Finite Group Databasebeta

← #6470733

Subgroups — #6470733

Order 512.

2322 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic, complemented
2C2×#3186C211yesmaximal, characteristic
3#37451C211yesmaximal, characteristic, complemented
4#34204C211yesmaximal, characteristic, complemented
5#33120C212yesmaximal, complemented
6#32917C212yesmaximal, complemented
7#29767C212yesmaximal, complemented
8#29727C212yesmaximal, complemented
9#25223C214yesmaximal, complemented
10#25200C214yesmaximal, complemented
11#25101C214yesmaximal, complemented
12#25090C214yesmaximal, complemented
13#21933C211yesmaximal, characteristic, complemented
14#20912C211yesmaximal, characteristic, complemented
15C2×#1873C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C2×#518C2211yescharacteristic, complemented
20C22×#117C2211yescharacteristic
21#3193—28no—
22#3186—28no—
23C2×#484C2212yescomplemented
24C2×#480C2212yes—
25C2×#480C2212yes—
26C2×#480C2212yescomplemented
27C2×#468C2214yes—
28C2×#466C2214yes—
29C2×#463C2214yescomplemented
30C2×#448C2212yescomplemented
31C2×#450C2212yescomplemented
32C2×#447C2212yes—
33C2×#447C2212yes—
34C2×#418C2212yescomplemented
35C22×#103C2212yescomplemented
36C2×#407C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2816—22no—
39#2626—22no—
40#2620C2212yes—
41#2620C2212yes—
42#2618—22no—
43#2502C2212yescomplemented
44#2501C2212yescomplemented
45#2436C2212yescomplemented
46#2435C2212yescomplemented
47#2405—22no—
48#2363C2214yes—
49#2361C2214yescomplemented
50#2342—22no—
51D4×C22⋊C4—22no—
52#2324—22no—
53#2316C2211yescharacteristic, complemented
54#2309C2211yescharacteristic, complemented
55C4×#121—22no—
56#2280C2211yescharacteristic
57#2274C2211yescharacteristic, complemented
58#2083C2214yescomplemented
59#2071C2214yescomplemented
60#2068C2214yescomplemented
61#2067C2214yescomplemented
62#2055C2214yescomplemented
63#2054C2214yescomplemented
64#2042C2214yescomplemented
65#2012C2214yescomplemented
66#1983C2214yescomplemented
67#1947C2214yescomplemented
68#1928C2214yescomplemented
69#1925C2214yescomplemented
70#1879C2214yescomplemented
71#1877C2214yescomplemented
72#1875C2212yes—
73#1875C2212yes—
74#1874C2214yescomplemented
75#1873C2214yescomplemented
76#1873C2212yescomplemented
77#1873C2212yescomplemented
78#1851C2214yescomplemented
79#1850C2214yescomplemented
80#1802C2214yescomplemented
81#1798C2212yescomplemented
82#1798C2212yescomplemented
83#1736C2212yescomplemented
84C2×#340C2212yescomplemented
85C23×Q8C2311yescharacteristic
86C23×D4—21no—
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×SD16C2314yes—
92C22×D8C2314yes—
93C22×M16C2312yes—
94C23×C8C2312yes—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×#123—24no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—24no—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121—24no—
110C2×#121—24no—
111C2×#121D418yes—
112C2×#121C2312yes—
113C2×#121C2312yes—
114C2×#121C2312yes—
115C2×#121C2312yes—
116#518—44no—
117C2×C4×D4—22no—
118C2×C4×D4—24no—
119C2×C4×D4C2314yes—
120C2×#117—21no—
121C2×#117—21no—
122C2×#117—44no—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—22no—
138C22×C42—21no—
139#484—28no—
140#480—28no—
141#480—28no—
142#480—28no—
143#468—216no—
144#466—216no—
145#463—48no—
146#450—28no—
147#448—28no—
148#447—28no—
149#447—28no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154#418—44no—
155C2×#103—44no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164#407—24no—
165C2×C22⋊C8—24no—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170#399—24no—
171#398—24no—
172#396—24no—
173#396—24no—
174#394—24no—
175#394—24no—
176#394—24no—
177#393—24no—
178#391C2312yes—
179#391C2312yes—
180#390—22no—
181#390—24no—
182#390C2314yes—
183#390C2312yes—
184#390C2312yes—
185#389—22no—
186#389C2314yes—
187#388—24no—
188#388—24no—
189#386—24no—
190#386—24no—
191#386—24no—
192#385—24no—
193#385—24no—
194#384C2312yes—
195#384C2312yes—
196#381D412yes—
197#380—22no—
198#380D412yes—
199#380D412yes—
200#379—22no—
201#379—24no—
202#379D412yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2314yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2314yes—
215C25—21no—
216C25—21no—
217C25C2×D411yescharacteristic
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8C2412yes—
224C22×D4—22no—
225C22×D4—28no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—42no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4—42no—
238C22×D4—44no—
239C22×D4—84no—
240C22×D4—84no—
241C22×D4C2×D414yes—
242C22×D4C2412yes—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D414yes—
256C23×C4#4114yes—
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×D8—28no—
264C2×D8—44no—
265C2×D8—44no—
266C2×M16—24no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8—24no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274#124—44no—
275#124—44no—
276#123—88no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283#121—28no—
284#121—28no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121—24no—
289#121—24no—
290#121—24no—
291#121—44no—
292#121—44no—
293#121C2×D418yes—
294#121C2×D418yes—
295C4×D4—28no—
296C4×D4—44no—
297C4×D4—48no—
298C4×D4—88no—
299#117—24no—
300#117—24no—
301#117—42no—
302#117—42no—
303#117—44no—
304#117—44no—
305#117—42no—
306#117—42no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4C2×D414yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D414yes—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—24no—
361C2×C42—44no—
362C2×C42C2412yes—
363C2×C42C2412yes—
364#107—28no—
365#107—28no—
366C4⋊C8—44no—
367C4⋊C8—44no—
368#103—28no—
369#103—28no—
370#103—44no—
371#102—28no—
372#102—28no—
373#102—28no—
374#102—28no—
375#102—44no—
376C22⋊C8—24no—
377C22⋊C8—24no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95C2×D414yes—
385#95C2×D414yes—
386C24—24no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—41no—
391C24—41no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—44no—
396C24—42no—
397C24—81no—
398C24—81no—
399C24—82no—
400C24—82no—
401C24—82no—
402C24#12114yes—
403C24C22×D411yescharacteristic
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—48no—
411C2×Q8—42no—
412C2×Q8—81no—
413C2×Q8—81no—
414C2×Q8—84no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—48no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—48no—
434C2×D4—42no—
435C2×D4—84no—
436C2×D4—81no—
437C2×D4—81no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—816no—
441C2×D4—816no—
442C2×D4C22×D414yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4#12114yes—
479C22×C4#12114yes—
480C22×C4#12114yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2511yescharacteristic
493SD16—88no—
494D8—88no—
495M16—84no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—84no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4C22×D414yes—
523C4⋊C4H(5,𝔽2)14yes—
524C22⋊C4—24no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C42—24no—
545C42—24no—
546C42—24no—
547C42—44no—
548C42—42no—
549C42—84no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—22no—
561C23—28no—
562C23—24no—
563C23—44no—
564C23—44no—
565C23—48no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—164no—
583C23—164no—
584C23—164no—
585C23—164no—
586C23C2×#12112yes—
587C23C2×#12112yes—
588C23C2×#12112yes—
589C23#57614yes—
590C23C23×D411yescharacteristic
591Q8—44no—
592Q8—44no—
593Q8—84no—
594D4—28no—
595D4—48no—
596D4—416no—
597D4—44no—
598D4—44no—
599D4—44no—
600D4—84no—
601D4—88no—
602D4—88no—
603D4—88no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—84no—
641C2×C4—88no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—168no—
646C2×C4C2×#12112yes—
647C2×C4C2×#12112yes—
648C2×C4C2×#12112yes—
649C2×C4C23×D411yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)12yes—
652C2×C4#58512yes—
653C8—44no—
654C8—44no—
655C22—22no—
656C22—24no—
657C22—22no—
658C22—24no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—88no—
673C22—84no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—82no—
679C22—84no—
680C22—168no—
681C22—168no—
682C22#304114yes—
683C22C22×#12111yescharacteristic
684C22#348411yescharacteristic
685C22#348511yescharacteristic
686C22C2×#57612yes—
687C22#358612yes—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#345912yes—
700C4#349612yes—
701C2—24no—
702C2—42no—
703C2—42no—
704C2—84no—
705C2—162no—
706C2#3498411yescharacteristic
707C2C2×#304111yescharacteristic
708C2#6042711yescharacteristic
709C2#6162412yes—
710C2#6175912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3186C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3193C2semidirect
3. #37451C2semidirect
4. #34204C2semidirect
5. #33120C2semidirect
6. #32917C2semidirect
7. #29767C2semidirect
8. #29727C2semidirect
9. #25223C2semidirect
10. #25200C2semidirect
11. #25101C2semidirect
12. #25090C2semidirect
13. #21933C2semidirect
14. #20912C2semidirect
15. C2×#1873C2semidirect
19. C2×#518C22semidirect
23. C2×#484C22semidirect
26. C2×#480C22semidirect
29. C2×#463C22semidirect
30. C2×#448C22semidirect
31. C2×#450C22semidirect
34. C2×#418C22semidirect
35. C22×#103C22semidirect
43. #2502C22semidirect
44. #2501C22semidirect
45. #2436C22semidirect
46. #2435C22semidirect
49. #2361C22semidirect
53. #2316C22semidirect
54. #2309C22semidirect
57. #2274C22semidirect
58. #2083C22semidirect
59. #2071C22semidirect
60. #2068C22semidirect
61. #2067C22semidirect
62. #2055C22semidirect
63. #2054C22semidirect
64. #2042C22semidirect
65. #2012C22semidirect
66. #1983C22semidirect
67. #1947C22semidirect
68. #1928C22semidirect
69. #1925C22semidirect
70. #1879C22semidirect
71. #1877C22semidirect
74. #1874C22semidirect
75. #1873C22semidirect
76. #1873C22semidirect
77. #1873C22semidirect
78. #1851C22semidirect
79. #1850C22semidirect
80. #1802C22semidirect
81. #1798C22semidirect
82. #1798C22semidirect
83. #1736C22semidirect
84. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#31862
#374512
#342042
#331202
#329172
#297672
#297272
#252232
#252002
#251012
#250902
#219332
#209122
C2×#18732