Finite Group Databasebeta

← #6453000

Subgroups — #6453000

Order 512.

3540 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2C2×#3094C211yesmaximal, characteristic, complemented
3C2×#2911C211yesmaximal, characteristic, complemented
4#34809C211yesmaximal, characteristic, complemented
5#24102C214yesmaximal, complemented
6#24101C214yesmaximal, complemented
7#24090C218yesmaximal, complemented
8#23553C212yesmaximal, complemented
9#23480C212yesmaximal, complemented
10#20973C211yesmaximal, characteristic, complemented
11#20898C211yesmaximal, characteristic, complemented
12#20873C212yesmaximal, complemented
13#20844C212yesmaximal, complemented
14#20701C211yesmaximal, characteristic, complemented
15C23×M16C2211yescharacteristic
16C2×#553—28no—
17C2×#553—22no—
18C2×#553—22no—
19C2×#553—28no—
20C2×#553C2214yes—
21C2×#553C2214yes—
22C22×#126—21no—
23C22×#125C2211yescharacteristic
24C22×#123—21no—
25C22×#121—21no—
26C22×#121C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C22×#117C2211yescharacteristic
33C23×C4⋊C4C2211yescharacteristic
34#3094—28no—
35C2×#481C2212yescomplemented
36C2×#482C2214yescomplemented
37C2×#480C2212yescomplemented
38#2911—28no—
39C2×#428C2212yes—
40C2×#428C2212yes—
41C2×#428C2212yes—
42C2×#428C2212yescomplemented
43C2×#428C2212yescomplemented
44C2×#427C2214yescomplemented
45C2×#418C2212yes—
46C2×#418C2212yes—
47C2×#407C2211yescharacteristic
48C2×#407C2212yes—
49#2487C2212yescomplemented
50#2486C2212yescomplemented
51#2485C2214yescomplemented
52#2484—21no—
53#2483—21no—
54#2482—21no—
55#2405—21no—
56#2395—22no—
57C4×#123—22no—
58#2296C2212yes—
59#2280C2212yes—
60#2280C2211yescharacteristic
61#2059C2218yescomplemented
62#2059C2218yescomplemented
63#2034C2218yescomplemented
64#2033C2218yescomplemented
65#1895C2214yescomplemented
66#1893C2214yescomplemented
67#1870C2214yescomplemented
68#1869C2214yescomplemented
69#1864C2218yescomplemented
70#1862C2218yescomplemented
71#1853C2218yescomplemented
72#1852C2218yescomplemented
73#1825C2214yescomplemented
74#1825C2214yescomplemented
75#1736C2212yes—
76#1736C2212yes—
77#1736C2212yes—
78#1728C2212yes—
79C23×D4—21no—
80C23×D4—21no—
81C23×D4D411yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C22×M16—24no—
88C22×M16—28no—
89C22×M16C2312yes—
90C22×M16C2312yes—
91C22×M16C2312yes—
92C23×C8C2312yes—
93#553—232no—
94#553—232no—
95#553—216no—
96#553—216no—
97#553—216no—
98#553—216no—
99C2×#126—216no—
100C2×#126—24no—
101C2×#126—24no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125—22no—
105C2×#125C2314yes—
106C2×#123—216no—
107C2×#123—22no—
108C2×#123—24no—
109C2×#123—24no—
110C2×#121—216no—
111C2×#121—24no—
112C2×#121—24no—
113C2×#121—24no—
114C2×#121—24no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—22no—
118C2×#121—24no—
119C2×#121D418yes—
120C2×#121C2314yes—
121C2×#121C2312yes—
122C2×#121C2312yes—
123C2×C4×D4—216no—
124C2×C4×D4—24no—
125C2×C4×D4—24no—
126C2×C4×D4—24no—
127C2×C4×D4—24no—
128C2×C4×D4—24no—
129C2×C4×D4—24no—
130C2×C4×D4—22no—
131C2×C4×D4D412yes—
132C2×C4×D4D412yes—
133C2×C4×D4C2314yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×#117—24no—
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117—21no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117D418yes—
144C2×#117C2312yes—
145C2×#117C2312yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C2×#117C2312yes—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—22no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4—24no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2311yescharacteristic
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4—21no—
167C22×C22⋊C4D411yescharacteristic
168C22×C22⋊C4D411yescharacteristic
169C22×C42—21no—
170C22×C42—21no—
171C22×C42C2311yescharacteristic
172#482—216no—
173#481—28no—
174#480—28no—
175#428—28no—
176#428—28no—
177#428—28no—
178#428—28no—
179#428—28no—
180#427—216no—
181C2×#107C2314yes—
182C2×#107C2314yes—
183C2×#107C2314yes—
184C2×#106C2314yes—
185C2×#106C2314yes—
186C2×#106C2314yes—
187#418—28no—
188#418—28no—
189C2×#102C2314yes—
190C2×#102C2314yes—
191C2×#102C2318yes—
192#407—24no—
193#407—28no—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2318yes—
197#400—22no—
198#399—22no—
199#398—22no—
200#397—22no—
201#397—24no—
202#394—24no—
203#389C2314yes—
204#389C2314yes—
205#388—24no—
206#387—24no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#381—21no—
212#381—22no—
213#381—22no—
214#380C2312yes—
215#380C2312yes—
216#380C2314yes—
217#380C2314yes—
218#379—22no—
219#379—21no—
220C4×C4⋊C4—24no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4—21no—
224#340C2318yes—
225#340C2318yes—
226#340C2318yes—
227#336C2318yes—
228C25—21no—
229C25C2×D411yescharacteristic
230C25C2×D411yescharacteristic
231C22×D4—28no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—28no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—44no—
242C22×D4—48no—
243C22×D4—84no—
244C22×D4C2×D414yes—
245C22×D4C2×D412yes—
246C22×D4C2×D412yes—
247C22×D4C2412yes—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—28no—
254C23×C4—28no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—84no—
264C23×C4—84no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D414yes—
271C23×C4C2×D414yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×M16—232no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16—24no—
282C2×M16—24no—
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—28no—
286C2×M16#4118yes—
287C22×C8—24no—
288C22×C8—28no—
289C22×C8C2414yes—
290C22×C8C2414yes—
291#126—232no—
292#126—232no—
293#125—216no—
294#125—28no—
295#125—28no—
296#123—232no—
297#123—232no—
298#123—48no—
299#121—232no—
300#121—232no—
301#121—216no—
302#121—28no—
303#121—28no—
304#121—28no—
305#121—24no—
306#121—24no—
307#121—28no—
308#121C2×D418yes—
309#121C2×D418yes—
310C4×D4—232no—
311C4×D4—232no—
312C4×D4—216no—
313C4×D4—28no—
314C4×D4—28no—
315C4×D4—28no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—28no—
321#117—28no—
322#117—28no—
323#117—24no—
324#117—24no—
325#117—24no—
326#117—24no—
327#117—24no—
328#117—24no—
329#117—24no—
330#117—24no—
331#117—44no—
332#117—44no—
333#117C2×D418yes—
334#117C2×D418yes—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—216no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—44no—
390C2×C22⋊C4C2×D412yes—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D412yes—
393C2×C22⋊C4C2×D412yes—
394C2×C42—28no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—44no—
403C2×C42C2×D412yes—
404C2×C42C2412yes—
405#107—28no—
406#107—28no—
407#107—28no—
408#106—28no—
409#106—28no—
410#106—28no—
411#102—28no—
412#102—28no—
413#102—216no—
414C22⋊C8—24no—
415C22⋊C8—24no—
416C22⋊C8—216no—
417#95—24no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—24no—
423C24—22no—
424C24—24no—
425C24—24no—
426C24—24no—
427C24—24no—
428C24—22no—
429C24—42no—
430C24—42no—
431C24—42no—
432C24—44no—
433C24—44no—
434C24—44no—
435C24#12014yes—
436C24C22×D411yescharacteristic
437C24C22×D411yescharacteristic
438C24C22×D412yes—
439C24C22×D411yescharacteristic
440C2×D4—216no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—28no—
447C2×D4—24no—
448C2×D4—216no—
449C2×D4—22no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—416no—
455C2×D4—432no—
456C2×D4—48no—
457C2×D4—48no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—84no—
462C2×D4—84no—
463C2×D4—816no—
464C2×D4C22×D414yes—
465C22×C4—24no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—216no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—216no—
486C22×C4—216no—
487C22×C4—24no—
488C22×C4—28no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—88no—
508C22×C4—82no—
509C22×C4—88no—
510C22×C4—816no—
511C22×C4—816no—
512C22×C4#12014yes—
513C22×C4#12014yes—
514C22×C4#12014yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C2×#4114yes—
528C22×C42−1+414yes—
529C22×C4C2511yescharacteristic
530M16—216no—
531M16—216no—
532M16—48no—
533C2×C8—216no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—24no—
539C2×C8C2×#4118yes—
540C4⋊C4—216no—
541C4⋊C4—216no—
542C4⋊C4—216no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—48no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—84no—
561C4⋊C4C22×D414yes—
562C4⋊C4C22×D414yes—
563C4⋊C4C22×D414yes—
564C22⋊C4—216no—
565C22⋊C4—216no—
566C22⋊C4—28no—
567C22⋊C4—28no—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C22⋊C4—28no—
571C22⋊C4—28no—
572C22⋊C4—24no—
573C22⋊C4—24no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—48no—
579C22⋊C4—82no—
580C22⋊C4—84no—
581C42—28no—
582C42—28no—
583C42—24no—
584C42—22no—
585C42—22no—
586C42—44no—
587C42—44no—
588C42—44no—
589C23—22no—
590C23—22no—
591C23—28no—
592C23—24no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—24no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—28no—
603C23—24no—
604C23—44no—
605C23—44no—
606C23—416no—
607C23—44no—
608C23—48no—
609C23—48no—
610C23—44no—
611C23—44no—
612C23—48no—
613C23—44no—
614C23—44no—
615C23—88no—
616C23—816no—
617C23C2×#12012yes—
618C23C2×#12012yes—
619C23C2×#12012yes—
620C23#54714yes—
621C23#57514yes—
622C23C23×D411yescharacteristic
623D4—28no—
624D4—48no—
625D4—48no—
626D4—432no—
627D4—48no—
628D4—42no—
629D4—42no—
630D4—88no—
631D4—88no—
632C2×C4—28no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—48no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—816no—
674C2×C4—816no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4C2×#12012yes—
678C2×C4C2×#12012yes—
679C2×C4C2×#12012yes—
680C2×C4#52514yes—
681C2×C4#53614yes—
682C2×C4#54714yes—
683C2×C4#54718yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C23×D411yescharacteristic
686C2×C4C22×#4112yes—
687C2×C4C2×2−1+412yes—
688C8—28no—
689C8—44no—
690C22—28no—
691C22—22no—
692C22—22no—
693C22—28no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—24no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—416no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—88no—
711C22—816no—
712C22—88no—
713C22#240614yes—
714C22C22×#12011yescharacteristic
715C22C2×#54712yes—
716C22C2×#57512yes—
717C22#358512yes—
718C4—22no—
719C4—22no—
720C4—42no—
721C4—42no—
722C4—44no—
723C4—42no—
724C4—44no—
725C4—44no—
726C4—84no—
727C4—82no—
728C4—82no—
729C4#240614yes—
730C4C2×#52511yescharacteristic
731C4C2×#53611yescharacteristic
732C4C2×#54712yes—
733C2—24no—
734C2—44no—
735C2—42no—
736C2—42no—
737C2—84no—
738C2C2×#240611yescharacteristic
739C2#6035911yescharacteristic
740C2#6036711yescharacteristic
741C2#6125514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3094C2semidirect
3. C2×#2911C2semidirect
4. #34809C2semidirect
5. #24102C2semidirect
6. #24101C2semidirect
7. #24090C2semidirect
8. #23553C2semidirect
9. #23480C2semidirect
10. #20973C2semidirect
11. #20898C2semidirect
12. #20873C2semidirect
13. #20844C2semidirect
14. #20701C2semidirect
35. C2×#481C22semidirect
36. C2×#482C22semidirect
37. C2×#480C22semidirect
42. C2×#428C22semidirect
43. C2×#428C22semidirect
44. C2×#427C22semidirect
49. #2487C22semidirect
50. #2486C22semidirect
51. #2485C22semidirect
61. #2059C22semidirect
62. #2059C22semidirect
63. #2034C22semidirect
64. #2033C22semidirect
65. #1895C22semidirect
66. #1893C22semidirect
67. #1870C22semidirect
68. #1869C22semidirect
69. #1864C22semidirect
70. #1862C22semidirect
71. #1853C22semidirect
72. #1852C22semidirect
73. #1825C22semidirect
74. #1825C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
C2×#30942
C2×#29112
#348092
#241022
#241012
#240902
#235532
#234802
#209732
#208982
#208732
#208442
#207012