Finite Group Databasebeta

← #7196936

Subgroups — #7196936

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#59821C211yesmaximal, characteristic, complemented
3#32675C214yesmaximal, complemented
4#32189C214yesmaximal, complemented
5#28415C212yesmaximal, complemented
6#28205C211yesmaximal, characteristic, complemented
7#28156C211yesmaximal, characteristic
8#27507C214yesmaximal, complemented
9#26328C214yesmaximal, complemented
10#24678C212yesmaximal, complemented
11#24631C212yesmaximal, complemented
12#23799C211yesmaximal, characteristic, complemented
13#22811C212yesmaximal, complemented
14#22311C211yesmaximal, characteristic, complemented
15#22220C211yesmaximal, characteristic, complemented
16C2×#556—21no—
17C2×#486C2211yescharacteristic
18C2×#475C2212yes—
19C2×#459C2212yes—
20C2×C4×SD16C2212yes—
21C2×C4×SD16C2212yes—
22#2808—22no—
23#2727—22no—
24#2592C2214yes—
25#2563—21no—
26#2548C2212yes—
27#2548C2212yes—
28#2547C2212yes—
29#2542C2212yes—
30#2538—21no—
31#2513—22no—
32#2512C2214yes—
33#2485—22no—
34#2448—22no—
35#2415—22no—
36#2355C2212yes—
37#2354—21no—
38#2354C2212yes—
39#2354C2212yes—
40Q8×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2096C2212yes—
47#2094C2212yes—
48#2094C2214yes—
49#2084C2212yes—
50#2072C2214yes—
51#2071C2212yes—
52#2055C2214yes—
53#2054C2214yes—
54#2043C2212yes—
55#2014C2212yes—
56#2013C2212yes—
57#2012C2214yes—
58#2012C2214yes—
59#2011C2212yes—
60#1990C2211yescharacteristic
61#1985C2214yes—
62#1983C2214yes—
63#1973C2214yes—
64#1972C2214yes—
65#1959C2214yes—
66#1957C2211yescharacteristic
67#1948C2214yes—
68#1939C2212yes—
69#1928C2214yes—
70#1927C2214yes—
71#1926C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1925C2214yes—
75#1917C2214yes—
76#1879C2214yes—
77#1862C2214yes—
78#1845C2211yescharacteristic
79#1831C2211yescharacteristic
80#1803C2212yes—
81#1802C2212yes—
82#1776C2214yes—
83#1770C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2312yes—
86C22×SD16C2312yes—
87C22×D8C2314yes—
88#556—48no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#126—24no—
95C2×#124—22no—
96C2×#124C2314yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C4×Q8—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#475—44no—
113#459—44no—
114C4×SD16—28no—
115C4×SD16—44no—
116C2×#107C2314yes—
117C2×#106C2312yes—
118C2×#106C2314yes—
119C2×#106C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#102C2314yes—
127C2×#102C2312yes—
128C2×#102C2312yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#97C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#399—24no—
136#397—24no—
137#397—24no—
138#397—24no—
139#396—24no—
140#394—24no—
141#394—24no—
142#393—24no—
143#391—21no—
144#391D412yes—
145#391C2312yes—
146#391C2314yes—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#390C2314yes—
157#390C2314yes—
158#390C2314yes—
159#389—24no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#388—24no—
168#388—24no—
169#388—24no—
170#388—24no—
171#388—24no—
172#387—24no—
173#384D411yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2314yes—
179#382—22no—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—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#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C2×SD16—24no—
215C2×SD16—42no—
216C2×SD16—42no—
217C2×SD16—42no—
218C2×SD16—42no—
219C2×SD16—42no—
220C2×D8—44no—
221C2×D8—44no—
222C2×D8—44no—
223C22×C8C2414yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C4⋊Q8—24no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—42no—
230#127—22no—
231#127—22no—
232#127—42no—
233#126—416no—
234#124—44no—
235#124—44no—
236#124—48no—
237C22⋊Q8—48no—
238C22⋊Q8—84no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—44no—
242C4×Q8—48no—
243C4×Q8—44no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—82no—
250C4×D4—82no—
251#117—48no—
252#117—84no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
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—24no—
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×D412yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2414yes—
304C2×C42C2411yescharacteristic
305#107—44no—
306#106—24no—
307#106—42no—
308#106—44no—
309C4⋊C8—24no—
310C4⋊C8—44no—
311#103—24no—
312#103—44no—
313#103—42no—
314#103—44no—
315#102—24no—
316#102—28no—
317#102—44no—
318#102—44no—
319#102—42no—
320#102—44no—
321#97—44no—
322C4×C8—22no—
323C4×C8#4114yes—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—22no—
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—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D414yes—
385C22×C4C22×D414yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2511yescharacteristic
391SD16—48no—
392SD16—48no—
393D8—88no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8C2×#4114yes—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—84no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—84no—
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—48no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—84no—
481D4—42no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—44no—
486D4—42no—
487D4—84no—
488D4—84no—
489D4—84no—
490D4—84no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
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—88no—
528C2×C4—88no—
529C2×C4#55012yes—
530C2×C4#55012yes—
531C2×C4C2×AGL(1,ℤ/8)12yes—
532C2×C4C2×AGL(1,ℤ/8)12yes—
533C2×C4C2×AGL(1,ℤ/8)12yes—
534C2×C4#57712yes—
535C2×C4C22×#4111yescharacteristic
536C2×C4C2×H(5,𝔽2)11yescharacteristic
537C2×C4#58512yes—
538C2×C4#58512yes—
539C8—24no—
540C8—44no—
541C8—42no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22C2×#55011yescharacteristic
557C22#348511yescharacteristic
558C22#348711yescharacteristic
559C22C22×AGL(1,ℤ/8)11yescharacteristic
560C22C2×#57711yescharacteristic
561C22#358511yescharacteristic
562C22#358611yescharacteristic
563C4—22no—
564C4—22no—
565C4—22no—
566C4—44no—
567C4—44no—
568C4—41no—
569C4—41no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4#339714yes—
577C4#349312yes—
578C4#354012yes—
579C2—162no—
580C2—162no—
581C2#4538811yescharacteristic
582C2C2×#339711yescharacteristic
583C2#6141511yescharacteristic
584C2#6171711yescharacteristic
585C2#6180811yescharacteristic
586C2#6189111yescharacteristic
587C2#6191611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28156C2
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. #59863C2semidirect
2. #59821C2semidirect
3. #32675C2semidirect
4. #32189C2semidirect
5. #28415C2semidirect
6. #28205C2semidirect
8. #27507C2semidirect
9. #26328C2semidirect
10. #24678C2semidirect
11. #24631C2semidirect
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
#598632
#598212
#326752
#321892
#284152
#282052
#281562
#275072
#263282
#246782
#246312
#237992
#228112
#223112
#222202