Finite Group Databasebeta

← #6513419

Subgroups — #6513419

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3215C211yesmaximal, characteristic, complemented
2C22×#463C211yesmaximal, characteristic
3#35240C211yesmaximal, characteristic, complemented
4#34377C211yesmaximal, characteristic, complemented
5#28262C214yesmaximal, complemented
6#28172C214yesmaximal, complemented
7#25381C214yesmaximal, complemented
8#25334C214yesmaximal, complemented
9#24532C212yesmaximal, complemented
10#24530C212yesmaximal, complemented
11#24357C212yesmaximal, complemented
12#24355C212yesmaximal
13#21977C211yesmaximal, characteristic, complemented
14#21885C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C23×D8C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3215—28no—
22C2×#497C2212yescomplemented
23C2×#489C2212yescomplemented
24C2×#464C2212yescomplemented
25C2×#464C2212yescomplemented
26C2×#464C2214yescomplemented
27C2×#463—28no—
28C2×#463C2214yes—
29C2×#463C2214yes—
30C2×#463C2214yes—
31C2×#451C2212yes—
32C2×#450C2212yes—
33C2×#432C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37#2848—21no—
38#2848—21no—
39#2633—22no—
40#2547C2212yescomplemented
41#2542C2212yescomplemented
42#2536—22no—
43#2502C2212yes—
44#2482—22no—
45#2435C2212yes—
46#2379—22no—
47#2367C2214yescomplemented
48#2342—22no—
49Q8×C22⋊C4C2212yes—
50D4×C22⋊C4C2212yes—
51#2337—22no—
52#2326C2214yescomplemented
53#2326C2214yescomplemented
54#2315—21no—
55#2312—21no—
56#2285C2211yescharacteristic
57#2283C2211yescharacteristic
58#2072C2214yescomplemented
59#2071C2214yescomplemented
60#2056C2214yescomplemented
61#2055C2214yescomplemented
62#2055C2214yescomplemented
63#2054C2214yes—
64#2015C2214yescomplemented
65#2012C2214yes—
66#1987C2212yescomplemented
67#1981C2212yescomplemented
68#1940C2214yescomplemented
69#1892C2212yes—
70#1892C2212yes—
71#1880C2214yescomplemented
72#1879C2214yes—
73#1878C2214yescomplemented
74#1877C2214yescomplemented
75#1850C2212yes—
76#1850C2212yes—
77#1843C2212yescomplemented
78#1832C2212yescomplemented
79#1824C2212yes—
80#1803C2214yescomplemented
81#1803C2214yescomplemented
82#1802C2212yescomplemented
83#1802C2212yescomplemented
84#1802C2214yescomplemented
85#1786C2212yes—
86#1762C2214yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×Q16C2312yes—
92C22×Q16C2312yes—
93C22×SD16—24no—
94C22×SD16—44no—
95C22×SD16C2314yes—
96C22×SD16C2312yes—
97C22×SD16C2312yes—
98C22×D8—24no—
99C22×D8—44no—
100C22×D8C2314yes—
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#533—24no—
104#533—24no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—21no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#124—24no—
111C2×#124D418yes—
112C2×#124C2314yes—
113C2×#124C2312yes—
114C2×#124C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4—24no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2314yes—
129C2×#117—21no—
130C2×#117—21no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#497—28no—
138#489—28no—
139#464—28no—
140#464—28no—
141#464—48no—
142#463—216no—
143#463—216no—
144#463—48no—
145#451—28no—
146#450—28no—
147#432—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×#103—24no—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102—24no—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2312yes—
168C2×#97C2312yes—
169#399—24no—
170#398—24no—
171#397—24no—
172#394—24no—
173#391—24no—
174#391C2314yes—
175#391C2314yes—
176#390C2314yes—
177#390C2314yes—
178#388—24no—
179#388—24no—
180#388—24no—
181#388—24no—
182#388—24no—
183#388—24no—
184#386—24no—
185#385—24no—
186#385—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#385—24no—
191#385—24no—
192#384C2312yes—
193#384C2314yes—
194#383D412yes—
195#382—21no—
196#382D412yes—
197#381D412yes—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380D411yescharacteristic
203#380D411yescharacteristic
204#380C2312yes—
205#379—22no—
206#379—22no—
207#379—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4C2312yes—
214#376C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C25—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8C2×D414yes—
229C22×Q8C2412yes—
230C22×Q8C2412yes—
231C22×D4—24no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4—42no—
240C22×D4C2×D414yes—
241C22×D4C2412yes—
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4—44no—
249C23×C4—84no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×Q16—24no—
254C2×Q16—24no—
255C2×Q16—44no—
256C2×Q16—44no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×SD16—416no—
265C2×D8—28no—
266C2×D8—28no—
267C2×D8—44no—
268C2×D8—416no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—42no—
272C22×C8—42no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281#125—44no—
282#124—28no—
283#124—28no—
284#124—28no—
285#124—24no—
286#124—24no—
287#124—44no—
288#124C2×D418yes—
289#124C2×D418yes—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—44no—
293C22⋊Q8—84no—
294C22⋊Q8—84no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308#117—44no—
309#117—82no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—84no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D414yes—
355C2×C42#4114yes—
356C2×C42C2414yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—22no—
363C4⋊C8—44no—
364C4⋊C8—44no—
365#103—28no—
366#103—24no—
367#103—24no—
368#103—44no—
369#103—44no—
370#102—28no—
371#102—24no—
372#102—24no—
373#102—28no—
374#102—44no—
375C22⋊C8—24no—
376#97—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D414yes—
386#95C2×D414yes—
387#95C2×D414yes—
388C24—22no—
389C24—22no—
390C24—24no—
391C24—44no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—82no—
398C24—82no—
399C24—82no—
400C24C22×D411yescharacteristic
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—82no—
414C2×Q8—84no—
415C2×Q8C22×D414yes—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—84no—
433C2×D4—82no—
434C2×D4C22×D414yes—
435C2×D4H(5,𝔽2)14yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—816no—
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
472Q16—84no—
473Q16—84no—
474SD16—416no—
475SD16—84no—
476SD16—84no—
477D8—416no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—44no—
487C2×C8—44no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C42—24no—
525C42—24no—
526C42—44no—
527C42—82no—
528C42—84no—
529C42#12114yes—
530C42#12114yes—
531C42C22×D412yes—
532C42C22×D412yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—48no—
551C23—84no—
552C23—84no—
553C23—88no—
554C23—88no—
555C23—84no—
556C23—84no—
557C23—164no—
558C23—164no—
559C23C2×#12112yes—
560C23#57712yes—
561C23#57812yes—
562C23C23×D411yescharacteristic
563Q8—24no—
564Q8—24no—
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—84no—
570Q8—84no—
571D4—24no—
572D4—24no—
573D4—24no—
574D4—24no—
575D4—48no—
576D4—42no—
577D4—42no—
578D4—44no—
579D4—84no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—168no—
615C2×C4C2×#12112yes—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C23×D411yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4#58512yes—
624C8—42no—
625C8—42no—
626C8—42no—
627C8—42no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22—22no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—816no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22—82no—
645C22—168no—
646C22#304614yes—
647C22C22×#12111yescharacteristic
648C22#348812yes—
649C22C2×#57711yescharacteristic
650C22C2×#57811yescharacteristic
651C22#358512yes—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—84no—
663C4—82no—
664C4#348411yescharacteristic
665C4#348511yescharacteristic
666C4#349412yes—
667C2—24no—
668C2—84no—
669C2—84no—
670C2#3551411yescharacteristic
671C2C2×#304611yescharacteristic
672C2#6042611yescharacteristic
673C2#6201712yes—
674C2#6201812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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×#3215C2semidirect
3. #35240C2semidirect
4. #34377C2semidirect
5. #28262C2semidirect
6. #28172C2semidirect
7. #25381C2semidirect
8. #25334C2semidirect
9. #24532C2semidirect
10. #24530C2semidirect
11. #24357C2semidirect
13. #21977C2semidirect
14. #21885C2semidirect
15. #20738C2semidirect
22. C2×#497C22semidirect
23. C2×#489C22semidirect
24. C2×#464C22semidirect
25. C2×#464C22semidirect
26. C2×#464C22semidirect
40. #2547C22semidirect
41. #2542C22semidirect
47. #2367C22semidirect
52. #2326C22semidirect
53. #2326C22semidirect
58. #2072C22semidirect
59. #2071C22semidirect
60. #2056C22semidirect
61. #2055C22semidirect
62. #2055C22semidirect
64. #2015C22semidirect
66. #1987C22semidirect
67. #1981C22semidirect
68. #1940C22semidirect
71. #1880C22semidirect
73. #1878C22semidirect
74. #1877C22semidirect
77. #1843C22semidirect
78. #1832C22semidirect
80. #1803C22semidirect
81. #1803C22semidirect
82. #1802C22semidirect
83. #1802C22semidirect
84. #1802C22semidirect
86. #1762C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32152
C22×#4632
#352402
#343772
#282622
#281722
#253812
#253342
#245322
#245302
#243572
#243552
#219772
#218852
#207382