Finite Group Databasebeta

← #6500779

Subgroups — #6500779

Order 512.

2480 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35513C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic, complemented
3#27584C212yesmaximal, complemented
4#27582C212yesmaximal, complemented
5#24886C214yesmaximal, complemented
6#24883C214yesmaximal, complemented
7#24874C214yesmaximal, complemented
8#24861C214yesmaximal, complemented
9#24015C212yesmaximal, complemented
10#24011C212yesmaximal, complemented
11#21947C212yesmaximal, complemented
12#21745C211yesmaximal, characteristic, complemented
13#20602C212yesmaximal, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#537C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#484C2212yescomplemented
19C2×#482C2214yescomplemented
20C2×#480C2212yescomplemented
21C2×#417C2212yes—
22C2×#418C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#408C2212yes—
26#2435—21no—
27#2433—21no—
28#2432—21no—
29#2431—21no—
30#2410—24no—
31#2408C2214yes—
32#2407C2214yes—
33#2379—21no—
34#2379—21no—
35#2369—21no—
36#2366—21no—
37#2354C2214yescomplemented
38D4×C4⋊C4C2214yescomplemented
39#2324—24no—
40#2311C2212yes—
41#2311C2211yescharacteristic
42#2311C2212yescomplemented
43#2297C2211yescharacteristic
44C2×#390C2211yescharacteristic
45#2285C2212yescomplemented
46#2095C2212yescomplemented
47#2094C2212yescomplemented
48#2093C2214yescomplemented
49#2072C2212yescomplemented
50#2071C2212yescomplemented
51#2071C2214yescomplemented
52#2036C2214yescomplemented
53#2034C2214yescomplemented
54#2033C2214yescomplemented
55#2033C2214yescomplemented
56#1927C2214yescomplemented
57#1925C2214yescomplemented
58#1925C2214yescomplemented
59#1925C2214yescomplemented
60#1875C2212yescomplemented
61#1874C2214yescomplemented
62#1873C2212yescomplemented
63#1870C2214yescomplemented
64#1869C2214yescomplemented
65#1854C2214yescomplemented
66#1854C2214yescomplemented
67#1852C2214yescomplemented
68#1852C2214yescomplemented
69#1817C2212yescomplemented
70#1809C2214yescomplemented
71#1808C2214yescomplemented
72#1788C2214yes—
73#1788C2214yes—
74#1786C2214yes—
75#1786C2214yes—
76#1780C2212yescomplemented
77C2×H(5,𝔽2)—28no—
78C2×H(5,𝔽2)—42no—
79C2×H(5,𝔽2)—44no—
80C2×H(5,𝔽2)—42no—
81C22×#41—21no—
82C22×#41C2312yes—
83C22×#41C2311yescharacteristic
84C22×#41C2311yescharacteristic
85C22×#41C2311yescharacteristic
86C23×D4—22no—
87C23×D4—21no—
88C23×D4C2312yes—
89C23×D4C2311yescharacteristic
90C22×M16C2312yes—
91C23×C8C2312yes—
92#537—42no—
93#537—42no—
94#535—44no—
95#532—28no—
96C2×C4⋊Q8D411yescharacteristic
97C2×#127—21no—
98C2×#127D411yescharacteristic
99C2×#124—21no—
100C2×#124D411yescharacteristic
101C2×#124D411yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×#121—22no—
105C2×#121—21no—
106C2×#121D414yes—
107C2×#121C2312yes—
108C2×#121C2314yes—
109C2×#120—22no—
110C2×#120—24no—
111C2×C4×D4—24no—
112C2×C4×D4C2314yes—
113C2×#117—22no—
114C2×#117C2312yes—
115C2×#117C2311yescharacteristic
116C2×#117C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—22no—
121C22×C42C2311yescharacteristic
122#484—44no—
123#482—48no—
124#480—44no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129#418—44no—
130#417—44no—
131#416—44no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102—44no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#408—28no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147#394—24no—
148#394—24no—
149#392—24no—
150#392—24no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391C2312yes—
155#390—22no—
156#390—21no—
157#390—22no—
158#390—24no—
159#390—21no—
160#390—21no—
161#390C2314yes—
162#390C2312yes—
163#389—24no—
164#388—24no—
165#388—24no—
166#386—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—24no—
173#385—24no—
174#385—24no—
175#385—24no—
176#384C2312yes—
177#384C2312yes—
178#382—22no—
179#380—22no—
180#380C2314yes—
181#380C2314yes—
182#380C2314yes—
183#380C2314yes—
184#379—24no—
185C4×C4⋊C4—24no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2312yes—
196#336C2312yes—
197C25—21no—
198C25—22no—
199H(5,𝔽2)—432no—
200H(5,𝔽2)—432no—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—24no—
209C2×#41—41no—
210C2×#41—41no—
211C2×#41—42no—
212C2×#41—42no—
213C2×#41—41no—
214C2×#41—41no—
215C2×#41—42no—
216C2×#41—42no—
217C2×#41—42no—
218C2×#41—44no—
219C2×#41—44no—
220C2×#41—44no—
221C2×#41#4118yes—
222C22×Q8C2×D411yescharacteristic
223C22×Q8C2×D411yescharacteristic
224C22×Q8C2411yescharacteristic
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—42no—
238C22×D4—41no—
239C22×D4—41no—
240C22×D4—42no—
241C22×D4—44no—
242C22×D4—48no—
243C22×D4—44no—
244C22×D4—48no—
245C22×D4—48no—
246C22×D4—44no—
247C22×D4C2×D414yes—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2411yescharacteristic
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—42no—
265C2×M16—42no—
266C22×C8—24no—
267C22×C8—42no—
268C22×C8—42no—
269C22×C8C2414yes—
270C22×C8C2414yes—
271C4⋊Q8—24no—
272#127—24no—
273#127—44no—
274#124—24no—
275#124—24no—
276#124—44no—
277C22⋊Q8—24no—
278C22⋊Q8—42no—
279C22⋊Q8—44no—
280#121—216no—
281#121—28no—
282#121—24no—
283#121—44no—
284#121—42no—
285#121—44no—
286#121—48no—
287#120—48no—
288#120—88no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—88no—
292#117—42no—
293#117—42no—
294#117—42no—
295#117—44no—
296#117—84no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—41no—
306C2×C4⋊C4—41no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—84no—
331C2×C22⋊C4C2×D414yes—
332C2×C22⋊C4C2×D414yes—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2412yes—
341#107—44no—
342#107—44no—
343#107—44no—
344#106—44no—
345#103—28no—
346#103—44no—
347#102—28no—
348#102—28no—
349#102—28no—
350#102—44no—
351#102—44no—
352#102—44no—
353C22⋊C8—28no—
354C22⋊C8—28no—
355C22⋊C8—42no—
356C22⋊C8—44no—
357C22⋊C8—42no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365C24—24no—
366C24—21no—
367C24—22no—
368C24—44no—
369C24—44no—
370C24—42no—
371C24—44no—
372C24—42no—
373C24—82no—
374C24—82no—
375C24—84no—
376C24—84no—
377C24C22×D412yes—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380#41—216no—
381#41—48no—
382#41—48no—
383#41—48no—
384#41—48no—
385#41—48no—
386#41—48no—
387#41—48no—
388#41—432no—
389#41—84no—
390#41—84no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8#12314yes—
400C2×Q8H(5,𝔽2)12yes—
401C2×D4—216no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—41no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—41no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—432no—
437C2×D4—464no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—48no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—84no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4#12318yes—
448C2×D4#12314yes—
449C2×D4H(5,𝔽2)12yes—
450C2×D4H(5,𝔽2)14yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4C2511yescharacteristic
491M16—84no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—84no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—42no—
535C42—48no—
536C42—82no—
537C42—82no—
538C23—22no—
539C23—24no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—24no—
553C23—416no—
554C23—48no—
555C23—42no—
556C23—44no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—44no—
562C23—42no—
563C23—88no—
564C23—164no—
565C23—168no—
566C23—164no—
567C23—164no—
568C23—168no—
569C23—162no—
570C23—162no—
571C23C2×#12314yes—
572C23C2×#12312yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—44no—
576Q8—42no—
577Q8—82no—
578D4—24no—
579D4—24no—
580D4—24no—
581D4—44no—
582D4—48no—
583D4—416no—
584D4—48no—
585D4—416no—
586D4—48no—
587D4—416no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—82no—
592D4—84no—
593D4—164no—
594D4—168no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—416no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—162no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4C2×#12312yes—
636C2×C4C2×#12314yes—
637C2×C4#57614yes—
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C8—44no—
643C8—44no—
644C22—22no—
645C22—24no—
646C22—24no—
647C22—28no—
648C22—41no—
649C22—42no—
650C22—41no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—48no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—168no—
670C22—1616no—
671C22C22×#12311yescharacteristic
672C22#348411yescharacteristic
673C22#348411yescharacteristic
674C22C2×#57612yes—
675C22#358612yes—
676C4—22no—
677C4—24no—
678C4—24no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—84no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4#349012yes—
687C4#349012yes—
688C2—42no—
689C2—44no—
690C2—82no—
691C2—82no—
692C2—84no—
693C2#3535011yescharacteristic
694C2#6052112yes—
695C2#6166111yescharacteristic
696C2#6167411yescharacteristic
697C2#6168412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35513C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34368C2semidirect
3. #27584C2semidirect
4. #27582C2semidirect
5. #24886C2semidirect
6. #24883C2semidirect
7. #24874C2semidirect
8. #24861C2semidirect
9. #24015C2semidirect
10. #24011C2semidirect
11. #21947C2semidirect
12. #21745C2semidirect
13. #20602C2semidirect
18. C2×#484C22semidirect
19. C2×#482C22semidirect
20. C2×#480C22semidirect
37. #2354C22semidirect
38. D4×C4⋊C4C22semidirect
42. #2311C22semidirect
45. #2285C22semidirect
46. #2095C22semidirect
47. #2094C22semidirect
48. #2093C22semidirect
49. #2072C22semidirect
50. #2071C22semidirect
51. #2071C22semidirect
52. #2036C22semidirect
53. #2034C22semidirect
54. #2033C22semidirect
55. #2033C22semidirect
56. #1927C22semidirect
57. #1925C22semidirect
58. #1925C22semidirect
59. #1925C22semidirect
60. #1875C22semidirect
61. #1874C22semidirect
62. #1873C22semidirect
63. #1870C22semidirect
64. #1869C22semidirect
65. #1854C22semidirect
66. #1854C22semidirect
67. #1852C22semidirect
68. #1852C22semidirect
69. #1817C22semidirect
70. #1809C22semidirect
71. #1808C22semidirect
76. #1780C22semidirect

Maximal subgroups

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

subgroupindex
#355132
#343682
#275842
#275822
#248862
#248832
#248742
#248612
#240152
#240112
#219472
#217452
#206022