Finite Group Databasebeta

← #6513464

Subgroups — #6513464

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic, complemented
2C2×#3039C211yesmaximal, characteristic
3#35200C211yesmaximal, characteristic
4#34371C211yesmaximal, characteristic, complemented
5#28261C212yesmaximal, complemented
6#28258C212yesmaximal, complemented
7#28170C212yesmaximal, complemented
8#28168C212yesmaximal, complemented
9#25370C212yesmaximal, complemented
10#25365C212yesmaximal, complemented
11#25324C212yesmaximal, complemented
12#25319C212yesmaximal, complemented
13#24540C214yesmaximal, complemented
14#24369C214yesmaximal
15#21975C211yesmaximal, characteristic, complemented
16#21892C211yesmaximal, characteristic, complemented
17#20736C211yesmaximal, characteristic
18C22×#137C2212yes—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22#3209—28no—
23C2×#494C2212yes—
24C2×#487C2212yes—
25#3039—28no—
26C2×#464C2214yes—
27C2×#462C2212yes—
28C2×#462C2212yes—
29C2×#462C2212yes—
30C2×#462C2212yes—
31C2×#462C2214yes—
32C2×#461C2212yes—
33C2×#461C2212yes—
34C2×#451C2214yes—
35C2×#431C2211yescharacteristic
36C2×#423C2211yescharacteristic
37C2×#419C2212yes—
38#2851—22no—
39#2638—22no—
40#2599—22no—
41#2567C2211yescharacteristic
42#2563C2211yescharacteristic
43#2547C2211yescharacteristic
44#2546C2211yescharacteristic
45#2499C2212yes—
46#2485—22no—
47#2432C2212yes—
48#2381—22no—
49#2365C2212yes—
50#2365C2212yes—
51#2345—22no—
52Q8×C22⋊C4C2214yes—
53#2331—22no—
54#2327C2212yes—
55#2327C2212yes—
56#2327C2212yes—
57#2326C2212yes—
58#2313—22no—
59#2283C2211yescharacteristic
60#2283C2211yescharacteristic
61#2074C2214yes—
62#2072C2214yes—
63#2056C2214yes—
64#2056C2214yes—
65#2056C2214yes—
66#2055C2214yes—
67#2016C2214yes—
68#2015C2214yes—
69#1987C2212yes—
70#1981C2212yes—
71#1939C2212yes—
72#1939C2212yes—
73#1894C2214yes—
74#1880C2214yes—
75#1878C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1851C2214yes—
79#1843C2212yes—
80#1832C2212yes—
81#1825C2212yes—
82#1805C2211yescharacteristic
83#1805C2211yescharacteristic
84#1805C2212yes—
85#1805C2212yes—
86#1805C2214yes—
87#1804C2211yescharacteristic
88#1804C2211yescharacteristic
89#1803C2214yes—
90#1788C2212yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93C22×#41C2311yescharacteristic
94C23×Q8C2312yes—
95C2×#137—28no—
96C2×#137—44no—
97C2×#137—44no—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102C22×M16C2312yes—
103#533—24no—
104#533—24no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—24no—
108C2×C4⋊Q8D418yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×#125—21no—
115C2×#124C2312yes—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C4×#41—24no—
119C4×#41—24no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2314yes—
125C2×C4×Q8C2314yes—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117—21no—
132C2×#117C2311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#494—28no—
138#487—28no—
139#464—48no—
140#462—28no—
141#462—28no—
142#462—28no—
143#462—28no—
144#462—48no—
145#461—28no—
146#461—28no—
147#451—216no—
148#431—24no—
149#423—24no—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2314yes—
156#419—28no—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C4×C8C2312yes—
169#398—22no—
170#397—24no—
171#397—22no—
172#395—22no—
173#394—24no—
174#393—22no—
175#391—24no—
176#391—24no—
177#391C2314yes—
178#391C2312yes—
179#391C2312yes—
180#391C2314yes—
181#391C2312yes—
182#390C2312yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#388—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#387—24no—
197#385—24no—
198#385—24no—
199#384C2312yes—
200#383—21no—
201#383D412yes—
202#382D412yes—
203#382C2312yes—
204#382C2311yescharacteristic
205#382C2311yescharacteristic
206#381—21no—
207#381D412yes—
208#380—22no—
209#380—22no—
210#380—24no—
211#380D412yes—
212#380C2312yes—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4C2312yes—
221#376—21no—
222#376C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2314yes—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—41no—
236C2×#41—41no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8C2×D418yes—
245C22×Q8C2412yes—
246C22×Q8C2414yes—
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257#137—432no—
258C2×Q16—28no—
259C2×Q16—28no—
260C2×Q16—28no—
261C2×Q16—28no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×SD16—28no—
268C2×SD16—28no—
269C2×SD16—44no—
270C2×M16—24no—
271C2×M16—42no—
272C2×M16—42no—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—24no—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—42no—
286C4⋊Q8C2×D418yes—
287C4⋊Q8C2×D418yes—
288#125—44no—
289#124—24no—
290#124—42no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—44no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—24no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—44no—
310C4×D4—44no—
311#117—22no—
312#117—22no—
313#117—44no—
314#117—44no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4C2×D414yes—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D414yes—
358C2×C42#4114yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C4⋊C8—24no—
364C4⋊C8—24no—
365C4⋊C8—22no—
366C4⋊C8—22no—
367C4⋊C8—44no—
368C4⋊C8—44no—
369#103—28no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—24no—
374#103—42no—
375#103—44no—
376#103—44no—
377#102—24no—
378#102—42no—
379C22⋊C8—24no—
380C4×C8—24no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95C2×D414yes—
396#95C2×D414yes—
397#95C2×D414yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—21no—
401C24C22×D411yescharacteristic
402#41—28no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—28no—
407C2×Q8—24no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—42no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—84no—
430C2×Q8—84no—
431C2×Q8C22×D418yes—
432C2×Q8H(5,𝔽2)12yes—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4H(5,𝔽2)12yes—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4#12114yes—
467C22×C4#12114yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478Q16—416no—
479Q16—88no—
480Q16—88no—
481SD16—416no—
482M16—48no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—48no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—84no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—44no—
536C42—84no—
537C42—84no—
538C42#12114yes—
539C42#12114yes—
540C42C22×D412yes—
541C42C22×D412yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C23—22no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—22no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23C2×#12112yes—
554C23C23×D411yescharacteristic
555Q8—28no—
556Q8—22no—
557Q8—28no—
558Q8—22no—
559Q8—41no—
560Q8—44no—
561Q8—44no—
562Q8—41no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—88no—
568Q8—84no—
569D4—22no—
570D4—22no—
571D4—48no—
572D4—41no—
573D4—41no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4#57814yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4#58511yescharacteristic
627C2×C4#58511yescharacteristic
628C8—44no—
629C8—42no—
630C8—42no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—82no—
641C22C22×#12111yescharacteristic
642C22#348811yescharacteristic
643C22#348811yescharacteristic
644C22C2×#57812yes—
645C22#358612yes—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—48no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#304414yes—
661C4#348411yescharacteristic
662C4#348511yescharacteristic
663C4#349411yescharacteristic
664C4#349411yescharacteristic
665C2—42no—
666C2—82no—
667C2#3551411yescharacteristic
668C2C2×#304411yescharacteristic
669C2#6042411yescharacteristic
670C2#6203412yes—
671C2#6203612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. C2×#3209C2semidirect
4. #34371C2semidirect
5. #28261C2semidirect
6. #28258C2semidirect
7. #28170C2semidirect
8. #28168C2semidirect
9. #25370C2semidirect
10. #25365C2semidirect
11. #25324C2semidirect
12. #25319C2semidirect
13. #24540C2semidirect
15. #21975C2semidirect
16. #21892C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C2×#30392
#352002
#343712
#282612
#282582
#281702
#281682
#253702
#253652
#253242
#253192
#245402
#243692
#219752
#218922
#207362