Finite Group Databasebeta

← #222520

Subgroups — #222520

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×SD16C211yesmaximal, characteristic
2#22770C211yesmaximal, characteristic
3#22525C211yesmaximal, characteristic, complemented
4D4×#103C211yesmaximal, characteristic
5C2×#1582C211yesmaximal, characteristic
6#9828C211yesmaximal, characteristic, complemented
7#9729C211yesmaximal, characteristic, complemented
8#9228C211yesmaximal, characteristic, complemented
9#9215C211yesmaximal, characteristic
10#8982C211yesmaximal, characteristic, complemented
11#8973C211yesmaximal, characteristic
12#8647C211yesmaximal, characteristic, complemented
13#8451C211yesmaximal, characteristic, complemented
14#8447C211yesmaximal, characteristic
15#8201C211yesmaximal, characteristic, complemented
16C2×Q82C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18Q8×SD16—24no—
19Q8×SD16C418yescomplemented
20C2×#498—21no—
21C2×#498C2211yescharacteristic
22C2×#475—21no—
23C2×#475C2211yescharacteristic
24C2×#476—21no—
25C2×#476C2211yescharacteristic
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×C4×SD16—21no—
29C2×C4×SD16C2211yescharacteristic
30C2×C8×D4C2211yescharacteristic
31C22×#103—21no—
32#2355C2211yescharacteristic
33D4×C4⋊C4C2211yescharacteristic
34#1993—21no—
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1988—21no—
38#1987C2211yescharacteristic
39#1981C2211yescharacteristic
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#1582—24no—
59C2×#334C2211yescharacteristic
60C2×#333C2211yescharacteristic
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—
73Q82—22no—
74Q82—22no—
75Q82—41no—
76Q82C2×C414yescomplemented
77D4×Q8—22no—
78D4×Q8—22no—
79D4×Q8—41no—
80D4×Q8C2×C414yes—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—41no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
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#498—28no—
98#498—22no—
99#498C2×C414yescomplemented
100#476—22no—
101#476—28no—
102#476C2×C414yes—
103#475—22no—
104#475—28no—
105#475C2×C414yescomplemented
106#446—22no—
107#446—41no—
108C8×Q8—22no—
109C8×Q8C2×C414yescomplemented
110C4×SD16—28no—
111C4×SD16C2×C414yes—
112C4×SD16D414yes—
113C8×D4—22no—
114C8×D4—41no—
115C2×#106—21no—
116C2×#106—21no—
117C2×#106—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—44no—
135C2×#103—41no—
136C2×#103D411yescharacteristic
137C2×#103D411yescharacteristic
138C2×#103D411yescharacteristic
139C2×#102—21no—
140C2×#102—21no—
141C2×#102D411yescharacteristic
142C2×C22⋊C8—21no—
143C2×#97—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#391—21no—
150#389—21no—
151#387—41no—
152#385—41no—
153#384C2311yescharacteristic
154#384C2311yescharacteristic
155#382—21no—
156#380—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2311yescharacteristic
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—41no—
171#340D411yescharacteristic
172#340D411yescharacteristic
173#336—21no—
174#334—22no—
175#333—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—41no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—81no—
190C22×Q8—81no—
191C22×Q8C2×D411yescharacteristic
192C22×Q8C2×D411yescharacteristic
193C22×D4C2×D411yescharacteristic
194C23×C4—21no—
195C23×C4—41no—
196C23×C4—41no—
197C2×SD16—22no—
198C2×SD16—22no—
199C2×SD16C22⋊C414yes—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—81no—
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214C4⋊Q8—24no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223C4⋊Q8—42no—
224C4⋊Q8—44no—
225C4⋊Q8—44no—
226C4⋊Q8C22×C412yes—
227C4⋊Q8C22×C412yes—
228C22⋊Q8—24no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231C22⋊Q8—42no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—42no—
239C4×Q8—42no—
240C4×Q8C22×C412yes—
241C4×Q8C22×C412yes—
242C4×Q8C2×D412yes—
243C4×Q8#4112yes—
244C4×D4—24no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—81no—
250C4×D4C22×C412yes—
251C4×D4C2×D412yes—
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—21no—
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#106—24no—
285#106—24no—
286#106—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—42no—
302#103—42no—
303#103—42no—
304#103—42no—
305#103—81no—
306#103—81no—
307#103—81no—
308#103—82no—
309#103C22⋊C414yes—
310#103SD1614yescomplemented
311#102—24no—
312#102—42no—
313#102C22⋊C414yes—
314C22⋊C8—42no—
315#97—42no—
316C4×C8—24no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—81no—
322C4×C8C22×C412yes—
323C4×C8C2×D412yes—
324#95—21no—
325#95—41no—
326#95—41no—
327#95—41no—
328#95—41no—
329C24—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—48no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—82no—
350C2×Q8—82no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×Q8—161no—
357C2×Q8—161no—
358C2×Q8C2×C22⋊C412yes—
359C2×Q8C4×D412yes—
360C2×Q8C2×SD1612yes—
361C2×Q8#13512yes—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4C2×C22⋊C412yes—
366C2×D4#13712yes—
367C2×D4#13712yes—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—44no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—161no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394SD16—28no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
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—84no—
420C2×C8C2×C22⋊C412yes—
421C2×C8C2×C22⋊C412yes—
422C2×C8C2×SD1612yes—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
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—82no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4C2×C22⋊C412yes—
456C4⋊C4C2×C22⋊C412yes—
457C4⋊C4C4×D412yes—
458C4⋊C4C2×SD1612yes—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—44no—
468C42—41no—
469C42—41no—
470C42—161no—
471C42C23×C411yescharacteristic
472C42C22×D411yescharacteristic
473C42C2×#4111yescharacteristic
474C42H(5,𝔽2)11yescharacteristic
475C23—22no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23D4211yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—42no—
491Q8—42no—
492Q8—88no—
493Q8—82no—
494Q8—81no—
495Q8—81no—
496Q8—161no—
497Q8—164no—
498Q8C4×SD1614yes—
499D4—22no—
500D4—22no—
501D4—22no—
502D4#41914yes—
503C2×C4—22no—
504C2×C4—22no—
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—41no—
514C2×C4—41no—
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—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—88no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—164no—
544C2×C4C22×C22⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4#51811yescharacteristic
547C2×C4D4211yescharacteristic
548C2×C4#54611yescharacteristic
549C2×C4#54611yescharacteristic
550C2×C4C22×SD1611yescharacteristic
551C2×C4C2×#13511yescharacteristic
552C2×C4C2×#13711yescharacteristic
553C2×C4C2×#13711yescharacteristic
554C2×C4#57711yescharacteristic
555C2×C4#57711yescharacteristic
556C2×C4#57711yescharacteristic
557C2×C4#57811yescharacteristic
558C8—22no—
559C8—22no—
560C8—22no—
561C8—42no—
562C8—41no—
563C8—81no—
564C8—81no—
565C8—81no—
566C8—81no—
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22D4×C22⋊C411yescharacteristic
575C22#326611yescharacteristic
576C22#327011yescharacteristic
577C22D4×SD1611yescharacteristic
578C22#328211yescharacteristic
579C22#328811yescharacteristic
580C22#328911yescharacteristic
581C4—24no—
582C4—21no—
583C4—21no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—82no—
591C4—82no—
592C4—161no—
593C4—161no—
594C4C2×#41911yescharacteristic
595C4#289811yescharacteristic
596C4C2×C4×SD1611yescharacteristic
597C4#294911yescharacteristic
598C4D4×SD1611yescharacteristic
599C4#328211yescharacteristic
600C4#331211yescharacteristic
601C4#332011yescharacteristic
602C2—44no—
603C2#1976511yescharacteristic
604C2#1976911yescharacteristic
605C2#1979611yescharacteristic
606C2#1979811yescharacteristic
607C2#2242511yescharacteristic
608C2C22⋊C4×SD1611yescharacteristic
609C2#2246411yescharacteristic
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. #22525C2semidirect
6. #9828C2semidirect
7. #9729C2semidirect
8. #9228C2semidirect
10. #8982C2semidirect
12. #8647C2semidirect
13. #8451C2semidirect
15. #8201C2semidirect
19. Q8×SD16C4semidirect
76. Q82C2×C4semidirect
99. #498C2×C4semidirect
105. #475C2×C4semidirect
109. C8×Q8C2×C4semidirect
310. #103SD16semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×SD162
#227702
#225252
D4×#1032
C2×#15822
#98282
#97292
#92282
#92152
#89822
#89732
#86472
#84512
#84472
#82012