Finite Group Databasebeta

← #6384537

Subgroups — #6384537

Order 512.

1787 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61303C211yesmaximal, characteristic
2#60218C211yesmaximal, characteristic, complemented
3Q8×#131C211yesmaximal, characteristic, complemented
4#23234C213yesmaximal, complemented
5#23080C213yesmaximal, complemented
6#23015C213yesmaximal, complemented
7#22783C213yesmaximal, complemented
8#22746C211yesmaximal, characteristic, complemented
9#22550C211yesmaximal, characteristic, complemented
10Q8×#104C211yesmaximal, characteristic, complemented
11#21658C213yesmaximal, complemented
12#21627C213yesmaximal, complemented
13#21517C213yesmaximal, complemented
14C4×C8×Q8C211yesmaximal, characteristic, complemented
15C4×#443C213yesmaximal, complemented
16#3480C2211yescharacteristic
17Q8×#41C2211yescharacteristic
18#3365C413yescomplemented
19#3364C413yescomplemented
20#3363C411yescharacteristic, complemented
21#3362C411yescharacteristic, complemented
22#3361C413yescomplemented
23#3360C413yescomplemented
24#3359C411yescharacteristic, complemented
25#3358C411yescharacteristic, complemented
26#3131C2213yes—
27#3076C2213yes—
28#3035C2213yes—
29#2971—21no—
30#2969—23no—
31#2966—23no—
32C8×#41—21no—
33Q8×M16—21no—
34Q8×M16C2211yescharacteristic
35Q8×M16C2211yescharacteristic
36#2962—23no—
37#2962C2213yes—
38#2960—23no—
39#2960C2213yes—
40C2×C8×Q8—21no—
41C2×C8×Q8C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C2×#443C2213yes—
44C2×#443C2213yes—
45C4×#135C2213yes—
46#2935C2213yes—
47#2931C2213yes—
48#2925C2213yes—
49#2906C2213yes—
50C4×#131C2213yes—
51C4×#131C2213yes—
52C42×Q8C2211yescharacteristic
53#1994C2213yes—
54#1986C2213yes—
55#1980C2213yes—
56#1962C2213yes—
57#1950C2213yes—
58#1950C2213yes—
59#1937C2216yes—
60#1937C2216yes—
61#1929C2216yes—
62#1922C2216yes—
63#1876C2216yes—
64#1866C2216yes—
65#1837C2213yes—
66#1806C2213yes—
67#1806C2213yes—
68C4×#108C2213yes—
69#1772C2213yes—
70#1772C2213yes—
71C8×C4⋊C4C2213yes—
72C8×C4⋊C4C2213yes—
73C4×C4⋊C8C2213yes—
74#1764C2216yes—
75#1764C2211yescharacteristic
76#1764C2213yes—
77#1764C2213yes—
78#1764C2213yes—
79C4×#104C2213yes—
80C4×#104C2213yes—
81C42×C8C2213yes—
82C2×#135C2313yes—
83C2×#131C2313yes—
84C2×#131C2313yes—
85C2×#131C2313yes—
86C2×#131C2311yescharacteristic
87#559C2×C413yes—
88Q82C2×C411yescharacteristic
89#557C2×C411yescharacteristic
90#557C2×C413yes—
91#556C2×C413yes—
92#554—23no—
93#554C2×C413yes—
94#553—23no—
95#550C2×C411yescharacteristic
96D4×Q8—21no—
97D4×Q8C2×C411yescharacteristic
98#548—21no—
99#533—23no—
100#533C2313yes—
101#532C2313yes—
102#529—23no—
103C4×#41—21no—
104C4×#41C2313yes—
105C4×#41C2313yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109#488C2×C416yes—
110#488C2×C416yes—
111#487C2×C413yes—
112#487C2×C413yes—
113#486C2×C413yes—
114#486C2×C413yes—
115#479C2×C416yes—
116#478C2×C416yes—
117#477C2×C413yes—
118#476C2×C413yes—
119#475C2×C413yes—
120#474C2×C413yes—
121#464C2×C416yes—
122#463C2×C416yes—
123#462C2×C413yes—
124#461C2×C413yes—
125#460C2×C413yes—
126#459C2×C413yes—
127#446—23no—
128#446—23no—
129#446—26no—
130#446—23no—
131#446—23no—
132#446—23no—
133C8×Q8—21no—
134C8×Q8—21no—
135C8×Q8—23no—
136C8×Q8—23no—
137C8×Q8—23no—
138C8×Q8C2×C411yescharacteristic
139C8×Q8C2×C411yescharacteristic
140C8×Q8C2×C413yes—
141C8×Q8C2×C413yes—
142C8×Q8D411yescharacteristic
143C8×Q8D413yes—
144C8×Q8D413yes—
145C8×Q8D411yescharacteristic
146#443—212no—
147#443—212no—
148C4×Q16C2×C413yes—
149C4×Q16C2×C413yes—
150C4×SD16C2×C413yes—
151C4×SD16C2×C413yes—
152C4×SD16C2×C413yes—
153C4×SD16C2×C413yes—
154C4×D8C2×C413yes—
155C4×D8C2×C413yes—
156#436—23no—
157#436—23no—
158#436—23no—
159#435—26no—
160C8×D4—23no—
161C8×D4—23no—
162C8×D4—23no—
163#433—26no—
164#432—23no—
165#431—23no—
166C2×#108C2313yes—
167C2×#108C2313yes—
168#427C2313yes—
169#424—26no—
170#424C2316yes—
171#423—23no—
172#423C2313yes—
173#423C2313yes—
174C2×C4⋊C8—23no—
175C2×C4⋊C8C2313yes—
176C2×C4⋊C8C2313yes—
177C2×#104C2313yes—
178C2×#104C2313yes—
179C2×#104C2313yes—
180C2×#104C2316yes—
181C2×#104C2311yescharacteristic
182#416C2316yes—
183#416C2313yes—
184#416C2311yescharacteristic
185#408C2316yes—
186#405—23no—
187#405—23no—
188#405—23no—
189#405—23no—
190#405C2311yescharacteristic
191#405C2313yes—
192#405C2313yes—
193#405C2313yes—
194C4×M16—23no—
195C4×M16—23no—
196C4×M16C2313yes—
197C4×M16C2313yes—
198C2×C4×C8—23no—
199C2×C4×C8—23no—
200C2×C4×C8C2313yes—
201C2×C4×C8C2313yes—
202C2×C4×C8C2311yescharacteristic
203C2×C4×C8C2313yes—
204C2×C4×C8C2313yes—
205C2×C4×C8C2313yes—
206C4×C4⋊C4—23no—
207C4×C4⋊C4C2313yes—
208C43C2313yes—
209#341C2313yes—
210#341C2316yes—
211#341C2311yescharacteristic
212#339C2313yes—
213#339C2316yes—
214#339C2313yes—
215#339C2313yes—
216#339C2311yescharacteristic
217#336C2316yes—
218C2×#41—21no—
219C2×#41C2413yes—
220C2×#41C2411yescharacteristic
221C22×Q8—21no—
222C22×Q8#4111yescharacteristic
223#135—26no—
224#135—26no—
225C2×Q16C22×C413yes—
226C2×SD16C22×C413yes—
227C2×SD16C22×C413yes—
228C2×D8C22×C413yes—
229#131—22no—
230#131—26no—
231#131—26no—
232#131—26no—
233#131—26no—
234#131—43no—
235#131—43no—
236#131#4114yescomplemented
237C2×M16—21no—
238C2×M16—23no—
239C2×M16—23no—
240C2×M16—23no—
241C2×M16C2413yes—
242C2×M16C2413yes—
243C2×M16C2411yescharacteristic
244C2×M16C2411yescharacteristic
245C22×C8—21no—
246C22×C8—23no—
247C22×C8—23no—
248C22×C8—23no—
249C22×C8C2413yes—
250C22×C8C2411yescharacteristic
251C22×C8C2413yes—
252C22×C8C2411yescharacteristic
253C4⋊Q8—23no—
254C4⋊Q8—23no—
255C4⋊Q8C22×C413yes—
256C4⋊Q8C22×C413yes—
257C4⋊Q8C22×C413yes—
258#127C22×C413yes—
259#126—26no—
260#125—23no—
261#125—26no—
262#125C22×C416yes—
263#124—23no—
264#124C22×C416yes—
265#123—26no—
266C22⋊Q8—23no—
267C22⋊Q8—23no—
268C22⋊Q8—23no—
269C22⋊Q8—23no—
270C22⋊Q8—23no—
271C22⋊Q8—26no—
272#121—23no—
273#121—23no—
274#121—23no—
275C4×Q8—28no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—23no—
279C4×Q8—23no—
280C4×Q8—23no—
281C4×Q8C22×C413yes—
282C4×Q8C22×C413yes—
283C4×Q8C22×C413yes—
284C4×Q8C22×C413yes—
285C4×Q8C22×C411yescharacteristic
286C4×Q8C22×C413yes—
287C4×Q8C2×D411yescharacteristic
288C4×Q8C2×D413yes—
289C4×D4—23no—
290C4×D4—23no—
291C4×D4—23no—
292C4×D4—23no—
293C4×D4—23no—
294C4×D4—23no—
295C4×D4—23no—
296C4×D4C22×C413yes—
297C4×D4C22×C413yes—
298C4×D4C22×C413yes—
299C4×D4C22×C413yes—
300#117—23no—
301#117—23no—
302#117—23no—
303#117C2411yescharacteristic
304#117C2413yes—
305C2×C4⋊C4—23no—
306C2×C4⋊C4—23no—
307C2×C4⋊C4#4113yes—
308C2×C42—23no—
309C2×C42—23no—
310C2×C42—23no—
311C2×C42—23no—
312C2×C42C2413yes—
313C2×C42C2411yescharacteristic
314C2×C42C2413yes—
315#108—26no—
316#108—26no—
317#107C22×C413yes—
318#107C22×C413yes—
319#106C22×C413yes—
320#106C22×C413yes—
321C4⋊C8—26no—
322C4⋊C8—26no—
323C4⋊C8—23no—
324C4⋊C8—23no—
325C4⋊C8—23no—
326C4⋊C8—23no—
327C4⋊C8C22×C413yes—
328C4⋊C8C22×C413yes—
329C4⋊C8C22×C416yes—
330C4⋊C8C2×D413yes—
331C4⋊C8C2×D413yes—
332C4⋊C8C2×D416yes—
333#104—26no—
334#104—26no—
335#104—26no—
336#104—46no—
337#104#4114yescomplemented
338#103C22×C411yescharacteristic
339#103C22×C411yescharacteristic
340#103C22×C413yes—
341#103C22×C413yes—
342#103C22×C416yes—
343#103C22×C416yes—
344#102C22×C411yescharacteristic
345#102C22×C411yescharacteristic
346#102C22×C413yes—
347#102C22×C413yes—
348#102C22×C416yes—
349#102C22×C416yes—
350C22⋊C8—26no—
351C22⋊C8—26no—
352#97—21no—
353#97—23no—
354#97—23no—
355#97—23no—
356#97—23no—
357#97—23no—
358C4×C8—23no—
359C4×C8—23no—
360C4×C8—21no—
361C4×C8—23no—
362C4×C8—26no—
363C4×C8—22no—
364C4×C8—23no—
365C4×C8—23no—
366C4×C8—26no—
367C4×C8—43no—
368C4×C8C22×C413yes—
369C4×C8C22×C413yes—
370C4×C8C22×C413yes—
371C4×C8C22×C413yes—
372C4×C8C2×D413yes—
373C4×C8C2×D413yes—
374C4×C8C2×D413yes—
375C4×C8C2×D413yes—
376#95—23no—
377#41—21no—
378#41—21no—
379#41—23no—
380#41—23no—
381#41—23no—
382#41—23no—
383#41—44no—
384#41C2×#4112yes—
385#41C2×#4112yes—
386C2×Q8—24no—
387C2×Q8—23no—
388C2×Q8—23no—
389C2×Q8—44no—
390C2×Q8C4×D413yes—
391C2×Q8C4×D411yescharacteristic
392C2×Q8C23×C413yes—
393C2×Q8C23×C411yescharacteristic
394C2×D4—21no—
395C2×D4—23no—
396C2×D4—23no—
397C2×D4—23no—
398C2×D4C23×C413yes—
399C2×D4C23×C411yescharacteristic
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4C2×#4113yes—
409C22×C4C2×#4113yes—
410C22×C4C2511yescharacteristic
411Q16—26no—
412SD16—26no—
413SD16—26no—
414D8—26no—
415M16—21no—
416M16—21no—
417M16—24no—
418M16—23no—
419M16—23no—
420M16—23no—
421M16—23no—
422M16—46no—
423M16—46no—
424M16—46no—
425M16C2×#4112yes—
426M16C2×#4112yes—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—23no—
430C2×C8—23no—
431C2×C8—23no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—23no—
435C2×C8—24no—
436C2×C8—23no—
437C2×C8—23no—
438C2×C8—46no—
439C2×C8—46no—
440C2×C8—46no—
441C2×C8C23×C411yescharacteristic
442C2×C8C23×C411yescharacteristic
443C2×C8C23×C413yes—
444C2×C8C23×C413yes—
445C2×C8C22×D413yes—
446C2×C8C22×D413yes—
447C2×C8C22×D411yescharacteristic
448C2×C8C22×D411yescharacteristic
449C2×C8C2×#4112yes—
450C2×C82−1+412yes—
451C4⋊C4—212no—
452C4⋊C4—26no—
453C4⋊C4—23no—
454C4⋊C4—23no—
455C4⋊C4—23no—
456C4⋊C4—23no—
457C4⋊C4—23no—
458C4⋊C4—23no—
459C4⋊C4—23no—
460C4⋊C4—23no—
461C4⋊C4—26no—
462C4⋊C4C4×D413yes—
463C4⋊C4C4×D413yes—
464C4⋊C4C4×D416yes—
465C4⋊C4C23×C413yes—
466C4⋊C4C23×C413yes—
467C4⋊C4C23×C411yescharacteristic
468C4⋊C4C23×C411yescharacteristic
469C22⋊C4—21no—
470C22⋊C4—23no—
471C22⋊C4—26no—
472C22⋊C4—23no—
473C22⋊C4—23no—
474C22⋊C4—23no—
475C22⋊C4—23no—
476C22⋊C4—26no—
477C42—212no—
478C42—23no—
479C42—23no—
480C42—23no—
481C42—23no—
482C42—21no—
483C42—23no—
484C42—21no—
485C42—23no—
486C42—23no—
487C42—46no—
488C42C23×C413yes—
489C42C23×C413yes—
490C42C22×D413yes—
491C42C22×D413yes—
492C42C2×#4112yes—
493C23—21no—
494C23—23no—
495C23C22×#4111yescharacteristic
496Q8—21no—
497Q8—26no—
498Q8—23no—
499Q8—23no—
500Q8—46no—
501Q8—83no—
502Q8#44314yescomplemented
503Q8C4×#4112yes—
504D4—22no—
505D4—22no—
506D4—23no—
507D4—21no—
508D4—23no—
509D4—43no—
510D4—43no—
511D4—46no—
512D4—83no—
513D4C4×#4112yes—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—26no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—26no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—23no—
525C2×C4—23no—
526C2×C4—23no—
527C2×C4—43no—
528C2×C4—43no—
529C2×C4—43no—
530C2×C4—46no—
531C2×C4—46no—
532C2×C4—43no—
533C2×C4—43no—
534C2×C4—83no—
535C2×C4—83no—
536C2×C4C2×C4×D413yes—
537C2×C4C2×C4×D413yes—
538C2×C4C2×C4×D413yes—
539C2×C4C2×C4×D413yes—
540C2×C4C24×C411yescharacteristic
541C2×C4C23×D411yescharacteristic
542C2×C4C22×#4111yescharacteristic
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×2−1+411yescharacteristic
545C2×C4#58511yescharacteristic
546C8—23no—
547C8—23no—
548C8—21no—
549C8—21no—
550C8—22no—
551C8—23no—
552C8—23no—
553C8—22no—
554C8—43no—
555C8—43no—
556C8#54812yes—
557C8#54812yes—
558C22—21no—
559C22—21no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—43no—
564C22—43no—
565C22—43no—
566C22C22×C4×D411yescharacteristic
567C22#348011yescharacteristic
568C22#348211yescharacteristic
569C4—24no—
570C4—23no—
571C4—23no—
572C4—23no—
573C4—22no—
574C4—46no—
575C4—41no—
576C4—43no—
577C4—41no—
578C4—43no—
579C4—43no—
580C4C2×#44316yes—
581C4C2×C4×#4111yescharacteristic
582C4C4×2−1+411yescharacteristic
583C4C2×#54811yescharacteristic
584C4D4×#4111yescharacteristic
585C2—21no—
586C2—21no—
587C2—42no—
588C2—83no—
589C2C4×#54811yescharacteristic
590C2C22×#44311yescharacteristic
591C2#6018311yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radical#61303C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60218C2semidirect
3. Q8×#131C2semidirect
4. #23234C2semidirect
5. #23080C2semidirect
6. #23015C2semidirect
7. #22783C2semidirect
8. #22746C2semidirect
9. #22550C2semidirect
10. Q8×#104C2semidirect
11. #21658C2semidirect
12. #21627C2semidirect
13. #21517C2semidirect
14. C4×C8×Q8C2semidirect
15. C4×#443C2semidirect
18. #3365C4semidirect
19. #3364C4semidirect
20. #3363C4semidirect
21. #3362C4semidirect
22. #3361C4semidirect
23. #3360C4semidirect
24. #3359C4semidirect
25. #3358C4semidirect
236. #131#41semidirect
337. #104#41semidirect
502. Q8#443semidirect

Maximal subgroups

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

subgroupindex
#613032
#602182
Q8×#1312
#232342
#230802
#230152
#227832
#227462
#225502
Q8×#1042
#216582
#216272
#215172
C4×C8×Q82
C4×#4432