Finite Group Databasebeta

← #6499900

Subgroups — #6499900

Order 512.

3652 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic, complemented
2C2×#3187C212yesmaximal, complemented
3C2×#2416C211yesmaximal, characteristic
4#25543C218yesmaximal, complemented
5#24429C218yesmaximal, complemented
6#24063C212yesmaximal, complemented
7#24003C212yesmaximal, complemented
8#23933C212yesmaximal, complemented
9#23919C212yesmaximal, complemented
10#21738C211yesmaximal, characteristic, complemented
11C2×#1872C212yesmaximal, complemented
12#3501—216no—
13C2×#554—24no—
14C2×#554C2214yescomplemented
15C2×D4×Q8C2214yescomplemented
16C2×#544—21no—
17C2×#543—21no—
18C2×#543—21no—
19C2×#533C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×#121—21no—
23C22×#121C2211yescharacteristic
24C22×#121C2211yescharacteristic
25C2×#518—21no—
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28#3187—216no—
29C2×#481C2214yescomplemented
30C2×#483C2214yes—
31C2×#465C2218yescomplemented
32C2×#448C2214yescomplemented
33C2×#448C2214yescomplemented
34C2×#450C2214yes—
35C22×#103C2214yescomplemented
36C22×C22⋊C8C2212yes—
37#2416—24no—
38#2416C2218yes—
39C2×#396—21no—
40C2×#394—21no—
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#386—21no—
44C2×#380C2211yescharacteristic
45#2058C2214yescomplemented
46#2043C2214yescomplemented
47#2035C2218yescomplemented
48#2023C2214yescomplemented
49#2013C2214yescomplemented
50#2000C2218yescomplemented
51#1880C2218yescomplemented
52#1874C2214yescomplemented
53#1872C2214yes—
54#1872C2214yes—
55#1872C2218yescomplemented
56#1869C2218yescomplemented
57#1865C2218yescomplemented
58#1853C2214yescomplemented
59#1817C2212yes—
60#1787C2214yescomplemented
61#1779C2212yes—
62C2×#340C2214yescomplemented
63C23×Q8C2312yes—
64C23×Q8C2312yes—
65C23×D4—21no—
66C23×D4—21no—
67C23×D4D411yescharacteristic
68C23×D4C2311yescharacteristic
69C24×C4—21no—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C22×SD16C2318yes—
73C23×C8C2314yes—
74#554—264no—
75#554—216no—
76#554—216no—
77D4×Q8—216no—
78D4×Q8—216no—
79#544—216no—
80#543—216no—
81#543—216no—
82#533—216no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125—24no—
87C2×#125—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8D4116yescomplemented
100C2×C22⋊Q8C2314yes—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×#121—28no—
104C2×#121—22no—
105C2×#121—22no—
106C2×#121—28no—
107C2×#121—44no—
108C2×#121—48no—
109C2×#121C2314yes—
110C2×#121C2314yes—
111#518—216no—
112C2×C4×Q8—22no—
113C2×C4×Q8C2314yes—
114C2×C4×D4—24no—
115C2×C4×D4—24no—
116C2×C4×D4—28no—
117C2×C4×D4D414yes—
118C2×C4×D4C2314yes—
119C2×#117—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—44no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4D411yescharacteristic
141C22×C22⋊C4D411yescharacteristic
142C22×C42C2311yescharacteristic
143#483—216no—
144#481—216no—
145#465—232no—
146#450—216no—
147#448—216no—
148#448—48no—
149C2×#106C2318yes—
150C2×#103—48no—
151C2×#103C2318yes—
152C2×#103C2318yes—
153C2×#103C2318yes—
154C2×#102C2318yes—
155C2×C22⋊C8—28no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160#396—28no—
161#394—28no—
162#390—22no—
163#390C2314yes—
164#389—22no—
165#389C2314yes—
166#386—28no—
167#380—22no—
168#380C2314yes—
169C2×#95—21no—
170C2×#95—21no—
171#340C2318yes—
172#340C2318yes—
173#336C2314yes—
174#336C2314yes—
175C25—21no—
176C25C2×D411yescharacteristic
177C25C2×D411yescharacteristic
178C22×Q8—28no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8C2×D418yescomplemented
188C22×Q8C2414yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—21no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—82no—
203C22×D4SD1618yescomplemented
204C22×D4C2×D414yes—
205C22×D4C2412yes—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—24no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—82no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D418yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C2×SD16—216no—
233C2×SD16—48no—
234C2×SD16—48no—
235C22×C8—28no—
236C22×C8—44no—
237C22×C8—44no—
238C22×C8C2418yes—
239C4⋊Q8—232no—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242#125—216no—
243#125—232no—
244C22⋊Q8—216no—
245C22⋊Q8—216no—
246C22⋊Q8—216no—
247C22⋊Q8—232no—
248C22⋊Q8—232no—
249C22⋊Q8—232no—
250C22⋊Q8—232no—
251C22⋊Q8—28no—
252C22⋊Q8—216no—
253C22⋊Q8—216no—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256C22⋊Q8—28no—
257C22⋊Q8C2×D4116yes—
258C22⋊Q8C2×D4116yescomplemented
259#121—28no—
260#121—28no—
261#121—28no—
262#121—44no—
263#121—48no—
264#121—432no—
265#121—48no—
266C4×Q8—216no—
267C4×Q8—28no—
268C4×Q8—28no—
269C4×D4—232no—
270C4×D4—232no—
271C4×D4—216no—
272C4×D4—28no—
273C4×D4—28no—
274#117—216no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D418yescomplemented
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—216no—
309C2×C22⋊C4—216no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331#106—216no—
332#103—216no—
333#103—216no—
334#103—48no—
335#102—216no—
336C22⋊C8—28no—
337C22⋊C8—28no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340#95—24no—
341#95—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—41no—
350C24—41no—
351C24—41no—
352C24—41no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24#12014yes—
361C24C2×SD1614yes—
362C24C2×SD1614yes—
363C24#13714yes—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366C24C22×D412yes—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—24no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—416no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8—82no—
380C2×Q8—88no—
381C2×Q8—816no—
382C2×Q8C22×D418yes—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—232no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—42no—
397C2×D4—88no—
398C2×D4—84no—
399C2×D4—816no—
400C2×D4—816no—
401C2×D4C2×SD16116yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—416no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—48no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—88no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—84no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—816no—
456C22×C4#12014yes—
457C22×C4#12018yes—
458C22×C4C2×SD1614yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C2511yescharacteristic
468SD16—816no—
469C2×C8—28no—
470C2×C8—28no—
471C2×C8—28no—
472C2×C8—88no—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4C22×D418yes—
493C4⋊C4H(5,𝔽2)14yes—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—84no—
504C42—216no—
505C42—24no—
506C42—24no—
507C23—21no—
508C23—22no—
509C23—21no—
510C23—22no—
511C23—21no—
512C23—22no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23#448116yescomplemented
549C23C2×#12012yes—
550C23C2×#12014yes—
551C23C22×SD1614yes—
552C23C22×SD1612yes—
553C23C22×SD1614yes—
554C23C2×#13712yes—
555C23C2×#13714yes—
556C23C23×D411yescharacteristic
557Q8—216no—
558Q8—48no—
559Q8—88no—
560Q8—88no—
561Q8—88no—
562D4—232no—
563D4—48no—
564D4—84no—
565D4—816no—
566D4—816no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—88no—
606C2×C4—88no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—816no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4C2×#12012yes—
613C2×C4C2×#12014yes—
614C2×C4C22×SD1614yes—
615C2×C4C23×D412yes—
616C2×C4C2×H(5,𝔽2)12yes—
617C2×C4#58512yes—
618C8—48no—
619C22—22no—
620C22—24no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—24no—
625C22—22no—
626C22—24no—
627C22—24no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—416no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—88no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—168no—
648C22C2×#44814yes—
649C22C2×#44818yes—
650C22C22×#12011yescharacteristic
651C22#348411yescharacteristic
652C22#348511yescharacteristic
653C22C23×SD1611yescharacteristic
654C22C22×#13711yescharacteristic
655C22#358511yescharacteristic
656C22#358611yescharacteristic
657C4—22no—
658C4—24no—
659C4—24no—
660C4—42no—
661C4—48no—
662C4—44no—
663C4—42no—
664C4—48no—
665C4—82no—
666C4—82no—
667C4#349614yes—
668C2—24no—
669C2—28no—
670C2—44no—
671C2—162no—
672C2#3534711yescharacteristic
673C2C22×#44811yescharacteristic
674C2#6037111yescharacteristic
675C2#6156011yescharacteristic
676C2#6157311yescharacteristic
677C2#6169511yescharacteristic
678C2#6170811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2416C2
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
1. C2×#3501C2semidirect
2. C2×#3187C2semidirect
4. #25543C2semidirect
5. #24429C2semidirect
6. #24063C2semidirect
7. #24003C2semidirect
8. #23933C2semidirect
9. #23919C2semidirect
10. #21738C2semidirect
11. C2×#1872C2semidirect
14. C2×#554C22semidirect
15. C2×D4×Q8C22semidirect
29. C2×#481C22semidirect
31. C2×#465C22semidirect
32. C2×#448C22semidirect
33. C2×#448C22semidirect
35. C22×#103C22semidirect
45. #2058C22semidirect
46. #2043C22semidirect
47. #2035C22semidirect
48. #2023C22semidirect
49. #2013C22semidirect
50. #2000C22semidirect
51. #1880C22semidirect
52. #1874C22semidirect
55. #1872C22semidirect
56. #1869C22semidirect
57. #1865C22semidirect
58. #1853C22semidirect
60. #1787C22semidirect
62. C2×#340C22semidirect
99. C2×C22⋊Q8D4semidirect
187. C22×Q8C2×D4semidirect
203. C22×D4SD16semidirect
258. C22⋊Q8C2×D4semidirect
303. C2×C4⋊C4C2×D4semidirect
548. C23#448semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#31872
C2×#24162
#255432
#244292
#240632
#240032
#239332
#239192
#217382
C2×#18722