Finite Group Databasebeta

← #6528933

Subgroups — #6528933

Order 512.

3246 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3507C211yesmaximal, characteristic, complemented
2C2×#3101C211yesmaximal, characteristic, complemented
3C2×#3100C211yesmaximal, characteristic
4C2×#2538C211yesmaximal, characteristic, complemented
5#24328C214yesmaximal, complemented
6#24327C214yesmaximal, complemented
7#24186C214yesmaximal, complemented
8#24185C214yesmaximal, complemented
9#24066C214yesmaximal, complemented
10#23936C214yesmaximal, complemented
11#21860C211yesmaximal, characteristic, complemented
12#21859C211yesmaximal, characteristic, complemented
13#21818C211yesmaximal, characteristic, complemented
14#3507—28no—
15#3507—28no—
16C2×#559C2212yescomplemented
17C2×#557C2212yescomplemented
18C2×#556—22no—
19C2×#556C2212yescomplemented
20C2×#554C2212yescomplemented
21C2×#552—22no—
22C2×#539—22no—
23C2×#536—21no—
24C2×#533C2211yescharacteristic
25C2×#532C2211yescharacteristic
26C2×#529C2214yes—
27C22×C22⋊Q8—21no—
28C2×#520—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31#3101—28no—
32#3100—28no—
33C2×#483C2212yes—
34C2×#480C2212yes—
35C2×#476C2212yes—
36C2×#474C2212yes—
37C2×#461C2212yes—
38C2×#459C2212yes—
39C2×#450C2212yes—
40C2×#447C2212yes—
41C2×C4×SD16C2214yes—
42C2×C4×D8C2214yes—
43C2×#431C2211yescharacteristic
44C2×#431C2211yescharacteristic
45#2538—22no—
46#2538—22no—
47#2538C2214yes—
48#2538C2214yes—
49C2×#396—22no—
50C2×#390C2212yes—
51C2×#389C2212yes—
52C2×#384C2211yescharacteristic
53C2×#380—21no—
54#2059C2218yescomplemented
55#2042C2218yescomplemented
56#2025C2218yes—
57#2012C2218yes—
58#1990C2212yescomplemented
59#1990C2212yescomplemented
60#1984C2214yescomplemented
61#1984C2214yescomplemented
62#1957C2212yescomplemented
63#1956C2212yescomplemented
64#1880C2218yes—
65#1879C2218yes—
66#1874C2214yes—
67#1872C2214yes—
68#1864C2218yescomplemented
69#1863C2218yescomplemented
70#1817C2214yes—
71#1803C2214yescomplemented
72#1802C2214yescomplemented
73C22×#41—21no—
74C23×Q8D411yescharacteristic
75C23×D4C2311yescharacteristic
76C24×C4—21no—
77C22×SD16C2314yes—
78C22×D8C2314yes—
79#559—28no—
80#559—28no—
81#557—28no—
82#557—28no—
83#556—232no—
84#556—28no—
85#556—28no—
86#554—28no—
87#554—28no—
88#552—232no—
89#539—232no—
90#536—216no—
91#533—24no—
92#533D418yes—
93#532—24no—
94#532D418yes—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100#529—216no—
101#529—216no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#125—24no—
105C2×#125—21no—
106C2×#125C2314yes—
107C2×#124—22no—
108C2×#124—21no—
109C2×#124C2314yes—
110C2×#123—24no—
111C2×#123—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—48no—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121C2312yes—
122#520—216no—
123C2×C4×Q8—22no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—21no—
127C2×C4×Q8D412yes—
128C2×C4×Q8D412yes—
129C2×C4×Q8C2314yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—24no—
134C2×C4×D4—22no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—24no—
139C2×C4×D4—24no—
140C2×C4×D4C2314yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×#117—21no—
145C2×#117—22no—
146C2×#117—24no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4—22no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4D412yes—
152C22×C4⋊C4D411yescharacteristic
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2312yes—
155C22×C22⋊C4—21no—
156C22×C22⋊C4—22no—
157C22×C22⋊C4—22no—
158C22×C42C2312yes—
159C22×C42C2311yescharacteristic
160#483—28no—
161#480—28no—
162#476—28no—
163#474—28no—
164#461—28no—
165#459—28no—
166#450—28no—
167#447—28no—
168C4×SD16—216no—
169C4×D8—216no—
170#431—24no—
171#431—24no—
172C2×#107C2314yes—
173C2×#106C2314yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176C2×#103C2314yes—
177C2×#103C2318yes—
178C2×#102C2314yes—
179C2×#102C2314yes—
180C2×#102C2314yes—
181C2×#102C2318yes—
182C2×C22⋊C8C2312yes—
183C2×C22⋊C8C2312yes—
184C2×C4×C8C2312yes—
185C2×C4×C8C2312yes—
186#396—28no—
187#396—28no—
188#390—24no—
189#390C2318yes—
190#389—24no—
191#389C2318yes—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2312yes—
196#380—24no—
197#380—24no—
198C2×#95—22no—
199#340C2318yes—
200#340C2318yes—
201#336C2318yes—
202#336C2318yes—
203C25—21no—
204C2×#41—28no—
205C2×#41—44no—
206C2×#41—44no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—21no—
211C22×Q8—21no—
212C22×Q8C2×D412yes—
213C22×Q8C2×D412yes—
214C22×Q8C2412yes—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—42no—
223C22×D4C2412yes—
224C22×D4C2412yes—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×SD16—28no—
240C2×SD16—28no—
241C2×SD16—44no—
242C2×D8—28no—
243C2×D8—28no—
244C2×D8—44no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—216no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8C2×D414yes—
252C4⋊Q8C2×D414yes—
253C4⋊Q8C2×D414yes—
254#127—22no—
255#127C2×D414yes—
256#126—232no—
257#126—232no—
258#125—232no—
259#125—28no—
260#125—28no—
261#125—28no—
262#124—216no—
263#124—28no—
264#124—28no—
265#124—28no—
266#123—232no—
267#123—232no—
268C22⋊Q8—216no—
269C22⋊Q8—216no—
270C22⋊Q8—232no—
271C22⋊Q8—28no—
272C22⋊Q8—24no—
273C22⋊Q8—416no—
274C22⋊Q8—416no—
275C22⋊Q8C2×D418yes—
276#121—28no—
277#121—24no—
278#121C2×D418yes—
279C4×Q8—216no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—28no—
284C4×Q8—28no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8C2×D418yes—
291C4×D4—232no—
292C4×D4—216no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4C2×D418yes—
304#117—216no—
305#117—232no—
306#117—28no—
307C2×C4⋊C4—216no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—28no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—84no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42C2×D414yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D414yes—
361C2×C42C2×D414yes—
362C2×C42C2414yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—28no—
366#106—28no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369#103—28no—
370#103—48no—
371#102—28no—
372#102—28no—
373#102—28no—
374#102—48no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C4×C8—24no—
378C4×C8—24no—
379#95—28no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24C22×D411yescharacteristic
390#41—432no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—416no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C22×D414yes—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4C22×D414yes—
431C2×D42−1+414yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—216no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—816no—
466C22×C4#12014yes—
467C22×C4#12014yes—
468C22×C4#13514yes—
469C22×C4AGL(1,ℤ/8)14yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480SD16—416no—
481D8—416no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—44no—
507C4⋊C4—416no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4C22×D414yes—
514C4⋊C4C22×D414yes—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—416no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—88no—
525C42—216no—
526C42—28no—
527C42—28no—
528C42—24no—
529C42—24no—
530C42#12014yes—
531C42#12014yes—
532C42C22×D412yes—
533C42C22×D412yes—
534C42C22×D412yes—
535C42C22×D412yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—24no—
541C23—24no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23C2×#12012yes—
562C23C2×#13512yes—
563C23C2×AGL(1,ℤ/8)12yes—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—44no—
567Q8—48no—
568Q8—48no—
569Q8—48no—
570Q8—84no—
571D4—24no—
572D4—28no—
573D4—24no—
574D4—48no—
575D4—416no—
576D4—44no—
577D4—42no—
578D4—42no—
579D4—48no—
580D4—84no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—416no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—816no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4#449116yes—
620C2×C4C2×#12012yes—
621C2×C4C2×#12012yes—
622C2×C4C2×#12012yes—
623C2×C4C2×#12012yes—
624C2×C4C2×#12012yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13512yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C23×D411yescharacteristic
631C2×C4C2×2−1+412yes—
632C2×C4#58512yes—
633C8—44no—
634C8—44no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—816no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C2×#44914yes—
657C22C22×#12011yescharacteristic
658C22#348611yescharacteristic
659C22#348711yescharacteristic
660C22C22×#13511yescharacteristic
661C22C22×AGL(1,ℤ/8)11yescharacteristic
662C22#358511yescharacteristic
663C22#358611yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—82no—
673C4—84no—
674C4C2×#44914yes—
675C4C2×#44914yes—
676C4#348912yes—
677C4#348912yes—
678C2—24no—
679C2—84no—
680C2—84no—
681C2#3565911yescharacteristic
682C2C22×#44911yescharacteristic
683C2#6037111yescharacteristic
684C2#6181911yescharacteristic
685C2#6182711yescharacteristic
686C2#6188211yescharacteristic
687C2#6189011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3507C2semidirect
2. C2×#3101C2semidirect
4. C2×#2538C2semidirect
5. #24328C2semidirect
6. #24327C2semidirect
7. #24186C2semidirect
8. #24185C2semidirect
9. #24066C2semidirect
10. #23936C2semidirect
11. #21860C2semidirect
12. #21859C2semidirect
13. #21818C2semidirect
16. C2×#559C22semidirect
17. C2×#557C22semidirect
19. C2×#556C22semidirect
20. C2×#554C22semidirect
54. #2059C22semidirect
55. #2042C22semidirect
58. #1990C22semidirect
59. #1990C22semidirect
60. #1984C22semidirect
61. #1984C22semidirect
62. #1957C22semidirect
63. #1956C22semidirect
68. #1864C22semidirect
69. #1863C22semidirect
71. #1803C22semidirect
72. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35072
C2×#31012
C2×#31002
C2×#25382
#243282
#243272
#241862
#241852
#240662
#239362
#218602
#218592
#218182