Finite Group Databasebeta

← #6496380

Subgroups — #6496380

Order 512.

1916 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×#3110C211yesmaximal, characteristic
2C22×#478C211yesmaximal, characteristic
3#37540C211yesmaximal, characteristic, complemented
4#34194C211yesmaximal, characteristic
5#33203C212yesmaximal, complemented
6#33040C212yesmaximal, complemented
7#29872C212yesmaximal, complemented
8#29836C212yesmaximal, complemented
9#24825C214yesmaximal, complemented
10#24690C214yesmaximal, complemented
11#24582C212yesmaximal, complemented
12#24579C212yesmaximal, complemented
13#24530C212yesmaximal
14#24527C212yesmaximal
15#21977C211yesmaximal, characteristic, complemented
16#21885C211yesmaximal, characteristic, complemented
17#21838C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3110—28no—
22C2×#484C2212yes—
23C2×#478—28no—
24C2×#478C2212yes—
25C2×#478C2212yes—
26C2×#478C2212yes—
27C2×#478C2212yes—
28C2×#474C2211yescharacteristic
29C2×#474C2211yescharacteristic
30C2×#461C2211yescharacteristic
31C2×#461C2211yescharacteristic
32C2×#450C2212yes—
33C2×#440C2212yes—
34C2×#440C2212yes—
35C2×#432C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41#2822—22no—
42#2660C2212yescomplemented
43#2642C2212yescomplemented
44#2634—22no—
45#2627—21no—
46#2627—21no—
47#2565C2214yescomplemented
48#2503C2212yes—
49#2502C2212yes—
50#2484—22no—
51#2354C2212yes—
52#2352—22no—
53D4×C4⋊C4C2212yes—
54#2345C2212yes—
55#2342—22no—
56D4×C22⋊C4C2212yes—
57#2319C2212yes—
58#2318—22no—
59#2285C2211yescharacteristic
60#2283C2211yescharacteristic
61#2280—21no—
62C2×#380—21no—
63#2275C2212yes—
64#2085C2214yescomplemented
65#2083C2214yescomplemented
66#2073C2214yes—
67#2071C2214yes—
68#2068C2214yescomplemented
69#2066C2214yescomplemented
70#2055C2214yes—
71#2055C2214yes—
72#1992C2214yescomplemented
73#1989C2212yescomplemented
74#1989C2212yescomplemented
75#1987C2212yescomplemented
76#1981C2212yescomplemented
77#1949C2214yescomplemented
78#1947C2214yescomplemented
79#1940C2214yescomplemented
80#1928C2214yescomplemented
81#1927C2214yescomplemented
82#1892C2212yes—
83#1892C2212yes—
84#1877C2214yes—
85#1877C2214yes—
86#1850C2212yes—
87#1850C2212yes—
88#1843C2212yescomplemented
89#1832C2212yescomplemented
90#1824C2212yes—
91#1823C2212yes—
92#1804C2214yes—
93#1803C2214yescomplemented
94#1802C2214yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C22×SD16C2312yes—
99C23×C8C2311yescharacteristic
100C23×C8C2311yescharacteristic
101#533—24no—
102#533—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×#125—21no—
107C2×#125—21no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125C2312yes—
111C2×#125C2312yes—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—24no—
119C2×C4×D4D418yes—
120C2×C4×D4C2314yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#484—28no—
138#478—28no—
139#478—28no—
140#478—28no—
141#478—28no—
142#474—24no—
143#474—24no—
144#461—24no—
145#461—24no—
146#450—28no—
147#440—28no—
148#440—28no—
149#432—24no—
150C2×#107—24no—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#106—24no—
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×C4⋊C8—24no—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2314yes—
164C2×C4⋊C8C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102—24no—
170C2×#102—24no—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×C22⋊C8C2312yes—
178C2×#97C2312yes—
179#400—24no—
180#400—24no—
181#400—22no—
182#400—22no—
183#398—24no—
184#398—24no—
185#398—24no—
186#398—24no—
187#395—22no—
188#395—22no—
189#391C2314yes—
190#391C2314yes—
191#390C2314yes—
192#389C2314yes—
193#388—24no—
194#388—24no—
195#387—24no—
196#387—24no—
197#387—24no—
198#386—24no—
199#385—22no—
200#385—22no—
201#385—24no—
202#384C2314yes—
203#384C2312yes—
204#383D412yes—
205#382—21no—
206#382D412yes—
207#382C2314yes—
208#381—22no—
209#381—22no—
210#381—22no—
211#381D412yes—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380D411yescharacteristic
218#380D411yescharacteristic
219#380C2312yes—
220#379—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2314yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—24no—
228C4×C22⋊C4C2312yes—
229#376C2312yes—
230C2×#95—22no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8C2412yes—
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4C2×D414yes—
247C22×D4C2411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—28no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—24no—
258C23×C4—84no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—28no—
265C2×SD16—44no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—42no—
269C22×C8—42no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278#125—24no—
279#125—24no—
280#125—24no—
281#125—24no—
282#125—44no—
283#125—44no—
284#125—44no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—44no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—44no—
299C4×D4C2×D414yes—
300C4×D4C2×D414yes—
301C4×D4C2×D414yes—
302C4×D4C2×D414yes—
303#117—44no—
304#117—82no—
305#117—84no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—84no—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—28no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—84no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D414yes—
359C2×C42#4114yes—
360C2×C42C2414yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—24no—
365#107—24no—
366#107—24no—
367#106—24no—
368#106—24no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373C4⋊C8—44no—
374C4⋊C8—44no—
375#103—24no—
376#103—24no—
377#103—44no—
378#103—44no—
379#102—24no—
380#102—24no—
381#102—24no—
382#102—24no—
383#102—24no—
384#102—24no—
385C22⋊C8—24no—
386#97—24no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D414yes—
400#95C2×D414yes—
401#95C2×D414yes—
402C24—21no—
403C24—21no—
404C24—24no—
405C24—44no—
406C24—42no—
407C24—42no—
408C24—82no—
409C24C22×D411yescharacteristic
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—42no—
415C2×Q8—82no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—44no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4C22×D412yes—
427C2×D4C22×D412yes—
428C2×D42−1+414yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—816no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4#12114yes—
461C22×C4#12114yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C2×#4112yes—
471C22×C4C2511yescharacteristic
472SD16—48no—
473SD16—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—44no—
483C2×C8—44no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4C22×D412yes—
514C4⋊C4C22×D412yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—24no—
527C42—24no—
528C42—44no—
529C42—82no—
530C42—84no—
531C42#12114yes—
532C42#12114yes—
533C42C22×D412yes—
534C42C22×D412yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—48no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—48no—
549C23—41no—
550C23—41no—
551C23—84no—
552C23—88no—
553C23—88no—
554C23C2×#12112yes—
555C23#57712yes—
556C23#57812yes—
557C23C23×D411yescharacteristic
558Q8—42no—
559Q8—42no—
560Q8—44no—
561Q8—84no—
562D4—22no—
563D4—22no—
564D4—22no—
565D4—24no—
566D4—48no—
567D4—42no—
568D4#53614yes—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—168no—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4C2×#12112yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×2−1+412yes—
619C2×C4#58512yes—
620C8—42no—
621C8—42no—
622C8—42no—
623C8—42no—
624C22—22no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—816no—
637C22—82no—
638C22—82no—
639C22#304614yes—
640C22C22×#12111yescharacteristic
641C22#348912yes—
642C22C2×#57711yescharacteristic
643C22C2×#57811yescharacteristic
644C22#358512yes—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4C2×#53611yescharacteristic
660C4#344911yescharacteristic
661C4#349512yes—
662C2—24no—
663C2—44no—
664C2—82no—
665C2#3530611yescharacteristic
666C2C2×#304611yescharacteristic
667C2#6042311yescharacteristic
668C2#6216412yes—
669C2#6216512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #37540C2semidirect
5. #33203C2semidirect
6. #33040C2semidirect
7. #29872C2semidirect
8. #29836C2semidirect
9. #24825C2semidirect
10. #24690C2semidirect
11. #24582C2semidirect
12. #24579C2semidirect
15. #21977C2semidirect
16. #21885C2semidirect
17. #21838C2semidirect
42. #2660C22semidirect
43. #2642C22semidirect
47. #2565C22semidirect
64. #2085C22semidirect
65. #2083C22semidirect
68. #2068C22semidirect
69. #2066C22semidirect
72. #1992C22semidirect
73. #1989C22semidirect
74. #1989C22semidirect
75. #1987C22semidirect
76. #1981C22semidirect
77. #1949C22semidirect
78. #1947C22semidirect
79. #1940C22semidirect
80. #1928C22semidirect
81. #1927C22semidirect
88. #1843C22semidirect
89. #1832C22semidirect
93. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C22×#4782
#375402
#341942
#332032
#330402
#298722
#298362
#248252
#246902
#245822
#245792
#245302
#245272
#219772
#218852
#218382