Finite Group Databasebeta

← #6470697

Subgroups — #6470697

Order 512.

2244 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3196C212yesmaximal, complemented
2#35032C211yesmaximal, characteristic
3C2×#2360C211yesmaximal, characteristic, complemented
4#33109C211yesmaximal, characteristic, complemented
5#33107C211yesmaximal, characteristic, complemented
6#32909C212yesmaximal, complemented
7#29761C212yesmaximal, complemented
8#29715C211yesmaximal, characteristic, complemented
9#29713C211yesmaximal, characteristic, complemented
10#25226C214yesmaximal, complemented
11#25224C214yesmaximal, complemented
12#25102C214yesmaximal, complemented
13#25100C214yesmaximal, complemented
14#21934C211yesmaximal, characteristic, complemented
15#20909C212yesmaximal, complemented
16C2×#525C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#3196—216no—
21C2×#484C2212yes—
22C2×#481C2212yes—
23C2×#485C2214yes—
24C2×#470C2214yes—
25C2×#468C2214yes—
26C2×#464C2214yescomplemented
27C2×#452C2214yes—
28C2×#451C2212yes—
29C2×#448C2212yes—
30C2×#417C2214yes—
31C2×#407C2212yes—
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2850C2212yes—
35#2617—21no—
36#2617—21no—
37#2602—22no—
38#2602—22no—
39#2500C2212yes—
40#2499C2212yes—
41#2432C2212yes—
42#2431C2212yes—
43#2360—22no—
44#2360—22no—
45#2360C2214yes—
46#2360C2214yes—
47#2325—21no—
48#2325—21no—
49C2×#391C2211yescharacteristic
50C2×#389—21no—
51C2×#389C2211yescharacteristic
52C2×#387—21no—
53C2×#385—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4—21no—
56C2×C4×C22⋊C4C2211yescharacteristic
57#2085C2214yescomplemented
58#2072C2212yescomplemented
59#2072C2212yescomplemented
60#2067C2214yescomplemented
61#2066C2214yescomplemented
62#2056C2214yescomplemented
63#2055C2214yescomplemented
64#2042C2214yescomplemented
65#2016C2212yescomplemented
66#2014C2212yescomplemented
67#1982C2212yescomplemented
68#1981C2212yescomplemented
69#1948C2212yescomplemented
70#1947C2212yescomplemented
71#1928C2214yescomplemented
72#1926C2214yescomplemented
73#1880C2214yescomplemented
74#1878C2214yescomplemented
75#1874C2214yes—
76#1874C2212yes—
77#1874C2212yes—
78#1872C2214yes—
79#1872C2212yes—
80#1872C2212yes—
81#1851C2214yes—
82#1851C2214yes—
83#1805C2214yescomplemented
84#1797C2211yescharacteristic
85#1797C2211yescharacteristic
86#1797C2212yes—
87#1736C2214yes—
88C22×#41C2312yes—
89C23×Q8C2311yescharacteristic
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94C22×M16C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#124C2312yes—
98#525—28no—
99#525—28no—
100C2×#123—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8D418yescomplemented
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—44no—
114C2×C4×Q8C2314yes—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×#117C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—44no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#485—216no—
136#484—28no—
137#481—28no—
138#470—216no—
139#468—216no—
140#464—48no—
141#452—216no—
142#451—28no—
143#448—28no—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149#417—48no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#407—28no—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2314yes—
162#397—22no—
163#397—22no—
164#394—22no—
165#394—22no—
166#394—22no—
167#394—22no—
168#392—22no—
169#392—22no—
170#391—21no—
171#391—21no—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2312yes—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—24no—
184#389—24no—
185#389C2314yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—24no—
193#387—24no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#385—24no—
199#385—24no—
200#385—22no—
201#385—22no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382C2312yes—
205#380—21no—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—24no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C4×C22⋊C4C2312yes—
217C2×#95—21no—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95D411yescharacteristic
222C2×#95D411yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230C25C2×D411yescharacteristic
231C2×#41—24no—
232C2×#41—24no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—42no—
238C22×Q8—42no—
239C22×Q8C2×D414yes—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—82no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D414yes—
265C23×C4C2×D411yescharacteristic
266C23×C4#4114yes—
267C23×C4C2411yescharacteristic
268C23×C4C2412yes—
269C2×Q16—28no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×SD16—28no—
273C2×SD16—44no—
274C2×SD16—44no—
275C2×M16—28no—
276C2×M16—44no—
277C2×M16—44no—
278C22×C8C2414yes—
279C22×C8C2414yes—
280C4⋊Q8—42no—
281#127—42no—
282#124—44no—
283#123—48no—
284C22⋊Q8—216no—
285C22⋊Q8—28no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8C2×D4116yes—
291#121—24no—
292#121—24no—
293#121—24no—
294#121—24no—
295#121—44no—
296#121—44no—
297#120—48no—
298C4×Q8—28no—
299C4×Q8—44no—
300C4×Q8—48no—
301C4×D4—48no—
302C4×D4—48no—
303#117—24no—
304#117—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—42no—
360C2×C42—42no—
361C2×C42—44no—
362C2×C42—82no—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—28no—
370#106—28no—
371C4⋊C8—42no—
372C4⋊C8—42no—
373C4⋊C8—44no—
374#103—28no—
375#103—28no—
376#103—28no—
377#103—28no—
378#103—44no—
379#102—28no—
380#102—28no—
381#102—44no—
382C22⋊C8—24no—
383C22⋊C8—24no—
384C22⋊C8—28no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24#12114yes—
410C24C22×D412yes—
411C24C22×D411yescharacteristic
412#41—416no—
413#41—88no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—48no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—48no—
426C2×Q8C22×D414yes—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4—82no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—48no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—48no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—84no—
492C22×C4—816no—
493C22×C4#12114yes—
494C22×C4#12118yes—
495C22×C4#13714yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D414yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4112yes—
507C22×C4C2×#4114yes—
508C22×C4C2511yescharacteristic
509Q16—88no—
510SD16—88no—
511M16—88no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C4⋊C4—24no—
519C4⋊C4—216no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C4⋊C4—84no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4C22×D414yes—
541C4⋊C4H(5,𝔽2)14yes—
542C22⋊C4—216no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—41no—
562C42—41no—
563C42—42no—
564C42—48no—
565C42—44no—
566C42—82no—
567C42—82no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—28no—
577C23—22no—
578C23—22no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23C2×#12114yes—
593C23C2×#12112yes—
594C23C2×#13712yes—
595C23#57512yes—
596C23C23×D411yescharacteristic
597Q8—28no—
598Q8—48no—
599Q8—44no—
600Q8—44no—
601Q8—44no—
602D4—44no—
603D4—44no—
604D4—88no—
605D4—88no—
606D4—88no—
607D4—162no—
608D4—162no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—44no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
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—88no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—816no—
665C2×C4—84no—
666C2×C4C2×#12114yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#13712yes—
669C2×C4C2×#13712yes—
670C2×C4C23×D411yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×H(5,𝔽2)12yes—
673C2×C4#58512yes—
674C8—44no—
675C8—44no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—416no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22#303714yes—
698C22C22×#12111yescharacteristic
699C22#348411yescharacteristic
700C22#348511yescharacteristic
701C22C22×#13711yescharacteristic
702C22C2×#57511yescharacteristic
703C22#358511yescharacteristic
704C22#358611yescharacteristic
705C4—22no—
706C4—22no—
707C4—24no—
708C4—22no—
709C4—44no—
710C4—41no—
711C4—41no—
712C4—42no—
713C4—44no—
714C4—42no—
715C4—44no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4#345912yes—
721C4#349612yes—
722C2—24no—
723C2—44no—
724C2—162no—
725C2#3498411yescharacteristic
726C2C2×#303711yescharacteristic
727C2#6042511yescharacteristic
728C2#6157611yescharacteristic
729C2#6158411yescharacteristic
730C2#6170911yescharacteristic
731C2#6171611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3196C2semidirect
3. C2×#2360C2semidirect
4. #33109C2semidirect
5. #33107C2semidirect
6. #32909C2semidirect
7. #29761C2semidirect
8. #29715C2semidirect
9. #29713C2semidirect
10. #25226C2semidirect
11. #25224C2semidirect
12. #25102C2semidirect
13. #25100C2semidirect
14. #21934C2semidirect
15. #20909C2semidirect
26. C2×#464C22semidirect
57. #2085C22semidirect
58. #2072C22semidirect
59. #2072C22semidirect
60. #2067C22semidirect
61. #2066C22semidirect
62. #2056C22semidirect
63. #2055C22semidirect
64. #2042C22semidirect
65. #2016C22semidirect
66. #2014C22semidirect
67. #1982C22semidirect
68. #1981C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
71. #1928C22semidirect
72. #1926C22semidirect
73. #1880C22semidirect
74. #1878C22semidirect
83. #1805C22semidirect
103. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31962
#350322
C2×#23602
#331092
#331072
#329092
#297612
#297152
#297132
#252262
#252242
#251022
#251002
#219342
#209092