Finite Group Databasebeta

← #6529924

Subgroups — #6529924

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2C2×#2986C211yesmaximal, characteristic, complemented
3#37619C211yesmaximal, characteristic
4#35179C211yesmaximal, characteristic, complemented
5#28270C212yesmaximal, complemented
6#28266C212yesmaximal, complemented
7#28192C212yesmaximal, complemented
8#28189C212yesmaximal, complemented
9#25479C212yesmaximal, complemented
10#25472C212yesmaximal, complemented
11#25303C212yesmaximal, complemented
12#25295C212yesmaximal, complemented
13#24618C214yesmaximal, complemented
14#24518C214yesmaximal, complemented
15#21985C211yesmaximal, characteristic, complemented
16#21863C211yesmaximal, characteristic
17#20735C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3103—28no—
22C2×#481C2212yes—
23C2×#475C2211yescharacteristic
24C2×#475C2211yescharacteristic
25C2×#460C2211yescharacteristic
26C2×#460C2211yescharacteristic
27C2×#448C2212yes—
28#2986—28no—
29C2×#446C2212yes—
30C2×#446C2212yes—
31C2×C8×Q8C2212yes—
32C2×C8×Q8C2212yes—
33C2×C4×SD16C2212yes—
34C2×C4×SD16C2212yes—
35C2×#433C2214yes—
36C2×#431C2211yescharacteristic
37C2×#431C2211yescharacteristic
38#2855—22no—
39#2816—22no—
40#2658—21no—
41#2640—21no—
42#2633—22no—
43#2627—22no—
44#2624C2214yes—
45#2582C2212yes—
46#2582C2212yes—
47#2581C2212yes—
48#2581C2212yes—
49#2580C2212yes—
50#2580C2212yes—
51#2568—21no—
52#2567—21no—
53#2565—22no—
54#2548C2212yes—
55#2546C2212yes—
56#2432—22no—
57#2317C2212yes—
58#2310C2212yes—
59#2285C2211yescharacteristic
60#2283C2211yescharacteristic
61#2088C2214yes—
62#2088C2214yes—
63#2068C2214yes—
64#2067C2214yes—
65#2066C2214yes—
66#2065C2214yes—
67#2055C2214yes—
68#2055C2214yes—
69#1990C2212yes—
70#1973C2214yes—
71#1968C2214yes—
72#1957C2212yes—
73#1948C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1941C2214yes—
77#1936C2214yes—
78#1935C2214yes—
79#1934C2214yes—
80#1933C2214yes—
81#1875C2214yes—
82#1854C2214yes—
83#1846C2212yes—
84#1835C2212yes—
85#1823C2212yes—
86#1805C2211yescharacteristic
87#1805C2211yescharacteristic
88#1805C2211yescharacteristic
89#1805C2211yescharacteristic
90#1804C2212yes—
91#1804C2212yes—
92#1787C2212yes—
93C4×#103C2212yes—
94C4×#103C2212yes—
95C22×#41—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98C22×SD16C2312yes—
99#533—24no—
100#533—24no—
101#532—24no—
102#532—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8D411yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125D412yes—
109C2×#124—22no—
110C2×#124—21no—
111C2×C22⋊Q8D412yes—
112C2×C22⋊Q8D412yes—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×#117C2312yes—
128C2×#117C2312yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133#481—28no—
134#475—24no—
135#475—24no—
136#460—24no—
137#460—24no—
138#448—28no—
139#446—28no—
140#446—28no—
141C8×Q8—28no—
142C8×Q8—28no—
143C4×SD16—28no—
144C4×SD16—28no—
145#433—216no—
146#431—24no—
147#431D418yes—
148C2×#107C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2311yescharacteristic
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170C2×#97C2314yes—
171C2×C4×C8C2312yes—
172C2×C4×C8C2314yes—
173C2×C4×C8C2312yes—
174#400—22no—
175#400—22no—
176#399—22no—
177#399—22no—
178#399—24no—
179#398—24no—
180#398—24no—
181#398—24no—
182#397—24no—
183#395—22no—
184#395—22no—
185#394—24no—
186#394—24no—
187#394—24no—
188#393—22no—
189#393—22no—
190#391—22no—
191#391—22no—
192#391—22no—
193#391—22no—
194#391—22no—
195#391C2314yes—
196#391C2314yes—
197#390—22no—
198#390C2314yes—
199#390C2314yes—
200#389—22no—
201#389—22no—
202#389C2314yes—
203#389C2314yes—
204#389C2314yes—
205#389C2314yes—
206#387—24no—
207#387—24no—
208#385—24no—
209#385—24no—
210#385—24no—
211#384—22no—
212#384C2312yes—
213#383—21no—
214#382—22no—
215#382C2312yes—
216#382C2311yescharacteristic
217#382C2311yescharacteristic
218#381—21no—
219#381—24no—
220#380—21no—
221#380—21no—
222#380C2312yes—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—21no—
227#376—21no—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236C2×#41—44no—
237C2×#41—84no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—41no—
245C22×Q8—41no—
246C22×Q8C2×D412yes—
247C22×Q8C2×D412yes—
248C22×Q8C2411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C2×SD16—42no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—84no—
275#127—22no—
276#127—22no—
277#125—28no—
278#124—44no—
279#124—84no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282#121—24no—
283#121—24no—
284#121—44no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—44no—
302#117—24no—
303#117—24no—
304#117—42no—
305#117—42no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4C2×D414yes—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D414yes—
352C2×C42#4114yes—
353C2×C42C2414yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#106—24no—
358#106—24no—
359#106—44no—
360C4⋊C8—28no—
361C4⋊C8—28no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8C2×D414yes—
365C4⋊C8C2×D414yes—
366#103—24no—
367#103—24no—
368#103—24no—
369#103—24no—
370#103—42no—
371#103—42no—
372#102—24no—
373#102—24no—
374#102—44no—
375#102—44no—
376C22⋊C8—24no—
377C22⋊C8C2×D418yes—
378#97—28no—
379C4×C8—28no—
380C4×C8—24no—
381C4×C8C2×D418yes—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398C24—22no—
399C24C22×D411yescharacteristic
400#41—816no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q82−1+412yes—
426C2×Q82−1+412yes—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—82no—
438C2×D4—84no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—84no—
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×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4C2511yescharacteristic
472SD16—48no—
473SD16—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8C22×D414yes—
485C2×C8C22×D414yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C22⋊C4—28no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—82no—
525C42—24no—
526C42—24no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42#12114yes—
534C42#12114yes—
535C42C22×D412yes—
536C42C22×D412yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23C2×#12112yes—
549C23C23×D411yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8—22no—
554Q8—48no—
555Q8—48no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—48no—
561Q8—84no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—84no—
567D4—88no—
568D4—84no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4#57714yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C2×C4#58511yescharacteristic
622C8—24no—
623C8—24no—
624C8—42no—
625C8—42no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—42no—
630C22—42no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—84no—
635C22—84no—
636C22C22×#12111yescharacteristic
637C22#348611yescharacteristic
638C22#348711yescharacteristic
639C22C2×#57712yes—
640C22#358412yes—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—81no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4—84no—
655C4#305714yes—
656C4#348911yescharacteristic
657C4#348911yescharacteristic
658C4#349511yescharacteristic
659C4#349511yescharacteristic
660C2—42no—
661C2—162no—
662C2#3566111yescharacteristic
663C2C2×#305711yescharacteristic
664C2#6045311yescharacteristic
665C2#6186512yes—
666C2#6192812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2986C2semidirect
4. #35179C2semidirect
5. #28270C2semidirect
6. #28266C2semidirect
7. #28192C2semidirect
8. #28189C2semidirect
9. #25479C2semidirect
10. #25472C2semidirect
11. #25303C2semidirect
12. #25295C2semidirect
13. #24618C2semidirect
14. #24518C2semidirect
15. #21985C2semidirect
17. #20735C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
C2×#29862
#376192
#351792
#282702
#282662
#281922
#281892
#254792
#254722
#253032
#252952
#246182
#245182
#219852
#218632
#207352