Finite Group Databasebeta

← #6519890

Subgroups — #6519890

Order 512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3255C211yesmaximal, characteristic, complemented
2C2×#2990C211yesmaximal, characteristic
3#35280C211yesmaximal, characteristic
4#34979C211yesmaximal, characteristic
5#28558C214yesmaximal, complemented
6#25479C212yesmaximal, complemented
7#25386C212yesmaximal, complemented
8#25341C212yesmaximal, complemented
9#25302C212yesmaximal, complemented
10#24838C212yesmaximal, complemented
11#24791C214yesmaximal, complemented
12#24752C212yesmaximal, complemented
13#24710C212yesmaximal, complemented
14#24660C212yesmaximal, complemented
15#21860C211yesmaximal, characteristic, complemented
16#21389C211yesmaximal, characteristic, complemented
17#21171C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3255—28no—
22C2×#475C2212yes—
23C2×#475C2212yes—
24C2×#470C2214yes—
25C2×#467C2214yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28#2990—28no—
29C2×#446C2212yes—
30C2×#435C2214yes—
31C2×C8×D4C2212yes—
32C2×#432C2212yes—
33C2×#431C2211yescharacteristic
34C2×#428C2212yes—
35C2×#423C2211yescharacteristic
36C22×C4⋊C8C2212yes—
37#2857C2212yes—
38#2857C2212yes—
39#2629—22no—
40#2611—22no—
41#2582—21no—
42#2582C2212yes—
43#2568C2211yescharacteristic
44#2547C2211yescharacteristic
45#2546C2211yescharacteristic
46#2542C2211yescharacteristic
47#2538—21no—
48#2538C2212yes—
49#2499—21no—
50#2432—21no—
51#2417—22no—
52#2417C2214yes—
53#2415C2212yes—
54#2415C2212yes—
55#2395—22no—
56#2316—21no—
57#2311—21no—
58C4×C4⋊Q8—21no—
59C4×#124—21no—
60C4×C22⋊Q8C2212yes—
61C4×C22⋊Q8C2212yes—
62C2×#384C2211yescharacteristic
63C4×#117C2211yescharacteristic
64#2088C2214yes—
65#2077C2214yes—
66#2068C2214yes—
67#2066C2214yes—
68#2056C2214yes—
69#2036C2214yes—
70#2035C2214yes—
71#2001C2214yes—
72#1990C2212yes—
73#1988C2211yescharacteristic
74#1988C2211yescharacteristic
75#1981C2211yescharacteristic
76#1981C2211yescharacteristic
77#1971C2214yes—
78#1956C2212yes—
79#1948C2212yes—
80#1948C2212yes—
81#1947C2212yes—
82#1947C2212yes—
83#1943C2214yes—
84#1935C2214yes—
85#1935C2214yes—
86#1892C2214yes—
87#1892C2214yes—
88#1873C2212yes—
89#1873C2212yes—
90#1872C2214yes—
91#1807C2214yes—
92#1803C2212yes—
93#1803C2212yes—
94C4×#103C2212yes—
95C4×#103C2212yes—
96#1736C2212yes—
97C22×#41D411yescharacteristic
98C22×Q16C2314yes—
99C22×M16C2312yes—
100C23×C8C2314yes—
101#533—24no—
102#533—24no—
103#533—24no—
104#533—24no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125—21no—
111C2×#125—21no—
112C2×#124—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2314yes—
121C4×#41—24no—
122C4×#41—24no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8D411yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4D412yes—
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117D411yescharacteristic
136C2×#117D412yes—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2312yes—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#475—28no—
145#475—28no—
146#470—216no—
147#467—48no—
148#462—28no—
149#462—28no—
150#446—28no—
151#435—216no—
152C8×D4—28no—
153#432—28no—
154#431—24no—
155#428—44no—
156C2×#107C2314yes—
157C2×#106C2314yes—
158C2×#106C2314yes—
159#423D418yes—
160C2×C4⋊C8—28no—
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2312yes—
174C2×C22⋊C8C2314yes—
175C2×C22⋊C8C2312yes—
176C2×#97C2314yes—
177C2×C4×C8C2312yes—
178#400—22no—
179#399—22no—
180#398—24no—
181#398—24no—
182#398—22no—
183#397—22no—
184#397—22no—
185#397—22no—
186#396—24no—
187#396—24no—
188#394—22no—
189#394—22no—
190#391—21no—
191#391—22no—
192#391—21no—
193#391—21no—
194#391—22no—
195#391C2312yes—
196#391C2314yes—
197#391C2312yes—
198#390—22no—
199#390—21no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389—22no—
205#389—22no—
206#389—22no—
207#389C2314yes—
208#389C2312yes—
209#389C2312yes—
210#388—22no—
211#388—22no—
212#387—22no—
213#387—22no—
214#386—24no—
215#385—22no—
216#385—22no—
217#384—21no—
218#384—21no—
219#384—21no—
220#384C2312yes—
221#384C2312yes—
222#382—21no—
223#380—22no—
224#380—22no—
225#380—22no—
226#380—24no—
227#380C2312yes—
228#380C2314yes—
229#380C2312yes—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235C4×C4⋊C4C2311yescharacteristic
236C4×C22⋊C4—21no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—21no—
240#376—21no—
241C43—21no—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—42no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8C2×D411yescharacteristic
256C22×Q8C2412yes—
257C22×Q8C2412yes—
258C22×D4C2×D412yes—
259C22×D4C2×D411yescharacteristic
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×M16—42no—
271C2×M16—42no—
272C2×M16C2×D418yes—
273C22×C8—28no—
274C22×C8—44no—
275C22×C8—44no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—44no—
288#125—44no—
289#125—44no—
290#124—44no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—42no—
298C22⋊Q8—42no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—84no—
304C4×Q8—24no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×D4—28no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—22no—
318C4×D4—22no—
319#117—24no—
320#117—28no—
321#117—22no—
322#117—22no—
323#117—41no—
324#117—41no—
325#117—44no—
326#117—44no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—82no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D414yes—
391C2×C42#4114yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—28no—
396#106—28no—
397#106—44no—
398C4⋊C8—28no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401C4⋊C8C2×D418yes—
402C4⋊C8C2×D418yes—
403#103—28no—
404#103—28no—
405#103—24no—
406#103—24no—
407#103—42no—
408#103—42no—
409#103—44no—
410#103—44no—
411C22⋊C8—24no—
412C22⋊C8—28no—
413#97—28no—
414C4×C8—24no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431C24C22×D411yescharacteristic
432C24C22×D412yes—
433#41—416no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4#13714yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4112yes—
505C22×C4C2511yescharacteristic
506Q16—88no—
507M16—48no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—88no—
517C2×C8C22×D418yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—24no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—84no—
554C4⋊C4—84no—
555C4⋊C4H(5,𝔽2)12yes—
556C4⋊C42−1+412yes—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—84no—
569C22⋊C4—84no—
570C42—24no—
571C42—24no—
572C42—22no—
573C42—22no—
574C42—42no—
575C42—42no—
576C42—44no—
577C42—44no—
578C42—42no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42#12114yes—
583C42#12114yes—
584C42C22×D412yes—
585C42C22×D412yes—
586C42C2×#4112yes—
587C42C2×#4112yes—
588C23—22no—
589C23—22no—
590C23—21no—
591C23—22no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—22no—
596C23—22no—
597C23—21no—
598C23—22no—
599C23—21no—
600C23C2×#12112yes—
601C23C2×#13714yes—
602C23C23×D411yescharacteristic
603Q8—44no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—84no—
613Q8—84no—
614D4—28no—
615D4—48no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—48no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—164no—
671C2×C4C2×#12112yes—
672C2×C4C2×#12112yes—
673C2×C4C2×#12112yes—
674C2×C4C2×#12112yes—
675C2×C4C2×#12112yes—
676C2×C4C2×#13712yes—
677C2×C4#57512yes—
678C2×C4#57712yes—
679C2×C4#57812yes—
680C2×C4C23×D411yescharacteristic
681C2×C4C22×#4111yescharacteristic
682C2×C4C2×H(5,𝔽2)11yescharacteristic
683C2×C4C2×2−1+411yescharacteristic
684C2×C4#58511yescharacteristic
685C2×C4#58511yescharacteristic
686C8—28no—
687C8—42no—
688C8—42no—
689C22—22no—
690C22—24no—
691C22—24no—
692C22—22no—
693C22—24no—
694C22—22no—
695C22—41no—
696C22—41no—
697C22—42no—
698C22—42no—
699C22—41no—
700C22—41no—
701C22—42no—
702C22—42no—
703C22C22×#12111yescharacteristic
704C22#348811yescharacteristic
705C22#348911yescharacteristic
706C22C22×#13711yescharacteristic
707C22C2×#57511yescharacteristic
708C22C2×#57711yescharacteristic
709C22C2×#57811yescharacteristic
710C4—22no—
711C4—22no—
712C4—22no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—82no—
719C4—81no—
720C4—84no—
721C4—82no—
722C4—81no—
723C4—82no—
724C4—82no—
725C4#305614yes—
726C4#306314yes—
727C4#349611yescharacteristic
728C4#349711yescharacteristic
729C4#351111yescharacteristic
730C4#351211yescharacteristic
731C2—42no—
732C2—44no—
733C2#3556411yescharacteristic
734C2C2×#305611yescharacteristic
735C2C2×#306311yescharacteristic
736C2#6198411yescharacteristic
737C2#6198611yescharacteristic
738C2#6213111yescharacteristic
739C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3255C2semidirect
5. #28558C2semidirect
6. #25479C2semidirect
7. #25386C2semidirect
8. #25341C2semidirect
9. #25302C2semidirect
10. #24838C2semidirect
11. #24791C2semidirect
12. #24752C2semidirect
13. #24710C2semidirect
14. #24660C2semidirect
15. #21860C2semidirect
16. #21389C2semidirect
17. #21171C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32552
C2×#29902
#352802
#349792
#285582
#254792
#253862
#253412
#253022
#248382
#247912
#247522
#247102
#246602
#218602
#213892
#211712