Finite Group Databasebeta

← #6508613

Subgroups — #6508613

Order 512.

3398 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3505C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic, complemented
3C22×#480C211yesmaximal, characteristic, complemented
4C2×#2486C211yesmaximal, characteristic, complemented
5#24094C214yesmaximal, complemented
6#24091C214yesmaximal, complemented
7#24066C214yesmaximal, complemented
8#24038C214yesmaximal, complemented
9#24035C214yesmaximal, complemented
10#24004C214yesmaximal, complemented
11#21759C211yesmaximal, characteristic, complemented
12#21758C211yesmaximal, characteristic, complemented
13#21738C211yesmaximal, characteristic, complemented
14#3505—28no—
15#3505—28no—
16C2×#553—22no—
17C2×#553C2212yes—
18C2×#552C2212yes—
19C2×#551—22no—
20C2×#548C2212yes—
21C2×#546C2212yes—
22C2×#539—21no—
23C2×#538—21no—
24C2×#534—21no—
25C2×#529C2212yes—
26C2×#525C2212yes—
27C22×#123—21no—
28C22×C22⋊Q8C2211yescharacteristic
29C22×#121C2211yescharacteristic
30C2×#518—21no—
31C22×C4×D4C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33C2×#483—28no—
34C2×#483C2214yescomplemented
35C2×#483C2214yescomplemented
36C2×#480—28no—
37C2×#480C2214yes—
38C2×#480C2214yescomplemented
39C22×#107C2212yescomplemented
40C22×#106C2212yescomplemented
41C22×#103C2212yescomplemented
42C22×#102C2212yescomplemented
43C22×C22⋊C8C2211yescharacteristic
44C22×C22⋊C8C2211yescharacteristic
45#2486—22no—
46#2486—22no—
47#2486C2214yes—
48#2486C2214yescomplemented
49C2×#398—21no—
50C2×#397—21no—
51C2×#389C2212yescomplemented
52C2×#381—21no—
53C2×#380C2211yescharacteristic
54C2×#380C2212yescomplemented
55#2059C2218yescomplemented
56#2042C2218yescomplemented
57#2036C2218yescomplemented
58#2034C2218yescomplemented
59#1893C2214yescomplemented
60#1892C2214yescomplemented
61#1874C2214yes—
62#1872C2214yescomplemented
63#1869C2214yes—
64#1869C2214yescomplemented
65#1864C2218yescomplemented
66#1863C2218yescomplemented
67#1854C2218yescomplemented
68#1824C2212yescomplemented
69#1823C2212yescomplemented
70#1817C2212yescomplemented
71#1788C2218yescomplemented
72#1787C2212yescomplemented
73#1786C2212yescomplemented
74#1779C2212yescomplemented
75C22×#41C2312yes—
76C23×Q8C2311yescharacteristic
77C23×D4—21no—
78C23×D4D411yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8C2312yes—
84C23×C8C2312yes—
85#553—232no—
86#553—28no—
87#553—28no—
88#552—28no—
89#552—28no—
90#551—232no—
91#548—28no—
92#548—28no—
93#546—28no—
94#546—28no—
95#539—216no—
96#538—216no—
97#534—216no—
98#529—28no—
99#529—28no—
100C2×#126—22no—
101C2×#126—22no—
102C2×#126—22no—
103C2×#125—22no—
104C2×#125C2312yes—
105C2×#124—22no—
106C2×#124C2312yes—
107#525—28no—
108#525—28no—
109C2×#123—22no—
110C2×#123—22no—
111C2×#123—22no—
112C2×#123—24no—
113C2×#123—24no—
114C2×#123—22no—
115C2×#123—22no—
116C2×#123—48no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8D418yes—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125C2×#121—22no—
126C2×#121—22no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121—24no—
130C2×#121—22no—
131C2×#121D418yes—
132C2×#121C2312yes—
133C2×#121C2312yes—
134C2×#121C2312yes—
135C2×#120—22no—
136C2×#120—22no—
137#518—216no—
138C2×C4×Q8C2312yes—
139C2×C4×D4—24no—
140C2×C4×D4—24no—
141C2×C4×D4—22no—
142C2×C4×D4—22no—
143C2×C4×D4—22no—
144C2×C4×D4—22no—
145C2×C4×D4—22no—
146C2×C4×D4D412yes—
147C2×C4×D4D412yes—
148C2×C4×D4C2312yes—
149C2×C4×D4C2312yes—
150C2×C4×D4C2312yes—
151C2×#117—22no—
152C2×#117—22no—
153C2×#117C2312yes—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—24no—
156C22×C4⋊C4—24no—
157C22×C4⋊C4—21no—
158C22×C4⋊C4—22no—
159C22×C4⋊C4—21no—
160C22×C4⋊C4—21no—
161C22×C4⋊C4D411yescharacteristic
162C22×C4⋊C4C2312yes—
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2311yescharacteristic
166C22×C4⋊C4C2311yescharacteristic
167C22×C4⋊C4C2312yes—
168C22×C22⋊C4—21no—
169C22×C22⋊C4—21no—
170C22×C22⋊C4—22no—
171C22×C22⋊C4—22no—
172C22×C22⋊C4—21no—
173C22×C22⋊C4D411yescharacteristic
174C22×C22⋊C4D411yescharacteristic
175C22×C42C2311yescharacteristic
176#483—216no—
177#483—216no—
178#480—216no—
179#480—216no—
180C2×#107—28no—
181C2×#107C2318yes—
182C2×#106—28no—
183C2×#106C2318yes—
184C2×#103—28no—
185C2×#103C2318yes—
186C2×#103C2318yes—
187C2×#102—28no—
188C2×#102C2318yes—
189C2×#102C2318yes—
190C2×C22⋊C8—24no—
191C2×C22⋊C8—24no—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2318yes—
197#398—24no—
198#398—24no—
199#397—24no—
200#397—24no—
201#389—24no—
202#389C2318yes—
203#381—24no—
204#381—24no—
205#380—21no—
206#380—21no—
207#380—24no—
208#380C2312yes—
209#380C2312yes—
210#380C2318yes—
211C2×#95—21no—
212C2×#95—22no—
213C2×#95—22no—
214#340C2318yes—
215#340C2318yes—
216#336C2318yes—
217C25—21no—
218C25C2×D411yescharacteristic
219C25C2×D411yescharacteristic
220C2×#41—24no—
221C2×#41—24no—
222C2×#41—48no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—44no—
226C22×Q8C2×D414yes—
227C22×Q8C2412yes—
228C22×D4—24no—
229C22×D4—28no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—44no—
239C22×D4—44no—
240C22×D4—44no—
241C22×D4C2×D412yes—
242C22×D4C2×D412yes—
243C22×D4C2×D414yes—
244C22×D4C2412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D414yes—
269C23×C4C2×D414yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C22×C8—24no—
275C22×C8—24no—
276C22×C8—44no—
277C22×C8—44no—
278C22×C8C2414yes—
279C22×C8C2414yes—
280#126—216no—
281#126—216no—
282#126—216no—
283#125—216no—
284#125—24no—
285#125—24no—
286#124—216no—
287#124—24no—
288#124—24no—
289#123—216no—
290#123—216no—
291#123—216no—
292#123—216no—
293#123—216no—
294#123—416no—
295#123—416no—
296C22⋊Q8—216no—
297C22⋊Q8—24no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8C2×D418yes—
304C22⋊Q8C2×D418yes—
305#121—216no—
306#121—216no—
307#121—216no—
308#121—24no—
309#121—28no—
310#121—28no—
311#121—24no—
312#121—24no—
313#121—24no—
314#121C2×D418yes—
315#121C2×D418yes—
316#120—216no—
317#120—216no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×D4—216no—
321C4×D4—216no—
322C4×D4—216no—
323C4×D4—216no—
324C4×D4—216no—
325C4×D4—24no—
326C4×D4—28no—
327C4×D4—28no—
328C4×D4—24no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—24no—
332C4×D4—24no—
333#117—216no—
334#117—216no—
335#117—24no—
336#117—24no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—28no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—48no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—44no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4C2×D412yes—
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D414yes—
368C2×C4⋊C4C2×D414yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C22⋊C4—216no—
374C2×C22⋊C4—28no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—28no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—28no—
391C2×C22⋊C4—48no—
392C2×C22⋊C4—44no—
393C2×C22⋊C4C2×D412yes—
394C2×C22⋊C4C2×D412yes—
395C2×C22⋊C4C2×D412yes—
396C2×C22⋊C4C2×D412yes—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42C2×D412yes—
402C2×C42C2412yes—
403#107—216no—
404#106—216no—
405#103—216no—
406#103—48no—
407#102—216no—
408#102—48no—
409C22⋊C8—24no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C22⋊C8—48no—
414#95—28no—
415#95—22no—
416#95—22no—
417#95—28no—
418C24—22no—
419C24—22no—
420C24—24no—
421C24—22no—
422C24—22no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—82no—
430C24—82no—
431C24#12014yes—
432C24#13514yes—
433C24AGL(1,ℤ/8)14yes—
434C24C22×D412yes—
435C24C22×D411yescharacteristic
436C24C22×D411yescharacteristic
437C24C22×D411yescharacteristic
438#41—88no—
439#41—88no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—28no—
443C2×Q8—48no—
444C2×Q8—82no—
445C2×Q8—84no—
446C2×Q8C22×D414yes—
447C2×D4—24no—
448C2×D4—28no—
449C2×D4—24no—
450C2×D4—28no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—28no—
456C2×D4—28no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—28no—
460C2×D4—48no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—416no—
466C2×D4—432no—
467C2×D4—84no—
468C2×D4—82no—
469C2×D4—84no—
470C2×D4—84no—
471C2×D4C22×D414yes—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—28no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4—84no—
519C22×C4—84no—
520C22×C4—82no—
521C22×C4—82no—
522C22×C4—816no—
523C22×C4#12014yes—
524C22×C4#12014yes—
525C22×C4#12014yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C22×D412yes—
535C22×C4C22×D412yes—
536C22×C4C22×D412yes—
537C22×C4C22×D412yes—
538C22×C4C2511yescharacteristic
539C2×C8—24no—
540C2×C8—24no—
541C2×C8—28no—
542C2×C8—28no—
543C2×C8—48no—
544C2×C8—48no—
545C4⋊C4—28no—
546C4⋊C4—28no—
547C4⋊C4—28no—
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—28no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—28no—
560C4⋊C4—44no—
561C4⋊C4—416no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4C22×D414yes—
566C4⋊C4C22×D414yes—
567C22⋊C4—28no—
568C22⋊C4—28no—
569C22⋊C4—28no—
570C22⋊C4—28no—
571C22⋊C4—28no—
572C22⋊C4—28no—
573C22⋊C4—24no—
574C22⋊C4—28no—
575C22⋊C4—28no—
576C22⋊C4—24no—
577C22⋊C4—28no—
578C22⋊C4—28no—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—416no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C42—28no—
585C42—28no—
586C42—24no—
587C42—22no—
588C42—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—24no—
596C23—22no—
597C23—24no—
598C23—24no—
599C23—22no—
600C23—22no—
601C23—44no—
602C23—44no—
603C23—48no—
604C23—48no—
605C23—48no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—42no—
610C23—42no—
611C23—48no—
612C23—82no—
613C23—84no—
614C23—82no—
615C23—84no—
616C23—84no—
617C23—164no—
618C23—164no—
619C23#449116yes—
620C23C2×#12012yes—
621C23C2×#12012yes—
622C23C2×#12012yes—
623C23C2×#13512yes—
624C23C2×#13512yes—
625C23C2×#13512yes—
626C23C2×AGL(1,ℤ/8)12yes—
627C23C2×AGL(1,ℤ/8)12yes—
628C23C2×AGL(1,ℤ/8)12yes—
629C23C23×D411yescharacteristic
630Q8—28no—
631Q8—44no—
632Q8—84no—
633D4—28no—
634D4—48no—
635D4—48no—
636D4—416no—
637D4—416no—
638D4—44no—
639D4—84no—
640D4—88no—
641D4—88no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—48no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—84no—
679C2×C4—82no—
680C2×C4—82no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—88no—
684C2×C4—88no—
685C2×C4—88no—
686C2×C4—88no—
687C2×C4—88no—
688C2×C4—88no—
689C2×C4C2×#12012yes—
690C2×C4C2×#12012yes—
691C2×C4C2×#12012yes—
692C2×C4C23×D411yescharacteristic
693C2×C4C23×D411yescharacteristic
694C2×C4#58514yes—
695C8—44no—
696C8—44no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—416no—
710C22—44no—
711C22—44no—
712C22—42no—
713C22—44no—
714C22—42no—
715C22—44no—
716C22—44no—
717C22—42no—
718C22—88no—
719C22—84no—
720C22—84no—
721C22—168no—
722C22C2×#44914yes—
723C22C2×#44914yes—
724C22C2×#44914yes—
725C22C22×#12011yescharacteristic
726C22#348512yes—
727C22C22×#13511yescharacteristic
728C22C22×AGL(1,ℤ/8)11yescharacteristic
729C22#358612yes—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—42no—
734C4—44no—
735C4—42no—
736C4—44no—
737C4—42no—
738C4—44no—
739C4—84no—
740C4—84no—
741C4#348512yes—
742C4#348512yes—
743C2—24no—
744C2—24no—
745C2—24no—
746C2—82no—
747C2—84no—
748C2#3544611yescharacteristic
749C2C22×#44911yescharacteristic
750C2#6037011yescharacteristic
751C2#6169612yes—
752C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3505C2
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. C22×#483C2semidirect
3. C22×#480C2semidirect
4. C2×#2486C2semidirect
5. #24094C2semidirect
6. #24091C2semidirect
7. #24066C2semidirect
8. #24038C2semidirect
9. #24035C2semidirect
10. #24004C2semidirect
11. #21759C2semidirect
12. #21758C2semidirect
13. #21738C2semidirect
34. C2×#483C22semidirect
35. C2×#483C22semidirect
38. C2×#480C22semidirect
39. C22×#107C22semidirect
40. C22×#106C22semidirect
41. C22×#103C22semidirect
42. C22×#102C22semidirect
48. #2486C22semidirect
51. C2×#389C22semidirect
54. C2×#380C22semidirect
55. #2059C22semidirect
56. #2042C22semidirect
57. #2036C22semidirect
58. #2034C22semidirect
59. #1893C22semidirect
60. #1892C22semidirect
62. #1872C22semidirect
64. #1869C22semidirect
65. #1864C22semidirect
66. #1863C22semidirect
67. #1854C22semidirect
68. #1824C22semidirect
69. #1823C22semidirect
70. #1817C22semidirect
71. #1788C22semidirect
72. #1787C22semidirect
73. #1786C22semidirect
74. #1779C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35052
C22×#4832
C22×#4802
C2×#24862
#240942
#240912
#240662
#240382
#240352
#240042
#217592
#217582
#217382