Finite Group Databasebeta

← #6479579

Subgroups — #6479579

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35254C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#33102C214yesmaximal, complemented
4#29910C212yesmaximal, complemented
5#29907C212yesmaximal, complemented
6#27869C214yesmaximal, complemented
7#27863C212yesmaximal, complemented
8#27857C212yesmaximal, complemented
9#27621C212yesmaximal, complemented
10#27612C212yesmaximal, complemented
11#27591C212yesmaximal, complemented
12#27585C212yesmaximal, complemented
13#21946C212yesmaximal, complemented
14#21770C212yesmaximal
15#20684C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#408C2212yes—
19#2854C2212yes—
20#2854C2212yes—
21#2624C2212yes—
22#2624C2212yes—
23#2448—22no—
24#2445—22no—
25#2418—22no—
26#2418—22no—
27#2355—22no—
28#2354—22no—
29#2352—22no—
30#2352—22no—
31#2316C2211yescharacteristic
32#2314C2212yes—
33#2314C2212yescomplemented
34#2311C2212yes—
35#2311C2211yescharacteristic
36#2309C2211yescharacteristic
37#2296C2211yescharacteristic
38C2×#390C2211yescharacteristic
39#2285C2212yes—
40#2285C2212yescomplemented
41#2285C2211yescharacteristic, complemented
42C2×#384C2211yescharacteristic, complemented
43#2274C2212yescomplemented
44#2101C2214yescomplemented
45#2100C2212yescomplemented
46#2100C2212yescomplemented
47#2094C2212yes—
48#2094C2212yes—
49#2092C2214yes—
50#2084C2212yescomplemented
51#2084C2214yescomplemented
52#2083C2212yescomplemented
53#2071C2212yes—
54#2071C2212yes—
55#2071C2214yes—
56#1926C2214yes—
57#1926C2214yes—
58#1926C2214yes—
59#1925C2214yes—
60#1920C2214yescomplemented
61#1920C2214yescomplemented
62#1919C2214yescomplemented
63#1918C2214yescomplemented
64#1895C2214yes—
65#1893C2214yes—
66#1875C2214yes—
67#1874C2214yes—
68#1854C2214yes—
69#1854C2214yescomplemented
70#1853C2214yes—
71#1853C2214yescomplemented
72#1825C2214yes—
73#1818C2212yes—
74#1818C2212yescomplemented
75#1817C2212yes—
76#1817C2212yescomplemented
77#1809C2214yescomplemented
78#1807C2214yescomplemented
79#1786C2214yes—
80#1729C2212yescomplemented
81#1728C2211yescharacteristic, complemented
82C2×#336C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C22×M16C2312yes—
86C23×C8C2312yes—
87#535—28no—
88C2×#127—21no—
89C2×#124—21no—
90C2×#123—22no—
91C2×#123—22no—
92C2×#121C2312yes—
93C2×#121C2312yes—
94C2×#120—21no—
95C2×#120—21no—
96C4×#41—28no—
97C2×C4×Q8D412yes—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124#408—28no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127#399—22no—
128#399—22no—
129#397—22no—
130#397—22no—
131#394—22no—
132#394—22no—
133#394—22no—
134#394—22no—
135#391—22no—
136#391—21no—
137#390—22no—
138#390—21no—
139#390—22no—
140#390—22no—
141#390C2312yes—
142#390C2312yes—
143#390C2312yes—
144#390C2312yes—
145#389—22no—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#388—24no—
151#388—24no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#387—24no—
157#387—24no—
158#386—22no—
159#386—22no—
160#386—21no—
161#386—21no—
162#386—22no—
163#386—22no—
164#385—22no—
165#384—22no—
166#384C2312yes—
167#384C2312yes—
168#384C2312yes—
169#382—22no—
170#382—22no—
171#382C2312yes—
172#380—22no—
173#380—22no—
174#380—22no—
175#380—22no—
176#380C2314yes—
177#380C2314yes—
178#380C2312yes—
179#380C2312yes—
180#379—22no—
181C4×C4⋊C4—24no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—24no—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203C25—21no—
204C2×#41—42no—
205C2×#41—41no—
206C2×#41—41no—
207C2×#41#4118yes—
208C22×Q8C2×D411yescharacteristic
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—42no—
214C22×D4—41no—
215C22×D4—41no—
216C22×D4C2×D411yescharacteristic
217C22×D4C2×D411yescharacteristic
218C22×D4C2×D411yescharacteristic
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C2×M16—24no—
229C2×M16—42no—
230C2×M16—42no—
231C22×C8—24no—
232C22×C8—42no—
233C22×C8—42no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236#127—44no—
237#124—44no—
238#123—84no—
239#123—84no—
240#121—28no—
241#121—28no—
242#120—44no—
243#120—44no—
244C4×Q8—28no—
245C4×D4—28no—
246C4×D4—28no—
247C4×D4—28no—
248#117—24no—
249#117—24no—
250#117—24no—
251#117—24no—
252#117—42no—
253#117—42no—
254#117—41no—
255#117—41no—
256#117—84no—
257#117—84no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—84no—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—84no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—44no—
318#106—44no—
319#106—44no—
320#106—44no—
321#102—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325C22⋊C8—28no—
326C22⋊C8—28no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338C24—22no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—82no—
343C24—82no—
344C24C22×D411yescharacteristic
345C24C22×D411yescharacteristic
346C24C22×D411yescharacteristic
347#41—48no—
348#41—48no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8#12314yes—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4#12314yes—
374C2×D4#12314yes—
375C2×D4#12314yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4114yes—
413C22×C4C2511yescharacteristic
414M16—84no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—84no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—44no—
472C42—42no—
473C42—42no—
474C42—84no—
475C42—84no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—84no—
496C23—88no—
497C23—164no—
498C23—162no—
499C23—162no—
500C23C2×#12312yes—
501C23C2×#12312yes—
502C23C2×#12312yes—
503C23C23×D411yescharacteristic
504Q8—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—44no—
510D4—84no—
511D4—164no—
512D4—164no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—162no—
555C2×C4—164no—
556C2×C4C2×#12312yes—
557C2×C4C2×#12312yes—
558C2×C4C2×#12312yes—
559C2×C4#57714yes—
560C2×C4C22×#4112yes—
561C2×C4#58511yescharacteristic
562C2×C4#58511yescharacteristic
563C2×C4#58512yes—
564C8—44no—
565C8—44no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—168no—
589C22C22×#12311yescharacteristic
590C22#344911yescharacteristic
591C22#344911yescharacteristic
592C22C2×#57712yes—
593C22#358512yes—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—24no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—84no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4#349112yes—
607C4#349112yes—
608C2—42no—
609C2—42no—
610C2—42no—
611C2—162no—
612C2#3509711yescharacteristic
613C2#6052512yes—
614C2#6098912yes—
615C2#6101211yescharacteristic
616C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #33102C2semidirect
4. #29910C2semidirect
5. #29907C2semidirect
6. #27869C2semidirect
7. #27863C2semidirect
8. #27857C2semidirect
9. #27621C2semidirect
10. #27612C2semidirect
11. #27591C2semidirect
12. #27585C2semidirect
13. #21946C2semidirect
15. #20684C2semidirect
33. #2314C22semidirect
40. #2285C22semidirect
41. #2285C22semidirect
42. C2×#384C22semidirect
43. #2274C22semidirect
44. #2101C22semidirect
45. #2100C22semidirect
46. #2100C22semidirect
50. #2084C22semidirect
51. #2084C22semidirect
52. #2083C22semidirect
60. #1920C22semidirect
61. #1920C22semidirect
62. #1919C22semidirect
63. #1918C22semidirect
69. #1854C22semidirect
71. #1853C22semidirect
74. #1818C22semidirect
76. #1817C22semidirect
77. #1809C22semidirect
78. #1807C22semidirect
80. #1729C22semidirect
81. #1728C22semidirect
82. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#341882
#331022
#299102
#299072
#278692
#278632
#278572
#276212
#276122
#275912
#275852
#219462
#217702
#206842