Finite Group Databasebeta

← #6508750

Subgroups — #6508750

Order 512.

2129 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3188C211yesmaximal, characteristic
2C22×#481C211yesmaximal, characteristic, complemented
3#35227C211yesmaximal, characteristic, complemented
4#34379C211yesmaximal, characteristic, complemented
5#28127C214yesmaximal, complemented
6#27654C214yesmaximal, complemented
7#25214C214yesmaximal, complemented
8#25202C214yesmaximal, complemented
9#24579C212yesmaximal, complemented
10#24578C212yesmaximal, complemented
11#24527C212yesmaximal, complemented
12#24526C212yesmaximal, complemented
13#21931C211yesmaximal, characteristic, complemented
14#21763C211yesmaximal, characteristic, complemented
15C2×#1823C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#3188—28no—
20C2×#484C2211yescharacteristic
21C2×#481—28no—
22C2×#481C2211yescharacteristic
23C2×#481C2214yes—
24C2×#481C2214yes—
25C2×#482C2212yes—
26C2×#466C2212yes—
27C2×#465C2212yes—
28C2×#449C2212yes—
29C2×#450C2211yescharacteristic
30C2×#447C2211yescharacteristic
31C22×#106C2212yes—
32C22×#102C2212yes—
33C22×C22⋊C8C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2849—22no—
36#2623—22no—
37#2504C2212yescomplemented
38#2503C2212yes—
39#2502C2212yescomplemented
40#2501C2212yes—
41#2487—22no—
42#2486—21no—
43#2486—21no—
44#2377—22no—
45#2361C2214yes—
46#2345C2212yes—
47D4×C22⋊C4—22no—
48D4×C22⋊C4C2212yes—
49#2325—22no—
50#2324C2214yes—
51#2317C2212yescomplemented
52#2315—21no—
53#2312—21no—
54#2311C2212yescomplemented
55#2280C2211yescharacteristic
56C2×#380C2211yescharacteristic
57#2085C2214yescomplemented
58#2084C2214yescomplemented
59#2073C2214yescomplemented
60#2071C2214yescomplemented
61#2066C2214yescomplemented
62#2065C2214yescomplemented
63#2055C2214yescomplemented
64#2054C2214yescomplemented
65#1949C2214yescomplemented
66#1948C2214yescomplemented
67#1927C2214yescomplemented
68#1926C2214yescomplemented
69#1893C2212yes—
70#1892C2212yes—
71#1879C2214yescomplemented
72#1877C2214yescomplemented
73#1874C2212yescomplemented
74#1872C2212yescomplemented
75#1850C2212yes—
76#1850C2212yes—
77#1850C2214yescomplemented
78#1825C2212yescomplemented
79#1823C2212yes—
80#1823C2212yes—
81#1823C2214yescomplemented
82#1804C2214yescomplemented
83#1802C2214yescomplemented
84#1798C2214yescomplemented
85#1788C2212yescomplemented
86C2×#340C2214yescomplemented
87C22×#41C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×SD16C2312yes—
93C22×D8C2312yes—
94C23×C8C2312yes—
95C23×C8C2312yes—
96#525—24no—
97#525—24no—
98C2×#123—21no—
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8C2311yescharacteristic
102C2×C22⋊Q8C2311yescharacteristic
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—24no—
106C2×#121D418yes—
107C2×#121C2311yescharacteristic
108C2×#121C2311yescharacteristic
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4C2314yes—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131#484—24no—
132#482—28no—
133#481—216no—
134#481—216no—
135#481—24no—
136#466—28no—
137#465—28no—
138#450—24no—
139#449—28no—
140#447—24no—
141C2×#107C2312yes—
142C2×#106—28no—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#102—28no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×C22⋊C8—24no—
158C2×C22⋊C8—24no—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2311yescharacteristic
163C2×C22⋊C8C2311yescharacteristic
164#400—24no—
165#399—24no—
166#398—22no—
167#398—22no—
168#397—22no—
169#397—22no—
170#391C2314yes—
171#390—24no—
172#390C2314yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#388—24no—
178#388—24no—
179#387—24no—
180#386—24no—
181#386—24no—
182#386—24no—
183#386—24no—
184#386—24no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—24no—
189#384C2314yes—
190#382C2314yes—
191#381—22no—
192#381—22no—
193#381D412yes—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380D411yescharacteristic
200#380D411yescharacteristic
201#380D412yes—
202#380C2312yes—
203#380C2312yes—
204#379—22no—
205#379D412yes—
206C4×C4⋊C4C2314yes—
207C4×C22⋊C4—22no—
208C4×C22⋊C4C2312yes—
209C4×C22⋊C4C2312yes—
210C2×#95—21no—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25—21no—
220C25C2×D411yescharacteristic
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—44no—
224C22×Q8C2411yescharacteristic
225C22×D4—22no—
226C22×D4—28no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—44no—
234C22×D4—84no—
235C22×D4C2×D414yes—
236C22×D4C2411yescharacteristic
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D414yes—
253C23×C4#4114yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×SD16—24no—
258C2×SD16—44no—
259C2×D8—24no—
260C2×D8—44no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—44no—
264C22×C8—44no—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268#123—44no—
269#123—84no—
270#123—84no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275#121—28no—
276#121—28no—
277#121—24no—
278#121—24no—
279#121—22no—
280#121—22no—
281#121—22no—
282#121—22no—
283#121C2×D418yes—
284#121C2×D418yes—
285#120—44no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—88no—
291#117—22no—
292#117—22no—
293#117—44no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—84no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D414yes—
344C2×C42—21no—
345C2×C42—24no—
346C2×C42C2414yes—
347#107—24no—
348#106—28no—
349#106—28no—
350#106—24no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353#103—24no—
354#103—44no—
355#102—28no—
356#102—28no—
357#102—24no—
358#102—24no—
359#102—24no—
360#102—44no—
361C22⋊C8—22no—
362C22⋊C8—24no—
363C22⋊C8—22no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95C2×D412yes—
375#95C2×D414yes—
376#95C2×D412yes—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—24no—
382C24—21no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—82no—
387C24—82no—
388C24#12114yes—
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391C24C22×D411yescharacteristic
392#41—84no—
393#41—84no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—24no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—84no—
416C2×D4—82no—
417C2×D4—84no—
418C2×D4—816no—
419C2×D4C22×D414yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—816no—
454C22×C4#12114yes—
455C22×C4#12114yes—
456C22×C4#12114yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2511yescharacteristic
469SD16—84no—
470D8—84no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—48no—
479C2×C8—84no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4C22×D414yes—
502C22⋊C4—24no—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—82no—
520C42—24no—
521C42—24no—
522C42—42no—
523C42—84no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—24no—
536C23—44no—
537C23—48no—
538C23—48no—
539C23—44no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—48no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—84no—
550C23—84no—
551C23—164no—
552C23—164no—
553C23C2×#12112yes—
554C23C2×#12112yes—
555C23C2×#12112yes—
556C23#57612yes—
557C23#57714yes—
558C23#57712yes—
559C23C23×D411yescharacteristic
560Q8—42no—
561Q8—42no—
562D4—24no—
563D4—24no—
564D4—416no—
565D4—42no—
566D4—44no—
567D4—42no—
568D4—84no—
569D4—88no—
570D4—84no—
571D4—84no—
572D4—84no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4—88no—
611C2×C4—168no—
612C2×C4—164no—
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4C2×#12112yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4#58514yes—
619C8—44no—
620C8—42no—
621C8—42no—
622C22—22no—
623C22—22no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—416no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—84no—
640C22—84no—
641C22—81no—
642C22—81no—
643C22—82no—
644C22—82no—
645C22—81no—
646C22—168no—
647C22#304214yes—
648C22#304514yes—
649C22C22×#12111yescharacteristic
650C22#348512yes—
651C22C2×#57611yescharacteristic
652C22C2×#57712yes—
653C22C2×#57711yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—81no—
661C4—84no—
662C4—81no—
663C4—84no—
664C4—81no—
665C4#348512yes—
666C4#349112yes—
667C2—24no—
668C2—24no—
669C2—42no—
670C2—84no—
671C2—161no—
672C2#3544711yescharacteristic
673C2C2×#304211yescharacteristic
674C2C2×#304511yescharacteristic
675C2#6176512yes—
676C2#6176612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3188C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#481C2semidirect
3. #35227C2semidirect
4. #34379C2semidirect
5. #28127C2semidirect
6. #27654C2semidirect
7. #25214C2semidirect
8. #25202C2semidirect
9. #24579C2semidirect
10. #24578C2semidirect
11. #24527C2semidirect
12. #24526C2semidirect
13. #21931C2semidirect
14. #21763C2semidirect
15. C2×#1823C2semidirect
37. #2504C22semidirect
39. #2502C22semidirect
51. #2317C22semidirect
54. #2311C22semidirect
57. #2085C22semidirect
58. #2084C22semidirect
59. #2073C22semidirect
60. #2071C22semidirect
61. #2066C22semidirect
62. #2065C22semidirect
63. #2055C22semidirect
64. #2054C22semidirect
65. #1949C22semidirect
66. #1948C22semidirect
67. #1927C22semidirect
68. #1926C22semidirect
71. #1879C22semidirect
72. #1877C22semidirect
73. #1874C22semidirect
74. #1872C22semidirect
77. #1850C22semidirect
78. #1825C22semidirect
81. #1823C22semidirect
82. #1804C22semidirect
83. #1802C22semidirect
84. #1798C22semidirect
85. #1788C22semidirect
86. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31882
C22×#4812
#352272
#343792
#281272
#276542
#252142
#252022
#245792
#245782
#245272
#245262
#219312
#217632
C2×#18232