Finite Group Databasebeta

← #6453001

Subgroups — #6453001

Order 512.

3302 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic, complemented
3C2×#2911C211yesmaximal, characteristic, complemented
4#34809C211yesmaximal, characteristic
5#24108C214yesmaximal, complemented
6#24107C214yesmaximal, complemented
7#24096C218yesmaximal, complemented
8#23552C212yesmaximal, complemented
9#23481C212yesmaximal, complemented
10#20972C211yesmaximal, characteristic, complemented
11#20898C211yesmaximal, characteristic
12#20874C212yesmaximal, complemented
13#20845C212yesmaximal, complemented
14#20701C211yesmaximal, characteristic, complemented
15C23×M16C2211yescharacteristic
16C2×#556—28no—
17C2×#556—22no—
18C2×#556—22no—
19C2×#556—28no—
20C2×#556C2214yes—
21C2×#556C2214yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#126—21no—
24C22×#126—21no—
25C22×#124—21no—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C22×#117C2211yescharacteristic
33C22×#117C2211yescharacteristic
34#3095—28no—
35C2×#484C2212yescomplemented
36C2×#485C2214yescomplemented
37C2×#483C2212yescomplemented
38#2911—28no—
39C2×#428C2212yes—
40C2×#428C2212yes—
41C2×#428C2212yes—
42C2×#428C2212yescomplemented
43C2×#428C2212yescomplemented
44C2×#427C2214yescomplemented
45C2×#419C2212yes—
46C2×#419C2212yes—
47C2×#407C2211yescharacteristic
48C2×#407C2212yes—
49#2487C2212yes—
50#2486C2212yes—
51#2485C2214yes—
52#2484—21no—
53#2483—21no—
54#2482—21no—
55#2405—21no—
56#2395—22no—
57C4×#123—22no—
58#2296C2212yes—
59#2280C2212yes—
60#2280C2211yescharacteristic
61#2058C2218yescomplemented
62#2057C2218yescomplemented
63#2036C2218yescomplemented
64#2035C2218yescomplemented
65#1894C2214yescomplemented
66#1892C2214yescomplemented
67#1870C2214yes—
68#1869C2214yes—
69#1865C2218yescomplemented
70#1863C2218yescomplemented
71#1854C2218yescomplemented
72#1854C2218yescomplemented
73#1825C2214yescomplemented
74#1825C2214yescomplemented
75#1736C2212yes—
76#1736C2212yes—
77#1736C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16—24no—
85C22×M16—28no—
86C22×M16C2312yes—
87C22×M16C2312yes—
88C22×M16C2312yes—
89C23×C8C2312yes—
90#556—232no—
91#556—232no—
92#556—216no—
93#556—216no—
94#556—216no—
95#556—216no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2314yes—
100C2×#126—216no—
101C2×#126—216no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#126—24no—
105C2×#126—24no—
106C2×#124—216no—
107C2×#124—24no—
108C2×#124—24no—
109C2×#123—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8D418yes—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—216no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2314yes—
127C2×C4×D4—24no—
128C2×C4×D4—24no—
129C2×C4×D4D412yes—
130C2×C4×D4D412yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—24no—
134C2×#117—22no—
135C2×#117—21no—
136C2×#117—21no—
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—24no—
140C2×#117—24no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117D418yes—
144C2×#117C2312yes—
145C2×#117C2314yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C2×#117C2312yes—
149C2×#117C2312yes—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4D411yescharacteristic
166C22×C22⋊C4D411yescharacteristic
167C22×C42—21no—
168C22×C42—21no—
169C22×C42—21no—
170C22×C42—21no—
171C22×C42C2311yescharacteristic
172#485—216no—
173#484—28no—
174#483—28no—
175#428—28no—
176#428—28no—
177#428—28no—
178#428—28no—
179#428—28no—
180#427—216no—
181C2×#107C2314yes—
182C2×#107C2314yes—
183C2×#107C2314yes—
184C2×#106C2314yes—
185C2×#106C2314yes—
186C2×#106C2314yes—
187#419—28no—
188#419—28no—
189C2×#103C2314yes—
190C2×#103C2314yes—
191C2×#103C2318yes—
192#407—24no—
193#407—28no—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2318yes—
197#400—22no—
198#399—22no—
199#398—22no—
200#397—22no—
201#397—24no—
202#394—24no—
203#389C2314yes—
204#389C2314yes—
205#388—24no—
206#387—24no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#381—21no—
212#381—22no—
213#381—22no—
214#380C2312yes—
215#380C2312yes—
216#380C2314yes—
217#380C2314yes—
218#379—22no—
219#379—21no—
220C4×C4⋊C4—24no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4—21no—
224#340C2318yes—
225#340C2318yes—
226#340C2318yes—
227#336C2318yes—
228C25C2×D411yescharacteristic
229C25C2×D411yescharacteristic
230C22×Q8—28no—
231C22×Q8—22no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—48no—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×D4—24no—
240C22×D4—44no—
241C22×D4C2×D412yes—
242C22×D4C2×D412yes—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D414yes—
262C23×C4C2×D414yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—232no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C2×M16—28no—
276C2×M16#4118yes—
277C22×C8—24no—
278C22×C8—28no—
279C22×C8C2414yes—
280C22×C8C2414yes—
281C4⋊Q8—216no—
282C4⋊Q8—28no—
283C4⋊Q8—28no—
284#126—232no—
285#126—232no—
286#126—232no—
287#126—232no—
288#124—232no—
289#124—232no—
290#123—48no—
291C22⋊Q8—216no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—28no—
298C22⋊Q8C2×D418yes—
299C22⋊Q8C2×D418yes—
300C4×Q8—232no—
301C4×Q8—232no—
302C4×Q8—216no—
303C4×Q8—28no—
304C4×Q8—28no—
305C4×D4—28no—
306C4×D4—28no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—24no—
311#117—216no—
312#117—28no—
313#117—28no—
314#117—24no—
315#117—24no—
316#117—28no—
317#117—28no—
318#117—24no—
319#117—24no—
320#117—24no—
321#117—24no—
322#117—24no—
323#117—24no—
324#117—44no—
325#117—44no—
326#117C2×D418yes—
327#117C2×D418yes—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—28no—
364C2×C22⋊C4—28no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—44no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C42—28no—
388C2×C42—28no—
389C2×C42—28no—
390C2×C42—24no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—24no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—44no—
400C2×C42C2×D412yes—
401C2×C42C2412yes—
402#107—28no—
403#107—28no—
404#107—28no—
405#106—28no—
406#106—28no—
407#106—28no—
408#103—28no—
409#103—28no—
410#103—216no—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C22⋊C8—216no—
414#95—24no—
415#95—24no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420C24—24no—
421C24—22no—
422C24—24no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24#12014yes—
428C24C22×D412yes—
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C24C22×D411yescharacteristic
432C2×Q8—216no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—216no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—432no—
439C2×Q8—48no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—48no—
444C2×Q8C22×D414yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—416no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—216no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—28no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—82no—
496C22×C4#12014yes—
497C22×C4#12014yes—
498C22×C4#12014yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C2×#4114yes—
512C22×C42−1+414yes—
513C22×C4C2511yescharacteristic
514M16—216no—
515M16—216no—
516M16—48no—
517C2×C8—216no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8C2×#4118yes—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—48no—
546C4⋊C4—84no—
547C4⋊C4C22×D414yes—
548C4⋊C4C22×D414yes—
549C4⋊C4C22×D414yes—
550C22⋊C4—216no—
551C22⋊C4—216no—
552C22⋊C4—28no—
553C22⋊C4—28no—
554C22⋊C4—28no—
555C22⋊C4—28no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—44no—
561C22⋊C4—48no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C22⋊C4—84no—
567C42—216no—
568C42—216no—
569C42—28no—
570C42—28no—
571C42—24no—
572C42—22no—
573C42—22no—
574C42—44no—
575C42—44no—
576C42—44no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—28no—
586C23—24no—
587C23—22no—
588C23—24no—
589C23—22no—
590C23—22no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—88no—
597C23—88no—
598C23C2×#12012yes—
599C23C2×#12012yes—
600C23C2×#12012yes—
601C23#54714yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—432no—
605Q8—48no—
606Q8—42no—
607Q8—42no—
608D4—48no—
609D4—48no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—48no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4C2×#12012yes—
656C2×C4C2×#12012yes—
657C2×C4C2×#12012yes—
658C2×C4#52514yes—
659C2×C4#53614yes—
660C2×C4#54714yes—
661C2×C4#54718yes—
662C2×C4#57514yes—
663C2×C4C23×D411yescharacteristic
664C2×C4C23×D411yescharacteristic
665C2×C4C22×#4112yes—
666C2×C4C2×2−1+412yes—
667C8—28no—
668C8—44no—
669C22—28no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—88no—
686C22—88no—
687C22#240614yes—
688C22C22×#12011yescharacteristic
689C22C2×#54712yes—
690C22C2×#57512yes—
691C22#358612yes—
692C4—22no—
693C4—22no—
694C4—24no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—44no—
699C4—42no—
700C4—44no—
701C4—84no—
702C4—82no—
703C4—82no—
704C4#240614yes—
705C4C2×#52511yescharacteristic
706C4C2×#53611yescharacteristic
707C4C2×#54712yes—
708C2—24no—
709C2—42no—
710C2—42no—
711C2—82no—
712C2C2×#240611yescharacteristic
713C2#6035911yescharacteristic
714C2#6036811yescharacteristic
715C2#6125714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
3. C2×#2911C2semidirect
5. #24108C2semidirect
6. #24107C2semidirect
7. #24096C2semidirect
8. #23552C2semidirect
9. #23481C2semidirect
10. #20972C2semidirect
12. #20874C2semidirect
13. #20845C2semidirect
14. #20701C2semidirect
35. C2×#484C22semidirect
36. C2×#485C22semidirect
37. C2×#483C22semidirect
42. C2×#428C22semidirect
43. C2×#428C22semidirect
44. C2×#427C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
63. #2036C22semidirect
64. #2035C22semidirect
65. #1894C22semidirect
66. #1892C22semidirect
69. #1865C22semidirect
70. #1863C22semidirect
71. #1854C22semidirect
72. #1854C22semidirect
73. #1825C22semidirect
74. #1825C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#30952
C2×#29112
#348092
#241082
#241072
#240962
#235522
#234812
#209722
#208982
#208742
#208452
#207012