Finite Group Databasebeta

← #6470399

Subgroups — #6470399

Order 512.

2862 conjugacy classes of subgroups.

The subgroup lattice has 777 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3202C211yesmaximal, characteristic, complemented
2C22×#408C211yesmaximal, characteristic
3#35305C212yesmaximal
4#32937C212yesmaximal, complemented
5#32931C212yesmaximal, complemented
6#29746C212yesmaximal, complemented
7#29739C212yesmaximal, complemented
8#24563C214yesmaximal, complemented
9#24554C214yesmaximal, complemented
10#24392C214yesmaximal, complemented
11#24380C214yesmaximal, complemented
12#21947C211yesmaximal, characteristic, complemented
13#21771C212yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C23×M16C2211yescharacteristic
16C24×C8C2211yescharacteristic
17C2×#536C2212yes—
18C2×#536C2212yes—
19C22×C22⋊Q8C2212yes—
20#3202—28no—
21C2×#485C2214yescomplemented
22C2×#470C2212yescomplemented
23C2×#470C2212yescomplemented
24C2×#467C2212yes—
25C2×#467C2212yes—
26C2×#467C2212yescomplemented
27C2×#467C2212yescomplemented
28C2×#451C2214yescomplemented
29C2×#408—28no—
30C2×#408C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#408C2211yescharacteristic
33C2×#408C2214yes—
34C2×#408C2211yescharacteristic
35C22×C22⋊C8C2214yes—
36#2848C2214yes—
37#2848C2214yes—
38#2616C2214yes—
39#2616C2214yes—
40#2435—22no—
41#2434—22no—
42#2432—22no—
43#2432—22no—
44#2406—22no—
45#2406—22no—
46#2405—22no—
47#2405—22no—
48#2311C2212yes—
49#2311C2212yes—
50#2280C2212yes—
51#2077C2214yescomplemented
52#2075C2214yescomplemented
53#2036C2214yescomplemented
54#2036C2214yescomplemented
55#2036C2214yescomplemented
56#2035C2214yescomplemented
57#2001C2214yescomplemented
58#2001C2214yescomplemented
59#1894C2214yescomplemented
60#1894C2214yescomplemented
61#1892C2214yes—
62#1892C2214yes—
63#1892C2214yescomplemented
64#1892C2214yescomplemented
65#1873C2214yescomplemented
66#1872C2214yescomplemented
67#1854C2214yescomplemented
68#1852C2214yescomplemented
69#1825C2214yescomplemented
70#1809C2212yescomplemented
71#1809C2212yescomplemented
72#1808C2212yes—
73#1808C2212yescomplemented
74#1808C2212yescomplemented
75#1807C2212yes—
76#1787C2214yescomplemented
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—42no—
80C22×#41—42no—
81C22×#41—42no—
82C22×#41—42no—
83C22×#41D411yescharacteristic
84C22×#41D412yes—
85C22×#41D411yescharacteristic
86C22×#41C2312yes—
87C22×#41C2311yescharacteristic
88C22×#41C2311yescharacteristic
89C23×Q8D411yescharacteristic
90C23×Q8C2312yes—
91C23×D4D411yescharacteristic
92C23×D4D412yes—
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C22×Q16C2312yes—
96C22×Q16C2312yes—
97C22×M16—24no—
98C22×M16—42no—
99C22×M16—42no—
100C22×M16C2312yes—
101C22×M16C2311yescharacteristic
102C22×M16C2311yescharacteristic
103C23×C8—24no—
104C23×C8—42no—
105C23×C8—42no—
106C23×C8C2312yes—
107C23×C8C2312yes—
108C23×C8C2312yes—
109C23×C8C2311yescharacteristic
110C23×C8C2311yescharacteristic
111#536—28no—
112#536—28no—
113#536—44no—
114#536—44no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—22no—
117C2×#124—22no—
118C2×#124—22no—
119C2×#123—22no—
120C2×#123—22no—
121C2×#123—22no—
122C2×#123—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—44no—
126C2×C22⋊Q8—44no—
127C2×C22⋊Q8C2314yes—
128C2×C22⋊Q8C2314yes—
129C2×C22⋊Q8C2314yes—
130C2×C22⋊Q8C2314yes—
131C2×C22⋊Q8C2314yes—
132C2×C22⋊Q8C2314yes—
133C2×#117C2312yes—
134C2×#117C2312yes—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2312yes—
137C22×C22⋊C4—22no—
138#485—216no—
139#470—28no—
140#470—44no—
141#467—28no—
142#467—44no—
143#467—44no—
144#467—44no—
145#451—216no—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158#408—24no—
159#408—24no—
160#408—24no—
161#408—216no—
162#408D418yes—
163C2×C22⋊C8—216no—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2314yes—
168#395—24no—
169#395—24no—
170#394—24no—
171#394—24no—
172#394—24no—
173#394—24no—
174#394—24no—
175#394—24no—
176#391—22no—
177#391—22no—
178#390—22no—
179#390—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#385—24no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—24no—
189#385—24no—
190#381—22no—
191#380C2314yes—
192#380C2314yes—
193#380C2314yes—
194#380C2314yes—
195#380C2314yes—
196#380C2314yes—
197#379—24no—
198#379—24no—
199#379—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208C25C2×D411yescharacteristic
209C25C2×D412yes—
210C2×#41—24no—
211C2×#41—28no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—22no—
215C2×#41—24no—
216C2×#41—22no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—42no—
220C2×#41—44no—
221C2×#41—42no—
222C2×#41—41no—
223C2×#41—41no—
224C2×#41—42no—
225C2×#41—42no—
226C2×#41—41no—
227C2×#41—41no—
228C2×#41—41no—
229C2×#41—41no—
230C2×#41—41no—
231C2×#41—41no—
232C2×#41—42no—
233C2×#41—42no—
234C2×#41—44no—
235C2×#41—416no—
236C2×#41—416no—
237C2×#41—416no—
238C2×#41—416no—
239C2×#41C2×D414yes—
240C2×#41#4114yes—
241C22×Q8—24no—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8—42no—
248C22×Q8—42no—
249C22×Q8—42no—
250C22×Q8—42no—
251C22×Q8C2×D412yes—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2414yes—
255C22×D4—24no—
256C22×D4—28no—
257C22×D4—42no—
258C22×D4—42no—
259C22×D4—44no—
260C22×D4—44no—
261C22×D4C2×D412yes—
262C22×D4C2×D412yes—
263C22×D4C2×D412yes—
264C22×D4C2×D412yes—
265C22×D4C2×D411yescharacteristic
266C22×D4C2×D411yescharacteristic
267C22×D4C2×D412yes—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—24no—
275C23×C4—24no—
276C23×C4—24no—
277C23×C4—24no—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—42no—
281C23×C4—42no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D412yes—
286C23×C4C2×D412yes—
287C23×C4C2×D412yes—
288C23×C4C2412yes—
289C23×C4C2412yes—
290C23×C4C2411yescharacteristic
291C23×C4C2411yescharacteristic
292C23×C4C2411yescharacteristic
293C2×Q16—24no—
294C2×Q16—24no—
295C2×Q16—42no—
296C2×Q16—42no—
297C2×Q16—42no—
298C2×Q16—42no—
299C2×M16—22no—
300C2×M16—24no—
301C2×M16—24no—
302C2×M16—41no—
303C2×M16—41no—
304C2×M16—42no—
305C2×M16—41no—
306C2×M16—41no—
307C2×M16—416no—
308C2×M16C2×D414yes—
309C22×C8—22no—
310C22×C8—24no—
311C22×C8—24no—
312C22×C8—24no—
313C22×C8—24no—
314C22×C8—24no—
315C22×C8—24no—
316C22×C8—41no—
317C22×C8—41no—
318C22×C8—42no—
319C22×C8—42no—
320C22×C8—41no—
321C22×C8—41no—
322C22×C8—42no—
323C22×C8—416no—
324C22×C8C2×D414yes—
325C22×C8C2412yes—
326C22×C8C2412yes—
327C22×C8C2412yes—
328C22×C8C2412yes—
329C4⋊Q8—48no—
330C4⋊Q8—48no—
331#124—48no—
332#124—48no—
333#123—48no—
334#123—48no—
335#123—48no—
336#123—48no—
337C22⋊Q8—28no—
338C22⋊Q8—28no—
339C22⋊Q8—28no—
340C22⋊Q8—28no—
341C22⋊Q8—28no—
342C22⋊Q8—28no—
343C22⋊Q8—28no—
344C22⋊Q8—28no—
345C22⋊Q8—44no—
346C22⋊Q8—44no—
347C22⋊Q8—44no—
348C22⋊Q8—44no—
349C22⋊Q8—48no—
350C22⋊Q8—48no—
351#117—24no—
352#117—24no—
353#117—42no—
354#117—42no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—48no—
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2414yes—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—48no—
383C2×C42—22no—
384C2×C42—22no—
385#107—24no—
386#107—42no—
387#107—42no—
388#107—42no—
389#106—24no—
390#106—42no—
391#103—28no—
392#103—28no—
393#103—44no—
394#103—44no—
395#103—44no—
396#103—44no—
397C22⋊C8—28no—
398C22⋊C8—28no—
399C22⋊C8—28no—
400C22⋊C8C2×D4116yes—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—24no—
411C24—24no—
412C24—24no—
413C24—28no—
414C24—42no—
415C24—42no—
416C24—44no—
417C24C22×D412yes—
418C24C22×D411yescharacteristic
419C24C22×D412yes—
420C24C22×D411yescharacteristic
421C24C22×D411yescharacteristic
422#41—44no—
423#41—44no—
424#41—44no—
425#41—44no—
426#41—48no—
427#41—48no—
428#41—48no—
429#41—48no—
430#41—416no—
431#41—432no—
432#41—416no—
433#41—44no—
434#41—44no—
435#41—44no—
436#41—44no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8—42no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—416no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—84no—
456C2×Q8—84no—
457C2×Q8—84no—
458C2×Q8—84no—
459C2×Q8#12114yes—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—24no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—42no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—416no—
480C2×D4—432no—
481C2×D4#12114yes—
482C2×D4#12118yes—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—28no—
486C22×C4—28no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—24no—
513C22×C4—24no—
514C22×C4—24no—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—24no—
525C22×C4—24no—
526C22×C4—24no—
527C22×C4—24no—
528C22×C4—42no—
529C22×C4—42no—
530C22×C4—42no—
531C22×C4—48no—
532C22×C4—48no—
533C22×C4—48no—
534C22×C4—48no—
535C22×C4—44no—
536C22×C4—44no—
537C22×C4—44no—
538C22×C4—44no—
539C22×C4—84no—
540C22×C4C22×D412yes—
541C22×C4C22×D411yescharacteristic
542C22×C4C22×D412yes—
543C22×C4C22×D414yes—
544C22×C4C22×D412yes—
545C22×C4C22×D411yescharacteristic
546C22×C4C22×D411yescharacteristic
547C22×C4C2×#4114yes—
548C22×C4C2×#4112yes—
549C22×C4H(5,𝔽2)12yes—
550C22×C4H(5,𝔽2)12yes—
551C22×C4H(5,𝔽2)12yes—
552C22×C4H(5,𝔽2)12yes—
553C22×C4C2511yescharacteristic
554Q16—84no—
555Q16—84no—
556M16—44no—
557M16—48no—
558M16—44no—
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—22no—
562C2×C8—22no—
563C2×C8—22no—
564C2×C8—22no—
565C2×C8—22no—
566C2×C8—22no—
567C2×C8—22no—
568C2×C8—22no—
569C2×C8—44no—
570C2×C8—48no—
571C2×C8—48no—
572C2×C8—48no—
573C2×C8—44no—
574C2×C8C22×D414yes—
575C2×C8C22×D414yes—
576C4⋊C4—24no—
577C4⋊C4—24no—
578C4⋊C4—24no—
579C4⋊C4—24no—
580C4⋊C4—24no—
581C4⋊C4—24no—
582C4⋊C4—44no—
583C4⋊C4—48no—
584C4⋊C4—48no—
585C4⋊C4—48no—
586C4⋊C4—48no—
587C4⋊C4—48no—
588C4⋊C4—48no—
589C22⋊C4—48no—
590C22⋊C4—44no—
591C22⋊C4—48no—
592C22⋊C4—44no—
593C22⋊C4—48no—
594C22⋊C4—48no—
595C22⋊C4—48no—
596C22⋊C4—48no—
597C22⋊C4—48no—
598C22⋊C4—84no—
599C22⋊C4—84no—
600C22⋊C4—84no—
601C22⋊C4—84no—
602C22⋊C4—84no—
603C42—44no—
604C42—44no—
605C23—22no—
606C23—21no—
607C23—21no—
608C23—28no—
609C23—22no—
610C23—22no—
611C23—24no—
612C23—21no—
613C23—22no—
614C23—22no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—22no—
619C23—24no—
620C23—22no—
621C23—24no—
622C23—22no—
623C23—22no—
624C23—22no—
625C23—24no—
626C23—48no—
627C23—48no—
628C23—48no—
629C23—416no—
630C23—44no—
631C23—44no—
632C23—48no—
633C23—84no—
634C23—88no—
635C23C2×#12112yes—
636C23C2×#12114yes—
637C23#53412yes—
638C23#53412yes—
639C23#57814yes—
640C23C23×D411yescharacteristic
641Q8—44no—
642Q8—48no—
643Q8—44no—
644Q8—44no—
645Q8—44no—
646Q8—44no—
647Q8—44no—
648Q8—88no—
649D4—44no—
650D4—48no—
651D4—48no—
652D4—48no—
653D4—44no—
654D4—48no—
655D4—48no—
656C2×C4—28no—
657C2×C4—28no—
658C2×C4—28no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—24no—
674C2×C4—24no—
675C2×C4—22no—
676C2×C4—22no—
677C2×C4—22no—
678C2×C4—22no—
679C2×C4—22no—
680C2×C4—22no—
681C2×C4—24no—
682C2×C4—24no—
683C2×C4—22no—
684C2×C4—22no—
685C2×C4—22no—
686C2×C4—48no—
687C2×C4—48no—
688C2×C4—48no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—88no—
706C2×C4—88no—
707C2×C4C2×#12114yes—
708C2×C4C2×#12112yes—
709C2×C4#53412yes—
710C2×C4#53412yes—
711C2×C4C23×D411yescharacteristic
712C2×C4C22×#4111yescharacteristic
713C2×C4C2×H(5,𝔽2)11yescharacteristic
714C2×C4C2×H(5,𝔽2)11yescharacteristic
715C2×C4C2×H(5,𝔽2)11yescharacteristic
716C2×C4C2×H(5,𝔽2)11yescharacteristic
717C8—24no—
718C8—24no—
719C8—42no—
720C8—42no—
721C22—24no—
722C22—22no—
723C22—24no—
724C22—22no—
725C22—24no—
726C22—28no—
727C22—24no—
728C22—22no—
729C22—22no—
730C22—24no—
731C22—416no—
732C22—42no—
733C22—41no—
734C22—41no—
735C22—42no—
736C22—42no—
737C22—41no—
738C22—41no—
739C22—42no—
740C22—44no—
741C22—44no—
742C22—42no—
743C22—42no—
744C22—44no—
745C22—42no—
746C22—816no—
747C22#320214yescomplemented
748C22C22×#12111yescharacteristic
749C22C2×#53411yescharacteristic
750C22C2×#53411yescharacteristic
751C22C2×#57812yes—
752C22#358412yes—
753C4—22no—
754C4—22no—
755C4—22no—
756C4—24no—
757C4—42no—
758C4—84no—
759C4—84no—
760C4—84no—
761C4—84no—
762C4#348411yescharacteristic
763C4#348411yescharacteristic
764C4#349011yescharacteristic
765C4#349011yescharacteristic
766C2—24no—
767C2—42no—
768C2—42no—
769C2—44no—
770C2—82no—
771C2#3498111yescharacteristic
772C2#6045912yes—
773C2C2×#320211yescharacteristic
774C2#6074911yescharacteristic
775C2#6077012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3202C2semidirect
4. #32937C2semidirect
5. #32931C2semidirect
6. #29746C2semidirect
7. #29739C2semidirect
8. #24563C2semidirect
9. #24554C2semidirect
10. #24392C2semidirect
11. #24380C2semidirect
12. #21947C2semidirect
13. #21771C2semidirect
21. C2×#485C22semidirect
22. C2×#470C22semidirect
23. C2×#470C22semidirect
26. C2×#467C22semidirect
27. C2×#467C22semidirect
28. C2×#451C22semidirect
51. #2077C22semidirect
52. #2075C22semidirect
53. #2036C22semidirect
54. #2036C22semidirect
55. #2036C22semidirect
56. #2035C22semidirect
57. #2001C22semidirect
58. #2001C22semidirect
59. #1894C22semidirect
60. #1894C22semidirect
63. #1892C22semidirect
64. #1892C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
67. #1854C22semidirect
68. #1852C22semidirect
69. #1825C22semidirect
70. #1809C22semidirect
71. #1809C22semidirect
73. #1808C22semidirect
74. #1808C22semidirect
76. #1787C22semidirect
747. C22#3202semidirect

Maximal subgroups

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

subgroupindex
C2×#32022
C22×#4082
#353052
#329372
#329312
#297462
#297392
#245632
#245542
#243922
#243802
#219472
#217712