Finite Group Databasebeta

← #6489536

Subgroups — #6489536

Order 512.

3197 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#3232C212yesmaximal, complemented
3#36621C211yesmaximal, characteristic
4#32984C212yesmaximal, complemented
5#32980C212yesmaximal
6#29642C211yesmaximal, characteristic, complemented
7#29641C211yesmaximal, characteristic, complemented
8#29639C211yesmaximal, characteristic, complemented
9#29638C211yesmaximal, characteristic, complemented
10#25906C214yesmaximal, complemented
11#25814C214yesmaximal, complemented
12#24306C214yesmaximal, complemented
13#24304C214yesmaximal, complemented
14#21836C212yesmaximal, complemented
15#20714C211yesmaximal, characteristic, complemented
16C2×#533—22no—
17C2×#533—22no—
18C2×#533—216no—
19C2×#533C2214yes—
20C2×#533C2214yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#529C2211yescharacteristic
24C22×#125C2212yes—
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—21no—
27C22×C4×Q8—21no—
28C22×C4×Q8—21no—
29C22×#117C2212yes—
30C23×C42C2211yescharacteristic
31#3232—216no—
32C2×#499C2214yescomplemented
33C2×#488C2214yescomplemented
34C2×#484C2212yes—
35C2×#481C2212yes—
36C2×#483C2212yes—
37C2×#480C2212yes—
38C2×#479C2214yescomplemented
39C2×#478C2214yescomplemented
40C2×#431C2212yes—
41#2855C2211yescharacteristic
42#2854C2211yescharacteristic
43#2853—21no—
44#2852—22no—
45#2851C2212yes—
46#2850C2212yes—
47#2849—21no—
48#2848C2211yescharacteristic
49#2847C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51C4×#123—21no—
52C4×#121C2212yes—
53C4×#120—21no—
54#2290—22no—
55#2283C2211yescharacteristic
56#2058C2214yescomplemented
57#2058C2214yescomplemented
58#2057C2214yescomplemented
59#2057C2214yescomplemented
60#2024C2214yescomplemented
61#2024C2214yescomplemented
62#2023C2214yescomplemented
63#2023C2214yescomplemented
64#1991C2218yescomplemented
65#1989C2218yescomplemented
66#1875C2218yes—
67#1873C2218yes—
68#1854C2218yes—
69#1854C2218yes—
70#1824C2212yes—
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1817C2211yescharacteristic
75#1817C2211yescharacteristic
76#1797C2214yes—
77#1779C2212yes—
78C4×C22⋊C8C2214yes—
79C23×Q8—21no—
80C23×Q8—21no—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84#533—232no—
85#533—232no—
86#533—232no—
87#533—232no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2314yes—
94#529—28no—
95C2×#126—21no—
96C2×#125—22no—
97C2×#125—22no—
98C2×#125—28no—
99C2×#125D4116yescomplemented
100C2×#125C2311yescharacteristic
101C2×#125C2314yes—
102C2×#125C2314yes—
103C2×#124C2311yescharacteristic
104C2×#123—21no—
105C2×#123—21no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—216no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×C22⋊Q8C2312yes—
114C2×#121C2311yescharacteristic
115C2×#121C2312yes—
116C2×#120—21no—
117C2×C4×Q8—216no—
118C2×C4×Q8—216no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4C2311yescharacteristic
127C2×#117—24no—
128C2×#117—28no—
129C2×#117—28no—
130C2×#117—21no—
131C2×#117C2318yes—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C22⋊C4—22no—
139C22×C42—22no—
140C22×C42—28no—
141C22×C42D412yes—
142C22×C42D411yescharacteristic
143C22×C42D411yescharacteristic
144C22×C42D412yes—
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149#499—216no—
150#488—216no—
151#484—28no—
152#483—28no—
153#481—28no—
154#480—28no—
155#479—216no—
156#478—216no—
157#431—28no—
158C2×#107C2314yes—
159C2×#107C2314yes—
160C2×#106C2314yes—
161C2×#106C2314yes—
162C2×C4⋊C8C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2318yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170C2×C4×C8C2314yes—
171#391C2312yes—
172#390C2312yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—22no—
184#386—21no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—21no—
189#385—22no—
190#385—21no—
191#385—22no—
192#385—22no—
193#385—22no—
194#385—21no—
195#383—21no—
196#382C2312yes—
197#381—21no—
198#381—21no—
199#380C2311yescharacteristic
200#380C2311yescharacteristic
201#380C2312yes—
202#380C2312yes—
203#379—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209#376—21no—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—28no—
218C22×Q8—28no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—44no—
222C22×Q8—48no—
223C22×Q8—44no—
224C22×Q8C2411yescharacteristic
225C22×D4—21no—
226C22×D4—22no—
227C22×D4C2411yescharacteristic
228C23×C4—24no—
229C23×C4—28no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—21no—
236C23×C4—44no—
237C23×C4—42no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—216no—
253C4⋊Q8—216no—
254C4⋊Q8—216no—
255C4⋊Q8—216no—
256#126—44no—
257#125—216no—
258#125—216no—
259#125—28no—
260#125—28no—
261#125—24no—
262#125C2×D4116yes—
263#125C2×D4116yes—
264#124—24no—
265#123—44no—
266#123—44no—
267C22⋊Q8—232no—
268C22⋊Q8—232no—
269C22⋊Q8—232no—
270C22⋊Q8—232no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273#121—24no—
274#121—28no—
275#120—44no—
276C4×Q8—232no—
277C4×Q8—232no—
278C4×Q8—232no—
279C4×Q8—232no—
280C4×Q8—24no—
281C4×D4—24no—
282C4×D4—48no—
283C4×D4—44no—
284#117—232no—
285#117—216no—
286#117—216no—
287#117—44no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—216no—
291C2×C4⋊C4—216no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4C2×D418yes—
310C2×C4⋊C4C2×D418yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2414yes—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C42—216no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—28no—
336C2×C42—24no—
337C2×C42—28no—
338C2×C42—28no—
339C2×C42—28no—
340C2×C42—28no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D418yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#107—28no—
356#106—28no—
357#106—28no—
358C4⋊C8—28no—
359#103—28no—
360#103—28no—
361#102—28no—
362#102—28no—
363C22⋊C8—24no—
364C22⋊C8—24no—
365C22⋊C8—216no—
366C4×C8—28no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—42no—
384C24C22×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—216no—
387C2×Q8—216no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—432no—
391C2×Q8—432no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C22×C4—232no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4#12014yes—
449C22×C4#12014yes—
450C22×C4#13514yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C42−1+418yes—
461C22×C4C2511yescharacteristic
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—28no—
465C2×C8—28no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4C22×D418yes—
488C4⋊C4C22×D418yes—
489C22⋊C4—232no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C42—232no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—44no—
511C42—42no—
512C42#12014yes—
513C42#12014yes—
514C42C22×D414yes—
515C42C22×D414yes—
516C23—22no—
517C23—22no—
518C23—28no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—48no—
526C23—48no—
527C23—44no—
528C23—44no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23C2×#12012yes—
535C23#53614yes—
536C23C2×#13512yes—
537C23#57512yes—
538C23C23×D411yescharacteristic
539Q8—432no—
540Q8—432no—
541Q8—44no—
542D4—44no—
543D4—88no—
544D4—84no—
545C2×C4—216no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—216no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—416no—
573C2×C4—416no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4C2×#12012yes—
592C2×C4C2×#12012yes—
593C2×C4C2×#12012yes—
594C2×C4C2×#12012yes—
595C2×C4C2×#12012yes—
596C2×C4#53614yes—
597C2×C4C2×#13514yes—
598C2×C4#57514yes—
599C2×C4C23×D412yes—
600C2×C4C2×2−1+414yes—
601C8—44no—
602C8—44no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22—82no—
619C22—81no—
620C22#300214yes—
621C22C22×#12011yescharacteristic
622C22C2×#53612yes—
623C22C22×#13511yescharacteristic
624C22C2×#57512yes—
625C22C2×#57511yescharacteristic
626C4—22no—
627C4—24no—
628C4—22no—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—48no—
635C4—48no—
636C4—81no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4—81no—
641C4#348614yes—
642C2—24no—
643C2—44no—
644C2—161no—
645C2#3524211yescharacteristic
646C2C2×#300211yescharacteristic
647C2#6035911yescharacteristic
648C2#6084612yes—
649C2#6086612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#533C2semidirect
2. C2×#3232C2semidirect
4. #32984C2semidirect
6. #29642C2semidirect
7. #29641C2semidirect
8. #29639C2semidirect
9. #29638C2semidirect
10. #25906C2semidirect
11. #25814C2semidirect
12. #24306C2semidirect
13. #24304C2semidirect
14. #21836C2semidirect
15. #20714C2semidirect
32. C2×#499C22semidirect
33. C2×#488C22semidirect
38. C2×#479C22semidirect
39. C2×#478C22semidirect
56. #2058C22semidirect
57. #2058C22semidirect
58. #2057C22semidirect
59. #2057C22semidirect
60. #2024C22semidirect
61. #2024C22semidirect
62. #2023C22semidirect
63. #2023C22semidirect
64. #1991C22semidirect
65. #1989C22semidirect
99. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#32322
#366212
#329842
#329802
#296422
#296412
#296392
#296382
#259062
#258142
#243062
#243042
#218362
#207142