Finite Group Databasebeta

← #6496381

Subgroups — #6496381

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3112C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic, complemented
3#37539C211yesmaximal, characteristic
4#34196C211yesmaximal, characteristic, complemented
5#33206C212yesmaximal, complemented
6#33039C212yesmaximal, complemented
7#29875C212yesmaximal, complemented
8#29835C212yesmaximal, complemented
9#24824C214yesmaximal, complemented
10#24689C214yesmaximal, complemented
11#24581C212yesmaximal, complemented
12#24576C212yesmaximal, complemented
13#24533C212yesmaximal, complemented
14#24528C212yesmaximal, complemented
15#21977C211yesmaximal, characteristic, complemented
16#21884C211yesmaximal, characteristic
17#21838C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3112—28no—
22C2×#480C2212yes—
23C2×#479—28no—
24C2×#479C2212yes—
25C2×#479C2212yes—
26C2×#479C2212yes—
27C2×#479C2212yes—
28C2×#477C2211yescharacteristic
29C2×#477C2211yescharacteristic
30C2×#460C2211yescharacteristic
31C2×#460C2211yescharacteristic
32C2×#448C2212yes—
33C2×#440C2212yes—
34C2×#440C2212yes—
35C2×#432C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#103C2211yescharacteristic
41#2820—22no—
42#2660C2212yes—
43#2641C2212yes—
44#2633—22no—
45#2618—21no—
46#2618—21no—
47#2566C2214yes—
48#2504C2212yes—
49#2501C2212yes—
50#2406—22no—
51#2355—22no—
52#2355C2212yes—
53Q8×C4⋊C4C2212yes—
54#2344—22no—
55#2344C2212yes—
56Q8×C22⋊C4C2212yes—
57#2320—22no—
58#2320C2212yes—
59#2285—21no—
60#2285C2211yescharacteristic
61#2283—21no—
62#2283C2211yescharacteristic
63#2276C2212yes—
64#2085C2214yes—
65#2083C2214yes—
66#2074C2214yes—
67#2072C2214yes—
68#2067C2214yes—
69#2065C2214yes—
70#2055C2214yes—
71#2055C2214yes—
72#1992C2214yes—
73#1989C2212yes—
74#1989C2212yes—
75#1988C2212yes—
76#1982C2212yes—
77#1949C2214yes—
78#1947C2214yes—
79#1940C2214yes—
80#1926C2214yes—
81#1925C2214yes—
82#1893C2212yes—
83#1893C2212yes—
84#1880C2214yes—
85#1880C2214yes—
86#1850C2212yes—
87#1850C2212yes—
88#1843C2212yes—
89#1832C2212yes—
90#1824C2212yes—
91#1823C2212yes—
92#1805C2214yes—
93#1803C2214yes—
94#1802C2214yes—
95C22×#41—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98C22×SD16C2312yes—
99C23×C8C2311yescharacteristic
100C23×C8C2311yescharacteristic
101#532—24no—
102#532—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#125—21no—
106C2×#125—21no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×#124—21no—
112C2×#123—22no—
113C2×#123—22no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—24no—
119C2×C4×Q8D418yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C2×#117—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#480—28no—
138#479—28no—
139#479—28no—
140#479—28no—
141#479—28no—
142#477—24no—
143#477—24no—
144#460—24no—
145#460—24no—
146#448—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⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2314yes—
164C2×C4⋊C8C2314yes—
165C2×#103—24no—
166C2×#103—24no—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2314yes—
176C2×#102C2314yes—
177C2×C22⋊C8C2312yes—
178C2×#97C2312yes—
179#399—24no—
180#398—22no—
181#398—22no—
182#397—24no—
183#396—24no—
184#396—24no—
185#395—24no—
186#394—24no—
187#394—22no—
188#394—22no—
189#391—24no—
190#391C2314yes—
191#390C2314yes—
192#390C2314yes—
193#389—24no—
194#389C2314yes—
195#387—24no—
196#387—24no—
197#387—24no—
198#386—22no—
199#386—22no—
200#385—24no—
201#385—24no—
202#384—22no—
203#384C2314yes—
204#384C2312yes—
205#383—22no—
206#383D412yes—
207#382—22no—
208#382—21no—
209#382—22no—
210#382D412yes—
211#382C2314yes—
212#381—22no—
213#381D412yes—
214#380—21no—
215#380—22no—
216#380D411yescharacteristic
217#380D411yescharacteristic
218#380C2312yes—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—24no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2314yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4C2312yes—
229#376—22no—
230#376C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—44no—
240C2×#41—82no—
241C2×#41—82no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—42no—
247C22×Q8C2×D414yes—
248C22×Q8C2411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—22no—
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—
276#127—22no—
277#127—22no—
278#125—24no—
279#125—24no—
280#125—24no—
281#125—24no—
282#125—44no—
283#125—44no—
284#124—44no—
285#123—84no—
286#123—84no—
287#121—24no—
288#121—24no—
289#121—44no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8—28no—
297C4×Q8—44no—
298C4×Q8C2×D414yes—
299C4×Q8C2×D414yes—
300C4×Q8C2×D414yes—
301C4×Q8C2×D414yes—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—44no—
305#117—84no—
306#117—82no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—28no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D414yes—
362C2×C42#4114yes—
363C2×C42C2414yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—24no—
367#107—24no—
368#107—24no—
369#107—24no—
370#106—24no—
371#106—24no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—22no—
375C4⋊C8—22no—
376C4⋊C8—44no—
377C4⋊C8—44no—
378#103—24no—
379#103—24no—
380#103—24no—
381#103—24no—
382#103—24no—
383#103—24no—
384#102—24no—
385#102—24no—
386#102—44no—
387#102—44no—
388C22⋊C8—24no—
389#97—24no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D414yes—
401#95C2×D414yes—
402#95C2×D414yes—
403C24—22no—
404C24C22×D411yescharacteristic
405#41—816no—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C2×Q8—42no—
417C2×Q8—81no—
418C2×Q8—81no—
419C2×Q8C22×D412yes—
420C2×Q8C22×D412yes—
421C2×Q82−1+414yes—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—82no—
432C2×D4—81no—
433C2×D4—81no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—82no—
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—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22×D412yes—
515C4⋊C4C22×D412yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—48no—
532C42—82no—
533C42—84no—
534C42#12114yes—
535C42#12114yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C42C2×#4112yes—
539C42C2×#4112yes—
540C23—22no—
541C23—22no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23C2×#12112yes—
548C23#57612yes—
549C23#57712yes—
550C23C23×D411yescharacteristic
551Q8—22no—
552Q8—24no—
553Q8—22no—
554Q8—22no—
555Q8—48no—
556Q8—42no—
557Q8—84no—
558Q8#53614yes—
559D4—42no—
560D4—42no—
561D4—44no—
562D4—84no—
563D4—84no—
564D4—88no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4C2×#12112yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×2−1+412yes—
617C2×C4#58512yes—
618C8—42no—
619C8—42no—
620C8—42no—
621C8—42no—
622C22—22no—
623C22—24no—
624C22—24no—
625C22—22no—
626C22—82no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22—82no—
631C22#304514yes—
632C22C22×#12111yescharacteristic
633C22#348912yes—
634C22C2×#57611yescharacteristic
635C22C2×#57711yescharacteristic
636C22#358612yes—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—84no—
652C4C2×#53611yescharacteristic
653C4#344911yescharacteristic
654C4#349512yes—
655C2—24no—
656C2—162no—
657C2#3530611yescharacteristic
658C2C2×#304511yescharacteristic
659C2#6042411yescharacteristic
660C2#6216712yes—
661C2#6217012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. C22×#479C2semidirect
4. #34196C2semidirect
5. #33206C2semidirect
6. #33039C2semidirect
7. #29875C2semidirect
8. #29835C2semidirect
9. #24824C2semidirect
10. #24689C2semidirect
11. #24581C2semidirect
12. #24576C2semidirect
13. #24533C2semidirect
14. #24528C2semidirect
15. #21977C2semidirect
17. #21838C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
C22×#4792
#375392
#341962
#332062
#330392
#298752
#298352
#248242
#246892
#245812
#245762
#245332
#245282
#219772
#218842
#218382