Finite Group Databasebeta

← #204553

Subgroups — #204553

Order 512.

1257 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34965C211yesmaximal, characteristic
2#20809C211yesmaximal, characteristic, complemented
3#20550C211yesmaximal, characteristic, complemented
4C2×#1495C211yesmaximal, characteristic, complemented
5C2×#1482C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic, complemented
7C2×#1478C211yesmaximal, characteristic, complemented
8#9678C212yesmaximal, complemented
9#9676C212yesmaximal, complemented
10#9675C212yesmaximal, complemented
11#9674C212yesmaximal, complemented
12C2×#529C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2603—21no—
19#2603—21no—
20#2566C412yes—
21#2544C412yes—
22#2504C411yescharacteristic
23#2502C411yescharacteristic
24#2435C411yescharacteristic
25#2433C411yescharacteristic
26#2411—22no—
27#2392—21no—
28#2392—21no—
29C4×#127C2212yes—
30C4×#124C2212yes—
31C4×#120—22no—
32C4×#117C2211yescharacteristic
33#1843—21no—
34#1832—21no—
35#1798—22no—
36#1770—21no—
37C8×C4⋊C4—21no—
38#1767C2212yescomplemented
39C4×C4⋊C8C2212yescomplemented
40C4×C4⋊C8C2212yescomplemented
41C4×C4⋊C8C2212yescomplemented
42#1753—22no—
43#1495—24no—
44#1486—24no—
45#1482—24no—
46#1478—24no—
47C2×Q8⋊C8C2211yescharacteristic, complemented
48C2×Q8⋊C8C2211yescharacteristic, complemented
49C2×Q8⋊C8C2212yescomplemented
50C2×D4⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52C2×D4⋊C8C2212yes—
53C2×D4⋊C8C2212yes—
54C2×D4⋊C8C2211yescharacteristic, complemented
55C2×D4⋊C8C2211yescharacteristic, complemented
56C2×D4⋊C8C2212yescomplemented
57C23×D4C2×C411yescharacteristic
58C24×C4—21no—
59C22×M16—21no—
60C2×#127C2×C412yes—
61#529—22no—
62#529—22no—
63#529—22no—
64#529—22no—
65C2×#126—21no—
66C2×#125C2×C411yescharacteristic
67C2×#124C2×C411yescharacteristic
68C2×#124C2×C412yes—
69C2×#123—21no—
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×#120—22no—
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4C2×C412yes—
84C2×C4×D4C2311yescharacteristic
85C2×C4×D4C2312yes—
86C2×#117—21no—
87C2×#117D412yes—
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C42C2311yescharacteristic
91#431—22no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431—22no—
96#431D414yescomplemented
97#423—22no—
98#423D414yescomplemented
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×#97—22no—
111C2×C4×C8—21no—
112C2×C4×C8—22no—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#398—21no—
117#398—21no—
118#397—21no—
119#397—21no—
120#396—22no—
121#396—22no—
122#394—22no—
123#394—22no—
124#394—21no—
125#394—21no—
126#392—21no—
127#392—21no—
128#391C2×C412yes—
129#390—22no—
130#390—22no—
131#390C2×C412yes—
132#390C2×C412yes—
133#390C2×C412yes—
134#388—22no—
135#388—22no—
136#386—22no—
137#386—22no—
138#386—22no—
139#385—22no—
140#385—22no—
141#385—22no—
142#384C2×C412yes—
143#382C2×C412yes—
144#379—22no—
145C4×C4⋊C4C2×C412yes—
146C4×C4⋊C4D411yescharacteristic
147C4×C4⋊C4D411yescharacteristic
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4C2×C411yescharacteristic
151C4×C22⋊C4C2×C411yescharacteristic
152C4×C22⋊C4D412yes—
153#376D411yescharacteristic
154#376D411yescharacteristic
155C43C2312yes—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162Q8⋊C8—24no—
163D4⋊C8—24no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166D4⋊C8—24no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—24no—
170C25—21no—
171C22×Q8C22×C411yescharacteristic
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—42no—
180C22×D4C22×C411yescharacteristic
181C22×D4C22×C412yes—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—81no—
189C23×C4—81no—
190C23×C4—81no—
191C23×C4—81no—
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—24no—
196C2×M16—81no—
197C2×M16—81no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—21no—
204#127—24no—
205#127—42no—
206#126—42no—
207#126—42no—
208#125—22no—
209#125—22no—
210#124—24no—
211#124—22no—
212#124—22no—
213#124—42no—
214#123—42no—
215#123—42no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218#121—22no—
219#121—22no—
220#120—48no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—82no—
242C4×D4—82no—
243C4×D4—82no—
244C4×D4—82no—
245#117—44no—
246#117—42no—
247#117—42no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4C22⋊C412yes—
280C2×C22⋊C4C22⋊C412yes—
281C2×C42—21no—
282C2×C42C22×C412yes—
283C2×C42C22×C412yes—
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42C2411yescharacteristic
296C4⋊C8—24no—
297C4⋊C8—22no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—44no—
301C4⋊C8—44no—
302C4⋊C8—42no—
303C4⋊C8—42no—
304C4⋊C8C2×D414yes—
305C4⋊C8C2×D412yes—
306C4⋊C8C2×D412yes—
307C22⋊C8—24no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310#97—82no—
311C4×C8—24no—
312C4×C8—22no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—22no—
318C4×C8—82no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95C22⋊C411yescharacteristic
324#95C22⋊C411yescharacteristic
325#95C22⋊C411yescharacteristic
326#95C22⋊C411yescharacteristic
327#95C22⋊C412yes—
328#95C22⋊C412yes—
329C24—22no—
330C24—21no—
331C24—44no—
332C24—44no—
333C24—42no—
334C24—81no—
335C24—81no—
336C24C2×C22⋊C411yescharacteristic
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×D4—28no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—82no—
362C2×D4—82no—
363C2×D4#13114yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C412yes—
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4#12412yes—
409C22×C4#12712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413M16—84no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—41no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—41no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—41no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—44no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—42no—
475C42—41no—
476C42—41no—
477C42C4×D412yes—
478C42C4×D412yes—
479C42#12112yes—
480C42#12112yes—
481C42#12412yes—
482C42#12712yes—
483C42C22×D411yescharacteristic
484C42C22×D411yescharacteristic
485C42C2×#4111yescharacteristic
486C42C2×#4111yescharacteristic
487C23—22no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—24no—
492C23—48no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—84no—
505C23—84no—
506C23—88no—
507C23—88no—
508C23#39012yes—
509C23C22×C22⋊C411yescharacteristic
510Q8—41no—
511Q8—41no—
512Q8—42no—
513D4—24no—
514D4—24no—
515D4—48no—
516D4—41no—
517D4—42no—
518D4—41no—
519D4—42no—
520D4—42no—
521D4—88no—
522D4—88no—
523D4—84no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—88no—
560C2×C4—84no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#39012yes—
570C2×C4#39012yes—
571C2×C4#39012yes—
572C2×C4#42114yes—
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×#12411yescharacteristic
578C2×C4C2×#12711yescharacteristic
579C2×C4C2×#13112yes—
580C2×C4#56812yes—
581C2×C4#57611yescharacteristic
582C2×C4#57611yescharacteristic
583C2×C4#57711yescharacteristic
584C2×C4#57711yescharacteristic
585C8—44no—
586C8—42no—
587C8—42no—
588C8—82no—
589C8—81no—
590C8—81no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—816no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—82no—
608C22—81no—
609C22C2×#39011yescharacteristic
610C22#288012yes—
611C22#289711yescharacteristic
612C22#289711yescharacteristic
613C22C2×#42112yes—
614C4—21no—
615C4—21no—
616C4—22no—
617C4—22no—
618C4—24no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4#292911yescharacteristic
628C4#293011yescharacteristic
629C4#293312yes—
630C4#304111yescharacteristic
631C4#304311yescharacteristic
632C4#304511yescharacteristic
633C4#304511yescharacteristic
634C2—42no—
635C2—84no—
636C2—161no—
637C2#1752112yes—
638C2#1752212yes—
639C2#2119111yescharacteristic
640C2#2125211yescharacteristic
641C2#2126311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2433C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20809C2semidirect
3. #20550C2semidirect
4. C2×#1495C2semidirect
5. C2×#1482C2semidirect
6. C2×#1486C2semidirect
7. C2×#1478C2semidirect
8. #9678C2semidirect
9. #9676C2semidirect
10. #9675C2semidirect
11. #9674C2semidirect
38. #1767C22semidirect
39. C4×C4⋊C8C22semidirect
40. C4×C4⋊C8C22semidirect
41. C4×C4⋊C8C22semidirect
47. C2×Q8⋊C8C22semidirect
48. C2×Q8⋊C8C22semidirect
49. C2×Q8⋊C8C22semidirect
54. C2×D4⋊C8C22semidirect
55. C2×D4⋊C8C22semidirect
56. C2×D4⋊C8C22semidirect
96. #431D4semidirect
98. #423D4semidirect

Maximal subgroups

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

subgroupindex
#349652
#208092
#205502
C2×#14952
C2×#14822
C2×#14862
C2×#14782
#96782
#96762
#96752
#96742