Finite Group Databasebeta

← #6393425

Subgroups — #6393425

Order 512.

3200 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3538C211yesmaximal, characteristic
2C2×#2978C211yesmaximal, characteristic
3#34201C211yesmaximal, characteristic
4#22499C214yesmaximal, complemented
5#22496C214yesmaximal, complemented
6#22430C214yesmaximal, complemented
7#22425C214yesmaximal, complemented
8#21873C212yesmaximal, complemented
9#21771C212yesmaximal, complemented
10#21768C212yesmaximal, complemented
11#21398C212yesmaximal, complemented
12#20913C212yesmaximal, complemented
13#20911C212yesmaximal, complemented
14#3538—28no—
15#3538C4116yescomplemented
16C2×#563—22no—
17C2×#552C2214yes—
18C2×#551—22no—
19C2×D4×Q8C2214yes—
20C2×#544—22no—
21C2×#541C2212yes—
22C2×#539—22no—
23C2×#536C2212yes—
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C2×#520C2211yescharacteristic
27C22×#117C2211yescharacteristic
28#2978—28no—
29C2×#436C2214yes—
30C2×#435C2214yes—
31C2×#433C2212yes—
32C2×#419C2214yes—
33C2×#419C2212yescomplemented
34C22×#103C2212yescomplemented
35C2×#408C2212yes—
36C2×#407C2212yes—
37#2363—22no—
38#2362—22no—
39#2354C2214yes—
40#2352C2214yes—
41#2321—22no—
42#2317C2211yescharacteristic
43#2311C2211yescharacteristic
44C4×#123—22no—
45#2280C2211yescharacteristic
46#2274C2211yescharacteristic
47#1992C2212yescomplemented
48#1991C2212yescomplemented
49#1989C2214yescomplemented
50#1981C2214yescomplemented
51#1960C2212yescomplemented
52#1956C2212yescomplemented
53#1894C2214yes—
54#1894C2214yes—
55#1894C2214yescomplemented
56#1892C2214yescomplemented
57#1873C2214yes—
58#1873C2214yes—
59#1873C2214yes—
60#1873C2214yes—
61#1873C2214yescomplemented
62#1872C2214yes—
63#1872C2214yes—
64#1872C2214yescomplemented
65#1825C2214yescomplemented
66#1824C2212yes—
67#1824C2212yes—
68#1805C2214yescomplemented
69#1803C2214yescomplemented
70#1787C2212yes—
71#1787C2212yes—
72#1787C2214yescomplemented
73#1763C2214yescomplemented
74C4×#103C2214yescomplemented
75#1736C2214yes—
76#1736C2212yescomplemented
77#1735C2212yescomplemented
78C22×#41C2311yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4C2311yescharacteristic
82C22×M16C2314yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85#563—232no—
86#552—216no—
87#552C2×C4132yescomplemented
88#551—232no—
89D4×Q8—216no—
90D4×Q8C2×C4132yescomplemented
91#544—232no—
92#541—28no—
93#541C2×C4116yescomplemented
94#539—232no—
95#536—28no—
96#536C2×C4116yes—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8C2312yes—
100C2×#126—24no—
101C2×#126—24no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125C2312yes—
105C2×#124—22no—
106#525C2×C418yes—
107#525D418yes—
108C2×#123—22no—
109C2×#123—22no—
110C2×#123—24no—
111C2×#123D412yes—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8C2×C4116yescomplemented
118C2×C22⋊Q8D412yes—
119C2×C22⋊Q8C2314yes—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2314yes—
122C2×#121D412yes—
123C2×#120D412yes—
124#520—44no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2314yes—
127C2×C4×D4—24no—
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4C2314yes—
131C2×#117—22no—
132C2×#117—24no—
133C2×#117—44no—
134C2×#117C2314yes—
135C2×#117C2312yes—
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—22no—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—22no—
146C22×C22⋊C4—21no—
147C22×C42—21no—
148#436—216no—
149#435—216no—
150#433—28no—
151C2×C4⋊C8C2314yes—
152#419—48no—
153#419—44no—
154C2×#103—44no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163#408—28no—
164#407—28no—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2314yes—
168C2×C22⋊C8C2314yes—
169C2×#97C2312yes—
170C2×C4×C8C2312yes—
171#391—21no—
172#390—21no—
173#390—24no—
174#389—22no—
175#388—24no—
176#388—24no—
177#388—24no—
178#387—24no—
179#386—22no—
180#386—24no—
181#385—22no—
182#384C2312yes—
183#382C2312yes—
184#381—21no—
185#380C2314yes—
186#380C2312yes—
187#380C2312yes—
188#379—21no—
189C4×C4⋊C4—24no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2314yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—24no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203C25C2×D411yescharacteristic
204C2×#41—42no—
205C2×#41—42no—
206C2×#41C22×C414yes—
207C2×#41C2×D414yes—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—42no—
213C22×Q8—44no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8—44no—
218C22×Q8C22×C418yes—
219C22×Q8C22×C418yes—
220C22×Q8C2×D411yescharacteristic
221C22×Q8C2414yes—
222C22×D4C2×D414yes—
223C22×D4C2×D412yes—
224C22×D4C2×D411yescharacteristic
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4C22×C414yes—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2412yes—
240C2×M16—28no—
241C2×M16—24no—
242C2×M16—44no—
243C2×M16—42no—
244C2×M16—44no—
245C2×M16—42no—
246C22×C8—24no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—216no—
252C4⋊Q8—232no—
253C4⋊Q8—24no—
254C4⋊Q8C22×C418yes—
255#126—232no—
256#126—232no—
257#125—232no—
258#125—232no—
259#125—24no—
260#125C22×C418yes—
261#124—216no—
262#123—216no—
263#123—216no—
264#123—232no—
265#123C22⋊C4116yes—
266C22⋊Q8—216no—
267C22⋊Q8—216no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—232no—
272C22⋊Q8—232no—
273C22⋊Q8C22⋊C4116yes—
274C22⋊Q8C22×C4116yes—
275C22⋊Q8C22×C4116yes—
276C22⋊Q8C22×C4116yes—
277#121C22⋊C4116yes—
278#120C22⋊C4116yes—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×Q8C22×C4116yes—
282C4×D4—232no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4C22×C4116yes—
286C4×D4C2×D4116yes—
287#117—24no—
288#117—232no—
289#117—28no—
290#117—24no—
291#117—44no—
292#117—42no—
293#117—42no—
294#117—42no—
295#117—48no—
296#117C22×C418yes—
297#117C22×C414yes—
298#117C2×D414yes—
299C2×C4⋊C4—216no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4C22×C418yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—216no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D414yes—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—44no—
349C2×C42C2412yes—
350C4⋊C8—28no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359C22⋊C8—28no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—28no—
363#97—24no—
364C4×C8—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370C24—41no—
371C24—41no—
372C24—42no—
373C24—42no—
374C24C2×C22⋊C414yes—
375C24C22×D412yes—
376C24C22×D411yescharacteristic
377C24C22×D412yes—
378#41—48no—
379#41—48no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—416no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×Q8—84no—
397C2×Q8—88no—
398C2×Q8—84no—
399C2×Q8C2×C22⋊C414yes—
400C2×Q8C23×C418yes—
401C2×D4—41no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4C2×C22⋊C4116yes—
408C2×D4C2×C22⋊C418yes—
409C2×D4C2×C22⋊C414yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—84no—
446C22×C4C2×C22⋊C418yes—
447C22×C4C2×C22⋊C414yes—
448C22×C4C23×C414yes—
449C22×C4C23×C414yes—
450C22×C4C23×C412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C2511yescharacteristic
459M16—88no—
460M16—84no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—84no—
468C4⋊C4—24no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—84no—
485C4⋊C4C2×C22⋊C418yes—
486C4⋊C4C2×C22⋊C4116yes—
487C4⋊C4C23×C418yes—
488C4⋊C4H(5,𝔽2)14yes—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4C2×C22⋊C4116yes—
501C22⋊C4C2×C22⋊C418yes—
502C22⋊C4C2×C22⋊C4116yes—
503C42—28no—
504C42—216no—
505C42—48no—
506C42—44no—
507C42—42no—
508C42—42no—
509C42—84no—
510C42C23×C414yes—
511C42C22×D414yes—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—44no—
527C23—48no—
528C23—84no—
529C23—82no—
530C23—82no—
531C23C22×C22⋊C412yes—
532C23C22×C22⋊C414yes—
533C23C22×C22⋊C414yes—
534C23C23×D411yescharacteristic
535Q8—28no—
536Q8—28no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—88no—
541Q8—88no—
542D4—416no—
543D4—44no—
544D4—48no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—88no—
582C2×C4—88no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—164no—
586C2×C4C22×C22⋊C418yes—
587C2×C4C22×C22⋊C414yes—
588C2×C4C22×C22⋊C414yes—
589C2×C4C22×C22⋊C412yes—
590C2×C4C24×C411yescharacteristic
591C2×C4C23×D411yescharacteristic
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4#58512yes—
594C8—44no—
595C8—44no—
596C22—24no—
597C22—28no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—24no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—88no—
614C22C23×C22⋊C411yescharacteristic
615C22#348411yescharacteristic
616C22#348511yescharacteristic
617C22#358612yes—
618C22#358612yes—
619C4—28no—
620C4—22no—
621C4—24no—
622C4—24no—
623C4—48no—
624C4—48no—
625C4—44no—
626C4—82no—
627C4—81no—
628C4—81no—
629C4—82no—
630C4C4×H(5,𝔽2)12yes—
631C4#349612yes—
632C2—44no—
633C2—44no—
634C2—42no—
635C2#3425611yescharacteristic
636C2#6008812yes—
637C2#6164911yescharacteristic
638C2#6165211yescharacteristic
639C2#6177912yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22499C2semidirect
5. #22496C2semidirect
6. #22430C2semidirect
7. #22425C2semidirect
8. #21873C2semidirect
9. #21771C2semidirect
10. #21768C2semidirect
11. #21398C2semidirect
12. #20913C2semidirect
13. #20911C2semidirect
15. #3538C4semidirect
33. C2×#419C22semidirect
34. C22×#103C22semidirect
47. #1992C22semidirect
48. #1991C22semidirect
49. #1989C22semidirect
50. #1981C22semidirect
51. #1960C22semidirect
52. #1956C22semidirect
55. #1894C22semidirect
56. #1892C22semidirect
61. #1873C22semidirect
64. #1872C22semidirect
65. #1825C22semidirect
68. #1805C22semidirect
69. #1803C22semidirect
72. #1787C22semidirect
73. #1763C22semidirect
74. C4×#103C22semidirect
76. #1736C22semidirect
77. #1735C22semidirect
87. #552C2×C4semidirect
90. D4×Q8C2×C4semidirect
93. #541C2×C4semidirect
117. C2×C22⋊Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35382
C2×#29782
#342012
#224992
#224962
#224302
#224252
#218732
#217712
#217682
#213982
#209132
#209112