Finite Group Databasebeta

← #6487536

Subgroups — #6487536

Order 512.

4526 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic, complemented
2C2×#3206C211yesmaximal, characteristic, complemented
3C2×#3100C211yesmaximal, characteristic, complemented
4#35165C211yesmaximal, characteristic, complemented
5#32880C214yesmaximal, complemented
6#29595C214yesmaximal, complemented
7#25882C214yesmaximal, complemented
8#25754C214yesmaximal, complemented
9#24237C218yesmaximal, complemented
10#21804C211yesmaximal, characteristic, complemented
11#21786C211yesmaximal, characteristic, complemented
12#20732C211yesmaximal, characteristic, complemented
13C23×#41C2212yes—
14C2×#547—28no—
15C2×#547—22no—
16C2×#547—22no—
17C2×#547—28no—
18C2×#547C2214yes—
19C2×#547C2214yes—
20C2×#535—21no—
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#123—22no—
24C22×C22⋊Q8C2212yes—
25C22×#121—21no—
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4—21no—
29#3206—28no—
30C2×#492C2212yescomplemented
31C2×#487C2212yescomplemented
32#3100—28no—
33C2×#480C2212yes—
34C2×#474C2212yescomplemented
35C2×#461C2214yescomplemented
36C2×#459C2212yescomplemented
37C2×#449C22116yescomplemented
38C2×#448C2214yes—
39C2×#447C2212yes—
40C2×C4×D8C2214yescomplemented
41C2×#431C2211yescharacteristic
42C2×#431C2211yescharacteristic
43#2847C2214yes—
44#2617—22no—
45#2547C2212yescomplemented
46#2544—22no—
47#2541C2212yescomplemented
48#2537C2214yescomplemented
49#2535—22no—
50#2311—21no—
51#2285C2211yescharacteristic
52#2036C22116yescomplemented
53#2000C22116yescomplemented
54#1981C2214yescomplemented
55#1956C2214yescomplemented
56#1954C2214yescomplemented
57#1954C2214yescomplemented
58#1823C2212yes—
59#1802C2214yescomplemented
60#1788C22116yescomplemented
61#1787C2214yes—
62#1786C2212yes—
63#1779C2214yes—
64C4×#102C2214yescomplemented
65C22×#41—28no—
66C22×#41—28no—
67C22×#41—24no—
68C22×#41—21no—
69C22×#41—416no—
70C22×#41C2318yes—
71C23×Q8C2312yes—
72C23×D4—22no—
73C23×D4D411yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4—22no—
76C24×C4—21no—
77C24×C4C2312yes—
78C22×SD16C2318yes—
79C22×D8C2314yes—
80#547—232no—
81#547—232no—
82#547—216no—
83#547—216no—
84#547—216no—
85#547—216no—
86#535—48no—
87#532—24no—
88#532—24no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8D418yescomplemented
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127—22no—
97C2×#127C2311yescharacteristic
98C2×#124—22no—
99C2×#124—21no—
100C2×#123—232no—
101C2×#123—28no—
102C2×#123—28no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8C2318yes—
107C2×#121—216no—
108C2×#121—24no—
109C2×#121—24no—
110C2×#121—24no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×#121C2318yes—
114C2×#120—24no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4—24no—
120C2×C4×D4—21no—
121C2×C4×D4D412yes—
122C2×C4×D4D412yes—
123C2×C4×D4D418yescomplemented
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C22×C4⋊C4—216no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—24no—
135C22×C4⋊C4—24no—
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—22no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4D412yes—
141C22×C42C2311yescharacteristic
142#492—28no—
143#487—28no—
144#480—28no—
145#474—28no—
146#461—216no—
147#459—28no—
148#449—432no—
149#448—216no—
150#447—28no—
151C4×D8—216no—
152#431—24no—
153#431—24no—
154C2×#107C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#103C2318yes—
158C2×#103C23116yes—
159C2×#102C2318yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C23116yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167#397—24no—
168#394—28no—
169#392—24no—
170#391—22no—
171#390—22no—
172#386—24no—
173#386—28no—
174#384C2312yes—
175#382—21no—
176#380—21no—
177#380—24no—
178#380C2312yes—
179#380C2318yes—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2312yes—
182#340C2318yes—
183C25—21no—
184C25C2×D412yes—
185C2×#41—232no—
186C2×#41—216no—
187C2×#41—216no—
188C2×#41—232no—
189C2×#41—416no—
190C2×#41—432no—
191C2×#41—464no—
192C2×#41—464no—
193C2×#41—44no—
194C2×#41—44no—
195C2×#41—44no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—21no—
199C22×Q8—48no—
200C22×Q8C2×D418yes—
201C22×Q8C2414yes—
202C22×D4—28no—
203C22×D4—216no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—28no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—416no—
215C22×D4C2×D412yes—
216C22×D4C2×D412yes—
217C22×D4C2×D414yes—
218C22×D4C2412yes—
219C22×D4C2412yes—
220C23×C4—216no—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—28no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—48no—
231C23×C4—416no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D412yes—
235C23×C4C2×D414yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C2×SD16—216no—
241C2×SD16—416no—
242C2×D8—28no—
243C2×D8—216no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8C2×D418yes—
253C4⋊Q8C2×D418yes—
254#127—22no—
255#127—22no—
256#127—48no—
257#124—48no—
258#124—44no—
259#123—264no—
260#123—264no—
261C22⋊Q8—232no—
262C22⋊Q8—216no—
263C22⋊Q8—216no—
264#121—232no—
265#121—232no—
266#121—24no—
267#121—24no—
268#121—416no—
269#121—44no—
270#121—416no—
271#120—416no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4C2×D418yes—
286C4×D4C2×D418yes—
287#117—44no—
288C2×C4⋊C4—216no—
289C2×C4⋊C4—232no—
290C2×C4⋊C4—232no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D414yes—
323C2×C42—22no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—28no—
331C4⋊C8—24no—
332C4⋊C8—24no—
333#103—216no—
334#103—416no—
335#102—28no—
336#102—216no—
337#102—28no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340C22⋊C8—416no—
341C4×C8—24no—
342C4×C8—24no—
343#95—22no—
344#95—22no—
345#95—28no—
346C24—28no—
347C24—24no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—44no—
354C24—44no—
355C24—44no—
356C24—42no—
357C24—42no—
358C24C22×D414yes—
359C24C22×D411yescharacteristic
360#41—464no—
361#41—4128no—
362#41—432no—
363#41—432no—
364#41—816no—
365C2×Q8—28no—
366C2×Q8—24no—
367C2×Q8—216no—
368C2×Q8—416no—
369C2×Q8—48no—
370C2×Q8—416no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8C22×D418yes—
375C2×D4—28no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—232no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—28no—
391C2×D4—432no—
392C2×D4—464no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—416no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—416no—
402C2×D4—82no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4C22×D414yes—
406C2×D4H(5,𝔽2)18yes—
407C22×C4—24no—
408C22×C4—232no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—216no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—432no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—464no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—432no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4#12014yes—
441C22×C4#12014yes—
442C22×C4#13514yes—
443C22×C4AGL(1,ℤ/8)14yes—
444C22×C4C22×D414yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2511yescharacteristic
453SD16—816no—
454D8—416no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—28no—
458C2×C8—28no—
459C4⋊C4—232no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—28no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—82no—
474C4⋊C4C22×D414yes—
475C22⋊C4—232no—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—28no—
479C22⋊C4—28no—
480C22⋊C4—416no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—88no—
485C42—44no—
486C42#12014yes—
487C42#12014yes—
488C42C22×D412yes—
489C42C22×D412yes—
490C42C22×D412yes—
491C42C22×D412yes—
492C23—22no—
493C23—22no—
494C23—216no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—416no—
503C23—416no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—44no—
508C23—48no—
509C23—42no—
510C23—42no—
511C23—48no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—88no—
518C23—88no—
519C23C2×#12012yes—
520C23C2×#13512yes—
521C23C2×AGL(1,ℤ/8)12yes—
522C23C23×D411yescharacteristic
523Q8—216no—
524Q8—416no—
525Q8—48no—
526Q8—84no—
527D4—216no—
528D4—24no—
529D4—28no—
530D4—464no—
531D4—44no—
532D4—44no—
533D4—48no—
534D4—48no—
535D4—84no—
536D4—88no—
537D4—84no—
538D4—1616no—
539D4#53518yescomplemented
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—44no—
550C2×C4—416no—
551C2×C4—48no—
552C2×C4—416no—
553C2×C4—464no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—88no—
569C2×C4#449116yescomplemented
570C2×C4C2×#12012yes—
571C2×C4C2×#12012yes—
572C2×C4C2×#12012yes—
573C2×C4C2×#12012yes—
574C2×C4C2×#12012yes—
575C2×C4C2×#13512yes—
576C2×C4C2×#13512yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C23×D411yescharacteristic
580C2×C4C23×D411yescharacteristic
581C2×C4C2×H(5,𝔽2)14yes—
582C8—44no—
583C8—44no—
584C22—24no—
585C22—24no—
586C22—22no—
587C22—22no—
588C22—28no—
589C22—432no—
590C22—48no—
591C22—48no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—816no—
600C22—84no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22C2×#44914yes—
606C22C22×#12011yescharacteristic
607C22#348412yes—
608C22C22×#13511yescharacteristic
609C22C22×AGL(1,ℤ/8)11yescharacteristic
610C22#358612yes—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—44no—
615C4—48no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4C2×#44914yes—
620C4C2×#44914yes—
621C4C2×#53512yes—
622C4#348412yes—
623C2—24no—
624C2—48no—
625C2—84no—
626C2—162no—
627C2#3522211yescharacteristic
628C2C22×#44911yescharacteristic
629C2#6036811yescharacteristic
630C2#6156512yes—
631C2#6158512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#547C2semidirect
2. C2×#3206C2semidirect
3. C2×#3100C2semidirect
4. #35165C2semidirect
5. #32880C2semidirect
6. #29595C2semidirect
7. #25882C2semidirect
8. #25754C2semidirect
9. #24237C2semidirect
10. #21804C2semidirect
11. #21786C2semidirect
12. #20732C2semidirect
30. C2×#492C22semidirect
31. C2×#487C22semidirect
34. C2×#474C22semidirect
35. C2×#461C22semidirect
36. C2×#459C22semidirect
37. C2×#449C22semidirect
40. C2×C4×D8C22semidirect
45. #2547C22semidirect
47. #2541C22semidirect
48. #2537C22semidirect
52. #2036C22semidirect
53. #2000C22semidirect
54. #1981C22semidirect
55. #1956C22semidirect
56. #1954C22semidirect
57. #1954C22semidirect
59. #1802C22semidirect
60. #1788C22semidirect
64. C4×#102C22semidirect
92. C2×C4⋊Q8D4semidirect
123. C2×C4×D4D4semidirect
539. D4#535semidirect
569. C2×C4#449semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#32062
C2×#31002
#351652
#328802
#295952
#258822
#257542
#242372
#218042
#217862
#207322