Finite Group Databasebeta

← #6629247

Subgroups — #6629247

Order 512.

4500 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic, complemented
2C22×#480C211yesmaximal, characteristic, complemented
3C2×#3006C211yesmaximal, characteristic
4#35262C211yesmaximal, characteristic, complemented
5#33012C218yesmaximal, complemented
6#33003C214yesmaximal, complemented
7#33002C214yesmaximal, complemented
8#20932C214yesmaximal, complemented
9#20881C214yesmaximal, complemented
10#20700C211yesmaximal, characteristic, complemented
11#20685C211yesmaximal, characteristic, complemented
12#20597C211yesmaximal, characteristic, complemented
13C23×#41C2211yescharacteristic
14C22×AGL(1,ℤ/8)C2212yes—
15C22×#121C2211yescharacteristic
16C2×#520—28no—
17C2×#520—22no—
18C2×#520—22no—
19C2×#520—28no—
20C2×#520C2214yes—
21C2×#520C2214yescomplemented
22C22×C4×Q8C2212yescomplemented
23C22×C4×D4—22no—
24C22×C4×D4C2212yescomplemented
25C22×#117—21no—
26C22×#117C2211yescharacteristic
27C23×C4⋊C4—21no—
28C23×C4⋊C4C2211yescharacteristic
29C2×#480—28no—
30C2×#480C2214yes—
31C2×#480C2214yescomplemented
32#3006—28no—
33C2×#449C2214yes—
34C2×#448C2212yes—
35C2×#447C2212yes—
36C22×#107C2212yescomplemented
37C2×#417C2212yes—
38C2×#416C2214yes—
39C22×#102C2212yescomplemented
40C2×#407C2211yescharacteristic
41C22×C22⋊C8C2211yescharacteristic
42#2854C2214yes—
43#2624C2214yescomplemented
44#2504—21no—
45#2501—21no—
46#2500—22no—
47#2482—22no—
48#2405—22no—
49#2317C2212yescomplemented
50#2309C2212yescomplemented
51#2280C2211yescharacteristic
52#1880C22116yescomplemented
53#1879C22116yescomplemented
54#1863C22116yescomplemented
55#1862C22116yescomplemented
56#1824C2214yescomplemented
57#1823C2214yescomplemented
58#1818C2214yescomplemented
59#1817C2214yescomplemented
60#1788C2214yescomplemented
61#1787C2212yescomplemented
62#1786C2212yescomplemented
63#1736C2214yescomplemented
64#1729C2214yescomplemented
65C22×#41—216no—
66C22×#41—24no—
67C22×#41—21no—
68C22×#41—21no—
69C22×#41D418yes—
70C22×#41C2312yes—
71C22×#41C2312yes—
72C23×Q8C2311yescharacteristic
73C23×D4—21no—
74C23×D4—21no—
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4—21no—
78C24×C4—21no—
79C24×C4—22no—
80C24×C4C2311yescharacteristic
81C2×AGL(1,ℤ/8)—28no—
82C2×AGL(1,ℤ/8)—216no—
83C22×SD16C2314yes—
84C22×D8C2314yes—
85C22×M16C2312yes—
86C23×C8C2312yes—
87C2×#123—24no—
88C2×#121—21no—
89C2×#121—21no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121C2312yes—
93C2×#121C2312yes—
94#520—232no—
95#520—232no—
96#520—216no—
97#520—216no—
98#520—216no—
99#520—216no—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—24no—
103C2×C4×Q8C2318yes—
104C2×C4×D4—232no—
105C2×C4×D4—28no—
106C2×C4×D4—28no—
107C2×C4×D4—28no—
108C2×C4×D4—28no—
109C2×C4×D4—24no—
110C2×C4×D4C2318yes—
111C2×#117—216no—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117—24no—
115C2×#117—24no—
116C2×#117D412yes—
117C2×#117D412yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—216no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—24no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D418yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4D412yes—
139C22×C42—22no—
140C22×C42C2312yes—
141#480—216no—
142#480—216no—
143#449—216no—
144#448—28no—
145#447—28no—
146C2×#107—28no—
147C2×#107C2318yes—
148#417—28no—
149#416—216no—
150C2×#103C2314yes—
151C2×#103C2318yes—
152C2×#102—28no—
153C2×#102C2318yes—
154C2×#102C2314yes—
155C2×#102C2318yes—
156#407—24no—
157C2×C22⋊C8—24no—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162#399—24no—
163#397—24no—
164#394—28no—
165#390—22no—
166#390—22no—
167#390C2318yes—
168#389C2318yes—
169#388—28no—
170#386—24no—
171#385—24no—
172#384—22no—
173#382—22no—
174#381—21no—
175#380C2312yes—
176#380C2312yes—
177#379—21no—
178#379—24no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181#340C23116yes—
182#336C23116yes—
183C25—21no—
184C25—21no—
185C25C2×D411yescharacteristic
186C2×#41—216no—
187C2×#41—264no—
188C2×#41—216no—
189C2×#41—264no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—28no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—28no—
197C2×#41C2×D418yes—
198C2×#41C2×D418yes—
199C2×#41#41116yes—
200C22×Q8—28no—
201C22×Q8—22no—
202C22×Q8—21no—
203C22×Q8C2×D414yes—
204C22×Q8C2412yes—
205C22×D4—216no—
206C22×D4—28no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—28no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—28no—
218C22×D4—44no—
219C22×D4—84no—
220C22×D4C2×D414yes—
221C22×D4C2412yes—
222C22×D4C2412yes—
223C23×C4—216no—
224C23×C4—21no—
225C23×C4—28no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—28no—
229C23×C4—22no—
230C23×C4—216no—
231C23×C4—28no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—28no—
235C23×C4—21no—
236C23×C4—88no—
237C23×C4—84no—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248AGL(1,ℤ/8)—432no—
249C2×SD16—28no—
250C2×SD16—216no—
251C2×D8—28no—
252C2×D8—216no—
253C2×M16—24no—
254C2×M16—28no—
255C22×C8—24no—
256C22×C8—28no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259#123—88no—
260#121—24no—
261#121—24no—
262#121—24no—
263#121—24no—
264#121—44no—
265#121—44no—
266C4×Q8—232no—
267C4×Q8—216no—
268C4×Q8—216no—
269C4×D4—264no—
270C4×D4—264no—
271C4×D4—232no—
272C4×D4—216no—
273C4×D4—216no—
274#117—232no—
275#117—232no—
276#117—28no—
277#117—28no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—28no—
283C2×C4⋊C4—216no—
284C2×C4⋊C4—232no—
285C2×C4⋊C4—232no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D418yes—
307C2×C4⋊C4C2×D418yes—
308C2×C4⋊C4C2×D418yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2414yes—
312C2×C22⋊C4—216no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×D414yes—
325C2×C42—216no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—28no—
329C2×C42—22no—
330C2×C42C2×D414yes—
331C2×C42C2414yes—
332#107—216no—
333#103—28no—
334#103—216no—
335#102—28no—
336#102—216no—
337#102—216no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342#95—22no—
343#95—22no—
344#95—28no—
345C24—28no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—42no—
352C24—44no—
353C24—44no—
354C24—42no—
355C24—44no—
356C24—82no—
357C24—82no—
358C24#12014yes—
359C24C22×D411yescharacteristic
360C24C22×D411yescharacteristic
361C24C22×D411yescharacteristic
362#41—264no—
363#41—232no—
364#41—264no—
365#41—216no—
366#41—216no—
367#41—216no—
368#41—232no—
369#41#123132yescomplemented
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8C22×D414yes—
377C2×Q8C2×#4118yes—
378C2×D4—264no—
379C2×D4—216no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—42no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—84no—
400C2×D4—816no—
401C2×D4C22×D414yes—
402C2×D4C2×#4118yes—
403C22×C4—28no—
404C22×C4—264no—
405C22×C4—216no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—216no—
412C22×C4—24no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—232no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—816no—
432C22×C4—816no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—82no—
436C22×C4—832no—
437C22×C4#12014yes—
438C22×C4#12014yes—
439C22×C4#12014yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4118yes—
452C22×C4C2511yescharacteristic
453SD16—416no—
454D8—416no—
455M16—48no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—28no—
459C2×C8—28no—
460C2×C8—48no—
461C4⋊C4—232no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—28no—
473C4⋊C4—42no—
474C4⋊C4—88no—
475C4⋊C4C22×D418yes—
476C22⋊C4—232no—
477C22⋊C4—28no—
478C22⋊C4—28no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—82no—
486C42—232no—
487C42—28no—
488C42—24no—
489C42—24no—
490C42—44no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—216no—
496C23—22no—
497C23—28no—
498C23—24no—
499C23—22no—
500C23—28no—
501C23—28no—
502C23—22no—
503C23—44no—
504C23—416no—
505C23—432no—
506C23—48no—
507C23—44no—
508C23—48no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23—168no—
518C23C2×#12012yes—
519C23C2×#12012yes—
520C23C2×#12012yes—
521C23#52514yes—
522C23#57614yes—
523C23C23×D411yescharacteristic
524Q8—216no—
525Q8—24no—
526Q8—28no—
527Q8C2×#12318yes—
528D4—232no—
529D4—216no—
530D4—232no—
531D4—24no—
532D4—28no—
533D4—44no—
534D4—44no—
535D4—88no—
536D4—88no—
537D4—84no—
538D4C2×#12318yes—
539C2×C4—216no—
540C2×C4—232no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—232no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—416no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—88no—
568C2×C4—88no—
569C2×C4—832no—
570C2×C4—84no—
571C2×C4C2×#12012yes—
572C2×C4C2×#12012yes—
573C2×C4C2×#12012yes—
574C2×C4C2×#12318yes—
575C2×C4#52514yes—
576C2×C4#52518yes—
577C2×C4C23×D411yescharacteristic
578C2×C4C23×D411yescharacteristic
579C2×C4C22×#4114yes—
580C8—44no—
581C8—44no—
582C22—22no—
583C22—28no—
584C22—28no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—28no—
590C22—22no—
591C22—22no—
592C22—416no—
593C22—432no—
594C22—44no—
595C22—416no—
596C22—44no—
597C22—44no—
598C22—82no—
599C22—88no—
600C22—82no—
601C22—82no—
602C22—168no—
603C22#284914yes—
604C22#300814yes—
605C22C22×#12011yescharacteristic
606C22C2×#52512yes—
607C22C2×#57612yes—
608C22#358612yes—
609C4—22no—
610C4—28no—
611C4—22no—
612C4—44no—
613C4—48no—
614C4—44no—
615C4—84no—
616C4—82no—
617C4#284914yes—
618C4C22×#12312yes—
619C4C2×#52512yes—
620C2—24no—
621C2—24no—
622C2—42no—
623C2—48no—
624C2—162no—
625C2C2×#284911yescharacteristic
626C2C2×#300811yescharacteristic
627C2#6036811yescharacteristic
628C2#6058414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3006C2
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×#520C2semidirect
2. C22×#480C2semidirect
4. #35262C2semidirect
5. #33012C2semidirect
6. #33003C2semidirect
7. #33002C2semidirect
8. #20932C2semidirect
9. #20881C2semidirect
10. #20700C2semidirect
11. #20685C2semidirect
12. #20597C2semidirect
21. C2×#520C22semidirect
22. C22×C4×Q8C22semidirect
24. C22×C4×D4C22semidirect
31. C2×#480C22semidirect
36. C22×#107C22semidirect
39. C22×#102C22semidirect
43. #2624C22semidirect
49. #2317C22semidirect
50. #2309C22semidirect
52. #1880C22semidirect
53. #1879C22semidirect
54. #1863C22semidirect
55. #1862C22semidirect
56. #1824C22semidirect
57. #1823C22semidirect
58. #1818C22semidirect
59. #1817C22semidirect
60. #1788C22semidirect
61. #1787C22semidirect
62. #1786C22semidirect
63. #1736C22semidirect
64. #1729C22semidirect
369. #41#123semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C22×#4802
C2×#30062
#352622
#330122
#330032
#330022
#209322
#208812
#207002
#206852
#205972