Finite Group Databasebeta

← #6526745

Subgroups — #6526745

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic, complemented
2C2×#3109C211yesmaximal, characteristic
3#35172C211yesmaximal, characteristic, complemented
4#34368C211yesmaximal, characteristic, complemented
5#28126C214yesmaximal, complemented
6#27654C214yesmaximal, complemented
7#25451C212yesmaximal, complemented
8#25450C212yesmaximal, complemented
9#25321C212yesmaximal, complemented
10#25318C212yesmaximal, complemented
11#24823C214yesmaximal, complemented
12#24688C214yesmaximal, complemented
13#21954C211yesmaximal, characteristic, complemented
14#21839C211yesmaximal, characteristic, complemented
15#20732C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3228—28no—
20C2×#500C2211yescharacteristic, complemented
21C2×#498C2211yescharacteristic, complemented
22C2×#486C2211yescharacteristic, complemented
23C2×#486C2211yescharacteristic, complemented
24#3109—28no—
25C2×#481C2212yes—
26C2×#482C2214yes—
27C2×#478C2214yescomplemented
28C2×#476C2212yes—
29C2×#459C2212yes—
30C2×#447C2212yes—
31C2×#442C2214yes—
32C2×#432C2211yescharacteristic
33C2×#431C2211yescharacteristic
34#2848—22no—
35#2626—22no—
36#2580—22no—
37#2566C2214yes—
38#2563C2212yes—
39#2542C2212yes—
40#2538—21no—
41#2538—21no—
42#2379—22no—
43#2369C2212yescomplemented
44#2366C2212yescomplemented
45#2354—21no—
46#2354C2212yescomplemented
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2212yescomplemented
49#2324—22no—
50#2324C2214yes—
51#2317C2212yes—
52#2311—22no—
53#2309C2212yes—
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56#2085C2214yescomplemented
57#2084C2214yescomplemented
58#2073C2214yescomplemented
59#2071C2214yescomplemented
60#2067C2214yes—
61#2065C2214yes—
62#2054C2214yes—
63#2054C2214yes—
64#1992C2212yescomplemented
65#1992C2212yescomplemented
66#1988C2212yescomplemented
67#1987C2212yescomplemented
68#1949C2214yescomplemented
69#1948C2214yescomplemented
70#1940C2214yescomplemented
71#1939C2212yescomplemented
72#1939C2212yescomplemented
73#1926C2214yes—
74#1925C2214yes—
75#1879C2214yes—
76#1879C2214yes—
77#1825C2214yes—
78#1823C2212yes—
79#1804C2214yescomplemented
80#1802C2212yes—
81#1802C2212yes—
82#1802C2214yescomplemented
83#1798C2214yes—
84#1786C2212yes—
85C4×#102C2212yes—
86C4×#102C2212yes—
87C22×#41—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×D8C2314yes—
91#533—24no—
92#533—24no—
93#532—24no—
94#532D418yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#125C2312yes—
100C2×#124—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2314yes—
119C2×#117—21no—
120C2×#117—22no—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C42C2311yescharacteristic
127#500—24no—
128#498—24no—
129#486—24no—
130#486—24no—
131#482—216no—
132#481—28no—
133#478—216no—
134#476—28no—
135#459—28no—
136#447—28no—
137#442—216no—
138#432—24no—
139#431—24no—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160#399—24no—
161#396—24no—
162#396—24no—
163#393—24no—
164#391—24no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—24no—
169#390C2314yes—
170#390C2314yes—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#388—24no—
177#388—24no—
178#386—24no—
179#386—24no—
180#385—24no—
181#385—24no—
182#385—24no—
183#385—24no—
184#385—24no—
185#384D411yescharacteristic
186#384D411yescharacteristic
187#384C2314yes—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#382—21no—
191#382D412yes—
192#382C2314yes—
193#380—21no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380D412yes—
199#380D412yes—
200#380C2312yes—
201#379—24no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2314yes—
218C25—21no—
219C2×#41—28no—
220C2×#41—82no—
221C2×#41—82no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×D4—22no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—42no—
233C22×D4C2412yes—
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×D8—28no—
247C2×D8—28no—
248C2×D8—44no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8C2×D414yes—
258#127—22no—
259#127C2×D414yes—
260#125—24no—
261#125—24no—
262#124—28no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C22⋊Q8—84no—
266C22⋊Q8—84no—
267#121—28no—
268#121—24no—
269#121C2×D418yes—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—84no—
283C4×D4C2×D418yes—
284#117—24no—
285#117—24no—
286#117—44no—
287#117—82no—
288#117—84no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—84no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D414yes—
338C2×C42#4114yes—
339C2×C42C2414yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—28no—
343#106—28no—
344#106—28no—
345C4⋊C8—24no—
346C4⋊C8—24no—
347C4⋊C8—44no—
348C4⋊C8—44no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—24no—
353#102—24no—
354#102—28no—
355#102—42no—
356#102—42no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359#97—24no—
360C4×C8—22no—
361C4×C8—22no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—81no—
376C24—81no—
377C24—81no—
378C24—81no—
379C24C22×D411yescharacteristic
380#41—816no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×Q8—81no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4C22×D414yes—
413C2×D42−1+414yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—816no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4#12114yes—
443C22×C4#12114yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4C2511yescharacteristic
454D8—416no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—84no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4C22×D414yes—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—82no—
509C42—24no—
510C42—24no—
511C42—44no—
512C42—44no—
513C42—84no—
514C42#12114yes—
515C42#12114yes—
516C42C22×D412yes—
517C42C22×D412yes—
518C42C2×#4112yes—
519C42C2×#4112yes—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—24no—
527C23—24no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23C2×#12112yes—
548C23C23×D411yescharacteristic
549Q8—84no—
550Q8—84no—
551Q8—84no—
552D4—24no—
553D4—28no—
554D4—24no—
555D4—48no—
556D4—42no—
557D4—44no—
558D4—42no—
559D4—48no—
560D4—84no—
561D4—88no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—44no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—88no—
600C2×C4—84no—
601C2×C4—88no—
602C2×C4—168no—
603C2×C4C2×#12112yes—
604C2×C4C2×#12112yes—
605C2×C4C2×#12112yes—
606C2×C4C2×#12112yes—
607C2×C4C2×#12112yes—
608C2×C4#57612yes—
609C2×C4#57712yes—
610C2×C4C23×D411yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×2−1+412yes—
613C2×C4#58512yes—
614C8—42no—
615C8—42no—
616C8—44no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—816no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22C22×#12111yescharacteristic
640C22#348511yescharacteristic
641C22#348611yescharacteristic
642C22C2×#57611yescharacteristic
643C22C2×#57711yescharacteristic
644C22#358511yescharacteristic
645C22#358511yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#304514yes—
659C4#348912yes—
660C4#349512yes—
661C2—42no—
662C2—84no—
663C2—84no—
664C2#3563311yescharacteristic
665C2C2×#304511yescharacteristic
666C2#6042311yescharacteristic
667C2#6180211yescharacteristic
668C2#6180811yescharacteristic
669C2#6184711yescharacteristic
670C2#6185011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3228C2semidirect
3. #35172C2semidirect
4. #34368C2semidirect
5. #28126C2semidirect
6. #27654C2semidirect
7. #25451C2semidirect
8. #25450C2semidirect
9. #25321C2semidirect
10. #25318C2semidirect
11. #24823C2semidirect
12. #24688C2semidirect
13. #21954C2semidirect
14. #21839C2semidirect
15. #20732C2semidirect
20. C2×#500C22semidirect
21. C2×#498C22semidirect
22. C2×#486C22semidirect
23. C2×#486C22semidirect
27. C2×#478C22semidirect
43. #2369C22semidirect
44. #2366C22semidirect
46. #2354C22semidirect
48. D4×C4⋊C4C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2073C22semidirect
59. #2071C22semidirect
64. #1992C22semidirect
65. #1992C22semidirect
66. #1988C22semidirect
67. #1987C22semidirect
68. #1949C22semidirect
69. #1948C22semidirect
70. #1940C22semidirect
71. #1939C22semidirect
72. #1939C22semidirect
79. #1804C22semidirect
82. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
C2×#31092
#351722
#343682
#281262
#276542
#254512
#254502
#253212
#253182
#248232
#246882
#219542
#218392
#207322