Finite Group Databasebeta

← #6459557

Subgroups — #6459557

Order 512.

2442 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3056C212yesmaximal, complemented
2#34633C211yesmaximal, characteristic
3C2×#2323C211yesmaximal, characteristic, complemented
4#28590C214yesmaximal, complemented
5#28589C214yesmaximal, complemented
6#28588C214yesmaximal, complemented
7#28587C214yesmaximal, complemented
8#23657C211yesmaximal, characteristic, complemented
9#23629C211yesmaximal, characteristic, complemented
10#23508C211yesmaximal, characteristic, complemented
11#23429C211yesmaximal, characteristic, complemented
12#21059C212yesmaximal, complemented
13#20971C212yesmaximal, complemented
14#20926C212yesmaximal, complemented
15#20335C211yesmaximal, characteristic, complemented
16C22×#137C2212yes—
17C23×M16C2212yes—
18C2×#532C2211yescharacteristic
19C2×#525C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#3056—216no—
24C2×#470C2214yes—
25C2×#468C2214yes—
26C2×#467C2214yes—
27C2×#465C2214yes—
28C2×#428C2212yes—
29C2×#417C2212yes—
30C2×#417C2212yes—
31C2×#417C2212yes—
32C2×#419C2212yes—
33#2621C2212yes—
34#2620C2212yes—
35#2620C2212yes—
36#2619C2212yes—
37#2618—21no—
38#2618—21no—
39#2617—21no—
40#2617—21no—
41#2392—22no—
42#2392—22no—
43#2323—22no—
44#2323—22no—
45#2323C2214yes—
46#2323C2214yes—
47#2297C2212yes—
48C2×#391C2211yescharacteristic
49C2×#389—21no—
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53C2×#387—21no—
54#2285C2211yescharacteristic
55C2×#380C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2085C2214yescomplemented
58#2084C2212yescomplemented
59#2083C2212yescomplemented
60#2074C2214yescomplemented
61#2072C2212yescomplemented
62#2072C2212yescomplemented
63#2043C2212yescomplemented
64#2042C2214yescomplemented
65#2041C2212yescomplemented
66#2016C2212yescomplemented
67#2015C2214yescomplemented
68#2014C2212yescomplemented
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1926C2214yescomplemented
72#1925C2214yescomplemented
73#1895C2214yes—
74#1894C2214yes—
75#1893C2214yes—
76#1892C2214yes—
77#1880C2214yescomplemented
78#1880C2214yescomplemented
79#1878C2214yescomplemented
80#1878C2214yescomplemented
81#1736C2212yes—
82#1736C2212yes—
83#1729C2214yes—
84#1729C2214yes—
85#1728C2212yes—
86#1728C2211yescharacteristic
87#1728C2211yescharacteristic
88C22×#41C2312yes—
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×#137—28no—
92C2×#137—28no—
93C2×#137—44no—
94C22×Q16C2314yes—
95C22×SD16C2314yes—
96C22×M16—28no—
97C22×M16—28no—
98C22×M16—44no—
99C22×M16C2312yes—
100C22×M16C2314yes—
101C22×M16C2312yes—
102C23×C8C2314yes—
103#532—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#124C2312yes—
107#525—28no—
108#525—28no—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8D418yescomplemented
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2314yes—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×#121C2312yes—
120C2×#120—22no—
121C2×C4×Q8—22no—
122C2×C4×Q8—44no—
123C2×C4×Q8C2314yes—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×#117—44no—
129C2×#117D412yes—
130C2×#117D412yes—
131C2×#117C2312yes—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4D411yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—21no—
143C22×C22⋊C4—21no—
144C22×C22⋊C4D411yescharacteristic
145C22×C22⋊C4D411yescharacteristic
146C22×C42—21no—
147C22×C42—21no—
148C22×C42C2311yescharacteristic
149C22×C42C2311yescharacteristic
150#470—216no—
151#468—216no—
152#467—216no—
153#465—216no—
154#428—28no—
155C2×#107C2314yes—
156C2×#106C2314yes—
157#419—28no—
158#417—28no—
159#417—28no—
160#417—44no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#396—22no—
177#394—22no—
178#394—22no—
179#394—22no—
180#394—22no—
181#394—22no—
182#394—22no—
183#392—22no—
184#392—22no—
185#391—21no—
186#391—21no—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—24no—
194#389—24no—
195#389—21no—
196#389—21no—
197#389C2312yes—
198#389C2312yes—
199#388—24no—
200#388—24no—
201#388—24no—
202#388—24no—
203#387—24no—
204#387—24no—
205#386—22no—
206#386—22no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382C2312yes—
210#380—22no—
211#380—21no—
212#380—21no—
213#380C2314yes—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235C25C2×D411yescharacteristic
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—24no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8C2×D414yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2412yes—
255C23×C4—24no—
256C23×C4—24no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D414yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4#4114yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272#137—432no—
273C2×Q16—28no—
274C2×Q16—28no—
275C2×Q16—44no—
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—44no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16—24no—
282C2×M16—28no—
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—28no—
286C2×M16—44no—
287C2×M16—432no—
288C2×M16#4118yes—
289C22×C8—28no—
290C22×C8—28no—
291C22×C8—44no—
292C22×C8C2414yes—
293C22×C8C2414yes—
294C4⋊Q8—42no—
295#127—42no—
296#124—42no—
297#124—42no—
298#123—48no—
299C22⋊Q8—216no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8C2×D4116yes—
306#121—24no—
307#121—24no—
308#121—24no—
309#121—24no—
310#121—44no—
311#121—44no—
312#120—48no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—48no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—48no—
319C4×D4—48no—
320#117—24no—
321#117—24no—
322#117—44no—
323#117—44no—
324#117—42no—
325#117—42no—
326#117—42no—
327#117—42no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—82no—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
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—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—82no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—24no—
383C2×C42—24no—
384C2×C42—22no—
385C2×C42—24no—
386C2×C42—24no—
387C2×C42—44no—
388C2×C42—82no—
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—28no—
396#106—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—44no—
402#102—28no—
403#102—28no—
404#102—44no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—41no—
423C24—41no—
424C24—41no—
425C24—41no—
426C24#12114yes—
427C24C22×D412yes—
428C24C22×D411yescharacteristic
429#41—48no—
430#41—48no—
431#41—48no—
432#41—48no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—216no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—48no—
447C2×Q8C22×D414yes—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4—82no—
466C2×D4—82no—
467C2×D4—82no—
468C2×D4—82no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—216no—
474C22×C4—216no—
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—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
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—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4#12114yes—
514C22×C4#12118yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C22×D414yes—
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D412yes—
525C22×C4C2×#4114yes—
526C22×C4C2×#4112yes—
527C22×C4C2×#4114yes—
528C22×C4H(5,𝔽2)14yes—
529C22×C4C2511yescharacteristic
530Q16—416no—
531SD16—416no—
532M16—216no—
533M16—416no—
534M16—48no—
535C2×C8—24no—
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8—416no—
541C2×C8C2×#4118yes—
542C4⋊C4—216no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—24no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—48no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—81no—
563C4⋊C4—81no—
564C4⋊C4—81no—
565C4⋊C4—81no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4C22×D414yes—
569C22⋊C4—216no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—84no—
586C42—22no—
587C42—22no—
588C42—24no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—48no—
594C42—44no—
595C42—84no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—28no—
607C23—44no—
608C23—44no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—82no—
617C23—82no—
618C23—82no—
619C23—82no—
620C23C2×#12114yes—
621C23C2×#12112yes—
622C23#53212yes—
623C23#53412yes—
624C23C23×D411yescharacteristic
625Q8—28no—
626Q8—44no—
627Q8—48no—
628Q8—42no—
629Q8—42no—
630Q8—42no—
631Q8—42no—
632D4—42no—
633D4—42no—
634D4—42no—
635D4—42no—
636D4—84no—
637D4—84no—
638D4—88no—
639D4—88no—
640D4—88no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—216no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—22no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—22no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—48no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—48no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—82no—
693C2×C4—82no—
694C2×C4—84no—
695C2×C4—84no—
696C2×C4—88no—
697C2×C4C2×#12114yes—
698C2×C4C2×#12112yes—
699C2×C4#53212yes—
700C2×C4#53412yes—
701C2×C4#54618yes—
702C2×C4#55318yes—
703C2×C4C23×D411yescharacteristic
704C2×C4C22×#4112yes—
705C2×C4C22×#4111yescharacteristic
706C2×C4C2×H(5,𝔽2)12yes—
707C8—28no—
708C8—44no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—24no—
716C22—24no—
717C22—24no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—88no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22#254314yes—
731C22C22×#12111yescharacteristic
732C22C2×#53211yescharacteristic
733C22C2×#53411yescharacteristic
734C22#358511yescharacteristic
735C22#358611yescharacteristic
736C22#358611yescharacteristic
737C22#358611yescharacteristic
738C4—24no—
739C4—44no—
740C4—44no—
741C4—42no—
742C4—41no—
743C4—41no—
744C4—42no—
745C4—41no—
746C4—41no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#254314yes—
753C4C2×#54612yes—
754C4C2×#55312yes—
755C2—24no—
756C2—44no—
757C2—162no—
758C2C2×#254311yescharacteristic
759C2#6042411yescharacteristic
760C2#6042511yescharacteristic
761C2#6070311yescharacteristic
762C2#6070511yescharacteristic
763C2#6075411yescharacteristic
764C2#6075511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34633C2
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
1. C2×#3056C2semidirect
3. C2×#2323C2semidirect
4. #28590C2semidirect
5. #28589C2semidirect
6. #28588C2semidirect
7. #28587C2semidirect
8. #23657C2semidirect
9. #23629C2semidirect
10. #23508C2semidirect
11. #23429C2semidirect
12. #21059C2semidirect
13. #20971C2semidirect
14. #20926C2semidirect
15. #20335C2semidirect
57. #2085C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #2074C22semidirect
61. #2072C22semidirect
62. #2072C22semidirect
63. #2043C22semidirect
64. #2042C22semidirect
65. #2041C22semidirect
66. #2016C22semidirect
67. #2015C22semidirect
68. #2014C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
71. #1926C22semidirect
72. #1925C22semidirect
77. #1880C22semidirect
78. #1880C22semidirect
79. #1878C22semidirect
80. #1878C22semidirect
112. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30562
#346332
C2×#23232
#285902
#285892
#285882
#285872
#236572
#236292
#235082
#234292
#210592
#209712
#209262
#203352