Finite Group Databasebeta

← #7196938

Subgroups — #7196938

Order 512.

1548 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#59875C211yesmaximal, characteristic, complemented
2#59822C211yesmaximal, characteristic
3#32680C214yesmaximal, complemented
4#32180C214yesmaximal, complemented
5#28420C212yesmaximal, complemented
6#28201C211yesmaximal, characteristic, complemented
7#28160C211yesmaximal, characteristic, complemented
8#27502C214yesmaximal, complemented
9#26334C214yesmaximal, complemented
10#24681C212yesmaximal, complemented
11#24628C212yesmaximal, complemented
12#23799C211yesmaximal, characteristic, complemented
13#22811C212yesmaximal, complemented
14#22311C211yesmaximal, characteristic, complemented
15#22220C211yesmaximal, characteristic, complemented
16C2×#555—21no—
17C2×#486C2211yescharacteristic
18C2×#474C2212yes—
19C2×#460C2212yes—
20C2×C4×SD16C2212yes—
21C2×C4×SD16C2212yes—
22#2810—22no—
23#2733—22no—
24#2568—21no—
25#2568C2212yes—
26#2568C2212yes—
27#2548C2214yes—
28#2541C2212yes—
29#2541C2212yes—
30#2537—21no—
31#2514—22no—
32#2514—22no—
33#2514C2214yes—
34#2414—22no—
35#2414—22no—
36#2355—21no—
37#2355C2212yes—
38#2355C2212yes—
39#2354C2212yes—
40#2354C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2096C2212yes—
47#2094C2212yes—
48#2092C2214yes—
49#2083C2212yes—
50#2073C2214yes—
51#2071C2212yes—
52#2055C2214yes—
53#2055C2214yes—
54#2041C2212yes—
55#2014C2214yes—
56#2014C2214yes—
57#2014C2212yes—
58#2013C2212yes—
59#2013C2212yes—
60#1990C2211yescharacteristic
61#1985C2214yes—
62#1982C2214yes—
63#1973C2214yes—
64#1973C2214yes—
65#1961C2214yes—
66#1957C2211yescharacteristic
67#1949C2214yes—
68#1939C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1926C2214yes—
72#1925C2214yes—
73#1925C2214yes—
74#1925C2214yes—
75#1917C2214yes—
76#1877C2214yes—
77#1862C2214yes—
78#1845C2211yescharacteristic
79#1831C2211yescharacteristic
80#1803C2212yes—
81#1802C2212yes—
82#1776C2214yes—
83#1770C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2312yes—
86C22×SD16C2312yes—
87C22×SD16C2314yes—
88#555—48no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#127C2314yes—
95C2×#124—24no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C2×#117—22no—
111#486—24no—
112#474—44no—
113#460—44no—
114C4×SD16—28no—
115C4×SD16—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2312yes—
119C2×#106C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#103C2312yes—
123C2×#103C2312yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2312yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×#97C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#401—24no—
136#400—24no—
137#400—24no—
138#400—24no—
139#394—24no—
140#394—24no—
141#392—24no—
142#392—24no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—21no—
148#391D412yes—
149#391C2312yes—
150#391C2314yes—
151#391C2314yes—
152#390—24no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#390C2314yes—
160#390C2314yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#388—24no—
168#388—24no—
169#387—24no—
170#387—24no—
171#387—24no—
172#384D411yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#383—24no—
178#382—22no—
179#382—22no—
180#382C2314yes—
181#380—22no—
182#380—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C43C2311yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4C2412yes—
213C22×D4C2412yes—
214C23×C4—22no—
215C23×C4—22no—
216C2×SD16—24no—
217C2×SD16—42no—
218C2×SD16—42no—
219C2×SD16—42no—
220C2×SD16—42no—
221C2×SD16—42no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C22×C8C2412yes—
226C22×C8C2414yes—
227C22×C8C2412yes—
228C4⋊Q8—24no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—42no—
232#127—22no—
233#127—22no—
234#127—42no—
235#127—44no—
236#127—44no—
237#124—416no—
238#121—48no—
239#121—48no—
240#121—84no—
241#121—84no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—44no—
245C4×Q8—44no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—82no—
252C4×D4—82no—
253#117—48no—
254#117—48no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2414yes—
300C2×C42C2411yescharacteristic
301#107—44no—
302#107—44no—
303#106—24no—
304#106—42no—
305C4⋊C8—24no—
306C4⋊C8—44no—
307#103—24no—
308#103—44no—
309#103—42no—
310#102—24no—
311#102—28no—
312#102—44no—
313#102—44no—
314#102—42no—
315#102—44no—
316#102—44no—
317#97—44no—
318C4×C8—22no—
319C4×C8#4114yes—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C24—22no—
335C24—22no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—48no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2511yescharacteristic
397SD16—48no—
398SD16—48no—
399SD16—88no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8C2×#4114yes—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—22no—
447C42—24no—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—41no—
452C42—41no—
453C42—48no—
454C42—44no—
455C42—44no—
456C42—84no—
457C42AGL(1,ℤ/8)14yes—
458C42C2×#4112yes—
459C42H(5,𝔽2)12yes—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23C23×D411yescharacteristic
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—84no—
480Q8—84no—
481D4—42no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—44no—
486D4—42no—
487D4—84no—
488D4—84no—
489D4—84no—
490D4—84no—
491D4—168no—
492D4—168no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
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—88no—
531C2×C4#55012yes—
532C2×C4#55012yes—
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)11yescharacteristic
539C2×C4#58512yes—
540C2×C4#58512yes—
541C8—24no—
542C8—42no—
543C8—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C2×#55011yescharacteristic
559C22#348511yescharacteristic
560C22#348711yescharacteristic
561C22C22×AGL(1,ℤ/8)11yescharacteristic
562C22C2×#57711yescharacteristic
563C22#358511yescharacteristic
564C22#358611yescharacteristic
565C4—22no—
566C4—22no—
567C4—22no—
568C4—44no—
569C4—41no—
570C4—44no—
571C4—41no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#339714yes—
579C4#349312yes—
580C4#354012yes—
581C2—162no—
582C2—162no—
583C2#4538811yescharacteristic
584C2C2×#339711yescharacteristic
585C2#6141511yescharacteristic
586C2#6172211yescharacteristic
587C2#6180511yescharacteristic
588C2#6189011yescharacteristic
589C2#6191111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59822C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59875C2semidirect
3. #32680C2semidirect
4. #32180C2semidirect
5. #28420C2semidirect
6. #28201C2semidirect
7. #28160C2semidirect
8. #27502C2semidirect
9. #26334C2semidirect
10. #24681C2semidirect
11. #24628C2semidirect
12. #23799C2semidirect
13. #22811C2semidirect
14. #22311C2semidirect
15. #22220C2semidirect

Maximal subgroups

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

subgroupindex
#598752
#598222
#326802
#321802
#284202
#282012
#281602
#275022
#263342
#246812
#246282
#237992
#228112
#223112
#222202