Finite Group Databasebeta

← #222584

Subgroups — #222584

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C211yesmaximal, characteristic
2#22770C211yesmaximal, characteristic
3#22547C211yesmaximal, characteristic, complemented
4D4×#103C211yesmaximal, characteristic
5C2×#1585C211yesmaximal, characteristic
6#9834C211yesmaximal, characteristic, complemented
7#9731C211yesmaximal, characteristic, complemented
8#9228C211yesmaximal, characteristic, complemented
9#9219C211yesmaximal, characteristic
10#8982C211yesmaximal, characteristic, complemented
11#8981C211yesmaximal, characteristic
12#8655C211yesmaximal, characteristic, complemented
13#8448C211yesmaximal, characteristic, complemented
14#8447C211yesmaximal, characteristic
15#8213C211yesmaximal, characteristic, complemented
16C2×#557C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18#3386—24no—
19#3386C418yescomplemented
20C2×#501—21no—
21C2×#500C2211yescharacteristic
22C2×#477—21no—
23C2×#475C2211yescharacteristic
24C2×#478—21no—
25C2×#474C2211yescharacteristic
26C2×#446C2211yescharacteristic
27C2×#446C2211yescharacteristic
28C2×#440—21no—
29C2×C4×SD16C2211yescharacteristic
30C2×C8×D4C2211yescharacteristic
31C22×#103—21no—
32#2355C2211yescharacteristic
33D4×C4⋊C4C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1981—21no—
37#1981C2211yescharacteristic
38#1981C2211yescharacteristic
39#1960—21no—
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1933—21no—
43#1919—21no—
44#1892—21no—
45#1877—21no—
46#1872—21no—
47#1834C2211yescharacteristic
48#1805—21no—
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53#1787—21no—
54C8×C4⋊C4C2211yescharacteristic
55#1763—21no—
56C4×#103C2211yescharacteristic
57C4×#102—21no—
58#1585—24no—
59C2×#335C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1277C2211yescharacteristic
70C23×Q8—21no—
71C22×SD16D411yescharacteristic
72C23×C8—21no—
73#557—22no—
74#557—22no—
75#557—41no—
76#557C2×C414yescomplemented
77D4×Q8—22no—
78D4×Q8—22no—
79D4×Q8—41no—
80D4×Q8C2×C414yes—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—21no—
86C2×#125—41no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—41no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8C2311yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4C2311yescharacteristic
96C22×C4⋊C4—21no—
97#501—28no—
98#500—22no—
99#500C2×C414yescomplemented
100#478—28no—
101#477—28no—
102#475—22no—
103#475C2×C414yescomplemented
104#474—22no—
105#474C2×C414yes—
106#446—22no—
107#446—22no—
108#446—41no—
109#446C2×C414yescomplemented
110#440—28no—
111C4×SD16C2×C414yes—
112C4×SD16D414yes—
113C8×D4—22no—
114C8×D4—41no—
115C2×#107—21no—
116C2×#107—21no—
117C2×#107—21no—
118C2×#106—21no—
119C2×#106D411yescharacteristic
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#103—21no—
133C2×#103—21no—
134C2×#103—21no—
135C2×#103—21no—
136C2×#103—21no—
137C2×#103—21no—
138C2×#103—44no—
139C2×#103D411yescharacteristic
140C2×#103D411yescharacteristic
141C2×#103D411yescharacteristic
142C2×#102—21no—
143C2×#102—21no—
144C2×#102D411yescharacteristic
145C2×C22⋊C8—21no—
146C2×#97—21no—
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152#391—21no—
153#389—21no—
154#387—41no—
155#385—41no—
156#384C2311yescharacteristic
157#384C2311yescharacteristic
158#382—21no—
159#380—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4C2311yescharacteristic
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—41no—
170#340—41no—
171#340D411yescharacteristic
172#340D411yescharacteristic
173#336—21no—
174#335—22no—
175#332—41no—
176#332—41no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179D4⋊C8—22no—
180#322—22no—
181C22×Q8—24no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—41no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—81no—
189C22×Q8—81no—
190C22×Q8C2×D411yescharacteristic
191C22×Q8C2×D411yescharacteristic
192C22×D4C2×D411yescharacteristic
193C23×C4—21no—
194C23×C4—41no—
195C23×C4—41no—
196C2×SD16—22no—
197C2×SD16—22no—
198C2×SD16C22⋊C414yes—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—81no—
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8C22×C412yes—
222C4⋊Q8C22×C412yes—
223#125—24no—
224#125—42no—
225#125—44no—
226#125—44no—
227C22⋊Q8—24no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230C22⋊Q8—42no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—42no—
238C4×Q8—42no—
239C4×Q8C22×C412yes—
240C4×Q8C22×C412yes—
241C4×Q8C2×D412yes—
242C4×Q8#4112yes—
243C4×D4—24no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—81no—
249C4×D4C22×C412yes—
250C4×D4C2×D412yes—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—81no—
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42C2411yescharacteristic
284#107—24no—
285#107—24no—
286#107—24no—
287#106—24no—
288#106C22⋊C414yes—
289C4⋊C8—24no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—42no—
298C4⋊C8C22×C412yes—
299#103—24no—
300#103—22no—
301#103—41no—
302#103—42no—
303#103—42no—
304#103—42no—
305#103—42no—
306#103—42no—
307#103—81no—
308#103—81no—
309#103—81no—
310#103—81no—
311#103—81no—
312#103—81no—
313#103C22⋊C414yes—
314#102—24no—
315#102—42no—
316#102C22⋊C414yes—
317C22⋊C8—42no—
318#97—24no—
319#97—42no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—81no—
325C4×C8C22×C412yes—
326C4×C8C2×D412yes—
327#95—21no—
328#95—41no—
329#95—41no—
330#95—41no—
331#95—41no—
332C24—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—44no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—48no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—82no—
358C2×Q8—82no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×Q8C2×C22⋊C412yes—
363C2×Q8C4×D412yes—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4C2×C22⋊C412yes—
368C2×D4#13712yes—
369C2×D4#13712yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—44no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—161no—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396SD16—28no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—84no—
423C2×C8C2×C22⋊C412yes—
424C2×C8C2×C22⋊C412yes—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—82no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4C2×C22⋊C412yes—
459C4⋊C4C2×C22⋊C412yes—
460C4⋊C4C4×D412yes—
461C4⋊C4AGL(1,ℤ/8)12yes—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—44no—
471C42—41no—
472C42—41no—
473C42—161no—
474C42C23×C411yescharacteristic
475C42C22×D411yescharacteristic
476C42C2×#4111yescharacteristic
477C42H(5,𝔽2)11yescharacteristic
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23D4211yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—88no—
499Q8—81no—
500Q8—81no—
501Q8—81no—
502Q8—81no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4#41914yes—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—88no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—164no—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4#51811yescharacteristic
551C2×C4D4211yescharacteristic
552C2×C4#54611yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
555C2×C4C2×#13711yescharacteristic
556C2×C4C2×#13711yescharacteristic
557C2×C4#57511yescharacteristic
558C2×C4#57611yescharacteristic
559C2×C4#57711yescharacteristic
560C2×C4#57711yescharacteristic
561C2×C4#57811yescharacteristic
562C8—22no—
563C8—22no—
564C8—42no—
565C8—42no—
566C8—41no—
567C8—41no—
568C8—81no—
569C8—81no—
570C8—81no—
571C22—24no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22D4×C22⋊C411yescharacteristic
579C22#326511yescharacteristic
580C22#326611yescharacteristic
581C22#326811yescharacteristic
582C22#327011yescharacteristic
583C22#329011yescharacteristic
584C22#329311yescharacteristic
585C4—21no—
586C4—21no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—82no—
595C4—82no—
596C4—161no—
597C4—161no—
598C4C2×#41911yescharacteristic
599C4#289811yescharacteristic
600C4C4×AGL(1,ℤ/8)11yescharacteristic
601C4#294611yescharacteristic
602C4#327411yescharacteristic
603C4#327511yescharacteristic
604C4#331211yescharacteristic
605C4#332011yescharacteristic
606C2—44no—
607C2#2003911yescharacteristic
608C2#2004311yescharacteristic
609C2#2007411yescharacteristic
610C2#2009411yescharacteristic
611C2#2242511yescharacteristic
612C2#2244311yescharacteristic
613C2#2246811yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22547C2semidirect
6. #9834C2semidirect
7. #9731C2semidirect
8. #9228C2semidirect
10. #8982C2semidirect
12. #8655C2semidirect
13. #8448C2semidirect
15. #8213C2semidirect
19. #3386C4semidirect
76. #557C2×C4semidirect
99. #500C2×C4semidirect
103. #475C2×C4semidirect
109. #446C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33862
#227702
#225472
D4×#1032
C2×#15852
#98342
#97312
#92282
#92192
#89822
#89812
#86552
#84482
#84472
#82132