Finite Group Databasebeta

← #6816639

Subgroups — #6816639

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3240C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic, complemented
3#36114C211yesmaximal, characteristic
4#33316C211yesmaximal, characteristic, complemented
5#31106C214yesmaximal, complemented
6#31101C214yesmaximal, complemented
7#29884C212yesmaximal, complemented
8#28726C211yesmaximal, characteristic, complemented
9#28693C211yesmaximal, characteristic, complemented
10#26047C212yesmaximal, complemented
11#26045C212yesmaximal, complemented
12#25817C212yesmaximal, complemented
13#25679C212yesmaximal, complemented
14#25015C212yesmaximal, complemented
15#25011C212yesmaximal
16C4×#489C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#529C2212yes—
20#3240—216no—
21C2×#501C2212yes—
22C2×#501C2212yes—
23C2×#501C2212yes—
24C2×#489C2212yes—
25C2×#489C2211yescharacteristic
26C2×#485C2214yes—
27C2×#482C2214yes—
28C2×#479C2214yes—
29C2×#478C2214yes—
30C2×#432C2212yes—
31#2868C2211yescharacteristic
32#2859C2211yescharacteristic
33#2857C2212yes—
34#2851C2212yes—
35#2552C2212yes—
36#2542—21no—
37#2538—21no—
38#2538C2211yescharacteristic
39#2537C2211yescharacteristic
40#2496—21no—
41#2470—22no—
42#2466—22no—
43#2441C2214yes—
44#2429—22no—
45#2423—22no—
46#2422—21no—
47#2421C2212yes—
48#2420C2212yes—
49#2365—21no—
50#2361—21no—
51#2354C2212yes—
52Q8×C4⋊C4—21no—
53Q8×C4⋊C4C2211yescharacteristic
54D4×C4⋊C4C2211yescharacteristic
55#2344—21no—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#124C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2088C2214yes—
60#2086C2214yes—
61#2085C2214yes—
62#2077C2214yes—
63#2076C2214yes—
64#2072C2212yes—
65#2071C2212yes—
66#2059C2214yes—
67#2059C2214yes—
68#2035C2214yes—
69#2033C2214yes—
70#1993C2212yes—
71#1993C2211yescharacteristic
72#1992C2212yes—
73#1959C2211yescharacteristic
74#1954C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1939C2212yes—
78#1936C2212yes—
79#1935C2212yes—
80#1924C2214yes—
81#1864C2214yes—
82#1863C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1847C2214yes—
86#1845C2212yes—
87#1844C2212yes—
88#1836C2211yescharacteristic
89#1803C2212yes—
90#1802C2212yes—
91#1767C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C4×#97C2211yescharacteristic
96#533—28no—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102#529—28no—
103#529—44no—
104C2×#126—22no—
105C2×#125C2312yes—
106C2×#125C2314yes—
107C2×#124C2311yescharacteristic
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117C2314yes—
128C22×C4⋊C4—21no—
129C22×C42C2312yes—
130#501—28no—
131#501—28no—
132#501—44no—
133#489—28no—
134#489D418yescomplemented
135#485—216no—
136#482—216no—
137#479—216no—
138#478—216no—
139#432—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×#97C2311yescharacteristic
159C2×#97C2311yescharacteristic
160C2×C4×C8C2312yes—
161#400—22no—
162#399—24no—
163#398—24no—
164#397—22no—
165#397—22no—
166#396—24no—
167#396—22no—
168#394—22no—
169#394—22no—
170#391—21no—
171#391—22no—
172#391C2311yescharacteristic
173#390—22no—
174#390—21no—
175#390C2311yescharacteristic
176#389—21no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—24no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2314yes—
190#388—24no—
191#388—22no—
192#388—21no—
193#388—21no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—24no—
198#386—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#384—22no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2312yes—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#382C2311yescharacteristic
212#382C2312yes—
213#381—22no—
214#380—22no—
215#380—21no—
216#380—21no—
217#380C2312yes—
218#380C2314yes—
219#380C2312yes—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C22⋊C4—21no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4C2312yes—
232C43C2311yescharacteristic
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8C2411yescharacteristic
246C22×D4—22no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8C2×D414yes—
263#126—48no—
264#125—28no—
265#125—24no—
266#125—24no—
267#125—44no—
268#124—22no—
269#124C2×D414yes—
270#123—48no—
271C22⋊Q8—24no—
272C22⋊Q8—42no—
273C22⋊Q8—48no—
274C22⋊Q8—48no—
275C22⋊Q8—84no—
276#121—24no—
277#121—42no—
278#121—44no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—48no—
283C4×Q8—48no—
284C4×Q8—44no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—48no—
292#117—28no—
293#117—48no—
294#117—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—28no—
361#107—28no—
362#107—44no—
363#106—28no—
364#106—28no—
365#106—44no—
366C4⋊C8—28no—
367C4⋊C8—24no—
368#103—28no—
369#103—24no—
370#103C2×D418yes—
371#102—28no—
372#102—24no—
373#102C2×D418yes—
374C22⋊C8—28no—
375C22⋊C8—28no—
376#97—24no—
377#97—24no—
378#97—22no—
379#97C2×D414yes—
380C4×C8—24no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24C22×D412yes—
400C2×Q8—24no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8C22×D412yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—82no—
419C2×D4C22×D412yes—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8C22×D414yes—
462C2×C82−1+414yes—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4C22×D414yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—82no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—41no—
524C42—41no—
525C42—48no—
526C42—44no—
527C42—42no—
528C42—44no—
529C42C22×D412yes—
530C42H(5,𝔽2)12yes—
531C422−1+414yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C23×D411yescharacteristic
547Q8—24no—
548Q8—48no—
549Q8—42no—
550Q8—88no—
551D4—24no—
552D4—44no—
553D4—42no—
554D4—88no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—88no—
605C2×C4—84no—
606C2×C4#53612yes—
607C2×C4#53612yes—
608C2×C4D4212yes—
609C2×C4D4212yes—
610C2×C4#57512yes—
611C2×C4#57512yes—
612C2×C4C23×D411yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×2−1+412yes—
615C2×C4C2×2−1+412yes—
616C8—24no—
617C8—24no—
618C8—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×#53611yescharacteristic
634C22C2×D4211yescharacteristic
635C22#349911yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57511yescharacteristic
638C22#358511yescharacteristic
639C22#358611yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4—81no—
655C4—81no—
656C4#324014yescomplemented
657C4#349912yes—
658C4#355212yes—
659C2—82no—
660C2—82no—
661C2#3936011yescharacteristic
662C2C2×#324011yescharacteristic
663C2#6088711yescharacteristic
664C2#6106711yescharacteristic
665C2#6107011yescharacteristic
666C2#6250811yescharacteristic
667C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3240C2semidirect
2. #36134C2semidirect
4. #33316C2semidirect
5. #31106C2semidirect
6. #31101C2semidirect
7. #29884C2semidirect
8. #28726C2semidirect
9. #28693C2semidirect
10. #26047C2semidirect
11. #26045C2semidirect
12. #25817C2semidirect
13. #25679C2semidirect
14. #25015C2semidirect
16. C4×#489C2semidirect
17. #21270C2semidirect
134. #489D4semidirect
656. C4#3240semidirect

Maximal subgroups

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

subgroupindex
C2×#32402
#361342
#361142
#333162
#311062
#311012
#298842
#287262
#286932
#260472
#260452
#258172
#256792
#250152
#250112
C4×#4892
#212702