Finite Group Databasebeta

← #6517058

Subgroups — #6517058

Order 512.

2492 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3215C211yesmaximal, characteristic, complemented
2C2×#2974C211yesmaximal, characteristic, complemented
3#35240C211yesmaximal, characteristic, complemented
4C2×#2540C211yesmaximal, characteristic
5#28610C214yesmaximal, complemented
6#28492C214yesmaximal, complemented
7#25391C214yesmaximal, complemented
8#25389C214yesmaximal, complemented
9#24386C212yesmaximal, complemented
10#24385C212yesmaximal, complemented
11#24380C212yesmaximal, complemented
12#24379C212yesmaximal, complemented
13#21806C211yesmaximal, characteristic, complemented
14#21385C211yesmaximal, characteristic, complemented
15#21363C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21#3215—28no—
22C2×#497C2212yescomplemented
23C2×#489C2212yescomplemented
24C2×#470C2214yescomplemented
25C2×#468C2214yescomplemented
26C2×#467C2214yescomplemented
27C2×#465C2214yescomplemented
28C2×#464C2214yescomplemented
29C2×#451C2212yes—
30C2×#450C2212yes—
31#2974—28no—
32C2×#436C2212yescomplemented
33C2×#435C2214yescomplemented
34C2×C8×D4C2212yescomplemented
35C2×#432C2211yescharacteristic
36C2×#431C2211yescharacteristic
37C2×#407C2212yes—
38C22×C22⋊C8C2212yes—
39#2848C2212yes—
40#2848C2212yes—
41#2633—21no—
42#2633—21no—
43#2547C2212yescomplemented
44#2542C2212yescomplemented
45#2540—22no—
46#2540—21no—
47#2540—21no—
48#2540C2212yes—
49#2540C2212yes—
50#2540C2214yes—
51#2536C2212yescomplemented
52#2536C2212yescomplemented
53#2502—21no—
54#2482—22no—
55#2435—21no—
56C2×#399—21no—
57C2×#392—21no—
58#2326—22no—
59C2×#390—22no—
60#2283C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62#2036C2214yescomplemented
63#2035C2214yescomplemented
64#2034C2214yescomplemented
65#2033C2214yescomplemented
66#2001C2214yescomplemented
67#2001C2214yescomplemented
68#2000C2214yescomplemented
69#2000C2214yescomplemented
70#1987C2212yescomplemented
71#1983C2214yescomplemented
72#1981C2212yescomplemented
73#1958C2214yescomplemented
74#1949C2214yescomplemented
75#1895C2214yescomplemented
76#1894C2214yescomplemented
77#1893C2214yescomplemented
78#1892C2214yescomplemented
79#1872C2212yes—
80#1872C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1787C2211yescharacteristic
84#1787C2211yescharacteristic
85#1786C2211yescharacteristic
86#1786C2211yescharacteristic
87#1776C2214yescomplemented
88C4×#103C2214yescomplemented
89C23×Q8C2311yescharacteristic
90C23×D4—22no—
91C23×D4D411yescharacteristic
92C23×D4C2311yescharacteristic
93C24×C4C2312yes—
94C22×Q16C2314yes—
95C22×SD16C2314yes—
96C22×M16C2314yes—
97C23×C8C2314yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#124—22no—
104C2×#124—21no—
105C2×#124—21no—
106C2×#124—24no—
107C2×#124—24no—
108C2×#124C2312yes—
109C2×#124C2312yes—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2314yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×#121—22no—
122C2×#121—22no—
123C2×#121—48no—
124C2×#121C2314yes—
125C2×#121C2314yes—
126C2×C4×Q8—21no—
127C2×C4×Q8C2312yes—
128C2×C4×D4—24no—
129C2×C4×D4—24no—
130C2×C4×D4D412yes—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4D411yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×C4×D4C2312yes—
135C2×C4×D4C2311yescharacteristic
136C2×#117—21no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4D411yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C22⋊C4—22no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—21no—
144C22×C22⋊C4D412yes—
145C22×C42—21no—
146C22×C42C2311yescharacteristic
147#497—28no—
148#489—28no—
149#470—48no—
150#468—48no—
151#467—48no—
152#465—48no—
153#464—216no—
154#451—28no—
155#450—28no—
156#436—28no—
157#435—216no—
158C8×D4—28no—
159#432D418yescomplemented
160#431—24no—
161C2×#107C2314yes—
162C2×#106C2314yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173#407—28no—
174C2×C22⋊C8—28no—
175C2×C22⋊C8C2311yescharacteristic
176C2×C22⋊C8C2314yes—
177C2×C22⋊C8C2311yescharacteristic
178C2×C22⋊C8C2314yes—
179C2×C22⋊C8C2312yes—
180C2×#97C2312yes—
181C2×C4×C8C2312yes—
182#399—24no—
183#399—22no—
184#399—22no—
185#399—22no—
186#399—22no—
187#398—22no—
188#398—22no—
189#397—22no—
190#397—22no—
191#394—22no—
192#394—22no—
193#392—22no—
194#392—24no—
195#392—22no—
196#391—22no—
197#390—22no—
198#390—24no—
199#390—24no—
200#390—24no—
201#390—24no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#385—22no—
207#385—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#384—22no—
213#383—21no—
214#382C2312yes—
215#381—21no—
216#380C2311yescharacteristic
217#380C2311yescharacteristic
218#380C2312yes—
219#380C2312yes—
220#380C2312yes—
221#380C2312yes—
222#379—22no—
223#379—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—21no—
230#376—21no—
231C2×#95—21no—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236C25—21no—
237C25C2×D412yes—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8C2412yes—
244C22×Q8C2412yes—
245C22×D4—24no—
246C22×D4—24no—
247C22×D4—24no—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—44no—
252C22×D4—48no—
253C22×D4—44no—
254C22×D4—48no—
255C22×D4—84no—
256C22×D4C2×D411yescharacteristic
257C22×D4C2×D412yes—
258C22×D4C2×D411yescharacteristic
259C22×D4C2412yes—
260C23×C4—22no—
261C23×C4—24no—
262C23×C4—24no—
263C23×C4—21no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2412yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C2×Q16—28no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×SD16—28no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×M16—28no—
281C2×M16—44no—
282C2×M16—44no—
283C22×C8—28no—
284C22×C8—44no—
285C22×C8—44no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2414yes—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293#125—44no—
294#124—24no—
295#124—24no—
296#124—24no—
297#124—24no—
298#124—48no—
299#124—44no—
300#124—44no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—28no—
304C22⋊Q8—28no—
305C22⋊Q8—44no—
306C22⋊Q8—44no—
307C22⋊Q8—48no—
308C22⋊Q8—48no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314#121—28no—
315#121—28no—
316#121—44no—
317#121—44no—
318#121—48no—
319#121—48no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—44no—
323C4×D4—28no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—24no—
332#117—44no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—48no—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2414yes—
355C2×C22⋊C4—28no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—48no—
376C2×C22⋊C4—48no—
377C2×C22⋊C4—44no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—82no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D414yes—
391C2×C42#4114yes—
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#107—28no—
395#106—28no—
396C4⋊C8—24no—
397C4⋊C8C2×D418yescomplemented
398#103—28no—
399#103—28no—
400#103—28no—
401#103—44no—
402#103—44no—
403#102—28no—
404#102—44no—
405#102—44no—
406C22⋊C8—22no—
407C22⋊C8—22no—
408C22⋊C8—28no—
409C22⋊C8—28no—
410C22⋊C8C2×D418yes—
411#97C2×D418yescomplemented
412C4×C8—24no—
413#95—22no—
414#95—22no—
415#95—21no—
416#95—21no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424C24—24no—
425C24—24no—
426C24—24no—
427C24—24no—
428C24—22no—
429C24—42no—
430C24—42no—
431C24—42no—
432C24—44no—
433C24—44no—
434C24—42no—
435C24—42no—
436C24—42no—
437C24C22×D412yes—
438C24C22×D412yes—
439C24C22×D411yescharacteristic
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—84no—
453C2×Q8—84no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—48no—
466C2×D4—48no—
467C2×D4—44no—
468C2×D4—416no—
469C2×D4—48no—
470C2×D4—48no—
471C2×D4—42no—
472C2×D4—84no—
473C2×D4—84no—
474C2×D4—84no—
475C2×D4—84no—
476C2×D4—832no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—28no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—42no—
509C22×C4—88no—
510C22×C4—88no—
511C22×C4#12114yes—
512C22×C4#12114yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D412yes—
521C22×C4C2×#4112yes—
522C22×C4C2511yescharacteristic
523Q16—88no—
524SD16—88no—
525M16—88no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—24no—
531C2×C8—24no—
532C2×C8—88no—
533C2×C8C22×D418yes—
534C4⋊C4—28no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4H(5,𝔽2)18yes—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—44no—
579C42—44no—
580C42—82no—
581C42—82no—
582C42#12114yes—
583C42#12114yes—
584C42C22×D412yes—
585C42C22×D412yes—
586C42C2×#4112yes—
587C42C2×#4112yes—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—48no—
604C23—48no—
605C23—48no—
606C23—48no—
607C23—42no—
608C23—44no—
609C23—44no—
610C23—42no—
611C23—42no—
612C23—84no—
613C23—84no—
614C23—84no—
615C23—88no—
616C23—84no—
617C23—84no—
618C23—88no—
619C23—84no—
620C23C2×#12112yes—
621C23#57514yes—
622C23#57712yes—
623C23#57812yes—
624C23C23×D411yescharacteristic
625Q8—44no—
626Q8—44no—
627Q8—44no—
628Q8—42no—
629Q8—42no—
630Q8—84no—
631Q8—84no—
632D4—28no—
633D4—48no—
634D4—44no—
635D4—44no—
636D4—42no—
637D4—42no—
638D4—84no—
639D4—816no—
640D4—816no—
641C2×C4—22no—
642C2×C4—28no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—28no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—44no—
659C2×C4—416no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4C2×#12112yes—
685C2×C4C2×#12112yes—
686C2×C4C2×#12112yes—
687C2×C4C2×#12112yes—
688C2×C4C2×#12112yes—
689C2×C4C23×D411yescharacteristic
690C2×C4C22×#4111yescharacteristic
691C2×C4C2×H(5,𝔽2)14yes—
692C8—28no—
693C8—42no—
694C8—42no—
695C22—22no—
696C22—24no—
697C22—24no—
698C22—22no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—816no—
714C22—82no—
715C22—82no—
716C22—84no—
717C22—84no—
718C22—82no—
719C22—168no—
720C22#306314yes—
721C22C22×#12111yescharacteristic
722C22#348412yes—
723C22C2×#57512yes—
724C22C2×#57711yescharacteristic
725C22C2×#57811yescharacteristic
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—48no—
732C4—42no—
733C4—42no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4#349812yes—
740C4#350812yes—
741C2—24no—
742C2—44no—
743C2—82no—
744C2—162no—
745C2#3552811yescharacteristic
746C2#6044411yescharacteristic
747C2C2×#306311yescharacteristic
748C2#6160112yes—
749C2#6160312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2540C2
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
1. C2×#3215C2semidirect
2. C2×#2974C2semidirect
3. #35240C2semidirect
5. #28610C2semidirect
6. #28492C2semidirect
7. #25391C2semidirect
8. #25389C2semidirect
9. #24386C2semidirect
10. #24385C2semidirect
11. #24380C2semidirect
12. #24379C2semidirect
13. #21806C2semidirect
14. #21385C2semidirect
15. #21363C2semidirect
22. C2×#497C22semidirect
23. C2×#489C22semidirect
24. C2×#470C22semidirect
25. C2×#468C22semidirect
26. C2×#467C22semidirect
27. C2×#465C22semidirect
28. C2×#464C22semidirect
32. C2×#436C22semidirect
33. C2×#435C22semidirect
34. C2×C8×D4C22semidirect
43. #2547C22semidirect
44. #2542C22semidirect
51. #2536C22semidirect
52. #2536C22semidirect
62. #2036C22semidirect
63. #2035C22semidirect
64. #2034C22semidirect
65. #2033C22semidirect
66. #2001C22semidirect
67. #2001C22semidirect
68. #2000C22semidirect
69. #2000C22semidirect
70. #1987C22semidirect
71. #1983C22semidirect
72. #1981C22semidirect
73. #1958C22semidirect
74. #1949C22semidirect
75. #1895C22semidirect
76. #1894C22semidirect
77. #1893C22semidirect
78. #1892C22semidirect
87. #1776C22semidirect
88. C4×#103C22semidirect
159. #432D4semidirect
397. C4⋊C8C2×D4semidirect
411. #97C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32152
C2×#29742
#352402
C2×#25402
#286102
#284922
#253912
#253892
#243862
#243852
#243802
#243792
#218062
#213852
#213632