Finite Group Databasebeta

← #6493660

Subgroups — #6493660

Order 512.

1747 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35328C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#33117C212yesmaximal, complemented
4#33116C212yesmaximal, complemented
5#29930C212yesmaximal, complemented
6#29929C212yesmaximal, complemented
7#27978C212yesmaximal, complemented
8#27915C212yesmaximal, complemented
9#27895C212yesmaximal, complemented
10#27894C212yesmaximal, complemented
11#27593C214yesmaximal, complemented
12#24017C214yesmaximal, complemented
13#21986C212yesmaximal, complemented
14#21809C212yesmaximal, complemented
15#20719C211yesmaximal, characteristic, complemented
16C2×#537C2212yes—
17C2×#532C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#485C2214yes—
20C2×#482C2214yes—
21C2×#432C2212yes—
22#2850C2212yes—
23#2850C2212yes—
24#2621C2211yescharacteristic
25#2621C2211yescharacteristic
26#2619C2211yescharacteristic
27#2619C2211yescharacteristic
28#2543—22no—
29#2543—22no—
30#2536—22no—
31#2535—22no—
32#2355—22no—
33#2354—22no—
34#2352—22no—
35#2352—22no—
36#2314C2212yes—
37#2314C2212yes—
38#2311C2212yes—
39#2311C2212yes—
40#2285C2212yes—
41#2285C2212yes—
42#2285C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45#2274C2212yes—
46#2097C2214yes—
47#2092C2214yes—
48#2072C2214yes—
49#2071C2214yes—
50#2036C2214yes—
51#2035C2214yes—
52#2034C2214yes—
53#2033C2214yes—
54#1982C2214yes—
55#1981C2214yes—
56#1959C2212yes—
57#1959C2212yes—
58#1959C2212yes—
59#1959C2212yes—
60#1949C2214yes—
61#1949C2214yes—
62#1928C2214yes—
63#1927C2214yes—
64#1926C2214yes—
65#1925C2214yes—
66#1874C2214yes—
67#1872C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1847C2212yes—
71#1847C2212yes—
72#1836C2212yes—
73#1836C2212yes—
74#1818C2212yes—
75#1798C2212yes—
76#1798C2212yes—
77#1788C2214yes—
78#1788C2214yes—
79#1780C2212yes—
80#1749C2212yes—
81#1749C2211yescharacteristic
82#1749C2211yescharacteristic
83C22×#41C2312yes—
84C22×#41C2311yescharacteristic
85#537—48no—
86#532—28no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127C2311yescharacteristic
89C2×#124C2312yes—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×#121—22no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C4×#41—28no—
100C2×C4×Q8D412yes—
101C2×C4×D4—22no—
102C2×C4×D4D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—22no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#485—48no—
118#482—48no—
119#432—28no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139#398—22no—
140#398—22no—
141#397—22no—
142#397—22no—
143#394—22no—
144#394—22no—
145#392—22no—
146#392—22no—
147#391—22no—
148#391C2312yes—
149#391C2312yes—
150#390—22no—
151#390C2312yes—
152#390C2312yes—
153#388—24no—
154#388—24no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#387—24no—
160#387—24no—
161#386—22no—
162#386—22no—
163#386—22no—
164#386—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#384—22no—
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#382—22no—
174#382—22no—
175#382—22no—
176#382C2312yes—
177#382C2312yes—
178#382C2311yescharacteristic
179#382C2311yescharacteristic
180#380—22no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380C2314yes—
186#380C2314yes—
187C4×C4⋊C4—24no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2314yes—
193C4×C22⋊C4—24no—
194C2×#95—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—28no—
205C2×#41—42no—
206C2×#41—44no—
207C2×#41—44no—
208C22×Q8C2×D411yescharacteristic
209C22×Q8C2412yes—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2×D411yescharacteristic
215C22×D4C2×D411yescharacteristic
216C22×D4C2×D411yescharacteristic
217C22×D4C2412yes—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—24no—
230#127—24no—
231#124—28no—
232C22⋊Q8—28no—
233C22⋊Q8—44no—
234C22⋊Q8—48no—
235#121—28no—
236#121—44no—
237#121—48no—
238#121—48no—
239#121—48no—
240#121—48no—
241C4×Q8—28no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—48no—
246#117—24no—
247#117—24no—
248#117—24no—
249#117—24no—
250#117—24no—
251#117—84no—
252#117—84no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—84no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2414yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42#4118yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#107—44no—
303#107—44no—
304#106—44no—
305#106—44no—
306C4⋊C8—28no—
307#103—44no—
308#103—44no—
309#103—44no—
310#103—44no—
311#102—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—28no—
316C22⋊C8—28no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319#97—24no—
320#97—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334C24—22no—
335C24C22×D411yescharacteristic
336C24C22×D411yescharacteristic
337C24C22×D411yescharacteristic
338#41—416no—
339#41—816no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—41no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—84no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4#12318yes—
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4114yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C4C2511yescharacteristic
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—84no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C42—24no—
451C42—24no—
452C42—22no—
453C42—22no—
454C42—44no—
455C42—84no—
456C42—84no—
457C42#12318yes—
458C42C2×#4114yes—
459C42C2×#4114yes—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23C2×#12312yes—
481C23#53512yes—
482C23C23×D411yescharacteristic
483Q8—44no—
484Q8—48no—
485D4—44no—
486D4—44no—
487D4—44no—
488D4—48no—
489D4—88no—
490D4—88no—
491D4—164no—
492D4—164no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—88no—
532C2×C4—164no—
533C2×C4—164no—
534C2×C4C2×#12314yes—
535C2×C4C2×#12312yes—
536C2×C4C2×#12314yes—
537C2×C4#53512yes—
538C2×C4C22×#4112yes—
539C2×C4C2×H(5,𝔽2)12yes—
540C2×C4#58512yes—
541C8—44no—
542C8—44no—
543C22—24no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—42no—
556C22—42no—
557C22—41no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22C22×#12311yescharacteristic
566C22C2×#53511yescharacteristic
567C22#344911yescharacteristic
568C22#358512yes—
569C22#358612yes—
570C4—24no—
571C4—24no—
572C4—22no—
573C4—22no—
574C4—42no—
575C4—42no—
576C4—41no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#349414yes—
583C2—42no—
584C2—42no—
585C2—81no—
586C2—162no—
587C2#3526211yescharacteristic
588C2#6051712yes—
589C2#6082912yes—
590C2#6101911yescharacteristic
591C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35328C2
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. #33117C2semidirect
4. #33116C2semidirect
5. #29930C2semidirect
6. #29929C2semidirect
7. #27978C2semidirect
8. #27915C2semidirect
9. #27895C2semidirect
10. #27894C2semidirect
11. #27593C2semidirect
12. #24017C2semidirect
13. #21986C2semidirect
14. #21809C2semidirect
15. #20719C2semidirect

Maximal subgroups

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

subgroupindex
#353282
#341882
#331172
#331162
#299302
#299292
#279782
#279152
#278952
#278942
#275932
#240172
#219862
#218092
#207192