Finite Group Databasebeta

← #6529066

Subgroups — #6529066

Order 512.

3080 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3C22×#477C211yesmaximal, characteristic, complemented
4#34952C211yesmaximal, characteristic
5#24333C214yesmaximal, complemented
6#24302C214yesmaximal, complemented
7#24155C214yesmaximal, complemented
8#24122C214yesmaximal, complemented
9#24098C214yesmaximal, complemented
10#23970C214yesmaximal, complemented
11#21885C211yesmaximal, characteristic, complemented
12#21847C211yesmaximal, characteristic
13#21834C211yesmaximal, characteristic, complemented
14#3503—28no—
15#3503—28no—
16C2×Q82C2212yes—
17C2×#557—21no—
18C2×#554C2214yes—
19C2×#554C2212yes—
20C2×D4×Q8—22no—
21C2×D4×Q8—21no—
22C2×#544—22no—
23C2×#544—21no—
24C2×#533C2212yes—
25C2×#533C2211yescharacteristic
26C22×C4⋊Q8C2212yes—
27C22×C4⋊Q8C2211yescharacteristic
28C2×#520—21no—
29C2×C4×#41C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31#3104—28no—
32C2×#484C2212yes—
33C2×#477—28no—
34C2×#477C2214yes—
35C2×#477C2214yes—
36C2×#477C2212yes—
37C2×#462C2212yes—
38C2×#451C2212yes—
39C2×C4×Q16C2214yes—
40C2×#431C2211yescharacteristic
41C22×#107C2212yes—
42C22×C4⋊C8C2211yescharacteristic
43C22×#103C2212yes—
44#2582C2212yes—
45#2581—21no—
46#2580—21no—
47#2579—22no—
48#2538C2212yes—
49#2487C2212yes—
50#2409C2212yes—
51#2396—22no—
52C4×C22⋊Q8C2214yes—
53#2297—21no—
54#2296—21no—
55#2285C2211yescharacteristic
56#2057C2218yes—
57#2036C2218yes—
58#2024C2218yes—
59#2001C2218yes—
60#1990C2214yes—
61#1987C2214yes—
62#1987C2212yes—
63#1984C2212yes—
64#1981C2212yes—
65#1954C2212yes—
66#1892C2214yes—
67#1878C2218yes—
68#1872C2218yes—
69#1869C2214yes—
70#1863C2218yes—
71#1852C2218yes—
72#1824C2214yes—
73#1803C2214yes—
74#1797C2214yes—
75C22×#41D411yescharacteristic
76C23×Q8—22no—
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C22×Q16C2314yes—
80C23×C8C2312yes—
81Q82—28no—
82Q82—28no—
83#557—216no—
84#554—28no—
85#554—216no—
86#554—28no—
87#554—216no—
88D4×Q8—232no—
89D4×Q8—216no—
90#544—232no—
91#544—216no—
92#533—28no—
93#533—28no—
94#533—24no—
95#533D418yes—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8D418yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#125—22no—
112C2×#125—24no—
113C2×#125—22no—
114C2×#125—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2312yes—
123#520—216no—
124C4×#41—24no—
125C4×#41—24no—
126C2×C4×Q8—22no—
127C2×C4×Q8—22no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—24no—
132C2×C4×Q8—24no—
133C2×C4×Q8D411yescharacteristic
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2314yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2311yescharacteristic
139C2×C4×D4—24no—
140C2×C4×D4—22no—
141C2×C4×D4D412yes—
142C2×C4×D4D411yescharacteristic
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C2×#117—22no—
146C2×#117—21no—
147C2×#117—21no—
148C2×#117D412yes—
149C2×#117D411yescharacteristic
150C22×C4⋊C4—22no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2314yes—
155C22×C4⋊C4C2312yes—
156C22×C42—21no—
157C22×C42C2312yes—
158C22×C42C2311yescharacteristic
159#484—28no—
160#477—216no—
161#477—216no—
162#477—28no—
163#462—28no—
164#451—28no—
165C4×Q16—216no—
166#431—24no—
167C2×#107—28no—
168C2×#107C2318yes—
169C2×#107C2314yes—
170C2×C4⋊C8—24no—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×#103—28no—
175C2×#103C2318yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2318yes—
179C2×C22⋊C8C2312yes—
180C2×C22⋊C8C2318yes—
181C2×C4×C8C2312yes—
182#400—22no—
183#399—22no—
184#398—24no—
185#396—24no—
186#395—22no—
187#393—22no—
188#391—22no—
189#391C2314yes—
190#390—22no—
191#389—22no—
192#389—22no—
193#389C2314yes—
194#387—24no—
195#386—22no—
196#386—22no—
197#385—24no—
198#385—22no—
199#385—22no—
200#384C2312yes—
201#382—21no—
202#380—21no—
203#380C2312yes—
204#380C2314yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—24no—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—24no—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2318yes—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41Q1618yescomplemented
216C22×Q8—216no—
217C22×Q8—28no—
218C22×Q8—22no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8—22no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—84no—
228C22×Q8—88no—
229C22×Q8C2×D411yescharacteristic
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C22×D4C2×D412yes—
233C22×D4C2×D411yescharacteristic
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C2×Q16—28no—
245C2×Q16—28no—
246C2×Q16—44no—
247C22×C8—24no—
248C22×C8—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—216no—
252C4⋊Q8—216no—
253C4⋊Q8—232no—
254C4⋊Q8—216no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8C2×D414yes—
264C4⋊Q8C2×D414yes—
265C4⋊Q8C2×D418yes—
266C4⋊Q8C2×D418yes—
267#125—216no—
268#125—232no—
269#125—216no—
270#125—28no—
271C22⋊Q8—216no—
272C22⋊Q8—232no—
273C22⋊Q8—232no—
274C22⋊Q8—232no—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C22⋊Q8—24no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—44no—
281C22⋊Q8C2×D418yes—
282C4×Q8—216no—
283C4×Q8—216no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×Q8C2×D418yes—
298C4×D4—232no—
299C4×D4—216no—
300C4×D4—28no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—22no—
306#117—216no—
307#117—28no—
308#117—24no—
309#117—28no—
310#117—44no—
311C2×C4⋊C4—216no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D418yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2414yes—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D414yes—
368C2×C42C2×D414yes—
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—216no—
373#107—28no—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377#103—28no—
378#103—216no—
379#103—28no—
380#103—48no—
381C22⋊C8—24no—
382C22⋊C8—48no—
383C4×C8—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—22no—
390C24C22×D412yes—
391C24C22×D411yescharacteristic
392#41—232no—
393C2×Q8—216no—
394C2×Q8—22no—
395C2×Q8—28no—
396C2×Q8—24no—
397C2×Q8—28no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—48no—
410C2×Q8—44no—
411C2×Q8—48no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—84no—
415C2×Q8—88no—
416C2×Q8—816no—
417C2×Q8—832no—
418C2×Q8C2×Q1614yes—
419C2×Q8C22×D414yes—
420C2×Q82−1+414yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4C2×Q1618yes—
426C2×D4C2×Q1614yes—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—216no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4#12014yes—
453C22×C4#12014yes—
454C22×C4C2×Q1618yes—
455C22×C4C2×Q1614yes—
456C22×C4#13714yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468Q16—416no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—28no—
474C2×C8—48no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4C22×D414yes—
505C4⋊C4C22×D418yes—
506C22⋊C4—216no—
507C22⋊C4—28no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—84no—
514C42—216no—
515C42—24no—
516C42—24no—
517C42—28no—
518C42—22no—
519C42—22no—
520C42—44no—
521C42—44no—
522C42—44no—
523C42#12014yes—
524C42#12014yes—
525C42C22×D412yes—
526C42C22×D412yes—
527C42C22×D412yes—
528C42C22×D412yes—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23C2×#12012yes—
536C23C22×Q1614yes—
537C23C22×Q1612yes—
538C23C2×#13712yes—
539C23C23×D411yescharacteristic
540Q8—28no—
541Q8—24no—
542Q8—24no—
543Q8—28no—
544Q8—48no—
545Q8—416no—
546Q8—48no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—88no—
552Q8—816no—
553D4—216no—
554D4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—216no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—88no—
590C2×C4—816no—
591C2×C4—84no—
592C2×C4#451116yes—
593C2×C4C2×#12012yes—
594C2×C4C2×#12012yes—
595C2×C4C2×#12012yes—
596C2×C4C2×#12012yes—
597C2×C4C2×#12012yes—
598C2×C4C22×Q1614yes—
599C2×C4C22×Q1612yes—
600C2×C4C22×Q1612yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C23×D411yescharacteristic
604C2×C4C23×D411yescharacteristic
605C2×C4C2×2−1+412yes—
606C2×C4#58512yes—
607C8—44no—
608C8—44no—
609C22—22no—
610C22—24no—
611C22—24no—
612C22—22no—
613C22—28no—
614C22—24no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22C2×#45114yes—
619C22C22×#12011yescharacteristic
620C22#348912yes—
621C22C23×Q1611yescharacteristic
622C22C22×#13711yescharacteristic
623C22#358512yes—
624C4—22no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—48no—
633C4—44no—
634C4—84no—
635C4—82no—
636C4C2×#45114yes—
637C4C2×#45114yes—
638C4#348611yescharacteristic
639C4#348711yescharacteristic
640C4#348912yes—
641C2—24no—
642C2—44no—
643C2#3565911yescharacteristic
644C2C22×#45111yescharacteristic
645C2#6036711yescharacteristic
646C2#6210012yes—
647C2#6211212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3104C2semidirect
3. C22×#477C2semidirect
5. #24333C2semidirect
6. #24302C2semidirect
7. #24155C2semidirect
8. #24122C2semidirect
9. #24098C2semidirect
10. #23970C2semidirect
11. #21885C2semidirect
13. #21834C2semidirect
215. C2×#41Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#31042
C22×#4772
#349522
#243332
#243022
#241552
#241222
#240982
#239702
#218852
#218472
#218342