Finite Group Databasebeta

← #205075

Subgroups — #205075

Order 512.

1017 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic, complemented
3#21816C211yesmaximal, characteristic
4#20818C211yesmaximal, characteristic, complemented
5#17562C211yesmaximal, characteristic, complemented
6#17561C211yesmaximal, characteristic
7#17323C211yesmaximal, characteristic, complemented
8#9722C214yesmaximal, complemented
9#9717C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3167—21no—
13#3167—21no—
14#3167—21no—
15#3167—21no—
16#3167C412yescomplemented
17#3167C412yescomplemented
18#3167C2212yes—
19#3167C2212yes—
20C2×#498C411yescharacteristic, complemented
21C2×#492C411yescharacteristic, complemented
22C2×#487C411yescharacteristic, complemented
23C2×#486C411yescharacteristic, complemented
24C2×#465—21no—
25C2×#465—21no—
26C2×C4×SD16—21no—
27C2×C4×SD16—21no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2283C2211yescharacteristic
32#2283C2211yescharacteristic
33#1995—22no—
34#1986C2214yescomplemented
35#1980C2214yes—
36#1963—22no—
37#1846—21no—
38#1835—21no—
39#1826—22no—
40#1808—21no—
41#1807—21no—
42#1789—22no—
43C4×#107C2212yescomplemented
44C4×#106C2212yescomplemented
45#1515—22no—
46#1514—22no—
47#1511—22no—
48#1510—22no—
49#1503C2214yescomplemented
50#1502C2214yes—
51#1452—21no—
52#1451—21no—
53#1451C2212yescomplemented
54#1450—21no—
55C8×M16—21no—
56C8×M16C2212yescomplemented
57C22×#41—21no—
58C22×SD16—21no—
59C22×SD16—21no—
60C23×C8D411yescharacteristic
61C23×C8D411yescharacteristic
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533—21no—
67#533—21no—
68#533C2×C412yes—
69#533C2312yes—
70#532—21no—
71#532—21no—
72#532—21no—
73#532—21no—
74#532—21no—
75#532—21no—
76#532C2×C412yes—
77#532C2312yes—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#127C2×C411yescharacteristic
82C2×#125—21no—
83C2×#124—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×#121—21no—
87C2×#121—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×#117—21no—
93C22×C42C2311yescharacteristic
94#498—21no—
95#498—21no—
96#498C2×C412yes—
97#498C2×C412yes—
98#492—21no—
99#492—21no—
100#492C2×C412yes—
101#492C2×C412yes—
102#487—21no—
103#487—21no—
104#487C2×C412yes—
105#487C2×C412yes—
106#486—21no—
107#486—21no—
108#486C2×C412yes—
109#486C2×C412yes—
110#465—22no—
111#465—22no—
112#465—22no—
113#465—22no—
114#465—22no—
115#465—22no—
116#465—22no—
117#465—22no—
118C4×SD16—22no—
119C4×SD16—22no—
120C4×SD16—22no—
121C4×SD16—22no—
122C4×SD16—22no—
123C4×SD16—22no—
124C4×SD16—22no—
125C4×SD16—22no—
126#432—21no—
127#432—21no—
128#432—21no—
129#432D412yes—
130#431—21no—
131#431—21no—
132#431—21no—
133#431C2312yes—
134C2×#107—22no—
135C2×#106—21no—
136C2×#106—21no—
137C2×#106—22no—
138#423—21no—
139#423D412yes—
140C2×#104—28no—
141C2×#103—21no—
142C2×#103—21no—
143C2×#102—21no—
144C2×#102—21no—
145#407D412yes—
146#407D412yes—
147C4×M16—21no—
148C4×M16C2312yes—
149C2×C4×C8C2×C411yescharacteristic
150C2×C4×C8C2×C411yescharacteristic
151C2×C4×C8C2×C411yescharacteristic
152C2×C4×C8C2×C411yescharacteristic
153C2×C4×C8D411yescharacteristic
154C2×C4×C8D411yescharacteristic
155C2×C4×C8D411yescharacteristic
156C2×C4×C8D411yescharacteristic
157C2×C4×C8D411yescharacteristic
158C2×C4×C8D411yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161#383—21no—
162#383—21no—
163#382—21no—
164#382—21no—
165#381—21no—
166#381—21no—
167#380—21no—
168#380—21no—
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2312yes—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173#376—21no—
174#376—21no—
175#340—22no—
176#340—22no—
177#339—28no—
178#328—28no—
179#327—28no—
180#322—22no—
181#322—22no—
182#322—22no—
183C82—22no—
184C2×#41—22no—
185C2×#41—22no—
186C2×#41—41no—
187C2×#41—41no—
188C2×#41—84no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—41no—
196C23×C4—41no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C2×SD16—22no—
201C2×SD16—22no—
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16—44no—
205C2×SD16—44no—
206C2×M16—22no—
207C2×M16C2×D414yes—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8C22⋊C411yescharacteristic
211C22×C8C22⋊C411yescharacteristic
212C22×C8C22⋊C411yescharacteristic
213C22×C8C22⋊C411yescharacteristic
214C22×C8C22⋊C411yescharacteristic
215C22×C8C22⋊C411yescharacteristic
216C22×C8C22⋊C411yescharacteristic
217C22×C8C22⋊C411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8C22×C412yes—
232C4⋊Q8C22×C412yes—
233C4⋊Q8C22×C412yes—
234#127—21no—
235#127—21no—
236#127—21no—
237#127C22×C412yes—
238#125—22no—
239#125—22no—
240#125—41no—
241#125—41no—
242#124—22no—
243#124—22no—
244#124—41no—
245#124—41no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—41no—
251C22⋊Q8—41no—
252C22⋊Q8—41no—
253C22⋊Q8—41no—
254#121—22no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—41no—
259#121—41no—
260#121—41no—
261#121—41no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—41no—
267C4×Q8—41no—
268C4×Q8—41no—
269C4×Q8—41no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278#117—22no—
279#117—22no—
280#117—41no—
281#117—41no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—41no—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42#4111yescharacteristic
308C2×C42#4111yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42#4111yescharacteristic
311C2×C42C2411yescharacteristic
312#107—44no—
313#106—22no—
314#106—22no—
315#106—22no—
316#106—22no—
317#106—44no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322#104—88no—
323#103—22no—
324#103—22no—
325#103—22no—
326#103—22no—
327#102—22no—
328#102—22no—
329#102—22no—
330#102—22no—
331C22⋊C8—22no—
332C22⋊C8—22no—
333C22⋊C8—22no—
334C22⋊C8C2×D414yes—
335#97—22no—
336#97—22no—
337C4×C8—22no—
338C4×C8—22no—
339C4×C8C22⋊C411yescharacteristic
340C4×C8C22⋊C411yescharacteristic
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×C8C2×D411yescharacteristic
352C4×C8C2×D411yescharacteristic
353C4×C8C2×D411yescharacteristic
354C4×C8C2×D411yescharacteristic
355#95—22no—
356#95—22no—
357#95—41no—
358#95—41no—
359#95—41no—
360#95—41no—
361#95—41no—
362#95—41no—
363C24—41no—
364C24#12711yescharacteristic
365#41—48no—
366#41—84no—
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—24no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—84no—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
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×C4#12711yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422SD16—44no—
423SD16—44no—
424SD16—44no—
425SD16—44no—
426M16—28no—
427M16—44no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8C2×C22⋊C411yescharacteristic
443C2×C8C2×C22⋊C411yescharacteristic
444C2×C8C2×C22⋊C411yescharacteristic
445C2×C8C2×C22⋊C411yescharacteristic
446C2×C8C2×C22⋊C411yescharacteristic
447C2×C8C2×C22⋊C411yescharacteristic
448C2×C8C2×C22⋊C411yescharacteristic
449C2×C8C2×C22⋊C411yescharacteristic
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C42—42no—
479C42—42no—
480C42—44no—
481C42—44no—
482C42C2×C22⋊C411yescharacteristic
483C42C2×C22⋊C411yescharacteristic
484C42C4×D411yescharacteristic
485C42C4×D411yescharacteristic
486C42C4×D411yescharacteristic
487C42C4×D411yescharacteristic
488C42#12111yescharacteristic
489C42#12111yescharacteristic
490C42#12111yescharacteristic
491C42#12111yescharacteristic
492C42#12411yescharacteristic
493C42#12411yescharacteristic
494C42C23×C411yescharacteristic
495C42C22×D411yescharacteristic
496C42C2×#4111yescharacteristic
497C42C2×#4111yescharacteristic
498C23—22no—
499C23—22no—
500C23—41no—
501C23—42no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23#39011yescharacteristic
506C23#39011yescharacteristic
507C23C2×#12711yescharacteristic
508Q8—42no—
509Q8—42no—
510Q8—44no—
511Q8—82no—
512Q8—82no—
513Q8—82no—
514Q8—82no—
515D4—42no—
516D4—42no—
517D4—44no—
518D4—82no—
519D4—82no—
520D4—82no—
521D4—82no—
522D4—84no—
523D4—84no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4#39011yescharacteristic
559C2×C4#39011yescharacteristic
560C2×C4#39011yescharacteristic
561C2×C4#39011yescharacteristic
562C2×C4#39011yescharacteristic
563C2×C4#39011yescharacteristic
564C2×C4C22×C22⋊C411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12411yescharacteristic
570C2×C4#57714yes—
571C8—21no—
572C8—21no—
573C8—21no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—44no—
578C8—44no—
579C8#42112yes—
580C8#42112yes—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—164no—
593C22#197612yes—
594C22C2×#39011yescharacteristic
595C22#315412yes—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—162no—
609C4—162no—
610C4C2×#42111yescharacteristic
611C4C2×#42111yescharacteristic
612C4#294912yes—
613C4#305712yes—
614C2—22no—
615C2—22no—
616C2—41no—
617C2—162no—
618C2#2123111yescharacteristic
619C2C2×#197611yescharacteristic
620C2#2125211yescharacteristic
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. #21845C2semidirect
4. #20818C2semidirect
5. #17562C2semidirect
7. #17323C2semidirect
8. #9722C2semidirect
9. #9717C2semidirect
16. #3167C4semidirect
17. #3167C4semidirect
20. C2×#498C4semidirect
21. C2×#492C4semidirect
22. C2×#487C4semidirect
23. C2×#486C4semidirect
34. #1986C22semidirect
43. C4×#107C22semidirect
44. C4×#106C22semidirect
49. #1503C22semidirect
53. #1451C22semidirect
56. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#218452
#218162
#208182
#175622
#175612
#173232
#97222
#97172