Finite Group Databasebeta

← #6374712

Subgroups — #6374712

Order 512.

2968 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62686C211yesmaximal, characteristic
2#60123C211yesmaximal, characteristic, complemented
3#34056C211yesmaximal, characteristic, complemented
4#21901C214yesmaximal, complemented
5#21858C214yesmaximal, complemented
6#21846C212yesmaximal, complemented
7#21817C212yesmaximal, complemented
8#21775C214yesmaximal, complemented
9#21418C212yesmaximal, complemented
10#21417C212yesmaximal, complemented
11#21383C214yesmaximal, complemented
12#20748C214yesmaximal, complemented
13#3461—22no—
14#3461C418yescomplemented
15#3460—22no—
16#3460C418yescomplemented
17#3459—22no—
18#3459—22no—
19#3458C2211yescharacteristic
20#3455—24no—
21#3455C2212yes—
22#3455C2212yes—
23#3449C2211yescharacteristic
24#3449C2212yes—
25#3444—24no—
26#3444C2212yes—
27#3444C2212yes—
28C2×C4×#41C2212yes—
29C2×C4×#41C2211yescharacteristic
30#2923—24no—
31#2922—24no—
32#2921C2212yes—
33#2921C2212yescomplemented
34#2920C2212yes—
35#2920C2212yescomplemented
36#2905C2212yescomplemented
37#2879C2212yes—
38C2×C4×M16C2212yescomplemented
39#2320—22no—
40#2319—22no—
41#2318—22no—
42#2318—22no—
43#2298C2212yes—
44#2298C2212yescomplemented
45#2297C2212yes—
46#2296C2212yescomplemented
47#2283C2212yes—
48#2283C2212yescomplemented
49C4×#117C2211yescharacteristic, complemented
50C2×#376C2211yescharacteristic, complemented
51#1995C2214yescomplemented
52#1994C2214yescomplemented
53#1986C2218yescomplemented
54#1980C2218yes—
55#1963C2214yes—
56#1962C2214yes—
57#1897C2218yescomplemented
58#1896C2218yescomplemented
59#1871C22116yescomplemented
60#1826C2218yes—
61#1806C22116yescomplemented
62#1789C2218yes—
63#1759C2218yescomplemented
64#1759C2218yescomplemented
65C22×#41C2312yes—
66C22×#41C2311yescharacteristic
67C22×M16C2314yes—
68#544—22no—
69#544C2×C418yescomplemented
70#543—22no—
71#542—28no—
72#542—28no—
73#542C2×C418yes—
74#542C2×C418yescomplemented
75#541—28no—
76#541C2×C414yes—
77#541C2×C414yescomplemented
78#540—28no—
79#540C2×C414yes—
80#540C2×C414yescomplemented
81#539—28no—
82#539—28no—
83#539—24no—
84#539—24no—
85#538—28no—
86#538—28no—
87#538—24no—
88#538—24no—
89#537—22no—
90#537C2×C418yes—
91#536—22no—
92#536C2×C414yes—
93#536C2×C414yescomplemented
94#535—22no—
95#535C2×C414yes—
96#535C2×C414yescomplemented
97#534—22no—
98#534—22no—
99#534—22no—
100#533—28no—
101#533—24no—
102#533C2×C414yes—
103#533C2×C414yescomplemented
104#533C2312yes—
105#533C2312yes—
106#532—24no—
107#532—28no—
108#532C2×C414yes—
109#532C2×C414yescomplemented
110#532C2312yes—
111#532C2312yes—
112#529—28no—
113#529—28no—
114#529—28no—
115#529—28no—
116#529—28no—
117#529—28no—
118#529—24no—
119#529—24no—
120#529—24no—
121#529—24no—
122#529C2×C418yes—
123#529C2×C418yescomplemented
124#525—24no—
125#525—24no—
126#525—24no—
127#525—28no—
128#525—28no—
129#525—24no—
130#525C2314yes—
131#525C2314yes—
132C4×#41—22no—
133C4×#41—22no—
134C4×#41—24no—
135C4×#41—24no—
136C4×#41—24no—
137C4×#41C2×C418yes—
138C4×#41C2×C418yescomplemented
139C4×#41D414yes—
140C4×#41D414yes—
141C4×#41C2314yes—
142C4×#41C2314yes—
143C2×C4×Q8C2×C414yescomplemented
144C2×C4×Q8D412yes—
145C2×C4×D4—24no—
146C2×C4×D4C2×C414yescomplemented
147C2×C4×D4D412yes—
148C2×C4×D4D412yes—
149C2×C4×D4D412yes—
150C2×#117—22no—
151C2×#117C2312yes—
152C2×#117C2312yes—
153C2×#117C2311yescharacteristic
154C2×#117C2311yescharacteristic
155C2×#117C2311yescharacteristic
156C22×C42—22no—
157C22×C42C2312yes—
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160C22×C42C2311yescharacteristic
161#433—28no—
162#433—28no—
163#432—24no—
164#432—24no—
165#431—24no—
166#431—24no—
167#424—28no—
168#423C2314yes—
169#423C2314yes—
170C2×#104C23116yes—
171C2×#104C23116yes—
172#408—28no—
173#407C2314yes—
174#407C2314yes—
175C4×M16—22no—
176C4×M16—22no—
177C4×M16C2314yes—
178C4×M16C2314yes—
179C2×#97—22no—
180C2×C4×C8—22no—
181#391—22no—
182#390—22no—
183#389—24no—
184#388—24no—
185#388—24no—
186#387—24no—
187#387—24no—
188#386—22no—
189#386—22no—
190#385—24no—
191#383—22no—
192#383—22no—
193#382—22no—
194#382—22no—
195#381—22no—
196#381—22no—
197#380—22no—
198#380—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2314yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4C2314yes—
204C4×C22⋊C4C2314yes—
205#376—22no—
206#376C2312yes—
207#376C2312yes—
208#376C2312yes—
209C2×#95—22no—
210C43C2312yes—
211#339C23116yes—
212#339C23116yes—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—44no—
218C2×#41—48no—
219C2×#41C22×C414yes—
220C2×#41C22×C414yes—
221C2×#41C2×D414yes—
222C2×#41C2414yes—
223C22×Q8C22×C412yes—
224C22×Q8C2×D411yescharacteristic
225C22×D4—22no—
226C22×D4C22×C412yes—
227C22×D4C2×D411yescharacteristic
228C22×D4C2×D411yescharacteristic
229C22×D4C2×D411yescharacteristic
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4C2×D412yes—
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×M16—24no—
240C2×M16—48no—
241C2×M16C2418yes—
242C22×C8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—28no—
245C4⋊Q8C22×C412yes—
246C4⋊Q8C22×C412yes—
247C4⋊Q8C22×C414yes—
248C4⋊Q8C22×C414yes—
249#127—24no—
250#127C22×C412yes—
251#127C22×C412yes—
252#126—28no—
253#126—28no—
254#126—28no—
255#126—28no—
256#126—28no—
257#126—28no—
258#125—28no—
259#125—28no—
260#125—28no—
261#125—24no—
262#125—24no—
263#125C22×C414yes—
264#125C22×C414yes—
265#124—24no—
266#124—24no—
267#124—28no—
268#124—28no—
269#124—24no—
270#124—24no—
271#124C22×C414yes—
272#124C22×C414yes—
273#123—24no—
274#123—24no—
275#123—28no—
276#123—28no—
277#123—24no—
278#123—24no—
279#123—28no—
280#123—28no—
281#123—28no—
282#123—28no—
283#123—28no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8C22×C418yes—
295C22⋊Q8C22×C418yes—
296#121—28no—
297#121—28no—
298#121—28no—
299#121—24no—
300#121—24no—
301#121—24no—
302#121—24no—
303#121—28no—
304#121—28no—
305#121—28no—
306#121C22×C418yes—
307#121C22×C418yes—
308#120—24no—
309#120—24no—
310#120—24no—
311#120—24no—
312#120—28no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8C22⋊C418yes—
320C4×Q8C22⋊C418yes—
321C4×Q8C22×C418yes—
322C4×Q8C22×C418yes—
323C4×D4—28no—
324C4×D4—28no—
325C4×D4—28no—
326C4×D4—28no—
327C4×D4—28no—
328C4×D4—28no—
329C4×D4—28no—
330C4×D4—28no—
331C4×D4—24no—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—24no—
335C4×D4—24no—
336C4×D4—24no—
337C4×D4C22⋊C418yes—
338C4×D4C22⋊C418yes—
339C4×D4C22⋊C418yes—
340C4×D4C22⋊C418yes—
341C4×D4C22⋊C418yes—
342C4×D4C22⋊C418yes—
343C4×D4C22×C418yes—
344C4×D4C22×C418yes—
345#117—24no—
346#117—24no—
347#117—22no—
348#117—22no—
349#117—28no—
350#117—28no—
351#117—28no—
352#117—24no—
353#117—24no—
354#117—44no—
355#117C22×C414yes—
356#117C22×C414yes—
357#117C22×C414yes—
358#117C22×C414yes—
359#117C2×D414yes—
360#117C2×D414yes—
361#117C2×D414yes—
362#117C2414yes—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4C22×C414yes—
368C2×C4⋊C4C2×D412yes—
369C2×C4⋊C4C2×D412yes—
370C2×C4⋊C4C2×D412yes—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D412yes—
379C2×C42—24no—
380C2×C42—22no—
381C2×C42—28no—
382C2×C42—24no—
383C2×C42—24no—
384C2×C42—22no—
385C2×C42—24no—
386C2×C42—44no—
387C2×C42C22×C414yes—
388C2×C42C22×C414yes—
389C2×C42C22×C414yes—
390C2×C42C2×D412yes—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2×D412yes—
396C2×C42C2×D412yes—
397C2×C42C2×D412yes—
398C2×C42C2×D412yes—
399C2×C42C2414yes—
400C2×C42C2414yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C4⋊C8—28no—
406C4⋊C8—28no—
407#104—416no—
408#104—416no—
409C22⋊C8—28no—
410C22⋊C8—28no—
411#97—24no—
412#97—24no—
413C4×C8—24no—
414C4×C8—24no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424C24—22no—
425C24C22×D411yescharacteristic
426C24C22×D411yescharacteristic
427C24C22×D411yescharacteristic
428#41—216no—
429#41—48no—
430#41—48no—
431#41—48no—
432#41—48no—
433#41—48no—
434#41—88no—
435#41—88no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8C2×C22⋊C414yes—
442C2×Q8C23×C414yes—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—48no—
455C2×D4C2×C22⋊C414yes—
456C2×D4C2×C22⋊C414yes—
457C2×D4C2×C22⋊C414yes—
458C2×D4C23×C414yes—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4C2×C22⋊C414yes—
478C22×C4C2×C22⋊C414yes—
479C22×C4C2×C22⋊C414yes—
480C22×C4C2×C22⋊C414yes—
481C22×C4C2×C22⋊C414yes—
482C22×C4C2×C22⋊C414yes—
483C22×C4C23×C412yes—
484C22×C4C23×C412yes—
485C22×C4C23×C412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C2511yescharacteristic
497M16—216no—
498M16—48no—
499C2×C8—28no—
500C2×C8—48no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—44no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4C2×C22⋊C418yes—
516C4⋊C4C2×C22⋊C418yes—
517C4⋊C4C2×C22⋊C418yes—
518C4⋊C4C23×C418yes—
519C22⋊C4—24no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—24no—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—44no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4C2×C22⋊C418yes—
533C22⋊C4C2×C22⋊C418yes—
534C22⋊C4C2×C22⋊C418yes—
535C42—28no—
536C42—24no—
537C42—28no—
538C42—24no—
539C42—28no—
540C42—28no—
541C42—44no—
542C42—84no—
543C42C2×C22⋊C414yes—
544C42C2×C22⋊C414yes—
545C42C2×C22⋊C414yes—
546C42C2×C22⋊C414yes—
547C42C2×C22⋊C414yes—
548C42C2×C22⋊C414yes—
549C42C23×C412yes—
550C42C23×C412yes—
551C42C23×C412yes—
552C42C23×C412yes—
553C42C22×D411yescharacteristic
554C42C22×D412yes—
555C42C22×D411yescharacteristic
556C42C22×D412yes—
557C42C22×D412yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—84no—
567C23C22×C22⋊C412yes—
568C23C22×C22⋊C412yes—
569C23C22×C22⋊C412yes—
570C23C23×D411yescharacteristic
571Q8—28no—
572Q8—44no—
573Q8—44no—
574Q8—84no—
575Q8—84no—
576D4—28no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—44no—
581D4—48no—
582D4—48no—
583D4—84no—
584D4—84no—
585D4—88no—
586D4—88no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—416no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4C22×C22⋊C412yes—
622C2×C4C22×C22⋊C412yes—
623C2×C4C22×C22⋊C414yes—
624C2×C4C22×C22⋊C412yes—
625C2×C4C22×C22⋊C412yes—
626C2×C4C22×C22⋊C412yes—
627C2×C4C22×C22⋊C414yes—
628C2×C4C22×C22⋊C414yes—
629C2×C4C24×C411yescharacteristic
630C2×C4C23×D411yescharacteristic
631C2×C4#58514yes—
632C8—48no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—44no—
641C22—48no—
642C22—44no—
643C22—82no—
644C22—82no—
645C22—88no—
646C22C23×C22⋊C411yescharacteristic
647C22#344912yes—
648C4—22no—
649C4—22no—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—48no—
657C4—44no—
658C4—84no—
659C4—84no—
660C4#342812yes—
661C4#344912yes—
662C2—22no—
663C2—42no—
664C2—42no—
665C2—42no—
666C2—84no—
667C2#3404411yescharacteristic
668C2#6009312yes—
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C22⋊C4
prime radical#62686C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60123C2semidirect
3. #34056C2semidirect
4. #21901C2semidirect
5. #21858C2semidirect
6. #21846C2semidirect
7. #21817C2semidirect
8. #21775C2semidirect
9. #21418C2semidirect
10. #21417C2semidirect
11. #21383C2semidirect
12. #20748C2semidirect
14. #3461C4semidirect
16. #3460C4semidirect
33. #2921C22semidirect
35. #2920C22semidirect
36. #2905C22semidirect
38. C2×C4×M16C22semidirect
44. #2298C22semidirect
46. #2296C22semidirect
48. #2283C22semidirect
49. C4×#117C22semidirect
50. C2×#376C22semidirect
51. #1995C22semidirect
52. #1994C22semidirect
53. #1986C22semidirect
57. #1897C22semidirect
58. #1896C22semidirect
59. #1871C22semidirect
61. #1806C22semidirect
63. #1759C22semidirect
64. #1759C22semidirect
69. #544C2×C4semidirect
74. #542C2×C4semidirect
77. #541C2×C4semidirect
80. #540C2×C4semidirect
93. #536C2×C4semidirect
96. #535C2×C4semidirect
103. #533C2×C4semidirect
109. #532C2×C4semidirect
123. #529C2×C4semidirect
138. C4×#41C2×C4semidirect
143. C2×C4×Q8C2×C4semidirect
146. C2×C4×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#626862
#601232
#340562
#219012
#218582
#218462
#218172
#217752
#214182
#214172
#213832
#207482