Finite Group Databasebeta

← #6508741

Subgroups — #6508741

Order 512.

1988 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C22×#485C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic
4#34380C211yesmaximal, characteristic, complemented
5#28130C214yesmaximal, complemented
6#27656C214yesmaximal, complemented
7#25224C214yesmaximal, complemented
8#25222C214yesmaximal, complemented
9#24581C212yesmaximal, complemented
10#24576C212yesmaximal, complemented
11#24533C212yesmaximal, complemented
12#24528C212yesmaximal, complemented
13#21933C211yesmaximal, characteristic, complemented
14#21758C211yesmaximal, characteristic
15#20702C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3193—28no—
20C2×#484C2212yes—
21C2×#485—28no—
22C2×#485C2214yes—
23C2×#485C2214yescomplemented
24C2×#480C2212yes—
25C2×#468C2212yes—
26C2×#468C2212yes—
27C2×#448C2212yes—
28C2×#450C2212yes—
29C22×#107C2211yescharacteristic, complemented
30C22×#106C2211yescharacteristic, complemented
31C22×#103C2211yescharacteristic, complemented
32C22×#103C2211yescharacteristic, complemented
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2854—21no—
36#2854—21no—
37#2624—22no—
38#2504C2212yes—
39#2501C2212yes—
40#2500C2214yes—
41#2482—22no—
42#2405—22no—
43#2378—22no—
44#2360C2214yescomplemented
45#2344C2212yescomplemented
46#2342—22no—
47Q8×C22⋊C4C2212yescomplemented
48#2329—22no—
49#2323C2214yes—
50#2317C2212yes—
51#2316—21no—
52#2313—21no—
53#2309C2212yes—
54#2280C2211yescharacteristic
55C2×#380C2211yescharacteristic
56#2085C2214yescomplemented
57#2083C2214yescomplemented
58#2074C2214yescomplemented
59#2072C2214yescomplemented
60#2067C2214yescomplemented
61#2065C2214yescomplemented
62#2055C2214yescomplemented
63#2055C2214yescomplemented
64#1949C2214yescomplemented
65#1947C2214yescomplemented
66#1926C2214yescomplemented
67#1925C2214yescomplemented
68#1893C2212yes—
69#1893C2212yes—
70#1880C2214yescomplemented
71#1880C2214yescomplemented
72#1874C2214yes—
73#1851C2212yescomplemented
74#1851C2212yescomplemented
75#1850C2212yescomplemented
76#1850C2212yescomplemented
77#1825C2214yes—
78#1824C2212yescomplemented
79#1823C2212yes—
80#1823C2212yescomplemented
81#1805C2214yescomplemented
82#1803C2214yescomplemented
83#1798C2214yes—
84#1786C2212yes—
85#1736C2212yescomplemented
86#1736C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×SD16C2312yes—
92C22×SD16C2312yes—
93C22×M16C2311yescharacteristic
94C22×M16C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8D418yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×C4×Q8C2314yes—
112C2×#117—21no—
113C2×#117—21no—
114C2×#117—22no—
115C2×#117—24no—
116C2×#117—24no—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42—21no—
131#485—216no—
132#485—216no—
133#484—28no—
134#480—28no—
135#468—28no—
136#468—28no—
137#450—28no—
138#448—28no—
139C2×#107—24no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#106—24no—
144C2×#106C2314yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157#407—24no—
158C2×C22⋊C8—24no—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163#399—24no—
164#397—24no—
165#394—24no—
166#394—24no—
167#391—24no—
168#391C2314yes—
169#390—22no—
170#390—22no—
171#390C2314yes—
172#390C2314yes—
173#389—24no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389C2314yes—
178#388—24no—
179#388—24no—
180#388—24no—
181#388—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#385—24no—
188#385—24no—
189#384C2314yes—
190#382C2314yes—
191#381—22no—
192#381D412yes—
193#380—21no—
194#380—21no—
195#380—22no—
196#380D411yescharacteristic
197#380D411yescharacteristic
198#380D412yes—
199#380C2312yes—
200#380C2312yes—
201#379—24no—
202#379D412yes—
203C4×C4⋊C4C2314yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4C2312yes—
207C4×C22⋊C4C2312yes—
208C2×#95—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217C25—21no—
218C25C2×D411yescharacteristic
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8C2×D414yes—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—44no—
233C22×D4—82no—
234C22×D4—82no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D414yes—
248C23×C4#4114yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—42no—
259C2×M16—42no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—42no—
263C22×C8—42no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268#123—84no—
269#123—84no—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8C2×D418yes—
275C22⋊Q8C2×D418yes—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—44no—
281#121—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—42no—
289#117—42no—
290#117—44no—
291#117—44no—
292#117—84no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
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—24no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—44no—
347C2×C42C2414yes—
348#107—28no—
349#107—24no—
350#107—24no—
351#106—28no—
352C4⋊C8—44no—
353C4⋊C8—44no—
354#103—28no—
355#103—28no—
356#103—24no—
357#103—24no—
358#102—24no—
359#102—24no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
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—22no—
380C24—42no—
381C24—42no—
382C24—44no—
383C24—42no—
384C24—81no—
385C24—81no—
386C24—82no—
387C24#12114yes—
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—84no—
399C2×Q8C22×D414yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4—84no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—816no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4#12114yes—
451C22×C4#12114yes—
452C22×C4#12114yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2511yescharacteristic
465SD16—84no—
466SD16—84no—
467M16—82no—
468M16—82no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—44no—
478C2×C8—44no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4C22×D414yes—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—82no—
518C42—24no—
519C42—24no—
520C42—44no—
521C42—44no—
522C42—84no—
523C42—84no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—28no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—164no—
548C23—164no—
549C23C2×#12112yes—
550C23C2×#12112yes—
551C23C2×#12112yes—
552C23#57612yes—
553C23#57712yes—
554C23C23×D411yescharacteristic
555Q8—24no—
556Q8—24no—
557Q8—44no—
558Q8—84no—
559D4—42no—
560D4—42no—
561D4—44no—
562D4—84no—
563D4—88no—
564D4—88no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
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—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—164no—
609C2×C4C2×#12112yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4C23×D411yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4#58512yes—
615C2×C4#58512yes—
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C22—22no—
621C22—24no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—88no—
635C22—82no—
636C22—82no—
637C22—84no—
638C22—84no—
639C22—82no—
640C22—168no—
641C22#304514yes—
642C22C22×#12111yescharacteristic
643C22#348512yes—
644C22C2×#57611yescharacteristic
645C22C2×#57711yescharacteristic
646C22#358612yes—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—84no—
658C4#348511yescharacteristic
659C4#348511yescharacteristic
660C4#349112yes—
661C2—24no—
662C2—42no—
663C2—42no—
664C2—162no—
665C2#3544711yescharacteristic
666C2C2×#304511yescharacteristic
667C2#6042411yescharacteristic
668C2#6175312yes—
669C2#6175912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×#485C2semidirect
4. #34380C2semidirect
5. #28130C2semidirect
6. #27656C2semidirect
7. #25224C2semidirect
8. #25222C2semidirect
9. #24581C2semidirect
10. #24576C2semidirect
11. #24533C2semidirect
12. #24528C2semidirect
13. #21933C2semidirect
15. #20702C2semidirect
23. C2×#485C22semidirect
29. C22×#107C22semidirect
30. C22×#106C22semidirect
31. C22×#103C22semidirect
32. C22×#103C22semidirect
44. #2360C22semidirect
45. #2344C22semidirect
47. Q8×C22⋊C4C22semidirect
56. #2085C22semidirect
57. #2083C22semidirect
58. #2074C22semidirect
59. #2072C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
62. #2055C22semidirect
63. #2055C22semidirect
64. #1949C22semidirect
65. #1947C22semidirect
66. #1926C22semidirect
67. #1925C22semidirect
70. #1880C22semidirect
71. #1880C22semidirect
73. #1851C22semidirect
74. #1851C22semidirect
75. #1850C22semidirect
76. #1850C22semidirect
78. #1824C22semidirect
80. #1823C22semidirect
81. #1805C22semidirect
82. #1803C22semidirect
85. #1736C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C22×#4852
#352622
#343802
#281302
#276562
#252242
#252222
#245812
#245762
#245332
#245282
#219332
#217582
#207022