Finite Group Databasebeta

← #205037

Subgroups — #205037

Order 512.

1091 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35241C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1521C211yesmaximal, characteristic
5C2×#1515C211yesmaximal, characteristic, complemented
6C2×#1512C211yesmaximal, characteristic
7#13781C211yesmaximal, characteristic, complemented
8#9716C212yesmaximal
9#9714C212yesmaximal, complemented
10#9713C212yesmaximal
11#9711C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#529C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2850C2211yescharacteristic
19#2850C2211yescharacteristic
20#2664—22no—
21#2568C411yescharacteristic, complemented
22#2565C411yescharacteristic, complemented
23#2565C411yescharacteristic, complemented
24#2547C411yescharacteristic, complemented
25#2546C411yescharacteristic, complemented
26#2545C411yescharacteristic, complemented
27#2545C411yescharacteristic, complemented
28#2542C411yescharacteristic, complemented
29#2498—22no—
30#2417—22no—
31#2323—22no—
32C2×#383C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1940—22no—
36#1939—22no—
37#1797C2212yes—
38#1797C2212yes—
39#1521—24no—
40#1518—24no—
41#1515—24no—
42#1512—24no—
43C2×#328C2212yes—
44C2×#328C2212yes—
45C2×#328C2212yes—
46C2×#328C2212yes—
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#327C2212yes—
50C2×#327C2212yes—
51#1436—22no—
52#1436—22no—
53#1434C2212yes—
54C4×#95C2212yes—
55#533—22no—
56#533—22no—
57#533—22no—
58#533—22no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63#529—22no—
64#529—22no—
65#529—22no—
66#529—22no—
67C2×#126—21no—
68C2×#125—21no—
69C2×#125C2×C411yescharacteristic
70C2×#124C2×C411yescharacteristic
71C2×#124C2×C411yescharacteristic
72C2×#124C2×C411yescharacteristic
73C2×#123—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8C2311yescharacteristic
77C2×C22⋊Q8C2311yescharacteristic
78C2×C22⋊Q8C2311yescharacteristic
79C2×#121C2311yescharacteristic
80C2×C4×Q8—21no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×#117—21no—
87C2×#117—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D412yes—
90C22×C42—22no—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#432—22no—
95#432—22no—
96#431—22no—
97#431—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C22⋊C8C2312yes—
103C2×C22⋊C8C2312yes—
104C2×#97—21no—
105C2×C4×C8—21no—
106#400—22no—
107#399—22no—
108#398—22no—
109#398—22no—
110#398—22no—
111#397—22no—
112#396—22no—
113#396—22no—
114#391—22no—
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#389—22no—
124#389—22no—
125#389—22no—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#387—22no—
131#387—22no—
132#386—22no—
133#386—22no—
134#384C2×C411yescharacteristic
135#384C2×C411yescharacteristic
136#383—22no—
137#383D414yes—
138#382—22no—
139#382C2×C411yescharacteristic
140#382C2×C411yescharacteristic
141#382D414yes—
142#380—22no—
143#380—22no—
144#380—22no—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4D412yes—
150C4×C4⋊C4D412yes—
151C2×#95—21no—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95D412yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160#336—22no—
161#336—22no—
162#328—24no—
163#328—24no—
164#328—24no—
165#328—24no—
166#328—24no—
167#328—24no—
168#327—24no—
169#327—24no—
170C22×Q8—21no—
171C22×Q8C22×C411yescharacteristic
172C22×Q8C22×C411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×D4—21no—
175C22×D4C22×C411yescharacteristic
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—81no—
191C4⋊Q8—81no—
192#126—42no—
193#126—42no—
194#125—22no—
195#125—22no—
196#125—42no—
197#125—42no—
198#124—22no—
199#124—22no—
200#124—81no—
201#124—81no—
202#123—42no—
203#123—42no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—48no—
217C22⋊Q8—42no—
218C22⋊Q8—42no—
219#121—21no—
220#121—21no—
221#121—21no—
222#121—21no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—42no—
234C4×D4—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—44no—
280C2×C42—81no—
281C2×C42—81no—
282C2×C42—82no—
283C2×C42—82no—
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42#4112yes—
297C2×C42#4112yes—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—82no—
301C4⋊C8—82no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306#97—42no—
307C4×C8—42no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—44no—
322#95—82no—
323#95—82no—
324#95C22⋊C412yes—
325#95C22⋊C412yes—
326#95C22⋊C412yes—
327#95C22⋊C412yes—
328#95C2×D414yes—
329#95C2×D414yes—
330C24—21no—
331C24C22×D411yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—44no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—42no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—42no—
353C2×D4—42no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C412yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—84no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C22⋊C4—44no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C42—42no—
461C42—42no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—88no—
470C42C4×D412yes—
471C42C4×D412yes—
472C42C4×D412yes—
473C42C4×D412yes—
474C42#12412yes—
475C42#12412yes—
476C42#12712yes—
477C42#12712yes—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23C22×C22⋊C411yescharacteristic
487C23C2×#12111yescharacteristic
488C23C2×#12111yescharacteristic
489C23#57712yes—
490C23#57812yes—
491Q8—81no—
492Q8—81no—
493Q8—81no—
494Q8—81no—
495Q8—81no—
496Q8—81no—
497Q8—84no—
498D4—81no—
499D4—81no—
500D4—81no—
501D4—81no—
502D4—82no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—84no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4#39012yes—
568C2×C4#39012yes—
569C2×C4#39012yes—
570C2×C4#39012yes—
571C2×C4#42114yes—
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×#12411yescharacteristic
575C2×C4C2×#12711yescharacteristic
576C8—82no—
577C8—82no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—81no—
587C22—82no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22C2×#39011yescharacteristic
592C22#288311yescharacteristic
593C22#288411yescharacteristic
594C22#289812yes—
595C22C2×#42112yes—
596C22#304211yescharacteristic
597C22#304411yescharacteristic
598C22#304612yes—
599C4—22no—
600C4—22no—
601C4—24no—
602C4—22no—
603C4—22no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—84no—
612C4—84no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—81no—
620C2—21no—
621C2—22no—
622C2—21no—
623C2—161no—
624C2#1759812yes—
625C2#1760112yes—
626C2#2120911yescharacteristic
627C2#2121411yescharacteristic
628C2#2126211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
3. C2×#1518C2semidirect
5. C2×#1515C2semidirect
7. #13781C2semidirect
9. #9714C2semidirect
11. #9711C2semidirect
21. #2568C4semidirect
22. #2565C4semidirect
23. #2565C4semidirect
24. #2547C4semidirect
25. #2546C4semidirect
26. #2545C4semidirect
27. #2545C4semidirect
28. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#352412
#219532
C2×#15182
C2×#15212
C2×#15152
C2×#15122
#137812
#97162
#97142
#97132
#97112