Finite Group Databasebeta

← #6477609

Subgroups — #6477609

Order 512.

2126 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic, complemented
2C22×#481C211yesmaximal, characteristic
3#35019C211yesmaximal, characteristic
4C2×#2345C211yesmaximal, characteristic, complemented
5#33110C214yesmaximal, complemented
6#29761C214yesmaximal, complemented
7#24583C214yesmaximal, complemented
8#24579C214yesmaximal, complemented
9#24483C214yesmaximal, complemented
10#24479C214yesmaximal, complemented
11#21930C211yesmaximal, characteristic, complemented
12#21759C211yesmaximal, characteristic, complemented
13#21758C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#484—28no—
18C2×#484C2214yes—
19C2×#484C2212yescomplemented
20C2×#484C2212yescomplemented
21C2×#481—28no—
22C2×#481C2214yes—
23C2×#481C2212yes—
24C2×#481C2212yes—
25C22×#107C2212yescomplemented
26C22×#106C2212yes—
27C22×#103C2212yescomplemented
28C22×#102C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2850C2214yes—
32#2639—22no—
33#2591—22no—
34#2561—22no—
35#2504C2212yes—
36#2503C2212yes—
37#2436C2212yes—
38#2435C2212yes—
39#2345—22no—
40#2345—22no—
41#2345C2214yes—
42#2345C2214yescomplemented
43#2308—21no—
44#2308—21no—
45C2×#389C2212yescomplemented
46C2×#387—22no—
47C2×#380—21no—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4C2211yescharacteristic
52#2085C2218yescomplemented
53#2067C2218yescomplemented
54#2066C2218yescomplemented
55#2042C2218yescomplemented
56#1983C2218yescomplemented
57#1949C2218yescomplemented
58#1927C2218yescomplemented
59#1925C2218yescomplemented
60#1893C2214yes—
61#1892C2214yes—
62#1874C2214yes—
63#1872C2214yescomplemented
64#1850C2214yes—
65#1850C2214yescomplemented
66#1824C2212yescomplemented
67#1823C2212yes—
68#1797C2214yescomplemented
69#1787C2212yescomplemented
70#1786C2212yes—
71C23×Q8C2311yescharacteristic
72C23×D4—21no—
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C24×C4C2311yescharacteristic
76C23×C8C2312yes—
77C23×C8C2312yes—
78C2×#124C2314yes—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8C2312yes—
84C2×C22⋊Q8C2312yes—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121C2312yes—
90C2×#121C2312yes—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4D411yescharacteristic
109C22×C22⋊C4D411yescharacteristic
110C22×C42—22no—
111C22×C42C2312yes—
112#484—216no—
113#484—28no—
114#484—28no—
115#481—216no—
116#481—28no—
117#481—28no—
118C2×#107—28no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106—28no—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2318yes—
125C2×#103—28no—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2318yes—
129C2×#102—28no—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2318yes—
133C2×C22⋊C8—24no—
134C2×C22⋊C8—24no—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139#400—24no—
140#397—24no—
141#394—24no—
142#393—24no—
143#391C2314yes—
144#390C2314yes—
145#389—24no—
146#389C2318yes—
147#388—24no—
148#388—24no—
149#387—28no—
150#387—28no—
151#386—24no—
152#386—24no—
153#386—22no—
154#386—22no—
155#386—22no—
156#386—22no—
157#382C2314yes—
158#381—22no—
159#381—22no—
160#380—21no—
161#380—21no—
162#380—24no—
163#380—24no—
164#380C2312yes—
165#380C2312yes—
166#379—22no—
167#379—22no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4C2318yes—
170C4×C22⋊C4—24no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4D412yes—
173C4×C22⋊C4D412yes—
174C4×C22⋊C4C2312yes—
175C4×C22⋊C4C2312yes—
176C2×#95—22no—
177C2×#95—22no—
178C2×#95—21no—
179C2×#95D412yes—
180#340C2318yes—
181#340C2318yes—
182#336C2318yes—
183C25—21no—
184C25C2×D411yescharacteristic
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—44no—
189C22×Q8C2412yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4C2412yes—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—28no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—44no—
210C23×C4—84no—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D414yes—
217C23×C4#4114yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227#124—48no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—44no—
233C22⋊Q8—44no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121—24no—
238#121—44no—
239#121—44no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—84no—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—28no—
293C2×C42—44no—
294C2×C42—84no—
295C2×C42C2×D414yes—
296C2×C42C2414yes—
297#107—28no—
298#107—28no—
299#106—28no—
300#106—28no—
301C4⋊C8—48no—
302#103—28no—
303#103—28no—
304#103—48no—
305#102—28no—
306#102—28no—
307#102—48no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—24no—
312#95—28no—
313#95—28no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—82no—
329C24—82no—
330C24#12114yes—
331C24#13514yes—
332C24#13714yes—
333C24C22×D411yescharacteristic
334C24C22×D411yescharacteristic
335C24C22×D411yescharacteristic
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—28no—
339C2×Q8—48no—
340C2×Q8—42no—
341C2×Q8—82no—
342C2×Q8—84no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—28no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—84no—
355C2×D4—82no—
356C2×D4—816no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—82no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—82no—
395C22×C4—816no—
396C22×C4#12114yes—
397C22×C4#12114yes—
398C22×C4#12114yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2511yescharacteristic
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—28no—
414C2×C8—28no—
415C2×C8—48no—
416C2×C8—48no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—28no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—88no—
434C4⋊C4C22×D414yes—
435C4⋊C4C22×D414yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C42—28no—
451C42—44no—
452C42—88no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—44no—
460C23—44no—
461C23—48no—
462C23—48no—
463C23—44no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—164no—
472C23—164no—
473C23#464116yes—
474C23C2×#12112yes—
475C23C2×#12112yes—
476C23C2×#12112yes—
477C23C2×#13512yes—
478C23C2×#13512yes—
479C23C2×#13512yes—
480C23C2×#13712yes—
481C23C2×#13712yes—
482C23C2×#13712yes—
483C23C23×D411yescharacteristic
484Q8—44no—
485Q8—44no—
486Q8—84no—
487D4—44no—
488D4—44no—
489D4—84no—
490D4—88no—
491D4—88no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—84no—
528C2×C4—88no—
529C2×C4—88no—
530C2×C4—88no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4C2×#12112yes—
535C2×C4C2×#12112yes—
536C2×C4C2×#12112yes—
537C2×C4C23×D411yescharacteristic
538C2×C4C22×#4111yescharacteristic
539C2×C4#58514yes—
540C8—44no—
541C8—44no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—416no—
555C22—84no—
556C22—82no—
557C22—84no—
558C22—82no—
559C22—82no—
560C22—168no—
561C22C2×#46414yes—
562C22C2×#46414yes—
563C22C2×#46414yes—
564C22C22×#12111yescharacteristic
565C22#348512yes—
566C22C22×#13511yescharacteristic
567C22C22×#13711yescharacteristic
568C22#358512yes—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—84no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4#344912yes—
581C4#349112yes—
582C2—24no—
583C2—24no—
584C2—24no—
585C2—162no—
586C2#3508911yescharacteristic
587C2C22×#46411yescharacteristic
588C2#6042311yescharacteristic
589C2#6169912yes—
590C2#6170912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#484C2semidirect
4. C2×#2345C2semidirect
5. #33110C2semidirect
6. #29761C2semidirect
7. #24583C2semidirect
8. #24579C2semidirect
9. #24483C2semidirect
10. #24479C2semidirect
11. #21930C2semidirect
12. #21759C2semidirect
19. C2×#484C22semidirect
20. C2×#484C22semidirect
25. C22×#107C22semidirect
27. C22×#103C22semidirect
42. #2345C22semidirect
45. C2×#389C22semidirect
49. C2×C4×C4⋊C4C22semidirect
52. #2085C22semidirect
53. #2067C22semidirect
54. #2066C22semidirect
55. #2042C22semidirect
56. #1983C22semidirect
57. #1949C22semidirect
58. #1927C22semidirect
59. #1925C22semidirect
63. #1872C22semidirect
65. #1850C22semidirect
66. #1824C22semidirect
68. #1797C22semidirect
69. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4812
#350192
C2×#23452
#331102
#297612
#245832
#245792
#244832
#244792
#219302
#217592
#217582