Finite Group Databasebeta

← #6500363

Subgroups — #6500363

Order 512.

2708 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic, complemented
2C2×#3007C211yesmaximal, characteristic
3#35305C211yesmaximal, characteristic
4#34370C211yesmaximal, characteristic, complemented
5#28133C214yesmaximal, complemented
6#25105C214yesmaximal, complemented
7#25095C214yesmaximal, complemented
8#24491C214yesmaximal, complemented
9#24369C214yesmaximal, complemented
10#24221C214yesmaximal, complemented
11#21935C211yesmaximal, characteristic, complemented
12#21768C211yesmaximal, characteristic, complemented
13#20600C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#137C2212yes—
16C2×#536C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19#3195—28no—
20C2×#483C2212yescomplemented
21C2×#483C2212yescomplemented
22C2×#470C2214yes—
23C2×#464C2214yescomplemented
24C2×#462C2214yescomplemented
25#3007—28no—
26C2×#452C2214yes—
27C2×#451C2212yes—
28C2×#451C2212yescomplemented
29C2×#451C2212yescomplemented
30C2×#450C2212yes—
31C2×#417C2212yes—
32C2×#419C2212yes—
33C2×#418C2212yes—
34C2×#407C2211yescharacteristic
35C2×#407C2211yescharacteristic
36#2848—22no—
37#2616—22no—
38#2435C2212yes—
39#2434C2212yes—
40#2432C2214yes—
41#2406—22no—
42#2405—21no—
43#2405—21no—
44#2378—22no—
45#2362C2214yes—
46#2345—22no—
47#2345C2212yescomplemented
48Q8×C22⋊C4C2212yescomplemented
49#2332—22no—
50#2323C2214yescomplemented
51#2312—21no—
52#2312—21no—
53#2311C2212yes—
54#2280C2211yescharacteristic
55#2280C2211yescharacteristic
56#2066C2214yescomplemented
57#2065C2214yescomplemented
58#2056C2214yescomplemented
59#2055C2214yescomplemented
60#2043C2214yescomplemented
61#2042C2214yescomplemented
62#2016C2214yescomplemented
63#2015C2214yescomplemented
64#1983C2214yescomplemented
65#1981C2214yescomplemented
66#1927C2214yescomplemented
67#1926C2214yescomplemented
68#1894C2214yescomplemented
69#1880C2214yescomplemented
70#1878C2214yescomplemented
71#1873C2214yes—
72#1851C2214yes—
73#1851C2214yescomplemented
74#1824C2212yescomplemented
75#1798C2214yescomplemented
76#1788C2214yes—
77#1788C2214yes—
78#1787C2212yescomplemented
79C22×#41—21no—
80C22×#41—21no—
81C22×#41—24no—
82C22×#41—24no—
83C22×#41—44no—
84C22×#41—44no—
85C22×#41C2312yes—
86C22×#41C2312yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4—21no—
93C24×C4C2311yescharacteristic
94C2×#137—28no—
95C2×#137—44no—
96C2×#137—44no—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100C22×M16C2312yes—
101C22×M16C2312yes—
102#536—44no—
103#536—44no—
104C2×C4⋊Q8C2314yes—
105C2×#124C2314yes—
106C2×#123—22no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
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×C4×Q8C2314yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130#483—28no—
131#483—28no—
132#470—216no—
133#464—48no—
134#462—48no—
135#452—216no—
136#451—28no—
137#451—28no—
138#451—28no—
139#450—28no—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143#419—44no—
144#418—44no—
145#417—28no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156#407—24no—
157#407—24no—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162#395—24no—
163#394—24no—
164#394—24no—
165#394—24no—
166#391—24no—
167#391C2314yes—
168#391C2314yes—
169#390C2314yes—
170#389—24no—
171#389C2314yes—
172#388—24no—
173#388—24no—
174#388—24no—
175#388—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#385—24no—
184#385—24no—
185#381—21no—
186#381D412yes—
187#380—24no—
188#380—22no—
189#380—22no—
190#380D412yes—
191#380D412yes—
192#380C2312yes—
193#380C2312yes—
194#379—21no—
195#379—22no—
196#379—22no—
197#379D411yescharacteristic
198#379D411yescharacteristic
199C4×C4⋊C4—24no—
200C4×C4⋊C4C2314yes—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4C2312yes—
206C4×C22⋊C4C2312yes—
207#340C2314yes—
208#340C2314yes—
209#336C2314yes—
210#336C2314yes—
211C25—21no—
212C25C2×D411yescharacteristic
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—42no—
222C2×#41—42no—
223C2×#41—42no—
224C2×#41—42no—
225C2×#41—48no—
226C2×#41—416no—
227C2×#41—48no—
228C2×#41—416no—
229C2×#41—432no—
230C2×#41—44no—
231C2×#41—44no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8C2×D414yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C22×D4—24no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—42no—
257C22×D4—48no—
258C22×D4—44no—
259C22×D4C2412yes—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4—42no—
268C23×C4—42no—
269C23×C4—48no—
270C23×C4—44no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D414yes—
275C23×C4#4114yes—
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279#137—432no—
280C2×Q16—28no—
281C2×Q16—28no—
282C2×Q16—28no—
283C2×Q16—44no—
284C2×Q16—44no—
285C2×Q16—44no—
286C2×SD16—28no—
287C2×SD16—28no—
288C2×SD16—44no—
289C2×M16—24no—
290C2×M16—24no—
291C2×M16—42no—
292C2×M16—42no—
293C2×M16—42no—
294C2×M16—42no—
295C22×C8C2414yes—
296C22×C8C2414yes—
297C4⋊Q8—28no—
298C4⋊Q8—44no—
299#124—28no—
300#124—44no—
301#123—48no—
302#123—48no—
303C22⋊Q8—28no—
304C22⋊Q8—28no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—416no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314C22⋊Q8C2×D418yes—
315C22⋊Q8C2×D418yes—
316C4×Q8—44no—
317C4×Q8—44no—
318#117—42no—
319#117—42no—
320#117—42no—
321#117—42no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—22no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42C2414yes—
365#106—28no—
366C4⋊C8—44no—
367C4⋊C8—44no—
368#103—28no—
369#103—28no—
370#103—28no—
371#103—28no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—28no—
376#102—28no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95C2×D414yes—
391#95C2×D414yes—
392C24—22no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—44no—
397C24—44no—
398C24—42no—
399C24—82no—
400C24#12114yes—
401C24C22×D411yescharacteristic
402C24C22×D411yescharacteristic
403C24C22×D411yescharacteristic
404#41—216no—
405#41—48no—
406#41—48no—
407#41—48no—
408#41—48no—
409#41—48no—
410#41—48no—
411#41—48no—
412#41—432no—
413#41—432no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—28no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—82no—
436C2×Q8—84no—
437C2×Q8—82no—
438C2×Q8—82no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8C22×D414yes—
442C2×Q8H(5,𝔽2)14yes—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—48no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—432no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4H(5,𝔽2)14yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—28no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—432no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4#12114yes—
497C22×C4#12114yes—
498C22×C4#12114yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2511yescharacteristic
511Q16—416no—
512Q16—88no—
513SD16—416no—
514M16—48no—
515M16—84no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4C22×D414yes—
539C22⋊C4—28no—
540C22⋊C4—28no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C42—24no—
557C42—24no—
558C42—84no—
559C42—82no—
560C42—84no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—48no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—84no—
580C23—84no—
581C23—88no—
582C23—816no—
583C23C2×#12112yes—
584C23C2×#12112yes—
585C23C2×#12112yes—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—24no—
589Q8—24no—
590Q8—48no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—44no—
596Q8—42no—
597Q8—42no—
598Q8—84no—
599Q8—84no—
600D4—24no—
601D4—24no—
602D4—48no—
603D4—416no—
604D4—416no—
605D4—42no—
606D4—42no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—416no—
632C2×C4—416no—
633C2×C4—48no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4C2×#12112yes—
650C2×C4C2×#12112yes—
651C2×C4C2×#12112yes—
652C2×C4C23×D411yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)12yes—
655C2×C4C2×H(5,𝔽2)12yes—
656C8—44no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—88no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22C22×#12111yescharacteristic
678C22#348412yes—
679C22#358612yes—
680C22#358612yes—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4#348412yes—
693C4#349011yescharacteristic
694C4#349011yescharacteristic
695C2—42no—
696C2—42no—
697C2—42no—
698C2—84no—
699C2#3534811yescharacteristic
700C2#6042411yescharacteristic
701C2#6042711yescharacteristic
702C2#6165514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
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×#3195C2semidirect
4. #34370C2semidirect
5. #28133C2semidirect
6. #25105C2semidirect
7. #25095C2semidirect
8. #24491C2semidirect
9. #24369C2semidirect
10. #24221C2semidirect
11. #21935C2semidirect
12. #21768C2semidirect
20. C2×#483C22semidirect
21. C2×#483C22semidirect
23. C2×#464C22semidirect
24. C2×#462C22semidirect
28. C2×#451C22semidirect
29. C2×#451C22semidirect
47. #2345C22semidirect
48. Q8×C22⋊C4C22semidirect
50. #2323C22semidirect
56. #2066C22semidirect
57. #2065C22semidirect
58. #2056C22semidirect
59. #2055C22semidirect
60. #2043C22semidirect
61. #2042C22semidirect
62. #2016C22semidirect
63. #2015C22semidirect
64. #1983C22semidirect
65. #1981C22semidirect
66. #1927C22semidirect
67. #1926C22semidirect
68. #1894C22semidirect
69. #1880C22semidirect
70. #1878C22semidirect
73. #1851C22semidirect
74. #1824C22semidirect
75. #1798C22semidirect
78. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#30072
#353052
#343702
#281332
#251052
#250952
#244912
#243692
#242212
#219352
#217682
#206002