Finite Group Databasebeta

← #6448702

Subgroups — #6448702

Order 512.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic, complemented
2C2×#3080C211yesmaximal, characteristic, complemented
3C2×#3038C211yesmaximal, characteristic, complemented
4#34065C211yesmaximal, characteristic, complemented
5#24164C218yesmaximal, complemented
6#24162C218yesmaximal, complemented
7#23480C212yesmaximal, complemented
8#23391C212yesmaximal, complemented
9#21314C211yesmaximal, characteristic, complemented
10#21260C211yesmaximal, characteristic, complemented
11#20925C212yesmaximal, complemented
12#20873C212yesmaximal, complemented
13#20627C211yesmaximal, characteristic, complemented
14C22×AGL(1,ℤ/8)C2212yes—
15C2×#540—28no—
16C2×#540—28no—
17C2×#540—22no—
18C2×#540—22no—
19C2×#540C2218yes—
20C22×#127C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×#525C2211yescharacteristic
23C22×#123—22no—
24C22×#121—22no—
25C22×#121C2212yes—
26C2×C4×#41C2211yescharacteristic
27C22×C4×D4C2212yes—
28C22×#117C2211yescharacteristic
29#3080—28no—
30C2×#479C2214yescomplemented
31C2×#478C2214yescomplemented
32#3038—28no—
33C2×#460C2214yescomplemented
34C2×#459C2214yescomplemented
35C2×#428C2212yes—
36C2×#423C2211yescharacteristic
37C2×#423C2211yescharacteristic
38C2×#417C2212yes—
39C2×#418C2212yes—
40C2×#418C2212yes—
41C2×#418C2212yes—
42C2×#407C2212yes—
43#2324C2214yescomplemented
44#2323C2214yescomplemented
45#2322—22no—
46#2321—22no—
47#2297C2211yescharacteristic
48#2296C2211yescharacteristic
49#2293—22no—
50#2290—22no—
51#2280C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2034C2218yescomplemented
54#2033C2218yescomplemented
55#2000C2218yescomplemented
56#1999C2218yescomplemented
57#1989C2218yescomplemented
58#1982C2218yescomplemented
59#1879C2218yescomplemented
60#1877C2218yescomplemented
61#1864C2218yescomplemented
62#1862C2218yescomplemented
63#1798C2218yescomplemented
64#1736C2212yes—
65#1736C2212yes—
66#1728C2212yes—
67C22×#41C2311yescharacteristic
68C23×D4—22no—
69C23×D4—22no—
70C23×D4C2312yes—
71C24×C4—22no—
72C24×C4C2311yescharacteristic
73C2×AGL(1,ℤ/8)—28no—
74C2×AGL(1,ℤ/8)—28no—
75C2×AGL(1,ℤ/8)—44no—
76C22×SD16C2314yes—
77C22×D8C2314yes—
78C22×M16C2312yes—
79C22×M16C2312yes—
80#540—232no—
81#540—232no—
82#540—232no—
83#540—232no—
84C2×#127—21no—
85C2×#127—21no—
86C2×#127—24no—
87C2×#127D418yescomplemented
88C2×#127C2314yes—
89C2×#125—21no—
90C2×#125—21no—
91C2×#125—24no—
92C2×#125D418yescomplemented
93C2×#125C2314yes—
94#525—24no—
95#525—24no—
96C2×#123—232no—
97C2×#123—21no—
98C2×#123—28no—
99C2×#123—28no—
100C2×C22⋊Q8C2312yes—
101C2×#121—232no—
102C2×#121—28no—
103C2×#121—24no—
104C2×#121—28no—
105C2×#121—28no—
106C2×#121—28no—
107C2×#121C2312yes—
108C2×#121C2318yes—
109C2×#120—21no—
110C4×#41—24no—
111C4×#41—24no—
112C2×C4×Q8C2312yes—
113C2×C4×D4—28no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—28no—
117C2×C4×D4—24no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2318yes—
120C2×#117—24no—
121C2×#117—24no—
122C2×#117—21no—
123C2×#117D412yes—
124C2×#117D412yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2314yes—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D412yes—
131C22×C4⋊C4C2312yes—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4D412yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#479—216no—
139#478—216no—
140#460—216no—
141#459—216no—
142#428—28no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#423—24no—
146#423—24no—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149#418—28no—
150#418—28no—
151#418—44no—
152#417—28no—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2318yes—
157C2×#102C2314yes—
158C2×#102C2318yes—
159#407—44no—
160C2×C22⋊C8C2318yes—
161#391C2312yes—
162#390—24no—
163#390C2312yes—
164#389—24no—
165#389C2314yes—
166#388—24no—
167#388—24no—
168#388—24no—
169#387—24no—
170#386—22no—
171#386—24no—
172#385—22no—
173#385—24no—
174#381—21no—
175#380C2314yes—
176#379—21no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182#376—21no—
183C43—21no—
184#340C2318yes—
185#340C2318yes—
186#336C2318yes—
187C25—22no—
188C25C2×D411yescharacteristic
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—22no—
194C2×#41#4114yes—
195C22×Q8C2411yescharacteristic
196C22×D4—21no—
197C22×D4—216no—
198C22×D4—22no—
199C22×D4—28no—
200C22×D4—216no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—88no—
210C22×D4—88no—
211C22×D4C2×D418yes—
212C22×D4C2414yes—
213C22×D4C2411yescharacteristic
214C23×C4—22no—
215C23×C4—216no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—28no—
224C23×C4—22no—
225C23×C4—88no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232AGL(1,ℤ/8)—432no—
233C2×SD16—28no—
234C2×SD16—28no—
235C2×SD16—44no—
236C2×D8—28no—
237C2×D8—28no—
238C2×D8—44no—
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—24no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247#127—28no—
248#127—28no—
249#127—28no—
250#127C2×D4116yes—
251#125—28no—
252#125—28no—
253#125—28no—
254#125C2×D4116yes—
255#123—264no—
256#123—264no—
257#123—28no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260#121—264no—
261#121—264no—
262#121—232no—
263#121—216no—
264#121—24no—
265#121—216no—
266#121—24no—
267#120—28no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×D4—232no—
271C4×D4—216no—
272C4×D4—216no—
273C4×D4—216no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—48no—
277#117—22no—
278#117—28no—
279#117—28no—
280#117—28no—
281#117—28no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—44no—
287#117#4114yes—
288C2×C4⋊C4—216no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—216no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D414yes—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334#107—28no—
335#106—28no—
336C4⋊C8—28no—
337C4⋊C8—28no—
338#103—28no—
339#102—28no—
340#102—28no—
341#102—28no—
342#102—216no—
343#102—48no—
344C22⋊C8—48no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349C24—22no—
350C24—28no—
351C24—21no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—44no—
356C24—48no—
357C24—48no—
358C24—44no—
359C24—44no—
360C24C22×D412yes—
361C24C22×D411yescharacteristic
362#41—28no—
363#41—28no—
364#41—44no—
365#41—44no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8C2×#4112yes—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—232no—
375C2×D4—28no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—24no—
387C2×D4—21no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—832no—
402C2×D4—832no—
403C2×D4C22×D418yes—
404C2×D4C2×#4112yes—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—232no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—832no—
440C22×C4#12018yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C22×C4C2511yescharacteristic
453SD16—416no—
454D8—416no—
455M16—48no—
456M16—48no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—44no—
476C4⋊C4—84no—
477C4⋊C4C22×D418yes—
478C4⋊C4C2×#4114yes—
479C22⋊C4—232no—
480C22⋊C4—232no—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C42—24no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—42no—
496C42—44no—
497C42—42no—
498C42—84no—
499C42#12014yes—
500C42#12014yes—
501C42C22×D414yes—
502C42C22×D414yes—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—416no—
512C23—48no—
513C23—416no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—44no—
518C23—416no—
519C23—41no—
520C23—44no—
521C23—41no—
522C23—44no—
523C23—84no—
524C23—84no—
525C23—816no—
526C23—816no—
527C23C2×#12012yes—
528C23#52512yes—
529C23#53712yes—
530C23C23×D411yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535D4—22no—
536D4—28no—
537D4—28no—
538D4—22no—
539D4—48no—
540D4—416no—
541D4—41no—
542D4—44no—
543D4—44no—
544D4—41no—
545D4—816no—
546D4—816no—
547D4—88no—
548D4—832no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—416no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—88no—
579C2×C4—832no—
580C2×C4C2×#12014yes—
581C2×C4C2×#12014yes—
582C2×C4C2×#12012yes—
583C2×C4#52512yes—
584C2×C4#53712yes—
585C2×C4#54614yes—
586C2×C4#54714yes—
587C2×C4#54814yes—
588C2×C4#55014yes—
589C2×C4C23×D411yescharacteristic
590C2×C4C23×D411yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×H(5,𝔽2)11yescharacteristic
594C2×C4C2×2−1+411yescharacteristic
595C8—44no—
596C8—44no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—48no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—48no—
607C22—44no—
608C22—48no—
609C22—44no—
610C22—88no—
611C22—832no—
612C22—88no—
613C22—82no—
614C22—82no—
615C22—88no—
616C22#240714yes—
617C22C22×#12011yescharacteristic
618C22C2×#52511yescharacteristic
619C22C2×#53711yescharacteristic
620C22#358514yes—
621C4—24no—
622C4—24no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—44no—
627C4—48no—
628C4—82no—
629C4—82no—
630C4#240714yes—
631C4C2×#54611yescharacteristic
632C4C2×#54711yescharacteristic
633C4C2×#54811yescharacteristic
634C4C2×#55011yescharacteristic
635C2—24no—
636C2—82no—
637C2—82no—
638C2—88no—
639C2C2×#240711yescharacteristic
640C2#6036712yes—
641C2#6056812yes—
642C2#6092012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#540C2semidirect
2. C2×#3080C2semidirect
3. C2×#3038C2semidirect
4. #34065C2semidirect
5. #24164C2semidirect
6. #24162C2semidirect
7. #23480C2semidirect
8. #23391C2semidirect
9. #21314C2semidirect
10. #21260C2semidirect
11. #20925C2semidirect
12. #20873C2semidirect
13. #20627C2semidirect
30. C2×#479C22semidirect
31. C2×#478C22semidirect
33. C2×#460C22semidirect
34. C2×#459C22semidirect
43. #2324C22semidirect
44. #2323C22semidirect
53. #2034C22semidirect
54. #2033C22semidirect
55. #2000C22semidirect
56. #1999C22semidirect
57. #1989C22semidirect
58. #1982C22semidirect
59. #1879C22semidirect
60. #1877C22semidirect
61. #1864C22semidirect
62. #1862C22semidirect
63. #1798C22semidirect
87. C2×#127D4semidirect
92. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5402
C2×#30802
C2×#30382
#340652
#241642
#241622
#234802
#233912
#213142
#212602
#209252
#208732
#206272