Finite Group Databasebeta

← #6473078

Subgroups — #6473078

Order 512.

2092 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C2×#3189C211yesmaximal, characteristic
3C2×#2621C211yesmaximal, characteristic
4#34203C211yesmaximal, characteristic
5#33119C212yesmaximal, complemented
6#32915C212yesmaximal, complemented
7#29764C212yesmaximal, complemented
8#29724C212yesmaximal, complemented
9#28588C214yesmaximal, complemented
10#28124C212yesmaximal, complemented
11#28121C212yesmaximal, complemented
12#27653C212yesmaximal, complemented
13#24764C214yesmaximal, complemented
14#24212C212yesmaximal, complemented
15#21768C211yesmaximal, characteristic
16#21762C211yesmaximal, characteristic
17#20628C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C22×#117C2211yescharacteristic
24#3195—28no—
25#3189—28no—
26C2×#484C2212yes—
27C2×#484C2212yes—
28C2×#483C2212yes—
29C2×#483C2212yes—
30C2×#470C2214yes—
31C2×#467C2214yes—
32C2×#462C2214yescomplemented
33C2×#451C2212yes—
34C2×#451C2212yes—
35C2×#451C2212yes—
36C2×#451C2212yes—
37C2×#424C2212yes—
38C2×#407C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40#2621—22no—
41#2621—22no—
42#2621C2212yes—
43#2621C2212yes—
44#2621C2212yes—
45#2621C2212yes—
46C2×#398—22no—
47C2×#397—22no—
48#2362—22no—
49#2360—22no—
50#2344—22no—
51#2342—22no—
52#2323C2212yes—
53#2323C2212yes—
54#2316C2211yescharacteristic
55#2310C2211yescharacteristic
56C4×C22⋊Q8C2212yes—
57C4×C22⋊Q8C2212yes—
58C2×#391C2211yescharacteristic
59C2×#391C2211yescharacteristic
60C2×#382C2211yescharacteristic
61#2280C2211yescharacteristic
62#2274C2211yescharacteristic
63#2085C2214yescomplemented
64#2074C2214yescomplemented
65#2068C2214yescomplemented
66#2066C2214yescomplemented
67#2056C2214yescomplemented
68#2056C2214yescomplemented
69#2043C2214yescomplemented
70#2016C2214yescomplemented
71#1981C2214yescomplemented
72#1949C2214yescomplemented
73#1928C2214yescomplemented
74#1927C2214yescomplemented
75#1894C2214yes—
76#1892C2214yes—
77#1878C2214yescomplemented
78#1878C2214yescomplemented
79#1873C2212yes—
80#1873C2212yes—
81#1873C2212yes—
82#1873C2212yes—
83#1824C2212yes—
84#1823C2212yes—
85#1805C2214yescomplemented
86#1798C2212yes—
87#1798C2212yes—
88#1797C2212yes—
89#1797C2212yes—
90#1787C2212yes—
91#1787C2212yes—
92#1729C2212yes—
93C23×Q8C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C23×Q8C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×M16C2312yes—
100C23×C8C2312yes—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8D418yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123#519—44no—
124C2×C4×Q8—22no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×#117—22no—
128C2×#117—44no—
129C2×#117D412yes—
130C2×#117D412yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4D412yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—21no—
146C22×C22⋊C4—21no—
147C22×C22⋊C4—21no—
148C22×C22⋊C4D412yes—
149C22×C42C2311yescharacteristic
150C22×C42C2311yescharacteristic
151#484—28no—
152#484—28no—
153#483—28no—
154#483—28no—
155#470—216no—
156#467—216no—
157#462—48no—
158#451—28no—
159#451—28no—
160#451—28no—
161#451—28no—
162C2×#107C2314yes—
163C2×#106C2314yes—
164#424—44no—
165C2×C4⋊C8C2314yes—
166C2×C4⋊C8C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175#407—24no—
176C2×C22⋊C8—24no—
177C2×C22⋊C8C2312yes—
178C2×C22⋊C8C2312yes—
179C2×C22⋊C8C2312yes—
180C2×C22⋊C8C2312yes—
181#398—24no—
182#398—24no—
183#398—24no—
184#398—24no—
185#397—24no—
186#397—24no—
187#397—24no—
188#397—24no—
189#391—22no—
190#391—22no—
191#391—22no—
192#391C2312yes—
193#391C2312yes—
194#391C2312yes—
195#391C2312yes—
196#391C2312yes—
197#391C2312yes—
198#389—22no—
199#389—22no—
200#389—22no—
201#389C2312yes—
202#389C2312yes—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#385—22no—
212#385—22no—
213#382—22no—
214#382C2312yes—
215#382C2312yes—
216#381—21no—
217#381—22no—
218#380—22no—
219#380C2312yes—
220#380C2312yes—
221#379—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—22no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95—22no—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C25C2×D411yescharacteristic
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—42no—
251C22×Q8—42no—
252C22×Q8—42no—
253C22×Q8—42no—
254C22×Q8—42no—
255C22×Q8—42no—
256C22×Q8C2×D414yes—
257C22×Q8C2412yes—
258C22×Q8C2412yes—
259C22×Q8C2412yes—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—42no—
267C23×C4—42no—
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D414yes—
274C23×C4#4114yes—
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×Q16—28no—
279C2×Q16—28no—
280C2×Q16—44no—
281C2×Q16—44no—
282C2×Q16—44no—
283C2×Q16—44no—
284C2×M16—24no—
285C2×M16—42no—
286C2×M16—42no—
287C22×C8—24no—
288C22×C8—42no—
289C22×C8—42no—
290C22×C8C2414yes—
291C22×C8C2414yes—
292C4⋊Q8—28no—
293C4⋊Q8—44no—
294C4⋊Q8—48no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
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⋊Q8—44no—
310C22⋊Q8—44no—
311C22⋊Q8C2×D418yes—
312C22⋊Q8C2×D418yes—
313C4×Q8—28no—
314C4×Q8—44no—
315C4×Q8—48no—
316#117—24no—
317#117—24no—
318#117—42no—
319#117—42no—
320#117—44no—
321#117—44no—
322#117—48no—
323#117—42no—
324#117—42no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—48no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—24no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—42no—
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#106—28no—
394C4⋊C8—28no—
395C4⋊C8—44no—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—44no—
401#103—44no—
402#103—44no—
403#103—44no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424C24—24no—
425C24—41no—
426C24—41no—
427C24—42no—
428C24#12114yes—
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C24C22×D411yescharacteristic
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—48no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—48no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—84no—
455C2×Q8—84no—
456C2×Q8—84no—
457C2×Q8—84no—
458C2×Q8C22×D414yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—82no—
502C22×C4#12114yes—
503C22×C4#12114yes—
504C22×C4#12114yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2511yescharacteristic
519Q16—88no—
520Q16—88no—
521M16—84no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—84no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—48no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—48no—
552C4⋊C4—44no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4C22×D414yes—
556C22⋊C4—28no—
557C22⋊C4—28no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—82no—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—42no—
575C42—42no—
576C42—42no—
577C42—44no—
578C42—44no—
579C42—82no—
580C42—82no—
581C42H(5,𝔽2)14yes—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—22no—
586C23—22no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—22no—
592C23—22no—
593C23—28no—
594C23—44no—
595C23—44no—
596C23—84no—
597C23—84no—
598C23C2×#12112yes—
599C23C2×#12112yes—
600C23C2×#12112yes—
601C23#57814yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—48no—
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—84no—
611Q8—84no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4C2×#12112yes—
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4#53414yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×H(5,𝔽2)12yes—
675C2×C4#58512yes—
676C8—44no—
677C8—44no—
678C22—22no—
679C22—24no—
680C22—24no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—88no—
693C22#304414yes—
694C22C22×#12111yescharacteristic
695C22C2×#53411yescharacteristic
696C22#344911yescharacteristic
697C22C2×#57812yes—
698C22#358612yes—
699C4—24no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—44no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#349612yes—
715C4#350512yes—
716C2—24no—
717C2—42no—
718C2—42no—
719C2#3500011yescharacteristic
720C2C2×#304411yescharacteristic
721C2#6042711yescharacteristic
722C2#6075912yes—
723C2#6098312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #33119C2semidirect
6. #32915C2semidirect
7. #29764C2semidirect
8. #29724C2semidirect
9. #28588C2semidirect
10. #28124C2semidirect
11. #28121C2semidirect
12. #27653C2semidirect
13. #24764C2semidirect
14. #24212C2semidirect
32. C2×#462C22semidirect
63. #2085C22semidirect
64. #2074C22semidirect
65. #2068C22semidirect
66. #2066C22semidirect
67. #2056C22semidirect
68. #2056C22semidirect
69. #2043C22semidirect
70. #2016C22semidirect
71. #1981C22semidirect
72. #1949C22semidirect
73. #1928C22semidirect
74. #1927C22semidirect
77. #1878C22semidirect
78. #1878C22semidirect
85. #1805C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#31892
C2×#26212
#342032
#331192
#329152
#297642
#297242
#285882
#281242
#281212
#276532
#247642
#242122
#217682
#217622
#206282