Finite Group Databasebeta

← #6400540

Subgroups — #6400540

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61303C211yesmaximal, characteristic
2#60253C211yesmaximal, characteristic, complemented
3#60218C211yesmaximal, characteristic, complemented
4#23234C213yesmaximal, complemented
5#23113C213yesmaximal, complemented
6#23080C213yesmaximal, complemented
7#22802C213yesmaximal, complemented
8Q8×#108C211yesmaximal, characteristic, complemented
9#22550C211yesmaximal, characteristic, complemented
10Q8×#104C211yesmaximal, characteristic, complemented
11#22234C213yesmaximal, complemented
12#22165C213yesmaximal, complemented
13#21627C213yesmaximal, complemented
14#21137C213yesmaximal, complemented
15#20569C211yesmaximal, characteristic, 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#2986—23no—
30#2985—23no—
31#2984—21no—
32#2982—21no—
33#2971—21no—
34#2969—23no—
35#2966—23no—
36C8×#41—21no—
37Q8×M16C2211yescharacteristic
38Q8×M16C2211yescharacteristic
39#2962C2213yes—
40#2960C2213yes—
41C2×C8×Q8C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C4×#135C2213yes—
44#2935C2213yes—
45#2931C2213yes—
46#2925C2213yes—
47#2925C2213yes—
48#2922C2213yes—
49C4×#131C2213yes—
50C42×Q8C2211yescharacteristic
51#1994C2213yes—
52#1986C2213yes—
53#1980C2213yes—
54#1974C2216yes—
55#1962C2213yes—
56#1950C2213yes—
57#1950C2213yes—
58#1937C2216yes—
59#1929C2213yes—
60#1929C2216yes—
61#1929C2216yes—
62#1929C2211yescharacteristic
63#1922C2213yes—
64#1922C2213yes—
65#1876C2213yes—
66#1876C2213yes—
67#1866C2216yes—
68#1866C2213yes—
69#1866C2213yes—
70#1848C2213yes—
71#1806C2213yes—
72#1806C2213yes—
73C4×#108C2213yes—
74#1772C2216yes—
75C8×C4⋊C4C2213yes—
76C8×C4⋊C4C2213yes—
77C4×C4⋊C8C2213yes—
78#1764C2216yes—
79C4×#104C2213yes—
80C4×#104C2213yes—
81#1744C2213yes—
82C2×#135C2313yes—
83C2×#131C2313yes—
84C2×#131C2313yes—
85#559C2×C413yes—
86Q82C2×C411yescharacteristic
87#557C2×C411yescharacteristic
88#557C2×C413yes—
89#556C2×C413yes—
90#554—23no—
91#554C2×C413yes—
92#553—23no—
93#550C2×C411yescharacteristic
94D4×Q8—21no—
95D4×Q8C2×C411yescharacteristic
96#548—21no—
97#533—23no—
98#533C2313yes—
99#532C2313yes—
100#529—23no—
101C4×#41—21no—
102C4×#41C2313yes—
103C4×#41C2313yes—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8C2311yescharacteristic
107#488C2×C416yes—
108#488C2×C416yes—
109#487C2×C413yes—
110#487C2×C413yes—
111#486C2×C413yes—
112#486C2×C413yes—
113#479C2×C416yes—
114#478C2×C416yes—
115#477C2×C413yes—
116#476C2×C413yes—
117#475C2×C413yes—
118#474C2×C413yes—
119#464C2×C416yes—
120#463C2×C416yes—
121#462C2×C413yes—
122#461C2×C413yes—
123#460C2×C413yes—
124#459C2×C413yes—
125#446—23no—
126#446—23no—
127#446—23no—
128#446—26no—
129#446—23no—
130#446—23no—
131C8×Q8—21no—
132C8×Q8—21no—
133C8×Q8—23no—
134C8×Q8—23no—
135C8×Q8—23no—
136C8×Q8C2×C411yescharacteristic
137C8×Q8C2×C411yescharacteristic
138C8×Q8C2×C413yes—
139C8×Q8C2×C413yes—
140C8×Q8D411yescharacteristic
141C8×Q8D413yes—
142C8×Q8D413yes—
143C8×Q8D411yescharacteristic
144C4×Q16C2×C413yes—
145C4×Q16C2×C413yes—
146C4×SD16C2×C413yes—
147C4×SD16C2×C413yes—
148C4×SD16C2×C413yes—
149C4×SD16C2×C413yes—
150C4×D8C2×C413yes—
151C4×D8C2×C413yes—
152#436—23no—
153#436—23no—
154#436—23no—
155#435—26no—
156C8×D4—23no—
157C8×D4—23no—
158C8×D4—23no—
159#433—26no—
160#433—26no—
161#433—23no—
162#433—23no—
163#432—23no—
164#432—23no—
165#431—23no—
166#431—23no—
167#431—23no—
168#431—23no—
169C2×#108C2311yescharacteristic
170C2×#108C2313yes—
171C2×#108C2316yes—
172#427C2313yes—
173#424C2313yes—
174#424C2313yes—
175#423C2313yes—
176#423C2313yes—
177C2×C4⋊C8C2313yes—
178C2×C4⋊C8C2313yes—
179C2×#104C2311yescharacteristic
180C2×#104C2313yes—
181C2×#104C2316yes—
182C2×#104C2313yes—
183C2×#104C2313yes—
184#416C2316yes—
185#416C2313yes—
186#416C2311yescharacteristic
187#408C2311yescharacteristic
188#408C2313yes—
189#408C2316yes—
190#405—23no—
191#405—23no—
192#405C2311yescharacteristic
193#405C2313yes—
194#405C2316yes—
195C4×M16—23no—
196C4×M16C2313yes—
197C4×M16C2313yes—
198C2×C4×C8—23no—
199C2×C4×C8C2316yes—
200C2×C4×C8C2313yes—
201C2×C4×C8C2313yes—
202C4×C4⋊C4—23no—
203C4×C4⋊C4C2313yes—
204C43C2313yes—
205#341C2313yes—
206#341C2313yes—
207#339C2313yes—
208#339C2316yes—
209#339C2313yes—
210#339C2313yes—
211#339C2311yescharacteristic
212#336C2311yescharacteristic
213#336C2313yes—
214#336C2313yes—
215#336C2313yes—
216C2×#41—21no—
217C2×#41C2411yescharacteristic
218C2×#41C2413yes—
219C22×Q8—21no—
220C22×Q8#4111yescharacteristic
221#135—43no—
222#135—43no—
223C2×Q16C22×C413yes—
224C2×SD16C22×C413yes—
225C2×SD16C22×C413yes—
226C2×D8C22×C413yes—
227#131—26no—
228#131—43no—
229#131—43no—
230#131—43no—
231C2×M16—23no—
232C2×M16—23no—
233C2×M16C2413yes—
234C2×M16C2411yescharacteristic
235C2×M16C2413yes—
236C2×M16C2411yescharacteristic
237C22×C8—23no—
238C22×C8—23no—
239C22×C8C2413yes—
240C22×C8C2411yescharacteristic
241C22×C8C2411yescharacteristic
242C22×C8C2413yes—
243C4⋊Q8—23no—
244C4⋊Q8—23no—
245C4⋊Q8C22×C413yes—
246C4⋊Q8C22×C413yes—
247C4⋊Q8C22×C413yes—
248#127C22×C413yes—
249#126—26no—
250#125—23no—
251#125—26no—
252#125C22×C416yes—
253#124—23no—
254#124C22×C416yes—
255#123—26no—
256C22⋊Q8—23no—
257C22⋊Q8—23no—
258C22⋊Q8—23no—
259C22⋊Q8—23no—
260C22⋊Q8—23no—
261C22⋊Q8—26no—
262#121—23no—
263#121—23no—
264#121—23no—
265C4×Q8—28no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—23no—
269C4×Q8—23no—
270C4×Q8—23no—
271C4×Q8C22×C413yes—
272C4×Q8C22×C413yes—
273C4×Q8C22×C413yes—
274C4×Q8C22×C413yes—
275C4×Q8C22×C411yescharacteristic
276C4×Q8C22×C413yes—
277C4×Q8C2×D411yescharacteristic
278C4×Q8C2×D413yes—
279C4×D4—23no—
280C4×D4—23no—
281C4×D4—23no—
282C4×D4—23no—
283C4×D4—23no—
284C4×D4—23no—
285C4×D4—23no—
286C4×D4C22×C413yes—
287C4×D4C22×C413yes—
288C4×D4C22×C413yes—
289C4×D4C22×C413yes—
290#117—23no—
291#117—23no—
292#117—23no—
293#117C2411yescharacteristic
294#117C2413yes—
295C2×C4⋊C4—23no—
296C2×C4⋊C4—23no—
297C2×C4⋊C4#4113yes—
298C2×C42—23no—
299C2×C42—23no—
300C2×C42—23no—
301C2×C42—23no—
302C2×C42C2411yescharacteristic
303C2×C42C2413yes—
304C2×C42C2413yes—
305#108—22no—
306#108—43no—
307#108—46no—
308#107C22×C413yes—
309#107C22×C413yes—
310#106C22×C413yes—
311#106C22×C413yes—
312C4⋊C8—23no—
313C4⋊C8—23no—
314C4⋊C8—23no—
315C4⋊C8—23no—
316C4⋊C8—23no—
317C4⋊C8—23no—
318C4⋊C8—23no—
319C4⋊C8—23no—
320C4⋊C8C22×C413yes—
321C4⋊C8C22×C413yes—
322C4⋊C8C22×C416yes—
323C4⋊C8C2×D413yes—
324C4⋊C8C2×D413yes—
325C4⋊C8C2×D416yes—
326#104—22no—
327#104—43no—
328#104—46no—
329#104—43no—
330#104—43no—
331#103C22×C411yescharacteristic
332#103C22×C411yescharacteristic
333#103C22×C413yes—
334#103C22×C413yes—
335#103C22×C416yes—
336#103C22×C416yes—
337#102C22×C411yescharacteristic
338#102C22×C411yescharacteristic
339#102C22×C413yes—
340#102C22×C413yes—
341#102C22×C416yes—
342#102C22×C416yes—
343C22⋊C8—23no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—23no—
347C22⋊C8—26no—
348C22⋊C8—26no—
349#97—21no—
350#97—23no—
351#97—26no—
352#97—23no—
353#97—23no—
354C4×C8—21no—
355C4×C8—23no—
356C4×C8—26no—
357C4×C8—23no—
358C4×C8—23no—
359C4×C8—46no—
360C4×C8C22×C413yes—
361C4×C8C22×C413yes—
362C4×C8C22×C413yes—
363C4×C8C22×C413yes—
364C4×C8C2×D413yes—
365C4×C8C2×D413yes—
366C4×C8C2×D413yes—
367C4×C8C2×D413yes—
368#95—23no—
369#41—21no—
370#41—23no—
371#41—21no—
372#41—23no—
373#41—23no—
374#41—23no—
375#41—21no—
376#41—21no—
377#41—44no—
378C2×Q8—24no—
379C2×Q8—23no—
380C2×Q8—23no—
381C2×Q8—44no—
382C2×Q8C4×D413yes—
383C2×Q8C4×D411yescharacteristic
384C2×Q8C23×C413yes—
385C2×Q8C23×C411yescharacteristic
386C2×D4—21no—
387C2×D4—23no—
388C2×D4—23no—
389C2×D4—23no—
390C2×D4C23×C413yes—
391C2×D4C23×C411yescharacteristic
392C22×C4—23no—
393C22×C4—21no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—21no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4C2×#4113yes—
401C22×C4C2×#4113yes—
402C22×C4C2511yescharacteristic
403Q16—43no—
404SD16—43no—
405SD16—43no—
406D8—43no—
407M16—21no—
408M16—21no—
409M16—21no—
410M16—23no—
411M16—23no—
412M16—23no—
413M16—23no—
414M16—46no—
415M16—46no—
416M162−1+412yes—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—23no—
420C2×C8—23no—
421C2×C8—23no—
422C2×C8—21no—
423C2×C8—23no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—23no—
427C2×C8—21no—
428C2×C8—23no—
429C2×C8—46no—
430C2×C8—46no—
431C2×C8C23×C411yescharacteristic
432C2×C8C23×C411yescharacteristic
433C2×C8C23×C413yes—
434C2×C8C23×C413yes—
435C2×C8C22×D413yes—
436C2×C8C22×D411yescharacteristic
437C2×C8C22×D411yescharacteristic
438C2×C8C22×D413yes—
439C4⋊C4—212no—
440C4⋊C4—26no—
441C4⋊C4—23no—
442C4⋊C4—23no—
443C4⋊C4—23no—
444C4⋊C4—23no—
445C4⋊C4—23no—
446C4⋊C4—23no—
447C4⋊C4—23no—
448C4⋊C4—23no—
449C4⋊C4—26no—
450C4⋊C4C4×D413yes—
451C4⋊C4C4×D413yes—
452C4⋊C4C4×D416yes—
453C4⋊C4C23×C413yes—
454C4⋊C4C23×C413yes—
455C4⋊C4C23×C411yescharacteristic
456C4⋊C4C23×C411yescharacteristic
457C22⋊C4—21no—
458C22⋊C4—23no—
459C22⋊C4—26no—
460C22⋊C4—23no—
461C22⋊C4—23no—
462C22⋊C4—23no—
463C22⋊C4—23no—
464C22⋊C4—26no—
465C42—212no—
466C42—23no—
467C42—23no—
468C42—23no—
469C42—23no—
470C42—21no—
471C42—23no—
472C42—21no—
473C42—23no—
474C42—21no—
475C42—23no—
476C42—83no—
477C42C23×C413yes—
478C42C23×C413yes—
479C42C22×D413yes—
480C42C22×D413yes—
481C23—21no—
482C23—23no—
483C23C22×#4111yescharacteristic
484Q8—26no—
485Q8—21no—
486Q8—23no—
487Q8—23no—
488Q8—21no—
489Q8—46no—
490Q8—83no—
491Q8#44314yes—
492D4—21no—
493D4—23no—
494D4—23no—
495D4—21no—
496D4—41no—
497D4—43no—
498D4—41no—
499D4—43no—
500D4—46no—
501D4—83no—
502C2×C4—26no—
503C2×C4—22no—
504C2×C4—26no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—23no—
509C2×C4—23no—
510C2×C4—23no—
511C2×C4—43no—
512C2×C4—41no—
513C2×C4—43no—
514C2×C4—41no—
515C2×C4—43no—
516C2×C4—46no—
517C2×C4—46no—
518C2×C4—43no—
519C2×C4—41no—
520C2×C4—43no—
521C2×C4—41no—
522C2×C4—83no—
523C2×C4—83no—
524C2×C4C2×C4×D413yes—
525C2×C4C2×C4×D413yes—
526C2×C4C2×C4×D413yes—
527C2×C4C2×C4×D413yes—
528C2×C4C24×C411yescharacteristic
529C2×C4C23×D411yescharacteristic
530C2×C4C2×2−1+411yescharacteristic
531C2×C4#58511yescharacteristic
532C2×C4#58511yescharacteristic
533C2×C4#58511yescharacteristic
534C8—23no—
535C8—23no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—23no—
541C8—23no—
542C8—22no—
543C8—43no—
544C8—43no—
545C8—41no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—43no—
550C22—41no—
551C22—43no—
552C22—43no—
553C22—41no—
554C22C22×C4×D411yescharacteristic
555C22#351811yescharacteristic
556C22#352411yescharacteristic
557C4—23no—
558C4—23no—
559C4—23no—
560C4—46no—
561C4—42no—
562C4—41no—
563C4—43no—
564C4—41no—
565C4—43no—
566C4—41no—
567C4—43no—
568C4C2×#44316yes—
569C4#342811yescharacteristic
570C4C4×2−1+411yescharacteristic
571C4#348711yescharacteristic
572C4#348911yescharacteristic
573C2—21no—
574C2—21no—
575C2—81no—
576C2—83no—
577C2#3429311yescharacteristic
578C2C22×#44311yescharacteristic
579C2#6018411yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radical#61303C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60253C2semidirect
3. #60218C2semidirect
4. #23234C2semidirect
5. #23113C2semidirect
6. #23080C2semidirect
7. #22802C2semidirect
8. Q8×#108C2semidirect
9. #22550C2semidirect
10. Q8×#104C2semidirect
11. #22234C2semidirect
12. #22165C2semidirect
13. #21627C2semidirect
14. #21137C2semidirect
15. #20569C2semidirect
18. #3365C4semidirect
19. #3364C4semidirect
20. #3363C4semidirect
21. #3362C4semidirect
22. #3361C4semidirect
23. #3360C4semidirect
24. #3359C4semidirect
25. #3358C4semidirect

Maximal subgroups

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

subgroupindex
#613032
#602532
#602182
#232342
#231132
#230802
#228022
Q8×#1082
#225502
Q8×#1042
#222342
#221652
#216272
#211372
#205692