Finite Group Databasebeta

← #6400730

Subgroups — #6400730

Order 512.

1721 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61303C211yesmaximal, characteristic
2#60274C211yesmaximal, characteristic, complemented
3Q8×#131C211yesmaximal, characteristic, complemented
4#23234C213yesmaximal, complemented
5#23015C213yesmaximal, complemented
6#22982C213yesmaximal, complemented
7Q8×#108C211yesmaximal, characteristic, complemented
8#22688C213yesmaximal, complemented
9#22550C211yesmaximal, characteristic, complemented
10Q8×#104C211yesmaximal, characteristic, complemented
11#22234C213yesmaximal, complemented
12#22130C213yesmaximal, complemented
13#21517C213yesmaximal, 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#2994—23no—
30#2993—23no—
31#2991—21no—
32#2988—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×#443C2216yes—
44C4×#135C2213yes—
45#2935C2213yes—
46#2931C2213yes—
47#2922C2213yes—
48#2906C2213yes—
49#2906C2213yes—
50C4×#131C2213yes—
51C42×Q8C2211yescharacteristic
52#1994C2213yes—
53#1986C2213yes—
54#1980C2213yes—
55#1962C2213yes—
56#1950C2213yes—
57#1950C2213yes—
58#1945C2216yes—
59#1937C2216yes—
60#1929C2216yes—
61#1929C2213yes—
62#1929C2213yes—
63#1922C2213yes—
64#1922C2213yes—
65#1876C2213yes—
66#1876C2213yes—
67#1866C2211yescharacteristic
68#1866C2213yes—
69#1866C2216yes—
70#1866C2216yes—
71#1848C2213yes—
72#1806C2213yes—
73#1806C2213yes—
74C4×#108C2213yes—
75C8×C4⋊C4C2213yes—
76C8×C4⋊C4C2213yes—
77C4×C4⋊C8C2213yes—
78#1764C2216yes—
79C4×#104C2213yes—
80C4×#104C2213yes—
81#1744C2213yes—
82C2×#135C2313yes—
83C2×#131C2311yescharacteristic
84C2×#131C2313yes—
85C2×#131C2316yes—
86#559C2×C413yes—
87Q82C2×C411yescharacteristic
88#557C2×C411yescharacteristic
89#557C2×C413yes—
90#556C2×C413yes—
91#554—23no—
92#554C2×C413yes—
93#553—23no—
94#550C2×C411yescharacteristic
95D4×Q8—21no—
96D4×Q8C2×C411yescharacteristic
97#548—21no—
98#533—23no—
99#533C2313yes—
100#532C2313yes—
101#529—23no—
102C4×#41—21no—
103C4×#41C2313yes—
104C4×#41C2313yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2311yescharacteristic
108#488C2×C416yes—
109#488C2×C416yes—
110#487C2×C413yes—
111#487C2×C413yes—
112#486C2×C413yes—
113#486C2×C413yes—
114#479C2×C416yes—
115#478C2×C416yes—
116#477C2×C413yes—
117#476C2×C413yes—
118#475C2×C413yes—
119#474C2×C413yes—
120#464C2×C416yes—
121#463C2×C416yes—
122#462C2×C413yes—
123#461C2×C413yes—
124#460C2×C413yes—
125#459C2×C413yes—
126#446—23no—
127#446—23no—
128#446—26no—
129#446—23no—
130#446—23no—
131#446—23no—
132C8×Q8—21no—
133C8×Q8—21no—
134C8×Q8—23no—
135C8×Q8—23no—
136C8×Q8—23no—
137C8×Q8C2×C411yescharacteristic
138C8×Q8C2×C411yescharacteristic
139C8×Q8C2×C413yes—
140C8×Q8C2×C413yes—
141C8×Q8D411yescharacteristic
142C8×Q8D413yes—
143C8×Q8D413yes—
144C8×Q8D411yescharacteristic
145#443—412no—
146C4×Q16C2×C413yes—
147C4×Q16C2×C413yes—
148C4×SD16C2×C413yes—
149C4×SD16C2×C413yes—
150C4×SD16C2×C413yes—
151C4×SD16C2×C413yes—
152C4×D8C2×C413yes—
153C4×D8C2×C413yes—
154#436—23no—
155#436—23no—
156#436—23no—
157#435—26no—
158C8×D4—23no—
159C8×D4—23no—
160C8×D4—23no—
161#433—23no—
162#433—23no—
163#431—23no—
164#431—23no—
165C2×#108C2311yescharacteristic
166C2×#108C2313yes—
167C2×#108C2316yes—
168#427C2313yes—
169#424—26no—
170#424—26no—
171#424C2316yes—
172#424C2313yes—
173#424C2311yescharacteristic
174#423—23no—
175#423—23no—
176#423C2313yes—
177#423C2313yes—
178C2×C4⋊C8—23no—
179C2×C4⋊C8—23no—
180C2×C4⋊C8C2313yes—
181C2×C4⋊C8C2313yes—
182C2×#104C2313yes—
183C2×#104C2311yescharacteristic
184C2×#104C2313yes—
185C2×#104C2316yes—
186C2×#104C2313yes—
187#416C2316yes—
188#416C2313yes—
189#416C2311yescharacteristic
190#408C2313yes—
191#408C2313yes—
192#405—23no—
193#405—23no—
194#405C2313yes—
195#405C2313yes—
196C4×M16—23no—
197C4×M16C2313yes—
198C4×M16C2313yes—
199C2×C4×C8—23no—
200C2×C4×C8C2316yes—
201C2×C4×C8C2313yes—
202C2×C4×C8C2313yes—
203C4×C4⋊C4—23no—
204C4×C4⋊C4C2313yes—
205C43C2313yes—
206#341C2313yes—
207#341C2313yes—
208#339C2313yes—
209#339C2316yes—
210#339C2313yes—
211#339C2311yescharacteristic
212#339C2313yes—
213#336C2313yes—
214#336C2313yes—
215#336C2311yescharacteristic
216#336C2313yes—
217C2×#41—21no—
218C2×#41C2413yes—
219C2×#41C2411yescharacteristic
220C22×Q8—21no—
221C22×Q8#4111yescharacteristic
222#135—26no—
223#135—26no—
224C2×Q16C22×C413yes—
225C2×SD16C22×C413yes—
226C2×SD16C22×C413yes—
227C2×D8C22×C413yes—
228#131—22no—
229#131—22no—
230#131—26no—
231#131—43no—
232#131—46no—
233#131—46no—
234C2×M16—21no—
235C2×M16—23no—
236C2×M16—26no—
237C2×M16C2413yes—
238C2×M16C2413yes—
239C2×M16C2411yescharacteristic
240C2×M16C2411yescharacteristic
241C22×C8—21no—
242C22×C8—23no—
243C22×C8—26no—
244C22×C8C2413yes—
245C22×C8C2411yescharacteristic
246C22×C8C2413yes—
247C22×C8C2411yescharacteristic
248C4⋊Q8—23no—
249C4⋊Q8—23no—
250C4⋊Q8C22×C413yes—
251C4⋊Q8C22×C413yes—
252C4⋊Q8C22×C413yes—
253#127C22×C413yes—
254#126—26no—
255#125—23no—
256#125—26no—
257#125C22×C416yes—
258#124—23no—
259#124C22×C416yes—
260#123—26no—
261C22⋊Q8—23no—
262C22⋊Q8—23no—
263C22⋊Q8—23no—
264C22⋊Q8—23no—
265C22⋊Q8—23no—
266C22⋊Q8—26no—
267#121—23no—
268#121—23no—
269#121—23no—
270C4×Q8—28no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—23no—
274C4×Q8—23no—
275C4×Q8—23no—
276C4×Q8C22×C413yes—
277C4×Q8C22×C413yes—
278C4×Q8C22×C413yes—
279C4×Q8C22×C413yes—
280C4×Q8C22×C411yescharacteristic
281C4×Q8C22×C413yes—
282C4×Q8C2×D411yescharacteristic
283C4×Q8C2×D413yes—
284C4×D4—23no—
285C4×D4—23no—
286C4×D4—23no—
287C4×D4—23no—
288C4×D4—23no—
289C4×D4—23no—
290C4×D4—23no—
291C4×D4C22×C413yes—
292C4×D4C22×C413yes—
293C4×D4C22×C413yes—
294C4×D4C22×C413yes—
295#117—23no—
296#117—23no—
297#117—23no—
298#117C2411yescharacteristic
299#117C2413yes—
300C2×C4⋊C4—23no—
301C2×C4⋊C4—23no—
302C2×C4⋊C4#4113yes—
303C2×C42—23no—
304C2×C42—23no—
305C2×C42—23no—
306C2×C42—23no—
307C2×C42C2411yescharacteristic
308C2×C42C2413yes—
309C2×C42C2413yes—
310#108—22no—
311#108—43no—
312#108—46no—
313#107C22×C413yes—
314#107C22×C413yes—
315#106C22×C413yes—
316#106C22×C413yes—
317C4⋊C8—23no—
318C4⋊C8—26no—
319C4⋊C8—23no—
320C4⋊C8—26no—
321C4⋊C8—21no—
322C4⋊C8—21no—
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—22no—
335#104—26no—
336#104—43no—
337#104—46no—
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—23no—
351C22⋊C8—23no—
352C22⋊C8—23no—
353C22⋊C8—23no—
354#97—23no—
355#97—23no—
356#97—23no—
357#97—23no—
358C4×C8—23no—
359C4×C8—23no—
360C4×C8—23no—
361C4×C8—23no—
362C4×C8—46no—
363C4×C8C22×C413yes—
364C4×C8C22×C413yes—
365C4×C8C22×C413yes—
366C4×C8C22×C413yes—
367C4×C8C2×D413yes—
368C4×C8C2×D413yes—
369C4×C8C2×D413yes—
370C4×C8C2×D413yes—
371#95—23no—
372#41—21no—
373#41—21no—
374#41—23no—
375#41—23no—
376#41—23no—
377#41—23no—
378#41—44no—
379#41H(5,𝔽2)12yes—
380#412−1+412yes—
381C2×Q8—24no—
382C2×Q8—23no—
383C2×Q8—23no—
384C2×Q8—44no—
385C2×Q8C4×D413yes—
386C2×Q8C4×D411yescharacteristic
387C2×Q8C23×C413yes—
388C2×Q8C23×C411yescharacteristic
389C2×D4—21no—
390C2×D4—23no—
391C2×D4—23no—
392C2×D4—23no—
393C2×D4C23×C413yes—
394C2×D4C23×C411yescharacteristic
395C22×C4—21no—
396C22×C4—23no—
397C22×C4—21no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4C2×#4113yes—
404C22×C4C2×#4113yes—
405C22×C4C2511yescharacteristic
406Q16—26no—
407SD16—26no—
408SD16—26no—
409D8—26no—
410M16—24no—
411M16—21no—
412M16—23no—
413M16—23no—
414M16—21no—
415M16—21no—
416M16—23no—
417M16—23no—
418M16—46no—
419M16—86no—
420M162−1+412yes—
421C2×C8—24no—
422C2×C8—21no—
423C2×C8—23no—
424C2×C8—23no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—23no—
429C2×C8—21no—
430C2×C8—23no—
431C2×C8—21no—
432C2×C8—23no—
433C2×C8—23no—
434C2×C8—46no—
435C2×C8—86no—
436C2×C8C23×C411yescharacteristic
437C2×C8C23×C411yescharacteristic
438C2×C8C23×C413yes—
439C2×C8C23×C413yes—
440C2×C8C22×D411yescharacteristic
441C2×C8C22×D413yes—
442C2×C8C22×D411yescharacteristic
443C2×C8C22×D413yes—
444C4⋊C4—212no—
445C4⋊C4—26no—
446C4⋊C4—23no—
447C4⋊C4—23no—
448C4⋊C4—23no—
449C4⋊C4—23no—
450C4⋊C4—23no—
451C4⋊C4—23no—
452C4⋊C4—23no—
453C4⋊C4—23no—
454C4⋊C4—26no—
455C4⋊C4C4×D413yes—
456C4⋊C4C4×D413yes—
457C4⋊C4C4×D416yes—
458C4⋊C4C23×C413yes—
459C4⋊C4C23×C413yes—
460C4⋊C4C23×C411yescharacteristic
461C4⋊C4C23×C411yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—23no—
464C22⋊C4—26no—
465C22⋊C4—23no—
466C22⋊C4—23no—
467C22⋊C4—23no—
468C22⋊C4—23no—
469C22⋊C4—26no—
470C42—212no—
471C42—23no—
472C42—23no—
473C42—23no—
474C42—23no—
475C42—23no—
476C42—21no—
477C42—23no—
478C42—21no—
479C42—23no—
480C42—21no—
481C42—83no—
482C42C23×C413yes—
483C42C23×C413yes—
484C42C22×D413yes—
485C42C22×D413yes—
486C23—21no—
487C23—23no—
488C23C22×#4111yescharacteristic
489Q8—21no—
490Q8—26no—
491Q8—23no—
492Q8—23no—
493Q8—46no—
494Q8—83no—
495Q8#44314yescomplemented
496Q8#52012yes—
497D4—22no—
498D4—22no—
499D4—23no—
500D4—21no—
501D4—23no—
502D4—43no—
503D4—43no—
504D4—46no—
505D4—83no—
506D4#52012yes—
507C2×C4—21no—
508C2×C4—26no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—26no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—23no—
516C2×C4—23no—
517C2×C4—23no—
518C2×C4—43no—
519C2×C4—41no—
520C2×C4—43no—
521C2×C4—41no—
522C2×C4—43no—
523C2×C4—46no—
524C2×C4—46no—
525C2×C4—43no—
526C2×C4—43no—
527C2×C4—83no—
528C2×C4—83no—
529C2×C4C2×C4×D413yes—
530C2×C4C2×C4×D413yes—
531C2×C4C2×C4×D413yes—
532C2×C4C2×C4×D413yes—
533C2×C4C24×C411yescharacteristic
534C2×C4C23×D411yescharacteristic
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4C2×2−1+411yescharacteristic
537C2×C4C2×2−1+411yescharacteristic
538C2×C4#58511yescharacteristic
539C8—23no—
540C8—23no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—22no—
545C8—21no—
546C8—23no—
547C8—23no—
548C8—41no—
549C8—43no—
550C8—43no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—43no—
557C22—43no—
558C22—43no—
559C22C22×C4×D411yescharacteristic
560C22#351611yescharacteristic
561C22#352211yescharacteristic
562C4—23no—
563C4—23no—
564C4—23no—
565C4—22no—
566C4—46no—
567C4—42no—
568C4—41no—
569C4—43no—
570C4—41no—
571C4—43no—
572C4—43no—
573C4C2×#44316yes—
574C4C2×#52011yescharacteristic
575C4C4×2−1+411yescharacteristic
576C4#348611yescharacteristic
577C4#348811yescharacteristic
578C2—21no—
579C2—21no—
580C2—42no—
581C2—83no—
582C2#3429411yescharacteristic
583C2C22×#44311yescharacteristic
584C2#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. Q8×#131C2semidirect
4. #23234C2semidirect
5. #23015C2semidirect
6. #22982C2semidirect
7. Q8×#108C2semidirect
8. #22688C2semidirect
9. #22550C2semidirect
10. Q8×#104C2semidirect
11. #22234C2semidirect
12. #22130C2semidirect
13. #21517C2semidirect
14. #21137C2semidirect
15. #20569C2semidirect
18. #3365C4semidirect
19. #3364C4semidirect
20. #3363C4semidirect
21. #3362C4semidirect
22. #3361C4semidirect
23. #3360C4semidirect
24. #3359C4semidirect
25. #3358C4semidirect
495. Q8#443semidirect

Maximal subgroups

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

subgroupindex
#613032
#602742
Q8×#1312
#232342
#230152
#229822
Q8×#1082
#226882
#225502
Q8×#1042
#222342
#221302
#215172
#211372
#205692