34. #2621C22semidirect
46. #2310C22semidirect
51. #2285C22semidirect
53. #2084C22semidirect
54. #2083C22semidirect
57. #2068C22semidirect
58. #2066C22semidirect
61. #2043C22semidirect
62. #2043C22semidirect
65. #1981C22semidirect
66. #1949C22semidirect
67. #1928C22semidirect
68. #1927C22semidirect
69. #1894C22semidirect
70. #1892C22semidirect
75. #1825C22semidirect
78. #1798C22semidirect
80. #1797C22semidirect
81. #1787C22semidirect
82. #1728C22semidirect
83. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32002
#353162
#342692
#331172
#329142
#297592
#297142
#285772
#285712
#281282
#276502
#247662
#242092
#217712
#206302