Finite Group Databasebeta

← #205503

Subgroups — #205503

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3131C211yesmaximal, characteristic
2#20476C211yesmaximal, characteristic, complemented
3#17793C211yesmaximal, characteristic, complemented
4#17562C211yesmaximal, characteristic, complemented
5#17561C211yesmaximal, characteristic
6#17496C211yesmaximal, characteristic, complemented
7#17495C211yesmaximal, characteristic
8#9754C214yesmaximal, complemented
9#9751C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3131—21no—
13#3131—21no—
14#3131—21no—
15#3131—21no—
16#3131C412yescomplemented
17#3131C412yescomplemented
18#3131C2212yes—
19#3131C2212yes—
20C2×#488—21no—
21C2×#488—21no—
22C2×#487C411yescharacteristic, complemented
23C2×#487C411yescharacteristic, complemented
24C2×#486C411yescharacteristic, complemented
25C2×#486C411yescharacteristic, complemented
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#416—21no—
30C2×#416—21no—
31C22×C4×C8C2211yescharacteristic
32#2283C2211yescharacteristic
33#1770C2212yescomplemented
34C8×C4⋊C4C2212yescomplemented
35#1752—21no—
36C8×C22⋊C4—21no—
37#1746—21no—
38#1726—21no—
39#1568—21no—
40#1568—21no—
41#1567—21no—
42#1566—21no—
43#1564C2212yescomplemented
44#1564C2212yescomplemented
45#1515—22no—
46#1514—22no—
47#1511—22no—
48#1510—22no—
49#1503C2214yescomplemented
50#1502C2214yes—
51#1497—22no—
52#1496—22no—
53#1491—22no—
54#1490—22no—
55#1481C2214yescomplemented
56#1480C2214yes—
57C22×#41—21no—
58C23×C8D411yescharacteristic
59C23×C8D411yescharacteristic
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533C2×C412yes—
67#533C2312yes—
68#532—21no—
69#532—21no—
70#532—21no—
71#532—21no—
72#532—21no—
73#532—21no—
74#532C2×C412yes—
75#532C2312yes—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#127C2×C411yescharacteristic
80C2×#125—21no—
81C2×#124—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×#121—21no—
85C2×#121—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×#117—21no—
91C22×C42C2311yescharacteristic
92#488—22no—
93#488—22no—
94#488—22no—
95#488—22no—
96#488—22no—
97#488—22no—
98#488—22no—
99#488—22no—
100#487—21no—
101#487—21no—
102#487—21no—
103#487—21no—
104#487C2×C412yes—
105#487C2×C412yes—
106#487C2×C412yes—
107#487C2×C412yes—
108#486—21no—
109#486—21no—
110#486—21no—
111#486—21no—
112#486C2×C412yes—
113#486C2×C412yes—
114#486C2×C412yes—
115#486C2×C412yes—
116#432—21no—
117#432—21no—
118#432—21no—
119#432—21no—
120#432—21no—
121#432—21no—
122#432D412yes—
123#432D412yes—
124#431—21no—
125#431—21no—
126#431C2312yes—
127#431C2312yes—
128#423—21no—
129#423—21no—
130C2×C4⋊C8C2312yes—
131#416—22no—
132#416—22no—
133#416—22no—
134#416—22no—
135#416—22no—
136#416—22no—
137#416—22no—
138#416—22no—
139C2×#103—21no—
140C2×#103—21no—
141C2×#103—21no—
142C2×#103—21no—
143C2×#102—21no—
144C2×#102—21no—
145C2×#102—21no—
146C2×#102—21no—
147#407—21no—
148#407—21no—
149C2×C22⋊C8—21no—
150C4×M16—21no—
151C4×M16—21no—
152C2×#97—21no—
153C2×#97—22no—
154C2×C4×C8—22no—
155C2×C4×C8C2×C411yescharacteristic
156C2×C4×C8C2×C411yescharacteristic
157C2×C4×C8C2×C411yescharacteristic
158C2×C4×C8C2×C411yescharacteristic
159C2×C4×C8D411yescharacteristic
160C2×C4×C8D411yescharacteristic
161C2×C4×C8D411yescharacteristic
162C2×C4×C8D411yescharacteristic
163C2×C4×C8D411yescharacteristic
164C2×C4×C8D411yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167#383D412yes—
168#382—21no—
169#381D412yes—
170#380—21no—
171C4×C4⋊C4C2312yes—
172C4×C22⋊C4—21no—
173#376—21no—
174#336—22no—
175#336—22no—
176#335—22no—
177#335—22no—
178#334—22no—
179#334—22no—
180#328—28no—
181#327—28no—
182Q8⋊C8—28no—
183D4⋊C8—28no—
184C2×#41—22no—
185C2×#41—22no—
186C2×#41—44no—
187C2×#41—44no—
188C2×#41—41no—
189C2×#41—41no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—41no—
197C23×C4—41no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4#4111yescharacteristic
201C2×M16—41no—
202C2×M16—41no—
203C22×C8—22no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8C22⋊C411yescharacteristic
207C22×C8C22⋊C411yescharacteristic
208C22×C8C22⋊C411yescharacteristic
209C22×C8C22⋊C411yescharacteristic
210C22×C8C22⋊C411yescharacteristic
211C22×C8C22⋊C411yescharacteristic
212C22×C8C22⋊C411yescharacteristic
213C22×C8C22⋊C411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8C22×C412yes—
228C4⋊Q8C22×C412yes—
229C4⋊Q8C22×C412yes—
230#127—21no—
231#127—21no—
232#127—21no—
233#127C22×C412yes—
234#125—22no—
235#125—22no—
236#125—41no—
237#125—41no—
238#124—22no—
239#124—22no—
240#124—41no—
241#124—41no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C22⋊Q8—41no—
249C22⋊Q8—41no—
250#121—22no—
251#121—22no—
252#121—22no—
253#121—22no—
254#121—41no—
255#121—41no—
256#121—41no—
257#121—41no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—41no—
263C4×Q8—41no—
264C4×Q8—41no—
265C4×Q8—41no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274#117—22no—
275#117—22no—
276#117—41no—
277#117—41no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—41no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C42—22no—
292C2×C42—41no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42C2411yescharacteristic
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—41no—
309C4⋊C8—41no—
310C4⋊C8C2×D414yes—
311#103—22no—
312#103—22no—
313#103—22no—
314#103—22no—
315#103—22no—
316#103—22no—
317#103—22no—
318#103—22no—
319#102—22no—
320#102—22no—
321#102—22no—
322#102—22no—
323#102—22no—
324#102—22no—
325#102—22no—
326#102—22no—
327C22⋊C8—22no—
328C22⋊C8—22no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—41no—
332C22⋊C8—41no—
333#97—22no—
334#97—22no—
335#97—44no—
336#97—41no—
337#97—41no—
338C4×C8—22no—
339C4×C8—22no—
340C4×C8—44no—
341C4×C8C22⋊C411yescharacteristic
342C4×C8C22⋊C411yescharacteristic
343C4×C8C22⋊C411yescharacteristic
344C4×C8C22⋊C411yescharacteristic
345C4×C8C22⋊C411yescharacteristic
346C4×C8C22⋊C411yescharacteristic
347C4×C8C22⋊C411yescharacteristic
348C4×C8C22⋊C411yescharacteristic
349C4×C8C22×C411yescharacteristic
350C4×C8C22×C411yescharacteristic
351C4×C8C22×C411yescharacteristic
352C4×C8C22×C411yescharacteristic
353C4×C8C2×D411yescharacteristic
354C4×C8C2×D411yescharacteristic
355C4×C8C2×D411yescharacteristic
356C4×C8C2×D411yescharacteristic
357#95—22no—
358#95—22no—
359#95—22no—
360#95C2×D414yes—
361C24—41no—
362C24#12111yescharacteristic
363#41—44no—
364#41—44no—
365#41—44no—
366#41—44no—
367#41—84no—
368#41—84no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—84no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—84no—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12411yescharacteristic
415C22×C4#12411yescharacteristic
416C22×C4#12711yescharacteristic
417C22×C4#12711yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421M16—82no—
422M16—82no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8C2×C22⋊C411yescharacteristic
441C2×C8C2×C22⋊C411yescharacteristic
442C2×C8C2×C22⋊C411yescharacteristic
443C2×C8C2×C22⋊C411yescharacteristic
444C2×C8C2×C22⋊C411yescharacteristic
445C2×C8C2×C22⋊C411yescharacteristic
446C2×C8C2×C22⋊C411yescharacteristic
447C2×C8C2×C22⋊C411yescharacteristic
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42C2×C22⋊C411yescharacteristic
477C42C2×C22⋊C411yescharacteristic
478C42C4×D411yescharacteristic
479C42C4×D411yescharacteristic
480C42C4×D411yescharacteristic
481C42C4×D411yescharacteristic
482C42#12111yescharacteristic
483C42#12111yescharacteristic
484C42#12411yescharacteristic
485C42#12411yescharacteristic
486C42#12711yescharacteristic
487C42#12711yescharacteristic
488C42C23×C411yescharacteristic
489C42C22×D411yescharacteristic
490C42C22×D411yescharacteristic
491C42C2×#4111yescharacteristic
492C23—22no—
493C23—22no—
494C23—41no—
495C23—42no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23#39011yescharacteristic
500C23#39011yescharacteristic
501C23C2×#12111yescharacteristic
502Q8—42no—
503Q8—42no—
504Q8—44no—
505Q8—82no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509D4—42no—
510D4—42no—
511D4—44no—
512D4—82no—
513D4—82no—
514D4—82no—
515D4—82no—
516D4—84no—
517D4—84no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—84no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—84no—
548C2×C4#39011yescharacteristic
549C2×C4#39011yescharacteristic
550C2×C4#39011yescharacteristic
551C2×C4#39011yescharacteristic
552C2×C4#39011yescharacteristic
553C2×C4#39011yescharacteristic
554C2×C4C22×C22⋊C411yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4C2×C4×D411yescharacteristic
557C2×C4C2×#12111yescharacteristic
558C2×C4C2×#12411yescharacteristic
559C2×C4C2×#12711yescharacteristic
560C2×C4#57512yes—
561C2×C4#57512yes—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—44no—
571C8—82no—
572C8—82no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—164no—
585C22#197612yes—
586C22C2×#39011yescharacteristic
587C22#304012yes—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—84no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4#289611yescharacteristic
602C4#289611yescharacteristic
603C4#295712yes—
604C4#304012yes—
605C2—22no—
606C2—22no—
607C2—41no—
608C2—162no—
609C2#2124011yescharacteristic
610C2C2×#197611yescharacteristic
611C2#2126111yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20476C2semidirect
3. #17793C2semidirect
4. #17562C2semidirect
6. #17496C2semidirect
8. #9754C2semidirect
9. #9751C2semidirect
16. #3131C4semidirect
17. #3131C4semidirect
22. C2×#487C4semidirect
23. C2×#487C4semidirect
24. C2×#486C4semidirect
25. C2×#486C4semidirect
33. #1770C22semidirect
34. C8×C4⋊C4C22semidirect
43. #1564C22semidirect
44. #1564C22semidirect
49. #1503C22semidirect
55. #1481C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31312
#204762
#177932
#175622
#175612
#174962
#174952
#97542
#97512