Finite Group Databasebeta

← #6526734

Subgroups — #6526734

Order 512.

2008 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
1C2×#3235C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3#35172C211yesmaximal, characteristic, complemented
4#34368C211yesmaximal, characteristic, complemented
5#28126C214yesmaximal, complemented
6#27655C214yesmaximal, complemented
7#25462C214yesmaximal, complemented
8#25330C214yesmaximal, complemented
9#24812C212yesmaximal, complemented
10#24811C212yesmaximal, complemented
11#24677C212yesmaximal, complemented
12#24676C212yesmaximal, complemented
13#21954C211yesmaximal, characteristic, complemented
14#21835C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3235—28no—
20C2×#501C2212yescomplemented
21C2×#490C2212yescomplemented
22#3100—28no—
23C2×#482C2214yes—
24C2×#480C2212yes—
25C2×#476C2212yescomplemented
26C2×#474C2212yes—
27C2×#474C2212yescomplemented
28C2×#459C2212yes—
29C2×#447C2212yes—
30C2×C4×D8C2212yes—
31C2×C4×D8C2212yes—
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#2083C2214yescomplemented
58#2073C2214yescomplemented
59#2071C2214yescomplemented
60#2067C2214yes—
61#2065C2214yes—
62#2054C2214yes—
63#2054C2214yes—
64#1990C2211yescharacteristic, complemented
65#1990C2211yescharacteristic, complemented
66#1990C2211yescharacteristic, complemented
67#1990C2211yescharacteristic, complemented
68#1989C2214yescomplemented
69#1949C2214yescomplemented
70#1947C2214yescomplemented
71#1940C2214yescomplemented
72#1939C2212yescomplemented
73#1939C2212yescomplemented
74#1926C2214yes—
75#1925C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1825C2214yes—
79#1824C2212yes—
80#1804C2214yescomplemented
81#1802C2212yes—
82#1802C2212yes—
83#1802C2214yescomplemented
84#1798C2214yes—
85#1786C2212yes—
86#1762C2214yes—
87C22×#41—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×D8C2312yes—
91C22×D8C2312yes—
92#533—24no—
93#533—24no—
94#532—24no—
95#532D418yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#125C2312yes—
101C2×#124—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4—24no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2314yes—
120C2×#117—21no—
121C2×#117—22no—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C42C2311yescharacteristic
128#501—28no—
129#490—28no—
130#482—216no—
131#480—28no—
132#476—28no—
133#474—28no—
134#474—28no—
135#459—28no—
136#447—28no—
137C4×D8—28no—
138C4×D8—28no—
139#432—24no—
140#431—24no—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×#97C2312yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162#399—24no—
163#396—24no—
164#396—24no—
165#393—24no—
166#391—24no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—24no—
171#390C2314yes—
172#390C2314yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#388—24no—
179#388—24no—
180#386—24no—
181#386—24no—
182#385—24no—
183#385—24no—
184#385—24no—
185#385—24no—
186#385—24no—
187#384D411yescharacteristic
188#384D411yescharacteristic
189#384C2314yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#382—21no—
193#382D412yes—
194#382C2314yes—
195#380—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380D412yes—
201#380D412yes—
202#380C2312yes—
203#379—24no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
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—24no—
247C2×D8—24no—
248C2×D8—24no—
249C2×D8—24no—
250C2×D8—42no—
251C2×D8—42no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8C2×D414yes—
261#127—22no—
262#127C2×D414yes—
263#125—24no—
264#125—24no—
265#124—28no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—84no—
269C22⋊Q8—84no—
270#121—28no—
271#121—24no—
272#121C2×D418yes—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—28no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—84no—
286C4×D4C2×D418yes—
287#117—24no—
288#117—24no—
289#117—44no—
290#117—82no—
291#117—84no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4C2×D414yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D414yes—
341C2×C42#4114yes—
342C2×C42C2414yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—24no—
347#107—28no—
348#106—28no—
349C4⋊C8—24no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353#102—24no—
354#102—24no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—24no—
359#102—24no—
360#102—44no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363#97—24no—
364C4×C8—22no—
365C4×C8—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—24no—
375C24—22no—
376C24—22no—
377C24—42no—
378C24—42no—
379C24—81no—
380C24—81no—
381C24—81no—
382C24—81no—
383C24C22×D411yescharacteristic
384#41—816no—
385C2×Q8—24no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4C22×D414yes—
419C2×D42−1+414yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—816no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4#12114yes—
449C22×C4#12114yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2511yescharacteristic
460D8—48no—
461D8—48no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—84no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4C22×D414yes—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—82no—
514C42—24no—
515C42—24no—
516C42—44no—
517C42—44no—
518C42—84no—
519C42#12114yes—
520C42#12114yes—
521C42C22×D412yes—
522C42C22×D412yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—24no—
532C23—24no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23C2×#12112yes—
553C23C23×D411yescharacteristic
554Q8—84no—
555Q8—84no—
556Q8—84no—
557D4—24no—
558D4—24no—
559D4—24no—
560D4—24no—
561D4—48no—
562D4—48no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—84no—
568D4—88no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—416no—
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—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4—88no—
609C2×C4—168no—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4#57612yes—
616C2×C4#57612yes—
617C2×C4C23×D411yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×2−1+412yes—
620C2×C4#58512yes—
621C8—44no—
622C8—42no—
623C8—42no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—816no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22C22×#12111yescharacteristic
647C22#348511yescharacteristic
648C22#348611yescharacteristic
649C22C2×#57611yescharacteristic
650C22C2×#57611yescharacteristic
651C22#358511yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#304114yes—
666C4#348912yes—
667C4#349512yes—
668C2—42no—
669C2—84no—
670C2—84no—
671C2#3563311yescharacteristic
672C2C2×#304111yescharacteristic
673C2#6042511yescharacteristic
674C2#6180211yescharacteristic
675C2#6180711yescharacteristic
676C2#6184711yescharacteristic
677C2#6184911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
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×#3235C2semidirect
3. #35172C2semidirect
4. #34368C2semidirect
5. #28126C2semidirect
6. #27655C2semidirect
7. #25462C2semidirect
8. #25330C2semidirect
9. #24812C2semidirect
10. #24811C2semidirect
11. #24677C2semidirect
12. #24676C2semidirect
13. #21954C2semidirect
14. #21835C2semidirect
15. #20738C2semidirect
20. C2×#501C22semidirect
21. C2×#490C22semidirect
25. C2×#476C22semidirect
27. C2×#474C22semidirect
43. #2369C22semidirect
44. #2366C22semidirect
46. #2354C22semidirect
48. D4×C4⋊C4C22semidirect
56. #2085C22semidirect
57. #2083C22semidirect
58. #2073C22semidirect
59. #2071C22semidirect
64. #1990C22semidirect
65. #1990C22semidirect
66. #1990C22semidirect
67. #1990C22semidirect
68. #1989C22semidirect
69. #1949C22semidirect
70. #1947C22semidirect
71. #1940C22semidirect
72. #1939C22semidirect
73. #1939C22semidirect
80. #1804C22semidirect
83. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32352
C2×#31002
#351722
#343682
#281262
#276552
#254622
#253302
#248122
#248112
#246772
#246762
#219542
#218352
#207382