Finite Group Databasebeta

← #6526690

Subgroups — #6526690

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic, complemented
2C2×#3101C211yesmaximal, characteristic
3C2×#2563C211yesmaximal, characteristic
4#34213C211yesmaximal, characteristic, complemented
5#28123C214yesmaximal
6#27653C214yesmaximal, complemented
7#25450C214yesmaximal, complemented
8#25319C214yesmaximal, complemented
9#24814C214yesmaximal, complemented
10#24679C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21834C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#3230—28no—
18C2×#500C2212yescomplemented
19C2×#487C2212yescomplemented
20#3101—28no—
21C2×#484C2214yes—
22C2×#483C2212yes—
23C2×#477C2214yescomplemented
24C2×#476C2212yes—
25C2×#461C2212yes—
26C2×#450C2212yes—
27C2×C4×SD16C2214yes—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30#2563—22no—
31#2563—22no—
32#2563C2214yes—
33#2563C2214yes—
34C2×#393—22no—
35#2370—22no—
36#2369C2212yescomplemented
37#2365C2212yescomplemented
38#2355—22no—
39#2355C2212yescomplemented
40Q8×C4⋊C4C2212yescomplemented
41#2329—22no—
42#2317—21no—
43C4×C22⋊Q8C2214yes—
44C2×#390C2212yes—
45#2285C2211yescharacteristic
46C2×#382C2212yes—
47C2×#380—21no—
48C2×#380—21no—
49#2274—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2085C2218yescomplemented
52#2074C2218yescomplemented
53#2067C2218yes—
54#2055C2218yes—
55#1990C2212yescomplemented
56#1990C2212yescomplemented
57#1987C2214yescomplemented
58#1949C2218yescomplemented
59#1939C2214yescomplemented
60#1939C2214yescomplemented
61#1925C2218yes—
62#1880C2218yes—
63#1824C2214yes—
64#1823C2212yes—
65#1805C2218yescomplemented
66#1802C2212yes—
67#1802C2212yes—
68#1797C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C23×Q8—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C22×SD16C2314yes—
75#533—24no—
76#533D418yes—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8C2312yes—
88C2×C4×Q8—21no—
89C2×C4×Q8C2312yes—
90C2×C4×Q8C2314yes—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C2×#117—21no—
98C2×#117—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109#500—28no—
110#487—28no—
111#484—216no—
112#483—28no—
113#477—216no—
114#476—28no—
115#461—28no—
116#450—28no—
117C4×SD16—216no—
118#431—24no—
119#431—24no—
120C2×#107C2318yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2318yes—
125C2×#103C2318yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2312yes—
134#393—28no—
135#393—28no—
136#391—24no—
137#391C2314yes—
138#390—24no—
139#390C2318yes—
140#389—22no—
141#389—22no—
142#389—24no—
143#389C2314yes—
144#387—24no—
145#387—24no—
146#387—24no—
147#385—24no—
148#384—22no—
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#382—22no—
152#382—24no—
153#382—21no—
154#382C2318yes—
155#380—21no—
156#380—24no—
157#380—24no—
158#380—24no—
159#380—24no—
160#380C2312yes—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4D412yes—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C22⋊C4—24no—
174C2×#95—22no—
175C2×#95—22no—
176C2×#95D412yes—
177#340C2318yes—
178#340C2314yes—
179#340C2314yes—
180#340C2318yes—
181#336C2318yes—
182C25—21no—
183C22×Q8—21no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—44no—
187C22×Q8—82no—
188C22×Q8—82no—
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—24no—
214C4⋊Q8—24no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—44no—
220C4⋊Q8—44no—
221C4⋊Q8C2×D414yes—
222C4⋊Q8C2×D414yes—
223#125—28no—
224C22⋊Q8—28no—
225C22⋊Q8—24no—
226C22⋊Q8C2×D418yes—
227C4×Q8—28no—
228C4×Q8—24no—
229C4×Q8—48no—
230C4×Q8C2×D418yes—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—44no—
236C4×D4—44no—
237#117—28no—
238#117—44no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—84no—
278C2×C22⋊C4—84no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42C2×D414yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D414yes—
288C2×C42#4114yes—
289C2×C42C2414yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292#107—216no—
293#106—28no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—48no—
297#103—28no—
298#103—216no—
299#102—28no—
300#102—28no—
301#102—44no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C4×C8—24no—
305C4×C8—24no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95C2×D414yes—
315#95C2×D414yes—
316C24—24no—
317C24—22no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24C22×D411yescharacteristic
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—81no—
332C2×Q8—81no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8—82no—
337C2×Q8—816no—
338C2×Q8C22×D414yes—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D42−1+414yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—816no—
379C22×C4#12114yes—
380C22×C4#12114yes—
381C22×C4#13514yes—
382C22×C4#13714yes—
383C22×C4C22×D414yes—
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4C2511yescharacteristic
393SD16—416no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4C22×D414yes—
427C22⋊C4—28no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C42—28no—
434C42—28no—
435C42—44no—
436C42—44no—
437C42—84no—
438C42—84no—
439C42#12114yes—
440C42#12114yes—
441C42C22×D412yes—
442C42C22×D412yes—
443C42C2×#4112yes—
444C42C2×#4112yes—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—24no—
450C23—48no—
451C23—48no—
452C23—48no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—84no—
460C23—84no—
461C23—84no—
462C23—84no—
463C23C2×#12112yes—
464C23C2×#13512yes—
465C23C2×#13712yes—
466C23C23×D411yescharacteristic
467Q8—28no—
468Q8—44no—
469Q8—84no—
470Q8—88no—
471Q8—88no—
472D4—24no—
473D4—24no—
474D4—48no—
475D4—42no—
476D4—42no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—88no—
513C2×C4—84no—
514C2×C4—88no—
515C2×C4—88no—
516C2×C4#464116yes—
517C2×C4C2×#12112yes—
518C2×C4C2×#12112yes—
519C2×C4C2×#12112yes—
520C2×C4C2×#12112yes—
521C2×C4C2×#12112yes—
522C2×C4C2×#13512yes—
523C2×C4C2×#13512yes—
524C2×C4C2×#13712yes—
525C2×C4C2×#13712yes—
526C2×C4C23×D411yescharacteristic
527C2×C4C22×#4111yescharacteristic
528C2×C4C2×2−1+412yes—
529C2×C4#58512yes—
530C8—44no—
531C8—44no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—816no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C2×#46414yes—
547C22C22×#12111yescharacteristic
548C22#348511yescharacteristic
549C22#348611yescharacteristic
550C22C22×#13511yescharacteristic
551C22C22×#13711yescharacteristic
552C22#358511yescharacteristic
553C22#358511yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—44no—
558C4—44no—
559C4—44no—
560C4—44no—
561C4—82no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4C2×#46414yes—
566C4C2×#46414yes—
567C4#348912yes—
568C4#349512yes—
569C2—24no—
570C2—84no—
571C2#3563311yescharacteristic
572C2C22×#46411yescharacteristic
573C2#6042611yescharacteristic
574C2#6170011yescharacteristic
575C2#6172111yescharacteristic
576C2#6181911yescharacteristic
577C2#6182611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3230C2semidirect
4. #34213C2semidirect
6. #27653C2semidirect
7. #25450C2semidirect
8. #25319C2semidirect
9. #24814C2semidirect
10. #24679C2semidirect
11. #21951C2semidirect
12. #21834C2semidirect
18. C2×#500C22semidirect
19. C2×#487C22semidirect
23. C2×#477C22semidirect
36. #2369C22semidirect
37. #2365C22semidirect
39. #2355C22semidirect
40. Q8×C4⋊C4C22semidirect
51. #2085C22semidirect
52. #2074C22semidirect
55. #1990C22semidirect
56. #1990C22semidirect
57. #1987C22semidirect
58. #1949C22semidirect
59. #1939C22semidirect
60. #1939C22semidirect
65. #1805C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#31012
C2×#25632
#342132
#281232
#276532
#254502
#253192
#248142
#246792
#219512
#218342
#207322