Finite Group Databasebeta

← #192950

Subgroups — #192950

Order 512.

878 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60268C211yesmaximal, characteristic
2#21899C211yesmaximal, characteristic
3#17817C211yesmaximal, characteristic, complemented
4#17803C211yesmaximal, characteristic, complemented
5#17458C211yesmaximal, characteristic
6#8905C212yesmaximal, complemented
7#8904C212yesmaximal, complemented
8#8903C212yesmaximal, complemented
9#8902C212yesmaximal, complemented
10C2×#1303C211yesmaximal, characteristic, complemented
11#7892C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2994—22no—
14#2993—22no—
15#2991—21no—
16#2990—21no—
17#2989—21no—
18#2988—21no—
19#2960—21no—
20#2933—22no—
21#2931—21no—
22#2925C2212yes—
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2906C2212yes—
28C2×#423C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1986—21no—
31#1980—21no—
32#1896—22no—
33#1806C2212yes—
34#1806C2212yes—
35#1574—22no—
36#1573—22no—
37#1572—21no—
38#1572—21no—
39#1571C2211yescharacteristic
40#1571C2211yescharacteristic
41#1570C2211yescharacteristic
42#1570C2211yescharacteristic
43#1567C2212yes—
44#1566C2212yes—
45#1564—21no—
46#1564—21no—
47#1482C412yescomplemented
48#1482C412yescomplemented
49#1481C411yescharacteristic, complemented
50#1481C411yescharacteristic, complemented
51#1480C411yescharacteristic, complemented
52#1480C411yescharacteristic, complemented
53#1476C2212yes—
54#1476C2212yes—
55#1303C2212yes—
56#1303C2212yes—
57#1279C2212yes—
58#1279C2212yes—
59C2×#131—22no—
60C22×M16—21no—
61#533C2×C411yescharacteristic
62#533D411yescharacteristic
63#533Q811yescharacteristic
64#532C2×C411yescharacteristic
65#532D411yescharacteristic
66#532Q811yescharacteristic
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71#529C2×C412yes—
72#529Q812yes—
73#525—21no—
74#525—21no—
75#525D412yes—
76C4×#41—21no—
77C4×#41—21no—
78C4×#41C2312yes—
79C22×C4⋊C4—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#446—22no—
83#446—22no—
84C8×Q8—22no—
85C8×Q8—22no—
86#436—22no—
87#436—22no—
88#436—24no—
89#435—24no—
90#435—24no—
91C8×D4—22no—
92C8×D4—22no—
93C8×D4—24no—
94#433—24no—
95#432—22no—
96#432—21no—
97#432—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—22no—
105#431—21no—
106#431—21no—
107#431C2×C411yescharacteristic
108#431C2×C411yescharacteristic
109#431C2×C411yescharacteristic
110#431C2×C411yescharacteristic
111#431C2×C412yes—
112#431D411yescharacteristic
113#431D411yescharacteristic
114#431D411yescharacteristic
115#431D411yescharacteristic
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#424—22no—
119#424—24no—
120#423—21no—
121#423—21no—
122#423—22no—
123#423D412yes—
124#423C2311yescharacteristic
125#423C2311yescharacteristic
126#423C2312yes—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—22no—
130C2×C4⋊C8C2×C411yescharacteristic
131C2×C4⋊C8C2×C411yescharacteristic
132C2×C4⋊C8D412yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×#104—22no—
136C2×#104—22no—
137#408—22no—
138#407—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142#405—22no—
143C4×M16—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8C2312yes—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2312yes—
151C2×#95—21no—
152#339—22no—
153#339—22no—
154#335—21no—
155#335—21no—
156#335—21no—
157#335—21no—
158#335—24no—
159#334—21no—
160#334—21no—
161#334—21no—
162#334—21no—
163#334—24no—
164Q8⋊C8C2×C412yescomplemented
165Q8⋊C8C2×C412yescomplemented
166Q8⋊C8C2×C412yescomplemented
167Q8⋊C8C2×C412yescomplemented
168D4⋊C8C2×C412yescomplemented
169D4⋊C8C2×C412yescomplemented
170D4⋊C8C2×C412yescomplemented
171D4⋊C8C2×C412yescomplemented
172C2×#41—21no—
173C2×#41—21no—
174C2×#41C2×D412yes—
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4#4111yescharacteristic
178C23×C4#4111yescharacteristic
179#131—82no—
180#131—82no—
181C2×M16—22no—
182C2×M16—24no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—81no—
187C2×M16—81no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—24no—
193C22×C8—22no—
194C22×C8—22no—
195C4⋊Q8C4⋊C411yescharacteristic
196C4⋊Q8C4⋊C411yescharacteristic
197C4⋊Q8C4⋊C411yescharacteristic
198#127C4⋊C411yescharacteristic
199#126—22no—
200#126—22no—
201#125—21no—
202#125—21no—
203#125C4⋊C412yes—
204#124—21no—
205#124—21no—
206#124C4⋊C412yes—
207#123—22no—
208#123—22no—
209#123—22no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8C4⋊C412yes—
215C22⋊Q8C4⋊C412yes—
216#121—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121C4⋊C412yes—
221#121C4⋊C412yes—
222#120—22no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8C22×C412yes—
228C4×Q8C2×Q812yes—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4C22×C412yes—
240C4×D4C2×Q812yes—
241#117—21no—
242#117—21no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117C2×D412yes—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—41no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—41no—
254C2×C42—81no—
255C2×C42—81no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×Q811yescharacteristic
264C2×C42C2×Q811yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—24no—
269C4⋊C8—24no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—22no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8C22×C411yescharacteristic
285C4⋊C8C22×C411yescharacteristic
286C4⋊C8C22×C411yescharacteristic
287C4⋊C8C22×C411yescharacteristic
288C4⋊C8C2×D411yescharacteristic
289C4⋊C8C2×D414yes—
290C4⋊C8C2×D411yescharacteristic
291C4⋊C8C2×D411yescharacteristic
292C4⋊C8C2×D411yescharacteristic
293#104—82no—
294#104—82no—
295C22⋊C8—24no—
296C22⋊C8—24no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—22no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310#97—24no—
311#97—41no—
312C4×C8—24no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—22no—
321C4×C8—21no—
322C4×C8—41no—
323C4×C8C22×C412yes—
324C4×C8C22×C412yes—
325#95—22no—
326#95—41no—
327C24#12311yescharacteristic
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—42no—
333#41—42no—
334#41—82no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8C2×C4⋊C412yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4C2×C4⋊C412yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—81no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4C2×C4⋊C411yescharacteristic
359C22×C4C2×C4⋊C411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4C22⋊Q811yescharacteristic
367C22×C4C22⋊Q811yescharacteristic
368C22×C4C22⋊Q811yescharacteristic
369C22×C4C22⋊Q811yescharacteristic
370C22×C4#12311yescharacteristic
371C22×C4#12311yescharacteristic
372C22×C4#12311yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376M16—42no—
377M16—42no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—82no—
383M16—84no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—24no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—82no—
404C2×C8—82no—
405C2×C8—84no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4C2×C4⋊C412yes—
420C4⋊C4C2×C4⋊C412yes—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—42no—
434C42—42no—
435C42—82no—
436C42C2×C4⋊C411yescharacteristic
437C42C2×C4⋊C411yescharacteristic
438C42C4×D411yescharacteristic
439C42C4×D411yescharacteristic
440C42C4×D411yescharacteristic
441C42C4×D411yescharacteristic
442C42#12011yescharacteristic
443C42#12011yescharacteristic
444C42C22⋊Q811yescharacteristic
445C42C22⋊Q811yescharacteristic
446C42C22⋊Q811yescharacteristic
447C42C22⋊Q811yescharacteristic
448C42C23×C411yescharacteristic
449C42C22×D411yescharacteristic
450C42C22×D411yescharacteristic
451C42C22×Q811yescharacteristic
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23#38511yescharacteristic
458C23#38511yescharacteristic
459C23C2×#12311yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—42no—
463Q8—42no—
464D4—22no—
465D4—22no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—44no—
470D4—42no—
471D4—82no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—88no—
506C2×C4—82no—
507C2×C4#38511yescharacteristic
508C2×C4#38511yescharacteristic
509C2×C4#38511yescharacteristic
510C2×C4#38511yescharacteristic
511C2×C4#38511yescharacteristic
512C2×C4#38511yescharacteristic
513C2×C4#44312yes—
514C2×C4#44412yes—
515C2×C4C22×C4⋊C411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×#12011yescharacteristic
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4#57611yescharacteristic
522C2×C4#57711yescharacteristic
523C2×C4#57711yescharacteristic
524C2×C4#57811yescharacteristic
525C8—42no—
526C8—42no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—44no—
534C8—41no—
535C8—41no—
536C8—81no—
537C8—81no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22#186612yes—
551C22C2×#38511yescharacteristic
552C22#309611yescharacteristic
553C22#309711yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—84no—
566C4#186612yes—
567C4#291211yescharacteristic
568C4#291311yescharacteristic
569C4C2×#44311yescharacteristic
570C4C2×#44411yescharacteristic
571C4#301211yescharacteristic
572C4#301311yescharacteristic
573C2—21no—
574C2—21no—
575C2—22no—
576C2—41no—
577C2—82no—
578C2#2084911yescharacteristic
579C2#2086711yescharacteristic
580C2C2×#186611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17817C2semidirect
4. #17803C2semidirect
6. #8905C2semidirect
7. #8904C2semidirect
8. #8903C2semidirect
9. #8902C2semidirect
10. C2×#1303C2semidirect
11. #7892C2semidirect
47. #1482C4semidirect
48. #1482C4semidirect
49. #1481C4semidirect
50. #1481C4semidirect
51. #1480C4semidirect
52. #1480C4semidirect
164. Q8⋊C8C2×C4semidirect
165. Q8⋊C8C2×C4semidirect
166. Q8⋊C8C2×C4semidirect
167. Q8⋊C8C2×C4semidirect
168. D4⋊C8C2×C4semidirect
169. D4⋊C8C2×C4semidirect
170. D4⋊C8C2×C4semidirect
171. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602682
#218992
#178172
#178032
#174582
#89052
#89042
#89032
#89022
C2×#13032
#78922