Finite Group Databasebeta

← #6404405

Subgroups — #6404405

Order 512.

1621 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61303C211yesmaximal, characteristic
2#60274C211yesmaximal, characteristic, complemented
3#60253C211yesmaximal, characteristic, complemented
4#23235C213yesmaximal, complemented
5#23114C213yesmaximal, complemented
6#22983C213yesmaximal, complemented
7#22782C213yesmaximal, complemented
8#22746C211yesmaximal, characteristic, complemented
9#22550C211yesmaximal, characteristic, complemented
10Q8×#104C211yesmaximal, characteristic, complemented
11#22166C213yesmaximal, complemented
12#22131C213yesmaximal, complemented
13#21660C213yesmaximal, complemented
14#21293C213yesmaximal, complemented
15Q8×#97C211yesmaximal, 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#2994—23no—
30#2993—23no—
31#2991—21no—
32#2988—21no—
33#2986—23no—
34#2985—23no—
35#2984—21no—
36#2982—21no—
37Q8×M16C2211yescharacteristic
38Q8×M16C2211yescharacteristic
39#2962C2213yes—
40#2960C2213yes—
41C2×C8×Q8C2211yescharacteristic
42C2×C8×Q8C2211yescharacteristic
43C4×#135C2213yes—
44#2935C2213yes—
45#2931C2213yes—
46#2925C2213yes—
47#2922C2213yes—
48#2922C2213yes—
49#2906C2213yes—
50C42×Q8C2211yescharacteristic
51#1994C2213yes—
52#1986C2213yes—
53#1980C2213yes—
54#1975C2213yes—
55#1975C2213yes—
56#1962C2213yes—
57#1950C2213yes—
58#1950C2213yes—
59#1946C2213yes—
60#1946C2213yes—
61#1938C2211yescharacteristic
62#1938C2213yes—
63#1938C2213yes—
64#1938C2213yes—
65#1938C2216yes—
66#1929C2216yes—
67#1922C2216yes—
68#1876C2216yes—
69#1866C2216yes—
70#1837C2213yes—
71#1806C2213yes—
72#1806C2213yes—
73C4×#108C2213yes—
74#1770C2213yes—
75#1770C2213yes—
76#1767C2213yes—
77#1765C2216yes—
78#1765C2216yes—
79C4×#104C2213yes—
80C4×#104C2213yes—
81C4×#97C2213yes—
82C2×#135C2313yes—
83C2×#131C2316yes—
84#559C2×C413yes—
85Q82C2×C411yescharacteristic
86#557C2×C411yescharacteristic
87#557C2×C413yes—
88#556C2×C413yes—
89#554—23no—
90#554C2×C413yes—
91#553—23no—
92#550C2×C411yescharacteristic
93D4×Q8—21no—
94D4×Q8C2×C411yescharacteristic
95#548—21no—
96#533—23no—
97#533C2313yes—
98#532C2313yes—
99#529—23no—
100C4×#41—21no—
101C4×#41C2313yes—
102C4×#41C2313yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2311yescharacteristic
106#488C2×C416yes—
107#488C2×C416yes—
108#487C2×C413yes—
109#487C2×C413yes—
110#486C2×C413yes—
111#486C2×C413yes—
112#479C2×C416yes—
113#478C2×C416yes—
114#477C2×C413yes—
115#476C2×C413yes—
116#475C2×C413yes—
117#474C2×C413yes—
118#464C2×C416yes—
119#463C2×C416yes—
120#462C2×C413yes—
121#461C2×C413yes—
122#460C2×C413yes—
123#459C2×C413yes—
124#446—23no—
125#446—23no—
126#446—23no—
127#446—26no—
128#446—23no—
129#446—23no—
130C8×Q8—21no—
131C8×Q8—21no—
132C8×Q8—23no—
133C8×Q8—23no—
134C8×Q8—23no—
135C8×Q8C2×C411yescharacteristic
136C8×Q8C2×C411yescharacteristic
137C8×Q8C2×C413yes—
138C8×Q8C2×C413yes—
139C8×Q8D411yescharacteristic
140C8×Q8D411yescharacteristic
141C8×Q8D413yes—
142C8×Q8D413yes—
143C4×Q16C2×C413yes—
144C4×Q16C2×C413yes—
145C4×SD16C2×C413yes—
146C4×SD16C2×C413yes—
147C4×SD16C2×C413yes—
148C4×SD16C2×C413yes—
149C4×D8C2×C413yes—
150C4×D8C2×C413yes—
151#436—23no—
152#436—23no—
153#436—23no—
154#435—26no—
155C8×D4—23no—
156C8×D4—23no—
157C8×D4—23no—
158#433—26no—
159#433—23no—
160#433—23no—
161#433—23no—
162#433—23no—
163#432—23no—
164#431—23no—
165#431—23no—
166#431—23no—
167#431—23no—
168#431—23no—
169C2×#108C2313yes—
170C2×#108C2313yes—
171#427C2313yes—
172#424—26no—
173#424C2313yes—
174#424C2313yes—
175#424C2311yescharacteristic
176#424C2313yes—
177#423—23no—
178#423C2313yes—
179#423C2313yes—
180C2×C4⋊C8—23no—
181C2×C4⋊C8C2313yes—
182C2×C4⋊C8C2313yes—
183C2×#104C2313yes—
184C2×#104C2311yescharacteristic
185C2×#104C2313yes—
186C2×#104C2316yes—
187C2×#104C2313yes—
188#416C2316yes—
189#416C2313yes—
190#416C2311yescharacteristic
191#408C2313yes—
192#408C2311yescharacteristic
193#408C2313yes—
194#408C2313yes—
195#405C2316yes—
196C4×M16C2313yes—
197C4×M16C2313yes—
198C2×#97C2311yescharacteristic
199C2×#97C2313yes—
200C2×#97C2313yes—
201C2×#97C2313yes—
202C2×C4×C8C2313yes—
203C2×C4×C8C2313yes—
204C4×C4⋊C4—23no—
205C4×C4⋊C4C2313yes—
206C43C2313yes—
207#341C2313yes—
208#341C2316yes—
209#341C2311yescharacteristic
210#339C2313yes—
211#339C2316yes—
212#339C2313yes—
213#339C2311yescharacteristic
214#339C2313yes—
215#336C2316yes—
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—46no—
228#131—46no—
229C2×M16—26no—
230C2×M16C2413yes—
231C2×M16C2411yescharacteristic
232C2×M16C2413yes—
233C2×M16C2411yescharacteristic
234C22×C8—26no—
235C22×C8C2413yes—
236C22×C8C2411yescharacteristic
237C22×C8C2413yes—
238C22×C8C2411yescharacteristic
239C4⋊Q8—23no—
240C4⋊Q8—23no—
241C4⋊Q8C22×C413yes—
242C4⋊Q8C22×C413yes—
243C4⋊Q8C22×C413yes—
244#127C22×C413yes—
245#126—26no—
246#125—23no—
247#125—26no—
248#125C22×C416yes—
249#124—23no—
250#124C22×C416yes—
251#123—26no—
252C22⋊Q8—23no—
253C22⋊Q8—23no—
254C22⋊Q8—23no—
255C22⋊Q8—23no—
256C22⋊Q8—23no—
257C22⋊Q8—26no—
258#121—23no—
259#121—23no—
260#121—23no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—23no—
264C4×Q8—23no—
265C4×Q8—23no—
266C4×Q8—44no—
267C4×Q8C22×C413yes—
268C4×Q8C22×C413yes—
269C4×Q8C22×C413yes—
270C4×Q8C22×C413yes—
271C4×Q8C22×C411yescharacteristic
272C4×Q8C22×C413yes—
273C4×Q8C2×D411yescharacteristic
274C4×Q8C2×D413yes—
275C4×D4—23no—
276C4×D4—23no—
277C4×D4—23no—
278C4×D4—23no—
279C4×D4—23no—
280C4×D4—23no—
281C4×D4—23no—
282C4×D4C22×C413yes—
283C4×D4C22×C413yes—
284C4×D4C22×C413yes—
285C4×D4C22×C413yes—
286#117—23no—
287#117—23no—
288#117—23no—
289#117C2411yescharacteristic
290#117C2413yes—
291C2×C4⋊C4—23no—
292C2×C4⋊C4—23no—
293C2×C4⋊C4#4113yes—
294C2×C42—23no—
295C2×C42—23no—
296C2×C42—23no—
297C2×C42—23no—
298C2×C42C2413yes—
299C2×C42C2411yescharacteristic
300C2×C42C2413yes—
301#108—43no—
302#108—43no—
303#107C22×C413yes—
304#107C22×C413yes—
305#106C22×C413yes—
306#106C22×C413yes—
307C4⋊C8—23no—
308C4⋊C8—23no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—23no—
312C4⋊C8—23no—
313C4⋊C8—23no—
314C4⋊C8—23no—
315C4⋊C8—23no—
316C4⋊C8—23no—
317C4⋊C8—23no—
318C4⋊C8—23no—
319C4⋊C8C22×C413yes—
320C4⋊C8C22×C413yes—
321C4⋊C8C22×C416yes—
322C4⋊C8C2×D413yes—
323C4⋊C8C2×D413yes—
324C4⋊C8C2×D416yes—
325#104—26no—
326#104—22no—
327#104—43no—
328#104—46no—
329#104—43no—
330#103C22×C411yescharacteristic
331#103C22×C411yescharacteristic
332#103C22×C413yes—
333#103C22×C413yes—
334#103C22×C416yes—
335#103C22×C416yes—
336#102C22×C411yescharacteristic
337#102C22×C411yescharacteristic
338#102C22×C413yes—
339#102C22×C413yes—
340#102C22×C416yes—
341#102C22×C416yes—
342C22⋊C8—23no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—23no—
346C22⋊C8—23no—
347C22⋊C8—23no—
348C22⋊C8—23no—
349C22⋊C8—23no—
350#97—26no—
351#97—23no—
352#97—23no—
353#97—22no—
354#97—43no—
355#97—43no—
356#97—43no—
357C4×C8—26no—
358C4×C8—23no—
359C4×C8—23no—
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—21no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—21no—
396C22×C4—23no—
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—23no—
409M16—21no—
410M16—23no—
411M16—23no—
412M16—21no—
413M16—21no—
414M16—23no—
415M16—86no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—23no—
421C2×C8—23no—
422C2×C8—23no—
423C2×C8—23no—
424C2×C8—21no—
425C2×C8—23no—
426C2×C8—21no—
427C2×C8—23no—
428C2×C8—86no—
429C2×C8C23×C411yescharacteristic
430C2×C8C23×C411yescharacteristic
431C2×C8C23×C413yes—
432C2×C8C23×C413yes—
433C2×C8C22×D413yes—
434C2×C8C22×D411yescharacteristic
435C2×C8C22×D411yescharacteristic
436C2×C8C22×D413yes—
437C4⋊C4—26no—
438C4⋊C4—23no—
439C4⋊C4—23no—
440C4⋊C4—23no—
441C4⋊C4—23no—
442C4⋊C4—23no—
443C4⋊C4—23no—
444C4⋊C4—23no—
445C4⋊C4—23no—
446C4⋊C4—26no—
447C4⋊C4—46no—
448C4⋊C4C4×D413yes—
449C4⋊C4C4×D413yes—
450C4⋊C4C4×D416yes—
451C4⋊C4C23×C413yes—
452C4⋊C4C23×C413yes—
453C4⋊C4C23×C411yescharacteristic
454C4⋊C4C23×C411yescharacteristic
455C22⋊C4—21no—
456C22⋊C4—23no—
457C22⋊C4—26no—
458C22⋊C4—23no—
459C22⋊C4—23no—
460C22⋊C4—23no—
461C22⋊C4—23no—
462C22⋊C4—26no—
463C42—23no—
464C42—23no—
465C42—23no—
466C42—23no—
467C42—23no—
468C42—21no—
469C42—21no—
470C42—23no—
471C42—23no—
472C42—46no—
473C42—83no—
474C42C23×C413yes—
475C42C23×C413yes—
476C42C22×D413yes—
477C42C22×D413yes—
478C422−1+412yes—
479C23—21no—
480C23—23no—
481C23C22×#4111yescharacteristic
482Q8—26no—
483Q8—21no—
484Q8—23no—
485Q8—23no—
486Q8—21no—
487Q8—46no—
488Q8—83no—
489Q8#44414yes—
490D4—21no—
491D4—23no—
492D4—23no—
493D4—21no—
494D4—41no—
495D4—43no—
496D4—41no—
497D4—43no—
498D4—46no—
499D4—83no—
500C2×C4—26no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—23no—
505C2×C4—23no—
506C2×C4—23no—
507C2×C4—22no—
508C2×C4—43no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—46no—
512C2×C4—46no—
513C2×C4—41no—
514C2×C4—43no—
515C2×C4—43no—
516C2×C4—41no—
517C2×C4—43no—
518C2×C4—43no—
519C2×C4—43no—
520C2×C4—83no—
521C2×C4—83no—
522C2×C4C2×C4×D413yes—
523C2×C4C2×C4×D413yes—
524C2×C4C2×C4×D413yes—
525C2×C4C2×C4×D413yes—
526C2×C4C24×C411yescharacteristic
527C2×C4C23×D411yescharacteristic
528C2×C4C2×2−1+411yescharacteristic
529C2×C4#58511yescharacteristic
530C2×C4#58511yescharacteristic
531C2×C4#58511yescharacteristic
532C8—23no—
533C8—23no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—21no—
538C8—23no—
539C8—23no—
540C8—41no—
541C8—43no—
542C8—43no—
543C8—41no—
544C22—21no—
545C22—21no—
546C22—43no—
547C22—41no—
548C22—41no—
549C22—43no—
550C22—41no—
551C22—43no—
552C22C22×C4×D411yescharacteristic
553C22#351611yescharacteristic
554C22#352411yescharacteristic
555C4—23no—
556C4—23no—
557C4—23no—
558C4—42no—
559C4—41no—
560C4—43no—
561C4—41no—
562C4—43no—
563C4—41no—
564C4—43no—
565C4—83no—
566C4C2×#44416yes—
567C4#342811yescharacteristic
568C4C4×2−1+411yescharacteristic
569C4#349711yescharacteristic
570C4#350011yescharacteristic
571C2—21no—
572C2—21no—
573C2—81no—
574C2—83no—
575C2#3431211yescharacteristic
576C2C22×#44411yescharacteristic
577C2#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. #60274C2semidirect
3. #60253C2semidirect
4. #23235C2semidirect
5. #23114C2semidirect
6. #22983C2semidirect
7. #22782C2semidirect
8. #22746C2semidirect
9. #22550C2semidirect
10. Q8×#104C2semidirect
11. #22166C2semidirect
12. #22131C2semidirect
13. #21660C2semidirect
14. #21293C2semidirect
15. Q8×#97C2semidirect
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
#602742
#602532
#232352
#231142
#229832
#227822
#227462
#225502
Q8×#1042
#221662
#221312
#216602
#212932
Q8×#972