Finite Group Databasebeta

← #7195463

Subgroups — #7195463

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#59822C211yesmaximal, characteristic
3#32012C214yesmaximal, complemented
4#31345C214yesmaximal, complemented
5#29276C211yesmaximal, characteristic, complemented
6#29254C211yesmaximal, characteristic, complemented
7#28607C212yesmaximal, complemented
8#26301C212yesmaximal, complemented
9#26232C212yesmaximal, complemented
10#26146C214yesmaximal, complemented
11#24104C214yesmaximal, complemented
12#23837C211yesmaximal, characteristic, complemented
13#22665C212yesmaximal, complemented
14#22264C211yesmaximal, characteristic, complemented
15#22220C211yesmaximal, characteristic, complemented
16C2×#559—21no—
17C2×#555C2212yes—
18C2×#486C2211yescharacteristic
19C2×#481C2214yes—
20C2×#436C2212yes—
21C2×#428C2214yes—
22#2733—22no—
23#2724—22no—
24#2582—21no—
25#2548—22no—
26#2547C2212yes—
27#2547C2212yes—
28#2541—21no—
29#2541C2212yes—
30#2537C2212yes—
31#2486C2214yes—
32#2447—22no—
33#2447—22no—
34#2414—22no—
35#2414C2214yes—
36#2354—21no—
37#2354C2212yes—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2212yes—
46#2093C2212yes—
47#2084C2212yes—
48#2084C2212yes—
49#2077C2214yes—
50#2076C2214yes—
51#2058C2212yes—
52#2058C2212yes—
53#2058C2212yes—
54#2034C2214yes—
55#2034C2214yes—
56#2034C2214yes—
57#2034C2214yes—
58#2023C2212yes—
59#1990C2211yescharacteristic
60#1988C2212yes—
61#1982C2212yes—
62#1957C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1947C2212yes—
66#1925C2214yes—
67#1920C2214yes—
68#1895C2214yes—
69#1895C2214yes—
70#1874C2214yes—
71#1870C2214yes—
72#1870C2214yes—
73#1862C2214yes—
74#1862C2214yes—
75#1854C2214yes—
76#1854C2214yes—
77#1852C2214yes—
78#1845C2211yescharacteristic
79#1831C2211yescharacteristic
80#1825C2214yes—
81#1809C2214yes—
82C4×#106C2212yes—
83C4×#106C2212yes—
84#1744C2211yescharacteristic
85C22×M16C2314yes—
86#559—48no—
87#555—48no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127—22no—
92C2×#127D411yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#124—22no—
95C2×#124—22no—
96C2×#124—24no—
97C2×#123—24no—
98C2×#121—22no—
99C2×#121—22no—
100C2×#121C2314yes—
101C2×#121C2314yes—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117C2314yes—
115C22×C4⋊C4C2314yes—
116#486—24no—
117#481—48no—
118#436—28no—
119#428—48no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2312yes—
123C2×#106C2314yes—
124C2×#106C2312yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#398—24no—
138#397—24no—
139#397—24no—
140#397—24no—
141#394—24no—
142#394—24no—
143#392—24no—
144#392—24no—
145#391—22no—
146#391—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#390—22no—
151#390—22no—
152#390—24no—
153#390—21no—
154#390—21no—
155#390D412yes—
156#390D412yes—
157#390D412yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#388—24no—
168#388—24no—
169#385—24no—
170#385—24no—
171#384D411yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#381—24no—
177#380—22no—
178#380C2314yes—
179#380C2314yes—
180#380C2314yes—
181#380C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2312yes—
191#340C2312yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2312yes—
196#340C2314yes—
197#340C2314yes—
198#340C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2×D412yes—
211C22×D4C2×D412yes—
212C22×D4C2×D412yes—
213C22×D4C2412yes—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4C2414yes—
217C2×M16—28no—
218C2×M16—44no—
219C2×M16—44no—
220C22×C8C2414yes—
221C22×C8C2412yes—
222C22×C8C2412yes—
223C4⋊Q8—24no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—42no—
227#127—24no—
228#127—22no—
229#127—22no—
230#127—84no—
231#124—48no—
232#124—48no—
233#124—416no—
234#123—416no—
235#121—28no—
236#121—44no—
237#121—48no—
238#121—44no—
239#121—44no—
240#121—84no—
241C4×Q8—44no—
242C4×Q8—44no—
243C4×Q8—82no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—48no—
250C4×D4—48no—
251C4×D4—44no—
252C4×D4—82no—
253#117—28no—
254#117—44no—
255#117—48no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4C2×D414yes—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D414yes—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—44no—
308#107—44no—
309#106—24no—
310#106—28no—
311#106—24no—
312#106—42no—
313#106—42no—
314C4⋊C8—24no—
315#102—28no—
316#102—44no—
317#102—44no—
318#102—44no—
319C22⋊C8—28no—
320C22⋊C8—44no—
321C22⋊C8—44no—
322C4×C8—22no—
323C4×C8#4114yes—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334C24—22no—
335C24C22×D414yes—
336C2×Q8—24no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C22×D414yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2511yescharacteristic
397M16—416no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8C2×#4114yes—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—84no—
433C4⋊C4—81no—
434C4⋊C4—82no—
435C4⋊C4—81no—
436C4⋊C4H(5,𝔽2)14yes—
437C22⋊C4—28no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—82no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—48no—
456C42—44no—
457C42—44no—
458C42—44no—
459C42AGL(1,ℤ/8)14yes—
460C42C2×#4112yes—
461C422−1+412yes—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C23×D411yescharacteristic
477Q8—84no—
478Q8—84no—
479Q8—84no—
480D4—48no—
481D4—48no—
482D4—44no—
483D4—44no—
484D4—84no—
485D4—84no—
486D4—84no—
487D4—168no—
488D4—168no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—88no—
530C2×C4—168no—
531C2×C4#54712yes—
532C2×C4#54712yes—
533C2×C4C2×AGL(1,ℤ/8)12yes—
534C2×C4C2×AGL(1,ℤ/8)12yes—
535C2×C4C2×AGL(1,ℤ/8)12yes—
536C2×C4#57712yes—
537C2×C4C22×#4111yescharacteristic
538C2×C4C2×H(5,𝔽2)12yes—
539C2×C4C2×2−1+411yescharacteristic
540C2×C4#58512yes—
541C8—24no—
542C8—44no—
543C8—42no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C2×#54711yescharacteristic
559C22#348411yescharacteristic
560C22#348511yescharacteristic
561C22C22×AGL(1,ℤ/8)11yescharacteristic
562C22C2×#57711yescharacteristic
563C22#358511yescharacteristic
564C22#358611yescharacteristic
565C4—22no—
566C4—22no—
567C4—22no—
568C4—44no—
569C4—41no—
570C4—41no—
571C4—44no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#335414yes—
579C4#350612yes—
580C4#355412yes—
581C2—84no—
582C2—162no—
583C2#4535411yescharacteristic
584C2C2×#335411yescharacteristic
585C2#6129111yescharacteristic
586C2#6158711yescharacteristic
587C2#6168011yescharacteristic
588C2#6171711yescharacteristic
589C2#6180511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59822C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
3. #32012C2semidirect
4. #31345C2semidirect
5. #29276C2semidirect
6. #29254C2semidirect
7. #28607C2semidirect
8. #26301C2semidirect
9. #26232C2semidirect
10. #26146C2semidirect
11. #24104C2semidirect
12. #23837C2semidirect
13. #22665C2semidirect
14. #22264C2semidirect
15. #22220C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598222
#320122
#313452
#292762
#292542
#286072
#263012
#262322
#261462
#241042
#238372
#226652
#222642
#222202