Finite Group Databasebeta

← #6472879

Subgroups — #6472879

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic, complemented
2#37451C211yesmaximal, characteristic
3#33054C214yesmaximal, complemented
4#30195C214yesmaximal, complemented
5#27721C214yesmaximal, complemented
6#27719C214yesmaximal, complemented
7#27285C214yesmaximal, complemented
8#27281C214yesmaximal, complemented
9#21768C211yesmaximal, characteristic, complemented
10#21767C211yesmaximal, characteristic
11#20973C211yesmaximal, characteristic
12#20972C211yesmaximal, characteristic, complemented
13#20386C211yesmaximal, characteristic, complemented
14C23×M16C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C2×#489C2214yescomplemented
18C2×#419C2212yescomplemented
19C2×#418C2212yes—
20C2×#407C2211yescharacteristic
21C2×#405C2212yescomplemented
22#2842—21no—
23#2840—21no—
24#2816—22no—
25#2665—22no—
26#2665—22no—
27#2628C2214yescomplemented
28#2626—22no—
29#2620C2214yes—
30#2618—22no—
31#2607—22no—
32#2502C2212yes—
33#2501C2212yes—
34#2439C2214yes—
35#2439C2214yescomplemented
36#2436C2212yes—
37#2435C2212yes—
38#2405—21no—
39#2405—21no—
40#2280C2211yescharacteristic
41#2280C2211yescharacteristic
42#2280C2211yescharacteristic
43#2100C2214yescomplemented
44#2099C2214yescomplemented
45#2074C2214yescomplemented
46#2073C2214yescomplemented
47#2042C2214yescomplemented
48#2042C2214yescomplemented
49#2024C2214yescomplemented
50#2023C2214yescomplemented
51#1990C2214yescomplemented
52#1958C2214yescomplemented
53#1936C2214yescomplemented
54#1935C2214yescomplemented
55#1934C2214yescomplemented
56#1933C2214yescomplemented
57#1895C2214yes—
58#1895C2214yes—
59#1894C2214yes—
60#1894C2214yescomplemented
61#1893C2212yes—
62#1893C2212yes—
63#1892C2212yescomplemented
64#1892C2212yescomplemented
65#1875C2214yes—
66#1873C2214yes—
67#1836C2214yescomplemented
68#1824C2212yescomplemented
69#1823C2212yes—
70#1787C2212yescomplemented
71#1786C2212yes—
72#1753C2214yes—
73#1753C2214yescomplemented
74#1736C2212yes—
75#1736C2212yescomplemented
76#1729C2212yescomplemented
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×M16—24no—
82C22×M16—44no—
83C22×M16C2312yes—
84C22×M16C2312yes—
85C22×M16C2312yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8C2314yes—
89C2×#124C2314yes—
90C2×#123—22no—
91C2×#123—22no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121—24no—
101C2×#121—24no—
102C2×#121C2312yes—
103C2×#121C2312yes—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112#489—48no—
113#419—44no—
114#418—44no—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123#407—24no—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126#405—44no—
127C2×#97C2314yes—
128C2×C4×C8C2314yes—
129#400—24no—
130#399—24no—
131#399—24no—
132#398—24no—
133#396—24no—
134#396—24no—
135#396—24no—
136#396—24no—
137#395—24no—
138#394—24no—
139#394—24no—
140#394—24no—
141#393—24no—
142#393—24no—
143#393—24no—
144#393—24no—
145#391—24no—
146#391—24no—
147#391C2314yes—
148#390C2314yes—
149#389C2314yes—
150#389C2314yes—
151#387—24no—
152#387—24no—
153#386—22no—
154#386—22no—
155#385—22no—
156#385—22no—
157#384C2314yes—
158#382C2314yes—
159#381—21no—
160#381—21no—
161#381D412yes—
162#380D412yes—
163#380D412yes—
164#380C2312yes—
165#380C2312yes—
166#380C2312yes—
167#380C2312yes—
168#379—21no—
169#379—21no—
170#379—22no—
171#379—22no—
172#379D411yescharacteristic
173#379D411yescharacteristic
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4C2312yes—
179C4×C22⋊C4C2312yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190C25—21no—
191C25C2×D411yescharacteristic
192C22×Q8—21no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8C2412yes—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—84no—
208C22×D4C2412yes—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4#4114yes—
218C23×C4#4114yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C2×M16—24no—
223C2×M16—24no—
224C2×M16—24no—
225C2×M16—24no—
226C2×M16—24no—
227C2×M16—42no—
228C2×M16—42no—
229C2×M16—42no—
230C2×M16—42no—
231C2×M16—416no—
232C22×C8—22no—
233C22×C8—24no—
234C22×C8—22no—
235C22×C8—42no—
236C22×C8C2412yes—
237C22×C8C2414yes—
238C22×C8C2412yes—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241#124—44no—
242#124—44no—
243#123—84no—
244#123—84no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251#121—24no—
252#121—24no—
253#121—24no—
254#121—24no—
255#121—44no—
256#121—44no—
257#117—42no—
258#117—42no—
259#117—42no—
260#117—42no—
261#117—42no—
262#117—42no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2414yes—
307#103—28no—
308#103—44no—
309#103—44no—
310#103—44no—
311#102—28no—
312#102—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317#97—28no—
318C4×C8—44no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—44no—
336C24—42no—
337C24—82no—
338C24—82no—
339C24#12414yes—
340C24C22×D411yescharacteristic
341C24C22×D411yescharacteristic
342C24C22×D411yescharacteristic
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—48no—
348C2×Q8—42no—
349C2×Q8—82no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—42no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—816no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4#12414yes—
395C22×C4#12414yes—
396C22×C4#12414yes—
397C22×C4C22×D414yes—
398C22×C4C22×D414yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2511yescharacteristic
409M16—48no—
410M16—48no—
411M16—48no—
412C2×C8—28no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—44no—
422C2×C8H(5,𝔽2)14yes—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C42—24no—
462C42—24no—
463C42—82no—
464C42—82no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—44no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23—88no—
486C23—168no—
487C23C2×#12412yes—
488C23C2×#12412yes—
489C23C2×#12412yes—
490C23C23×D411yescharacteristic
491Q8—44no—
492Q8—42no—
493Q8—42no—
494Q8—84no—
495Q8—168no—
496D4—44no—
497D4—42no—
498D4—42no—
499D4—84no—
500D4—88no—
501D4—88no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4C2×#12412yes—
540C2×C4C2×#12412yes—
541C2×C4C2×#12412yes—
542C2×C4C22×#4111yescharacteristic
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×H(5,𝔽2)12yes—
545C2×C4#58512yes—
546C8—24no—
547C8—44no—
548C8—42no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—88no—
562C22—84no—
563C22—84no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—168no—
568C22C22×#12411yescharacteristic
569C22#349612yes—
570C22#358512yes—
571C22#358612yes—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—44no—
576C4—44no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#345912yes—
583C4#349011yescharacteristic
584C4#349111yescharacteristic
585C2—42no—
586C2—42no—
587C2—42no—
588C2—162no—
589C2#3499311yescharacteristic
590C2#6061911yescharacteristic
591C2#6062011yescharacteristic
592C2#6229414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37480C2semidirect
3. #33054C2semidirect
4. #30195C2semidirect
5. #27721C2semidirect
6. #27719C2semidirect
7. #27285C2semidirect
8. #27281C2semidirect
9. #21768C2semidirect
12. #20972C2semidirect
13. #20386C2semidirect
17. C2×#489C22semidirect
18. C2×#419C22semidirect
21. C2×#405C22semidirect
27. #2628C22semidirect
35. #2439C22semidirect
43. #2100C22semidirect
44. #2099C22semidirect
45. #2074C22semidirect
46. #2073C22semidirect
47. #2042C22semidirect
48. #2042C22semidirect
49. #2024C22semidirect
50. #2023C22semidirect
51. #1990C22semidirect
52. #1958C22semidirect
53. #1936C22semidirect
54. #1935C22semidirect
55. #1934C22semidirect
56. #1933C22semidirect
60. #1894C22semidirect
63. #1892C22semidirect
64. #1892C22semidirect
67. #1836C22semidirect
68. #1824C22semidirect
70. #1787C22semidirect
73. #1753C22semidirect
75. #1736C22semidirect
76. #1729C22semidirect

Maximal subgroups

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

subgroupindex
#374802
#374512
#330542
#301952
#277212
#277192
#272852
#272812
#217682
#217672
#209732
#209722
#203862