Finite Group Databasebeta

← #6499926

Subgroups — #6499926

Order 512.

3478 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic, complemented
2C2×#3193C212yesmaximal, complemented
3C2×#2414C211yesmaximal, characteristic
4#25559C218yesmaximal, complemented
5#24445C218yesmaximal, complemented
6#24064C212yesmaximal, complemented
7#24006C212yesmaximal, complemented
8#23933C212yesmaximal, complemented
9#23919C212yesmaximal, complemented
10#21742C211yesmaximal, characteristic, complemented
11#20911C212yesmaximal, 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×#121C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C2×#518—21no—
25C22×C4×D4C2211yescharacteristic
26C22×#117C2211yescharacteristic
27#3193—216no—
28C2×#484C2214yes—
29C2×#480C2214yescomplemented
30C2×#468C2218yescomplemented
31C2×#448C2214yescomplemented
32C2×#448C2214yescomplemented
33C2×#450C2214yes—
34C2×#419C2214yescomplemented
35C2×#407C2212yes—
36#2414—24no—
37#2414C2218yes—
38C2×#394—21no—
39C2×#392—21no—
40C2×#390—21no—
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#380C2211yescharacteristic
45#2057C2214yescomplemented
46#2043C2214yescomplemented
47#2036C2218yescomplemented
48#2023C2214yescomplemented
49#2013C2214yescomplemented
50#2000C2218yescomplemented
51#1880C2218yescomplemented
52#1874C2214yescomplemented
53#1873C2218yescomplemented
54#1872C2214yes—
55#1872C2214yes—
56#1870C2218yescomplemented
57#1863C2218yescomplemented
58#1852C2214yescomplemented
59#1817C2212yes—
60#1787C2214yescomplemented
61#1779C2212yes—
62#1736C2214yescomplemented
63C23×Q8C2312yes—
64C23×Q8C2312yes—
65C23×D4—21no—
66C23×D4—21no—
67C23×D4D411yescharacteristic
68C23×D4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C22×SD16C2318yes—
72C22×M16C2314yes—
73#554—264no—
74#554—216no—
75#554—216no—
76D4×Q8—216no—
77D4×Q8—216no—
78#544—216no—
79#543—216no—
80#543—216no—
81#533—216no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#125—24no—
86C2×#125—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8D4116yescomplemented
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2314yes—
101C2×C22⋊Q8C2314yes—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121—28no—
105C2×#121—44no—
106C2×#121C2314yes—
107C2×#121C2314yes—
108#518—216no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2314yes—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4—28no—
114C2×C4×D4D414yes—
115C2×C4×D4C2314yes—
116C2×#117—22no—
117C2×#117—22no—
118C2×#117—44no—
119C2×#117C2314yes—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4D411yescharacteristic
138C22×C22⋊C4D411yescharacteristic
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141#484—216no—
142#480—216no—
143#468—232no—
144#450—216no—
145#448—216no—
146#448—48no—
147C2×#107C2318yes—
148#419—48no—
149C2×#103C2318yes—
150C2×#103C2318yes—
151C2×#103C2318yes—
152C2×#102C2318yes—
153#407—28no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#394—28no—
159#392—28no—
160#390—28no—
161#390—22no—
162#390C2314yes—
163#389—22no—
164#389C2314yes—
165#388—28no—
166#380—22no—
167#380C2314yes—
168C2×#95—21no—
169C2×#95—21no—
170#340C2318yes—
171#340C2318yes—
172#336C2314yes—
173#336C2314yes—
174C25—21no—
175C25C2×D411yescharacteristic
176C25C2×D411yescharacteristic
177C22×Q8—28no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8C2×D418yescomplemented
187C22×Q8C2414yes—
188C22×D4—24no—
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—82no—
201C22×D4—82no—
202C22×D4—82no—
203C22×D4C2×D414yes—
204C22×D4C2412yes—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D412yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D418yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C2×SD16—216no—
228C2×SD16—48no—
229C2×SD16—48no—
230C2×M16—28no—
231C2×M16—44no—
232C2×M16—44no—
233C22×C8C2418yes—
234C4⋊Q8—232no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237#125—216no—
238#125—232no—
239C22⋊Q8—216no—
240C22⋊Q8—216no—
241C22⋊Q8—216no—
242C22⋊Q8—232no—
243C22⋊Q8—232no—
244C22⋊Q8—232no—
245C22⋊Q8—232no—
246C22⋊Q8—28no—
247C22⋊Q8—216no—
248C22⋊Q8—216no—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C22⋊Q8—28no—
252C22⋊Q8C2×D4116yes—
253C22⋊Q8C2×D4116yescomplemented
254#121—28no—
255#121—28no—
256#121—28no—
257#121—44no—
258#121—48no—
259#121—48no—
260C4×Q8—216no—
261C4×Q8—28no—
262C4×Q8—28no—
263C4×D4—232no—
264C4×D4—232no—
265C4×D4—216no—
266C4×D4—28no—
267C4×D4—28no—
268#117—216no—
269#117—28no—
270#117—44no—
271#117—48no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D418yescomplemented
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—216no—
301C2×C22⋊C4—216no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327#107—216no—
328#103—216no—
329#103—216no—
330#103—48no—
331#102—216no—
332C22⋊C8—28no—
333C22⋊C8—28no—
334C22⋊C8—44no—
335C22⋊C8—44no—
336#95—24no—
337#95—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—41no—
350C24—41no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24#12014yes—
359C24#13714yes—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C22×D411yescharacteristic
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—24no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—416no—
370C2×Q8—82no—
371C2×Q8—82no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—88no—
377C2×Q8—816no—
378C2×Q8C22×D418yes—
379C2×D4—232no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—48no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—42no—
392C2×D4—88no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—816no—
397C2×D4—816no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—416no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—88no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4#12014yes—
449C22×C4#12018yes—
450C22×C4#13714yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C2511yescharacteristic
460SD16—816no—
461M16—88no—
462C2×C8—28no—
463C2×C8—28no—
464C2×C8—28no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4C22×D418yes—
484C4⋊C4H(5,𝔽2)14yes—
485C22⋊C4—216no—
486C22⋊C4—216no—
487C22⋊C4—216no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C42—216no—
497C42—24no—
498C42—24no—
499C42—48no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—28no—
517C23—28no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23C2×#12014yes—
545C23C2×#12012yes—
546C23C2×#13714yes—
547C23C2×#13712yes—
548C23C2×#13712yes—
549C23C23×D411yescharacteristic
550Q8—216no—
551Q8—48no—
552Q8—88no—
553Q8—88no—
554Q8—88no—
555D4—232no—
556D4—48no—
557D4—84no—
558D4—816no—
559D4—816no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—88no—
599C2×C4—88no—
600C2×C4—88no—
601C2×C4—88no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4C2×#12012yes—
605C2×C4C2×#12014yes—
606C2×C4C2×#13714yes—
607C2×C4C23×D412yes—
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4#58512yes—
610C8—48no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—88no—
632C22—88no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—168no—
641C22#300414yes—
642C22C22×#12011yescharacteristic
643C22#348411yescharacteristic
644C22#348511yescharacteristic
645C22C22×#13711yescharacteristic
646C22C22×#13711yescharacteristic
647C22#358511yescharacteristic
648C22#358511yescharacteristic
649C4—22no—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—48no—
654C4—44no—
655C4—42no—
656C4—48no—
657C4—82no—
658C4—82no—
659C4#349614yes—
660C2—24no—
661C2—44no—
662C2—44no—
663C2—162no—
664C2#3534711yescharacteristic
665C2C2×#300411yescharacteristic
666C2#6036911yescharacteristic
667C2#6157011yescharacteristic
668C2#6158811yescharacteristic
669C2#6170811yescharacteristic
670C2#6172111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2414C2
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×#3193C2semidirect
4. #25559C2semidirect
5. #24445C2semidirect
6. #24064C2semidirect
7. #24006C2semidirect
8. #23933C2semidirect
9. #23919C2semidirect
10. #21742C2semidirect
11. #20911C2semidirect
14. C2×#554C22semidirect
15. C2×D4×Q8C22semidirect
29. C2×#480C22semidirect
30. C2×#468C22semidirect
31. C2×#448C22semidirect
32. C2×#448C22semidirect
34. C2×#419C22semidirect
45. #2057C22semidirect
46. #2043C22semidirect
47. #2036C22semidirect
48. #2023C22semidirect
49. #2013C22semidirect
50. #2000C22semidirect
51. #1880C22semidirect
52. #1874C22semidirect
53. #1873C22semidirect
56. #1870C22semidirect
57. #1863C22semidirect
58. #1852C22semidirect
60. #1787C22semidirect
62. #1736C22semidirect
98. C2×C22⋊Q8D4semidirect
186. C22×Q8C2×D4semidirect
253. C22⋊Q8C2×D4semidirect
295. C2×C4⋊C4C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#31932
C2×#24142
#255592
#244452
#240642
#240062
#239332
#239192
#217422
#209112