Finite Group Databasebeta

← #6473082

Subgroups — #6473082

Order 512.

2222 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×#3193C211yesmaximal, characteristic
2C2×#3189C211yesmaximal, characteristic, complemented
3C2×#2620C211yesmaximal, characteristic
4#34203C211yesmaximal, characteristic, complemented
5#33120C212yesmaximal, complemented
6#32918C212yesmaximal, complemented
7#29767C212yesmaximal, complemented
8#29728C212yesmaximal, complemented
9#28590C214yesmaximal, complemented
10#28123C212yesmaximal, complemented
11#28121C212yesmaximal, complemented
12#27648C212yesmaximal, complemented
13#24765C214yesmaximal, complemented
14#24217C212yesmaximal, complemented
15#21768C211yesmaximal, characteristic, complemented
16#21761C211yesmaximal, characteristic
17#20628C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C2×#519C2211yescharacteristic, complemented
23C22×#117C2211yescharacteristic
24#3193—28no—
25#3189—28no—
26C2×#484C2212yes—
27C2×#484C2212yes—
28C2×#484C2212yescomplemented
29C2×#480C2212yes—
30C2×#468C2214yes—
31C2×#467C2214yescomplemented
32C2×#464C2214yescomplemented
33C2×#451C2212yes—
34C2×#451C2212yescomplemented
35C2×#448C2212yes—
36C2×#450C2212yes—
37C2×#424C2212yescomplemented
38C2×#407C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40#2620—22no—
41#2620—22no—
42#2620C2212yes—
43#2620C2212yes—
44#2620C2212yes—
45#2620C2212yes—
46C2×#396—22no—
47C2×#394—22no—
48#2362—22no—
49#2360—22no—
50#2344—22no—
51#2342—22no—
52#2323C2212yescomplemented
53#2323C2212yescomplemented
54#2316C2211yescharacteristic, complemented
55#2310C2211yescharacteristic, complemented
56C4×C22⋊Q8C2212yes—
57C4×C22⋊Q8C2212yes—
58C2×#391C2211yescharacteristic
59C2×#390C2211yescharacteristic
60C2×#384C2211yescharacteristic
61#2280C2211yescharacteristic
62#2274C2211yescharacteristic, complemented
63#2083C2214yescomplemented
64#2072C2214yescomplemented
65#2068C2214yescomplemented
66#2067C2214yescomplemented
67#2056C2214yescomplemented
68#2055C2214yescomplemented
69#2042C2214yescomplemented
70#2015C2214yescomplemented
71#1983C2214yescomplemented
72#1947C2214yescomplemented
73#1928C2214yescomplemented
74#1925C2214yescomplemented
75#1894C2214yescomplemented
76#1893C2214yes—
77#1880C2214yescomplemented
78#1878C2214yescomplemented
79#1875C2212yes—
80#1875C2212yes—
81#1873C2212yescomplemented
82#1873C2212yescomplemented
83#1824C2212yes—
84#1824C2212yes—
85#1803C2214yescomplemented
86#1798C2212yescomplemented
87#1798C2212yescomplemented
88#1797C2212yes—
89#1797C2212yes—
90#1787C2212yes—
91#1786C2212yes—
92#1729C2212yescomplemented
93C23×Q8C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C23×D4—21no—
96C23×D4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100C22×M16C2312yes—
101C23×C8C2312yes—
102C2×#124—22no—
103C2×#124—44no—
104C2×#124C2312yes—
105C2×#124C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8D418yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×#121—21no—
120C2×#121—24no—
121C2×#121—24no—
122C2×#121C2312yes—
123C2×#121C2312yes—
124#519—44no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×#117—22no—
129C2×#117—44no—
130C2×#117D412yes—
131C2×#117D412yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4D412yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—21no—
146C22×C22⋊C4—21no—
147C22×C22⋊C4—22no—
148C22×C22⋊C4—21no—
149C22×C22⋊C4—22no—
150C22×C22⋊C4D412yes—
151C22×C42C2311yescharacteristic
152C22×C42C2311yescharacteristic
153#484—28no—
154#484—28no—
155#484—28no—
156#480—28no—
157#468—216no—
158#467—216no—
159#464—48no—
160#451—28no—
161#451—28no—
162#450—28no—
163#448—28no—
164C2×#107C2314yes—
165C2×#107C2314yes—
166#424—44no—
167C2×C4⋊C8C2314yes—
168C2×C4⋊C8C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#102C2314yes—
175C2×#102C2314yes—
176C2×#102C2314yes—
177#407—24no—
178C2×C22⋊C8—24no—
179C2×C22⋊C8C2312yes—
180C2×C22⋊C8C2312yes—
181C2×C22⋊C8C2312yes—
182C2×C22⋊C8C2312yes—
183#396—24no—
184#396—24no—
185#396—24no—
186#396—24no—
187#394—24no—
188#394—24no—
189#394—24no—
190#394—24no—
191#391—22no—
192#391—22no—
193#391C2312yes—
194#391C2312yes—
195#391C2312yes—
196#391C2312yes—
197#390—22no—
198#390C2312yes—
199#390C2312yes—
200#389—22no—
201#389—22no—
202#389—22no—
203#389C2312yes—
204#389C2312yes—
205#388—22no—
206#388—22no—
207#388—22no—
208#388—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#385—22no—
214#385—22no—
215#384—22no—
216#384C2312yes—
217#384C2312yes—
218#381—21no—
219#381—22no—
220#380—22no—
221#380C2312yes—
222#380C2312yes—
223#379—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4—22no—
234C2×#95—22no—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C25—21no—
242C25C2×D411yescharacteristic
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—42no—
249C22×Q8—42no—
250C22×Q8—42no—
251C22×Q8—42no—
252C22×Q8C2×D414yes—
253C22×Q8C2412yes—
254C22×Q8C2412yes—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—42no—
261C22×D4—42no—
262C22×D4—44no—
263C22×D4—84no—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—42no—
271C23×C4—42no—
272C23×C4C2×D412yes—
273C23×C4C2×D412yes—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D414yes—
278C23×C4#4114yes—
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C23×C4C2411yescharacteristic
282C2×Q16—28no—
283C2×Q16—44no—
284C2×Q16—44no—
285C2×SD16—28no—
286C2×SD16—44no—
287C2×SD16—44no—
288C2×M16—24no—
289C2×M16—42no—
290C2×M16—42no—
291C22×C8—24no—
292C22×C8—42no—
293C22×C8—42no—
294C22×C8C2414yes—
295C22×C8C2414yes—
296#124—28no—
297#124—44no—
298#124—48no—
299C22⋊Q8—28no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309C22⋊Q8C2×D418yes—
310C22⋊Q8C2×D418yes—
311#121—24no—
312#121—24no—
313#121—24no—
314#121—24no—
315#121—44no—
316#121—44no—
317C4×Q8—28no—
318C4×Q8—44no—
319C4×Q8—48no—
320#117—24no—
321#117—24no—
322#117—42no—
323#117—42no—
324#117—44no—
325#117—44no—
326#117—48no—
327#117—42no—
328#117—42no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—84no—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—44no—
377C2×C22⋊C4—48no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4—44no—
380C2×C22⋊C4—84no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C42—24no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—42no—
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—28no—
400#107—28no—
401C4⋊C8—28no—
402C4⋊C8—44no—
403#103—28no—
404#103—28no—
405#103—28no—
406#103—44no—
407#103—44no—
408#102—28no—
409#102—44no—
410#102—44no—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C22⋊C8—24no—
414C22⋊C8—24no—
415#95—24no—
416#95—24no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427C24—24no—
428C24—22no—
429C24—41no—
430C24—41no—
431C24—42no—
432C24—42no—
433C24—44no—
434C24—42no—
435C24—42no—
436C24—44no—
437C24#12114yes—
438C24C22×D411yescharacteristic
439C24C22×D411yescharacteristic
440C24C22×D411yescharacteristic
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—48no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—48no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—84no—
457C2×Q8—84no—
458C2×Q8C22×D414yes—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—84no—
469C2×D4—84no—
470C2×D4—84no—
471C2×D4—84no—
472C2×D4—816no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—41no—
494C22×C4—42no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—82no—
512C22×C4#12114yes—
513C22×C4#12114yes—
514C22×C4#12114yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C2×#4112yes—
526C22×C4C2×#4112yes—
527C22×C4C2×#4112yes—
528C22×C4C2511yescharacteristic
529Q16—88no—
530SD16—88no—
531M16—84no—
532C2×C8—24no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—84no—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—84no—
562C4⋊C4—84no—
563C4⋊C4C22×D414yes—
564C22⋊C4—28no—
565C22⋊C4—28no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—48no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—48no—
579C22⋊C4—84no—
580C22⋊C4—84no—
581C22⋊C4—82no—
582C42—24no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—42no—
587C42—42no—
588C42—42no—
589C42—44no—
590C42—44no—
591C42—82no—
592C42—82no—
593C42H(5,𝔽2)14yes—
594C23—22no—
595C23—21no—
596C23—21no—
597C23—22no—
598C23—22no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—22no—
604C23—22no—
605C23—28no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—44no—
613C23—84no—
614C23—84no—
615C23—84no—
616C23—84no—
617C23—88no—
618C23—88no—
619C23—84no—
620C23C2×#12112yes—
621C23C2×#12112yes—
622C23C2×#12112yes—
623C23#57814yes—
624C23C23×D411yescharacteristic
625Q8—28no—
626Q8—48no—
627Q8—44no—
628Q8—44no—
629Q8—44no—
630Q8—84no—
631D4—44no—
632D4—44no—
633D4—84no—
634D4—88no—
635D4—88no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—48no—
665C2×C4—48no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—84no—
684C2×C4—84no—
685C2×C4—84no—
686C2×C4—84no—
687C2×C4—84no—
688C2×C4C2×#12112yes—
689C2×C4C2×#12112yes—
690C2×C4C2×#12112yes—
691C2×C4#53414yes—
692C2×C4C23×D411yescharacteristic
693C2×C4C22×#4111yescharacteristic
694C2×C4C2×H(5,𝔽2)12yes—
695C2×C4#58512yes—
696C8—44no—
697C8—44no—
698C22—22no—
699C22—24no—
700C22—24no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—88no—
713C22—82no—
714C22—82no—
715C22—84no—
716C22—84no—
717C22—82no—
718C22—168no—
719C22#304414yes—
720C22C22×#12111yescharacteristic
721C22C2×#53411yescharacteristic
722C22#344911yescharacteristic
723C22C2×#57812yes—
724C22#358512yes—
725C4—24no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—44no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—44no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4#349612yes—
740C4#350512yes—
741C2—24no—
742C2—42no—
743C2—42no—
744C2—162no—
745C2#3500011yescharacteristic
746C2C2×#304411yescharacteristic
747C2#6042611yescharacteristic
748C2#6076412yes—
749C2#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×#3189C2semidirect
4. #34203C2semidirect
5. #33120C2semidirect
6. #32918C2semidirect
7. #29767C2semidirect
8. #29728C2semidirect
9. #28590C2semidirect
10. #28123C2semidirect
11. #28121C2semidirect
12. #27648C2semidirect
13. #24765C2semidirect
14. #24217C2semidirect
15. #21768C2semidirect
17. #20628C2semidirect
22. C2×#519C22semidirect
28. C2×#484C22semidirect
31. C2×#467C22semidirect
32. C2×#464C22semidirect
34. C2×#451C22semidirect
37. C2×#424C22semidirect
52. #2323C22semidirect
53. #2323C22semidirect
54. #2316C22semidirect
55. #2310C22semidirect
62. #2274C22semidirect
63. #2083C22semidirect
64. #2072C22semidirect
65. #2068C22semidirect
66. #2067C22semidirect
67. #2056C22semidirect
68. #2055C22semidirect
69. #2042C22semidirect
70. #2015C22semidirect
71. #1983C22semidirect
72. #1947C22semidirect
73. #1928C22semidirect
74. #1925C22semidirect
75. #1894C22semidirect
77. #1880C22semidirect
78. #1878C22semidirect
81. #1873C22semidirect
82. #1873C22semidirect
85. #1803C22semidirect
86. #1798C22semidirect
87. #1798C22semidirect
92. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#31892
C2×#26202
#342032
#331202
#329182
#297672
#297282
#285902
#281232
#281212
#276482
#247652
#242172
#217682
#217612
#206282