Finite Group Databasebeta

← #6798587

Subgroups — #6798587

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#36136C211yesmaximal, characteristic
3#33230C211yesmaximal, characteristic, complemented
4#31840C214yesmaximal, complemented
5#31838C214yesmaximal, complemented
6#29815C212yesmaximal, complemented
7#29225C211yesmaximal, characteristic, complemented
8#28447C211yesmaximal, characteristic, complemented
9#27978C212yesmaximal, complemented
10#25604C212yesmaximal, complemented
11#25579C214yesmaximal, complemented
12#25336C214yesmaximal, complemented
13#21867C212yesmaximal, complemented
14#21657C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic, complemented
16C2×#554C2212yes—
17C2×#529C2212yes—
18C2×#497C2212yes—
19C2×#491C2211yescharacteristic
20C2×#489C2214yes—
21C2×#464C2214yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2858C2211yescharacteristic
25#2857C2212yes—
26#2850C2212yes—
27#2567C2212yes—
28#2554—22no—
29#2552—22no—
30#2540C2212yes—
31#2502C2214yes—
32#2479—22no—
33#2456—22no—
34#2440—22no—
35#2440—22no—
36#2429C2212yes—
37#2428C2212yes—
38#2367C2212yes—
39#2363—22no—
40D4×C4⋊C4—22no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#125C2211yescharacteristic
43C2×#384C2212yes—
44C42×D4C2211yescharacteristic
45#2100C2214yes—
46#2072C2214yes—
47#2071C2214yes—
48#2056C2214yes—
49#2055C2214yes—
50#2016C2212yes—
51#2014C2212yes—
52#2001C2214yes—
53#2001C2214yes—
54#2000C2214yes—
55#2000C2214yes—
56#1992C2212yes—
57#1992C2211yescharacteristic
58#1988C2214yes—
59#1960C2211yescharacteristic
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1957C2212yes—
64#1956C2212yes—
65#1955C2212yes—
66#1940C2212yes—
67#1933C2214yes—
68#1923C2214yes—
69#1878C2214yes—
70#1877C2214yes—
71#1874C2214yes—
72#1874C2214yes—
73#1872C2214yes—
74#1872C2214yes—
75#1836C2212yes—
76#1836C2211yescharacteristic
77#1803C2214yes—
78#1802C2214yes—
79#1767C2211yescharacteristic
80#1763C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83C22×Q16C2314yes—
84C22×SD16C2314yes—
85#554—48no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91#529—28no—
92#529—44no—
93C2×#126—22no—
94C2×#125—22no—
95C2×#125C2311yescharacteristic
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2314yes—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4D412yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2311yescharacteristic
113C2×#117—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4C2314yes—
117C22×C42C2312yes—
118#497—44no—
119#491—24no—
120#489—48no—
121#464—48no—
122#432—44no—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2311yescharacteristic
136C2×#97C2312yes—
137C2×#97C2312yes—
138C2×#97C2311yescharacteristic
139C2×C4×C8C2312yes—
140#399—22no—
141#398—22no—
142#398—24no—
143#398—24no—
144#396—22no—
145#395—22no—
146#395—22no—
147#394—22no—
148#392—22no—
149#392—22no—
150#391C2312yes—
151#391C2314yes—
152#391C2312yes—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390C2312yes—
158#389—24no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#387—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#385—22no—
174#385—24no—
175#385—24no—
176#384—22no—
177#384—22no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2314yes—
181#384C2312yes—
182#384C2312yes—
183#382C2311yescharacteristic
184#380—22no—
185#380—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2311yescharacteristic
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4D412yes—
194C4×C22⋊C4C2312yes—
195C43C2311yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2312yes—
204C22×Q8—22no—
205C22×Q8C2412yes—
206C22×Q8C2412yes—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2×D411yescharacteristic
211C22×D4C2412yes—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2412yes—
217C2×Q16—44no—
218C2×Q16—44no—
219C2×Q16—44no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C22×C8C2414yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C4⋊Q8—22no—
227C4⋊Q8—28no—
228C4⋊Q8—22no—
229C4⋊Q8—42no—
230C4⋊Q8—84no—
231#127—42no—
232#126—48no—
233#125—22no—
234#125—48no—
235#125#4114yes—
236#124—24no—
237#124—24no—
238#124—28no—
239#123—48no—
240C22⋊Q8—24no—
241C22⋊Q8—416no—
242C22⋊Q8—42no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246#121—24no—
247#121—42no—
248#121—48no—
249#121—44no—
250C4×Q8—24no—
251C4×Q8—28no—
252C4×Q8—42no—
253C4×D4—28no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—42no—
257C4×D4—48no—
258C4×D4—48no—
259#117—48no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—84no—
280C2×C4⋊C4—84no—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
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—22no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2412yes—
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C4⋊C8—24no—
316C4⋊C8—44no—
317#103—28no—
318#103—44no—
319#103—44no—
320#103—44no—
321#103—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325C22⋊C8—44no—
326C22⋊C8—44no—
327#97—22no—
328#97—22no—
329#97—24no—
330#97—24no—
331C4×C8—42no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95C2×D411yescharacteristic
341#95C2×D411yescharacteristic
342#95C2×D412yes—
343#95C2×D412yes—
344C24—22no—
345C24C22×D412yes—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—44no—
355C2×Q8—84no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—82no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C2511yescharacteristic
401Q16—88no—
402SD16—88no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C4⋊C4—22no—
413C4⋊C4—28no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—84no—
436C4⋊C4C2×#4114yes—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C22⋊C4#13718yes—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—42no—
459C42—44no—
460C42—44no—
461C42—82no—
462C42C2×#4112yes—
463C42H(5,𝔽2)14yes—
464C42H(5,𝔽2)12yes—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23C2×#13714yes—
479C23C23×D411yescharacteristic
480Q8—44no—
481Q8—44no—
482Q8—44no—
483Q8—44no—
484Q8—164no—
485D4—28no—
486D4—44no—
487D4—44no—
488D4—88no—
489D4—164no—
490D4—164no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—88no—
527C2×C4—168no—
528C2×C4#53512yes—
529C2×C4#53512yes—
530C2×C4#54614yes—
531C2×C4C2×#13714yes—
532C2×C4C2×#13714yes—
533C2×C4C22×#4111yescharacteristic
534C2×C4C2×H(5,𝔽2)12yes—
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4#58512yes—
537C8—42no—
538C8—42no—
539C8—44no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C2×#53511yescharacteristic
555C22C2×#54611yescharacteristic
556C22#350011yescharacteristic
557C22C22×#13712yes—
558C22#358512yes—
559C4—24no—
560C4—22no—
561C4—22no—
562C4—24no—
563C4—24no—
564C4—42no—
565C4—42no—
566C4—44no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4#351712yes—
573C4#354612yes—
574C2—44no—
575C2—162no—
576C2#3895611yescharacteristic
577C2#6080011yescharacteristic
578C2#6083011yescharacteristic
579C2#6116212yes—
580C2#6258212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36136C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36180C2semidirect
3. #33230C2semidirect
4. #31840C2semidirect
5. #31838C2semidirect
6. #29815C2semidirect
7. #29225C2semidirect
8. #28447C2semidirect
9. #27978C2semidirect
10. #25604C2semidirect
11. #25579C2semidirect
12. #25336C2semidirect
13. #21867C2semidirect
14. #21657C2semidirect
15. #21271C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#361362
#332302
#318402
#318382
#298152
#292252
#284472
#279782
#256042
#255792
#253362
#218672
#216572
#212712