Finite Group Databasebeta

← #6344656

Subgroups — #6344656

Order 512.

1841 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60558C211yesmaximal, characteristic
2#60294C211yesmaximal, characteristic, complemented
3C2×#2972C211yesmaximal, characteristic, complemented
4#17924C214yesmaximal, complemented
5#17922C214yesmaximal, complemented
6#17914C214yesmaximal, complemented
7#17910C214yesmaximal, complemented
8#17873C214yesmaximal, complemented
9#17852C214yesmaximal, complemented
10#17810C211yesmaximal, characteristic, complemented
11#17462C211yesmaximal, characteristic, complemented
12C2×#1490C211yesmaximal, characteristic, complemented
13#17327C211yesmaximal, characteristic, complemented
14#3444C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16#3114C414yes—
17#3110C412yes—
18#3109C412yes—
19#3105C2214yes—
20#3101C2212yes—
21#3100C2212yes—
22#3092C412yes—
23#3077C412yes—
24#3036C412yes—
25#3003C412yes—
26#2990—22no—
27#2989C2212yescomplemented
28#2988C2212yescomplemented
29#2985C2212yescomplemented
30#2981C2212yescomplemented
31#2977—22no—
32#2972—22no—
33#2972—22no—
34#2972C2214yes—
35#2972C2212yes—
36#2972C2212yes—
37#2968—22no—
38#2966C2214yescomplemented
39C4×AGL(1,ℤ/8)C2214yes—
40C2×C8×D4—22no—
41C2×C8×D4C2214yes—
42#2925—22no—
43#2920C2211yescharacteristic
44C2×#432C2211yescharacteristic
45C2×#431C2211yescharacteristic
46C2×#431C2211yescharacteristic
47C22×C22⋊C8—22no—
48C2×C4×M16C2211yescharacteristic
49#1586C2218yescomplemented
50#1585C2218yescomplemented
51#1582C2218yescomplemented
52#1581C2218yescomplemented
53#1580C2218yescomplemented
54SD16⋊C8C2218yescomplemented
55C8×SD16C2218yescomplemented
56C8×D8C2218yescomplemented
57#1573C2214yescomplemented
58#1572C2214yescomplemented
59#1565C2214yescomplemented
60#1490C2214yes—
61#1490C2212yes—
62#1490C2212yes—
63#1485C2212yescomplemented
64#1484C2212yescomplemented
65#1483C2212yescomplemented
66#1483C2212yescomplemented
67#1476C2214yescomplemented
68C2×D4⋊C8C2214yes—
69#1453C2214yescomplemented
70#1451C2212yescomplemented
71#1451C2212yescomplemented
72C8×M16C2212yescomplemented
73C8×M16C2212yescomplemented
74C23×D4C2×C411yescharacteristic
75C24×C4—21no—
76C2×AGL(1,ℤ/8)C2×C414yes—
77C22×M16—21no—
78C23×C8—24no—
79#533C2×C411yescharacteristic
80#533C2×C411yescharacteristic
81#533C2311yescharacteristic
82#532C2×C411yescharacteristic
83#532C2×C411yescharacteristic
84#532C2311yescharacteristic
85#529—21no—
86#529—21no—
87#529—21no—
88#529—21no—
89#529C2×C412yes—
90#529C2312yes—
91#525—21no—
92#525—21no—
93#525C2×C412yes—
94C4×#41—21no—
95C4×#41—21no—
96C4×#41C2312yes—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4C2×C412yes—
104C2×C4×D4C2×C412yes—
105C2×C4×D4C2×C412yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C22×C4⋊C4C2×C411yescharacteristic
110C22×C22⋊C4—21no—
111C22×C42C2311yescharacteristic
112#484C2×C414yes—
113#483C2×C414yes—
114#481C2×C414yes—
115#480C2×C414yes—
116#478C2×C418yes—
117#476C2×C414yes—
118#474C2×C414yes—
119#463C2×C418yes—
120#461C2×C414yes—
121#459C2×C414yes—
122#450C2×C414yes—
123#450C2×C414yes—
124#447C2×C414yes—
125#447C2×C414yes—
126#446—24no—
127#446C2314yescomplemented
128C8×Q8C2314yescomplemented
129#442C2×C418yes—
130#440C2×C418yes—
131C4×SD16C2318yes—
132C4×D8C2318yes—
133#436C2314yescomplemented
134#435—24no—
135#435—24no—
136#435—24no—
137C8×D4—24no—
138C8×D4—24no—
139C8×D4—24no—
140C8×D4—28no—
141C8×D4—24no—
142C8×D4—24no—
143C8×D4—24no—
144C8×D4C2318yes—
145C8×D4C2318yes—
146C8×D4C2314yescomplemented
147#433—24no—
148#433—24no—
149#432—21no—
150#432—21no—
151#432—22no—
152#432C2×C412yes—
153#432C2×C412yes—
154#432C2312yes—
155#432C2312yes—
156#431—21no—
157#431—21no—
158#431—22no—
159#431D412yes—
160#431D412yes—
161#431C2312yes—
162#431C2312yes—
163#431C2312yes—
164#431C2312yes—
165#431C2312yes—
166#431C2311yescharacteristic
167#431C2311yescharacteristic
168#428C2×C414yes—
169#423—22no—
170#423C2×C412yes—
171C2×C4⋊C8—22no—
172C2×C4⋊C8D412yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175#418C2×C414yes—
176#417C2×C414yes—
177#408—24no—
178#407C2×C412yes—
179C2×C22⋊C8—21no—
180C2×C22⋊C8—21no—
181C2×C22⋊C8—24no—
182C2×C22⋊C8—28no—
183C2×C22⋊C8—24no—
184C2×C22⋊C8—24no—
185C2×C22⋊C8D412yes—
186#405—24no—
187C4×M16—21no—
188C4×M16—21no—
189C4×M16C2311yescharacteristic
190C4×M16C2311yescharacteristic
191C4×M16C2312yes—
192C4×M16C2312yes—
193C2×#97—21no—
194C2×#97—21no—
195C2×C4×C8—21no—
196C2×C4×C8C2312yes—
197C2×C4×C8C2312yes—
198#335C2318yescomplemented
199#334C2318yescomplemented
200Q8⋊C8C2314yescomplemented
201Q8⋊C8C2314yescomplemented
202D4⋊C8C2318yes—
203D4⋊C8C2318yes—
204D4⋊C8C2314yescomplemented
205D4⋊C8C2314yescomplemented
206#322C2314yescomplemented
207#322C2314yescomplemented
208#322C2314yescomplemented
209C82C2314yescomplemented
210C25—21no—
211C2×#41—21no—
212C2×#41—21no—
213C2×#41C22×C412yes—
214C22×D4—28no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C22×C412yes—
219C22×D4C22×C412yes—
220C22×D4C22×C412yes—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—84no—
229C23×C4C22×C411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4#4111yescharacteristic
232AGL(1,ℤ/8)—216no—
233C2×SD16C22×C418yes—
234C2×D8C22×C418yes—
235C2×M16—21no—
236C2×M16—22no—
237C2×M16—21no—
238C2×M16—24no—
239C2×M16—44no—
240C2×M16C22×C414yes—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—28no—
245C22×C8—28no—
246C22×C8—24no—
247C22×C8—88no—
248C22×C8C2×D414yes—
249C4⋊Q8C22×C411yescharacteristic
250C4⋊Q8C22×C411yescharacteristic
251C4⋊Q8C22×C411yescharacteristic
252#127C22×C411yescharacteristic
253#126—22no—
254#126—22no—
255#125—21no—
256#125—21no—
257#125C22×C412yes—
258#124—21no—
259#124—21no—
260#124C22×C412yes—
261#123—22no—
262#123—22no—
263#123—22no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8C22×C412yes—
269C22⋊Q8C22×C412yes—
270#121—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121C22×C412yes—
275#121C22×C412yes—
276#120—22no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×Q8C22×C412yes—
282C4×Q8C2412yes—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—21no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4C22×C414yes—
306C4×D4C22×C414yes—
307C4×D4C22×C414yes—
308C4×D4C22×C412yes—
309C4×D4C2414yes—
310C4×D4C2412yes—
311#117—21no—
312#117—21no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117C22×C412yes—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C22×C412yes—
321C2×C4⋊C4C22×C412yes—
322C2×C4⋊C4C22×C412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42—22no—
330C2×C42C22×C411yescharacteristic
331C2×C42C22×C411yescharacteristic
332C2×C42C22×C411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4111yescharacteristic
337C2×C42#4111yescharacteristic
338C2×C42#4111yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107C22×C418yes—
343#106C22×C418yes—
344C4⋊C8—24no—
345C4⋊C8—24no—
346C4⋊C8—22no—
347C4⋊C8—22no—
348C4⋊C8C22×C414yes—
349C4⋊C8C22×C414yes—
350C4⋊C8C2×D412yes—
351C4⋊C8C2×D412yes—
352C4⋊C8C2×D414yes—
353C4⋊C8C2414yes—
354C4⋊C8C2414yes—
355#103C22×C418yes—
356#102C22×C418yes—
357#102C22×C418yes—
358#102C22×C418yes—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8—28no—
364C22⋊C8—28no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8C22×C414yes—
368C22⋊C8C22×C414yes—
369C22⋊C8C2×D414yes—
370C22⋊C8C2×D412yes—
371C22⋊C8C2×D412yes—
372#97—24no—
373#97—22no—
374#97—22no—
375#97—22no—
376#97—21no—
377#97—21no—
378#97C22×C414yes—
379C4×C8—24no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8C2×D414yes—
384C4×C8C2412yes—
385C4×C8C2412yes—
386C4×C8C2414yes—
387C4×C8C2412yes—
388C4×C8C2412yes—
389C24—28no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—82no—
394C24—82no—
395C24C4×D411yescharacteristic
396#41—28no—
397#41—82no—
398#41—82no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C23×C412yes—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—24no—
416C2×D4#13118yes—
417C2×D4C23×C412yes—
418C2×D4C23×C414yes—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C4×D411yescharacteristic
446C22×C4C4×D411yescharacteristic
447C22×C4C4×D411yescharacteristic
448C22×C4C23×C411yescharacteristic
449C22×C4C23×C411yescharacteristic
450C22×C4C23×C411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
457SD16—216no—
458D8—216no—
459M16—24no—
460M16—28no—
461M16—44no—
462M16—42no—
463M16—84no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—44no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8—84no—
472C2×C8—88no—
473C2×C8—88no—
474C2×C8—82no—
475C2×C8C23×C418yes—
476C2×C8C22×D414yes—
477C2×C8C22×D414yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4C23×C412yes—
492C4⋊C4C23×C412yes—
493C4⋊C4C23×C414yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42C4×D411yescharacteristic
511C42C4×D411yescharacteristic
512C42C4×D411yescharacteristic
513C42C4×D411yescharacteristic
514C42C4×D411yescharacteristic
515C42C4×D411yescharacteristic
516C42C23×C411yescharacteristic
517C42C23×C411yescharacteristic
518C42C23×C411yescharacteristic
519C42C22×D411yescharacteristic
520C42C22×D411yescharacteristic
521C42C22×D411yescharacteristic
522C42C2×#4111yescharacteristic
523C42C2×#4111yescharacteristic
524C42C2×#4111yescharacteristic
525C42C2511yescharacteristic
526C23—28no—
527C23—28no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—28no—
532C23—22no—
533C23—41no—
534C23—44no—
535C23—44no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—816no—
541C23C2×C4×D411yescharacteristic
542C23C2×C4×D411yescharacteristic
543C23C2×C4×D411yescharacteristic
544Q8—24no—
545Q8—81no—
546Q8—81no—
547D4—28no—
548D4—28no—
549D4—28no—
550D4—24no—
551D4—24no—
552D4—44no—
553D4—44no—
554D4—81no—
555D4—81no—
556D4—82no—
557D4C2×#13118yes—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×#13114yes—
600C2×C4C24×C411yescharacteristic
601C2×C4C23×D411yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C8—24no—
604C8—24no—
605C8—24no—
606C8—28no—
607C8—42no—
608C8—44no—
609C8—44no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—28no—
618C22—44no—
619C22—416no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—88no—
626C22—88no—
627C22#292912yes—
628C22#295612yes—
629C22C22×C4×D411yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4#292912yes—
644C4C22×#13112yes—
645C4#358512yes—
646C2—22no—
647C2—22no—
648C2—41no—
649C2—48no—
650C2—82no—
651C2C2×#292911yescharacteristic
652C2#6016811yescharacteristic
653C2C2×#295611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60558C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60294C2semidirect
3. C2×#2972C2semidirect
4. #17924C2semidirect
5. #17922C2semidirect
6. #17914C2semidirect
7. #17910C2semidirect
8. #17873C2semidirect
9. #17852C2semidirect
10. #17810C2semidirect
11. #17462C2semidirect
12. C2×#1490C2semidirect
13. #17327C2semidirect
27. #2989C22semidirect
28. #2988C22semidirect
29. #2985C22semidirect
30. #2981C22semidirect
38. #2966C22semidirect
49. #1586C22semidirect
50. #1585C22semidirect
51. #1582C22semidirect
52. #1581C22semidirect
53. #1580C22semidirect
54. SD16⋊C8C22semidirect
55. C8×SD16C22semidirect
56. C8×D8C22semidirect
57. #1573C22semidirect
58. #1572C22semidirect
59. #1565C22semidirect
63. #1485C22semidirect
64. #1484C22semidirect
65. #1483C22semidirect
66. #1483C22semidirect
67. #1476C22semidirect
69. #1453C22semidirect
70. #1451C22semidirect
71. #1451C22semidirect
72. C8×M16C22semidirect
73. C8×M16C22semidirect
127. #446C23semidirect
128. C8×Q8C23semidirect
133. #436C23semidirect
146. C8×D4C23semidirect
198. #335C23semidirect
199. #334C23semidirect
200. Q8⋊C8C23semidirect
201. Q8⋊C8C23semidirect
204. D4⋊C8C23semidirect
205. D4⋊C8C23semidirect
206. #322C23semidirect
207. #322C23semidirect
208. #322C23semidirect
209. C82C23semidirect

Maximal subgroups

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

subgroupindex
#605582
#602942
C2×#29722
#179242
#179222
#179142
#179102
#178732
#178522
#178102
#174622
C2×#14902
#173272