Finite Group Databasebeta

← #6477731

Subgroups — #6477731

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C2×#2924C211yesmaximal, characteristic, complemented
3#35311C211yesmaximal, characteristic, complemented
4C2×#2624C211yesmaximal, characteristic
5#33142C214yesmaximal, complemented
6#29787C214yesmaximal, complemented
7#24612C214yesmaximal, complemented
8#24606C212yesmaximal, complemented
9#24602C212yesmaximal, complemented
10#24518C214yesmaximal, complemented
11#24507C212yesmaximal, complemented
12#24501C212yesmaximal, complemented
13#21944C211yesmaximal, characteristic, complemented
14#21761C211yesmaximal, characteristic
15#21759C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3193—28no—
20C2×#484C2212yes—
21C2×#480C2212yes—
22C2×#468C2214yes—
23C2×#448C2212yes—
24C2×#450C2212yes—
25#2924—28no—
26C2×#433C2214yescomplemented
27C2×#432C2212yes—
28C2×#431C2212yes—
29C2×#424C2212yescomplemented
30C2×#407C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C2×#405C2212yescomplemented
33#2855C2212yes—
34#2855C2212yes—
35#2625C2212yescomplemented
36#2625C2212yescomplemented
37#2624—21no—
38#2624—21no—
39#2624—22no—
40#2624C2214yes—
41#2624C2212yes—
42#2624C2212yes—
43#2503—21no—
44#2502—21no—
45#2499—22no—
46#2484—21no—
47#2484—21no—
48#2483—22no—
49C2×#399—21no—
50C2×#397—21no—
51C2×#394—22no—
52#2317C2212yescomplemented
53#2310C2212yescomplemented
54C2×#390C2212yes—
55C2×#389C2212yes—
56#2280C2211yescharacteristic
57C2×#380C2211yescharacteristic
58#2087C2214yescomplemented
59#2086C2214yescomplemented
60#2068C2214yescomplemented
61#2067C2214yescomplemented
62#2066C2214yescomplemented
63#2065C2214yescomplemented
64#2055C2214yescomplemented
65#2055C2214yescomplemented
66#1973C2214yescomplemented
67#1968C2214yescomplemented
68#1944C2214yescomplemented
69#1942C2214yescomplemented
70#1936C2214yescomplemented
71#1935C2214yescomplemented
72#1934C2214yescomplemented
73#1933C2214yescomplemented
74#1893C2214yes—
75#1892C2214yescomplemented
76#1875C2214yes—
77#1872C2212yes—
78#1872C2212yes—
79#1854C2214yescomplemented
80#1853C2212yes—
81#1852C2212yes—
82#1824C2211yescharacteristic
83#1824C2211yescharacteristic
84#1824C2212yescomplemented
85#1809C2214yescomplemented
86#1787C2212yescomplemented
87#1786C2211yescharacteristic
88#1786C2211yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C23×C8C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—24no—
105C2×#121—24no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117D411yescharacteristic
111C2×#117D411yescharacteristic
112C2×#117D412yes—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4D412yes—
130C22×C42C2312yes—
131#484—28no—
132#480—28no—
133#468—216no—
134#450—28no—
135#448—28no—
136#433—216no—
137#432—28no—
138#431—28no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142#424—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#407D418yes—
152C2×C22⋊C8—24no—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158#405—28no—
159C2×#97C2314yes—
160C2×C4×C8C2314yes—
161#400—22no—
162#400—22no—
163#399—24no—
164#399—22no—
165#399—22no—
166#398—22no—
167#398—22no—
168#397—24no—
169#397—22no—
170#397—22no—
171#397—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#394—24no—
176#394—24no—
177#394—24no—
178#394—24no—
179#391—22no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#390—22no—
186#390—22no—
187#390C2314yes—
188#390C2314yes—
189#389—22no—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2314yes—
196#389C2314yes—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#384—22no—
206#382—22no—
207#381—22no—
208#381—22no—
209#381—21no—
210#381—22no—
211#381—22no—
212#380—21no—
213#380—21no—
214#380C2312yes—
215#380C2312yes—
216#380C2311yescharacteristic
217#380C2311yescharacteristic
218#380C2312yes—
219#379—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95—21no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230C25—21no—
231C25C2×D411yescharacteristic
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—44no—
236C22×Q8C2412yes—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—84no—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—44no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4#4114yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×SD16—28no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×M16—44no—
265C2×M16C2×D418yes—
266C22×C8—24no—
267C22×C8—44no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121—44no—
282#121—44no—
283#117—24no—
284#117—24no—
285#117—28no—
286#117—22no—
287#117—22no—
288#117—24no—
289#117—24no—
290#117—22no—
291#117—22no—
292#117—44no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—84no—
342C2×C22⋊C4—82no—
343C2×C22⋊C4—84no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355#107—28no—
356#107—44no—
357#106—44no—
358C4⋊C8—28no—
359C4⋊C8—28no—
360#103—28no—
361#103—44no—
362#103—44no—
363#102—28no—
364#102—44no—
365#102—44no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—24no—
369C22⋊C8C2×D418yes—
370C22⋊C8C2×D418yes—
371#97—28no—
372C4×C8—28no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—44no—
398C24—44no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24#12114yes—
403C24C22×D411yescharacteristic
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—84no—
413C2×Q8—84no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—816no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4#12114yes—
457C22×C4#12114yes—
458C22×C4#12114yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473SD16—88no—
474M16—48no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—48no—
482C2×C8C22×D418yes—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—44no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—88no—
556C23—88no—
557C23—84no—
558C23C2×#12112yes—
559C23C2×#12112yes—
560C23C2×#12112yes—
561C23#57712yes—
562C23#57812yes—
563C23C23×D411yescharacteristic
564Q8—44no—
565Q8—44no—
566Q8—84no—
567D4—44no—
568D4—42no—
569D4—42no—
570D4—84no—
571D4—88no—
572D4—88no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4#58514yes—
626C8—28no—
627C8—42no—
628C8—42no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—24no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—88no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—168no—
650C22#306314yes—
651C22C22×#12111yescharacteristic
652C22#344912yes—
653C22C2×#57711yescharacteristic
654C22C2×#57811yescharacteristic
655C22#358512yes—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—84no—
668C4—82no—
669C4#348512yes—
670C4#349112yes—
671C2—24no—
672C2—42no—
673C2—42no—
674C2—162no—
675C2#3508911yescharacteristic
676C2C2×#306311yescharacteristic
677C2#6045411yescharacteristic
678C2#6097912yes—
679C2#6098112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2924C2semidirect
3. #35311C2semidirect
5. #33142C2semidirect
6. #29787C2semidirect
7. #24612C2semidirect
8. #24606C2semidirect
9. #24602C2semidirect
10. #24518C2semidirect
11. #24507C2semidirect
12. #24501C2semidirect
13. #21944C2semidirect
15. #21759C2semidirect
26. C2×#433C22semidirect
29. C2×#424C22semidirect
32. C2×#405C22semidirect
35. #2625C22semidirect
36. #2625C22semidirect
52. #2317C22semidirect
53. #2310C22semidirect
58. #2087C22semidirect
59. #2086C22semidirect
60. #2068C22semidirect
61. #2067C22semidirect
62. #2066C22semidirect
63. #2065C22semidirect
64. #2055C22semidirect
65. #2055C22semidirect
66. #1973C22semidirect
67. #1968C22semidirect
68. #1944C22semidirect
69. #1942C22semidirect
70. #1936C22semidirect
71. #1935C22semidirect
72. #1934C22semidirect
73. #1933C22semidirect
75. #1892C22semidirect
79. #1854C22semidirect
84. #1824C22semidirect
85. #1809C22semidirect
86. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#29242
#353112
C2×#26242
#331422
#297872
#246122
#246062
#246022
#245182
#245072
#245012
#219442
#217612
#217592