Finite Group Databasebeta

← #6529835

Subgroups — #6529835

Order 512.

2062 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic, complemented
3#35179C211yesmaximal, characteristic, complemented
4C2×#2538C211yesmaximal, characteristic
5#28272C214yesmaximal, complemented
6#28195C214yesmaximal, complemented
7#25477C214yesmaximal, complemented
8#25301C214yesmaximal, complemented
9#24609C214yesmaximal, complemented
10#24507C214yesmaximal
11#21982C211yesmaximal, characteristic, complemented
12#21860C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C22⋊Q8—21no—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3101—28no—
19C2×#483C2212yes—
20C2×#476C2212yes—
21C2×#461C2212yes—
22C2×#450C2212yes—
23#2983—28no—
24C2×#446C2214yescomplemented
25C2×#446C2214yescomplemented
26C2×C4×SD16C2214yes—
27C2×#432C2214yes—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30#2855C2214yes—
31#2627—22no—
32#2582C2212yescomplemented
33#2582C2212yescomplemented
34#2581—22no—
35#2568C2212yescomplemented
36#2567C2212yescomplemented
37#2565—22no—
38#2538—22no—
39#2538—22no—
40#2538C2214yes—
41#2538C2214yes—
42C2×#396—22no—
43#2317—21no—
44#2310—21no—
45C2×#390C2212yes—
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50#2088C2218yescomplemented
51#2068C2218yescomplemented
52#2067C2218yes—
53#2055C2218yes—
54#1990C2212yescomplemented
55#1973C2218yes—
56#1956C2212yescomplemented
57#1947C2214yescomplemented
58#1943C2218yescomplemented
59#1936C2218yes—
60#1935C2218yescomplemented
61#1872C2214yes—
62#1852C2214yes—
63#1845C2212yescomplemented
64#1834C2212yescomplemented
65#1823C2212yes—
66#1803C2214yescomplemented
67#1802C2212yes—
68#1802C2212yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C23×Q8D411yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C22×SD16C2314yes—
75#533—24no—
76#533—24no—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×#125—21no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—48no—
84C2×C22⋊Q8C2312yes—
85C2×C4×Q8—24no—
86C2×C4×Q8—21no—
87C2×C4×Q8—24no—
88C2×C4×Q8D412yes—
89C2×C4×Q8D412yes—
90C2×C4×Q8C2312yes—
91C2×C4×Q8C2312yes—
92C2×C4×Q8C2312yes—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4C2312yes—
98C2×C4×D4C2312yes—
99C2×#117—21no—
100C2×#117—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D412yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#483—28no—
114#476—28no—
115#461—28no—
116#450—28no—
117#446—216no—
118#446—216no—
119C4×SD16—216no—
120#432—216no—
121#431—24no—
122#431D418yes—
123C2×#107C2318yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2318yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2318yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2318yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#398—24no—
140#398—24no—
141#396—28no—
142#396—28no—
143#395—24no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391C2314yes—
149#391C2314yes—
150#390—24no—
151#390C2318yes—
152#389—22no—
153#389—22no—
154#389—24no—
155#389C2314yes—
156#389C2314yes—
157#389C2318yes—
158#387—24no—
159#387—24no—
160#385—24no—
161#384—22no—
162#384—21no—
163#384—21no—
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#384C2312yes—
167#384C2312yes—
168#382—22no—
169#382—21no—
170#381—24no—
171#380—21no—
172#380—24no—
173#380—24no—
174#380C2312yes—
175C4×C4⋊C4—21no—
176C4×C4⋊C4C2312yes—
177C2×#95—22no—
178#340C2318yes—
179#340C2314yes—
180#340C2314yes—
181#340C2318yes—
182C25—21no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—21no—
187C22×Q8C2×D412yes—
188C22×Q8C2×D412yes—
189C22×Q8C2412yes—
190C22×D4—24no—
191C22×D4—21no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—42no—
195C22×D4C2412yes—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2411yescharacteristic
208C2×SD16—28no—
209C2×SD16—28no—
210C2×SD16—44no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217#125—44no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—416no—
221C22⋊Q8—416no—
222C22⋊Q8—44no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—44no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—44no—
237C4×D4—44no—
238#117—44no—
239#117—44no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—84no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42C2×D414yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D414yes—
292C2×C42#4114yes—
293C2×C42C2414yes—
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—48no—
297#106—28no—
298C4⋊C8—216no—
299C4⋊C8—24no—
300C4⋊C8C2×D418yes—
301#103—28no—
302#103—48no—
303#102—28no—
304#102—28no—
305#102—44no—
306C22⋊C8—24no—
307C22⋊C8C2×D418yes—
308#97—216no—
309C4×C8—24no—
310C4×C8C2×D418yes—
311#95—28no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320C24—24no—
321C24—22no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24C22×D411yescharacteristic
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—42no—
340C2×Q8—416no—
341C2×Q8—81no—
342C2×Q8—81no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D42−1+414yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—816no—
383C22×C4#12114yes—
384C22×C4#12114yes—
385C22×C4AGL(1,ℤ/8)14yes—
386C22×C4#13714yes—
387C22×C4C22×D414yes—
388C22×C4C22×D414yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2511yescharacteristic
397SD16—416no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8C22×D418yes—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—28no—
415C4⋊C4—44no—
416C4⋊C4—416no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C22⋊C4—416no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—88no—
439C42—28no—
440C42—24no—
441C42—24no—
442C42—44no—
443C42—44no—
444C42#12114yes—
445C42#12114yes—
446C42C22×D412yes—
447C42C22×D412yes—
448C42C2×#4112yes—
449C42C2×#4112yes—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—48no—
456C23—48no—
457C23—48no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23C2×#12112yes—
469C23C2×AGL(1,ℤ/8)12yes—
470C23C2×#13712yes—
471C23C23×D411yescharacteristic
472Q8—48no—
473Q8—48no—
474Q8—44no—
475Q8—44no—
476Q8—84no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—42no—
481D4—42no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—88no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4#468116yes—
522C2×C4C2×#12112yes—
523C2×C4C2×#12112yes—
524C2×C4C2×#12112yes—
525C2×C4C2×#12112yes—
526C2×C4C2×#12112yes—
527C2×C4C2×AGL(1,ℤ/8)12yes—
528C2×C4C2×AGL(1,ℤ/8)12yes—
529C2×C4C2×#13712yes—
530C2×C4C2×#13712yes—
531C2×C4C23×D411yescharacteristic
532C2×C4C22×#4111yescharacteristic
533C2×C4C2×2−1+412yes—
534C2×C4#58512yes—
535C8—28no—
536C8—44no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—816no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C2×#46814yes—
552C22C22×#12111yescharacteristic
553C22#348611yescharacteristic
554C22#348711yescharacteristic
555C22C22×AGL(1,ℤ/8)11yescharacteristic
556C22C22×#13711yescharacteristic
557C22#358511yescharacteristic
558C22#358611yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—44no—
563C4—44no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4C2×#46814yes—
571C4C2×#46814yes—
572C4#348912yes—
573C4#349512yes—
574C2—24no—
575C2—84no—
576C2#3566111yescharacteristic
577C2C22×#46811yescharacteristic
578C2#6045611yescharacteristic
579C2#6182611yescharacteristic
580C2#6182711yescharacteristic
581C2#6188911yescharacteristic
582C2#6189011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2983C2semidirect
3. #35179C2semidirect
5. #28272C2semidirect
6. #28195C2semidirect
7. #25477C2semidirect
8. #25301C2semidirect
9. #24609C2semidirect
11. #21982C2semidirect
12. #21860C2semidirect
24. C2×#446C22semidirect
25. C2×#446C22semidirect
32. #2582C22semidirect
33. #2582C22semidirect
35. #2568C22semidirect
36. #2567C22semidirect
50. #2088C22semidirect
51. #2068C22semidirect
54. #1990C22semidirect
56. #1956C22semidirect
57. #1947C22semidirect
58. #1943C22semidirect
60. #1935C22semidirect
63. #1845C22semidirect
64. #1834C22semidirect
66. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#29832
#351792
C2×#25382
#282722
#281952
#254772
#253012
#246092
#245072
#219822
#218602
#207322