Finite Group Databasebeta

← #203015

Subgroups — #203015

Order 512.

712 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#21418C211yesmaximal, characteristic
2#20431C211yesmaximal, characteristic, complemented
3#17807C211yesmaximal, characteristic, complemented
4#17737C211yesmaximal, characteristic
5#17466C211yesmaximal, characteristic
6#9584C211yesmaximal, characteristic, complemented
7#9583C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9581C211yesmaximal, characteristic, complemented
10#9580C211yesmaximal, characteristic, complemented
11#9579C211yesmaximal, characteristic, complemented
12#9578C211yesmaximal, characteristic, complemented
13#9577C211yesmaximal, characteristic, complemented
14#8078C211yesmaximal, characteristic, complemented
15#7868C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#405—21no—
23C4×#117C2211yescharacteristic
24#1994C2211yescharacteristic
25#1994C2211yescharacteristic
26#1962C2211yescharacteristic
27#1962C2211yescharacteristic
28#1897—21no—
29#1897—21no—
30#1768—21no—
31#1768—21no—
32#1767C2211yescharacteristic
33#1767C2211yescharacteristic
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36#1759—21no—
37#1759—21no—
38#1729—21no—
39#1573—22no—
40#1571C2211yescharacteristic
41#1571C2211yescharacteristic
42#1570C2211yescharacteristic
43#1570C2211yescharacteristic
44#1565—22no—
45#1561C412yescomplemented
46#1560C412yescomplemented
47#1559C412yescomplemented
48#1557C412yescomplemented
49#1542C2211yescharacteristic
50#1542C2211yescharacteristic
51#1540C2211yescharacteristic
52#1540C2211yescharacteristic
53#1483—22no—
54#1481C2211yescharacteristic
55#1481C2211yescharacteristic
56#1480C2211yescharacteristic
57#1480C2211yescharacteristic
58#1477—22no—
59#1301C2211yescharacteristic
60#1301C2211yescharacteristic
61#1301C2211yescharacteristic
62#1301C2211yescharacteristic
63#1276C2211yescharacteristic
64#1276C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69#533C2×C411yescharacteristic
70#533C2×C411yescharacteristic
71#533C2311yescharacteristic
72#532C2×C411yescharacteristic
73#532C2×C411yescharacteristic
74#532C2311yescharacteristic
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#525—21no—
82#525—21no—
83#525—21no—
84C4×#41—21no—
85C4×#41—21no—
86C4×#41—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C42C2311yescharacteristic
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432C2×C412yes—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104#423—21no—
105#423—21no—
106#423—21no—
107#423C2×C412yes—
108#423C2×C412yes—
109#423C2×C412yes—
110#423C2311yescharacteristic
111#423C2311yescharacteristic
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8D412yes—
117C2×C4⋊C8D412yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×#104—21no—
123C2×#104—21no—
124C2×#104—21no—
125C2×#104—21no—
126#407—21no—
127#407—21no—
128#407—21no—
129#405—44no—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16—21no—
133C2×#97—21no—
134C2×#97—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141#376D411yescharacteristic
142#376D411yescharacteristic
143C43C2311yescharacteristic
144C43C2311yescharacteristic
145#339—21no—
146#339—21no—
147#339—21no—
148#339—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#335—22no—
154#335—22no—
155#334—22no—
156#334—22no—
157#333C2×C412yes—
158#333C2×C412yes—
159#332C2×C412yes—
160#332C2×C412yes—
161#332C2×C412yes—
162#332C2×C412yes—
163#331C2×C412yes—
164#331C2×C412yes—
165Q8⋊C8—22no—
166Q8⋊C8—22no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C23×C4#4111yescharacteristic
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—41no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—42no—
191C4⋊Q8C22×C411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193C4⋊Q8C22×C411yescharacteristic
194#127C22×C411yescharacteristic
195#126—41no—
196#126—41no—
197#125—21no—
198#125—21no—
199#125—21no—
200#124—21no—
201#124—21no—
202#124—21no—
203#123—41no—
204#123—41no—
205#123—41no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212#121—21no—
213#121—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121—21no—
218#120—41no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237#117—22no—
238#117—22no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—41no—
243#117—41no—
244#117—41no—
245#117—41no—
246#117—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—41no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42C2411yescharacteristic
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—41no—
288C4⋊C8C22×C412yes—
289C4⋊C8C22×C412yes—
290C4⋊C8C22×C412yes—
291C4⋊C8C22×C412yes—
292C4⋊C8C2×D412yes—
293C4⋊C8C2×D412yes—
294#104—42no—
295#104—42no—
296#104—42no—
297#104—42no—
298C22⋊C8—22no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305C22⋊C8—41no—
306#97—22no—
307#97—42no—
308#97—41no—
309#97—41no—
310#97—42no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—42no—
316C4×C8—41no—
317C4×C8—42no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322C24#12311yescharacteristic
323#41—44no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—82no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—81no—
351C22×C4#11711yescharacteristic
352C22×C4#11711yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12311yescharacteristic
360C22×C4#12311yescharacteristic
361C22×C4#12311yescharacteristic
362C22×C4#12411yescharacteristic
363C22×C4#12411yescharacteristic
364C22×C4#12611yescharacteristic
365C22×C4#12611yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369M16—44no—
370M16—44no—
371M16—44no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—82no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C22⋊C4—24no—
412C22⋊C4—42no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—42no—
429C42—42no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—81no—
434C42—81no—
435C42#11711yescharacteristic
436C42#11711yescharacteristic
437C42C4×D411yescharacteristic
438C42C4×D411yescharacteristic
439C42C4×D411yescharacteristic
440C42C4×D411yescharacteristic
441C42#12111yescharacteristic
442C42#12111yescharacteristic
443C42#12411yescharacteristic
444C42#12411yescharacteristic
445C42#12611yescharacteristic
446C42#12611yescharacteristic
447C42AGL(1,ℤ/8)12yes—
448C42#13712yes—
449C42C23×C411yescharacteristic
450C42C22×D411yescharacteristic
451C42C2×#4111yescharacteristic
452C42C2×#4111yescharacteristic
453C23—21no—
454C23—21no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23#38811yescharacteristic
459C23#38811yescharacteristic
460C23C2×#12311yescharacteristic
461Q8—41no—
462Q8—41no—
463Q8—42no—
464Q8—81no—
465D4—41no—
466D4—41no—
467D4—42no—
468D4—81no—
469D4—82no—
470D4—82no—
471D4—81no—
472D4—81no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4#38811yescharacteristic
515C2×C4#38811yescharacteristic
516C2×C4#38811yescharacteristic
517C2×C4#38811yescharacteristic
518C2×C4#38811yescharacteristic
519C2×C4#38811yescharacteristic
520C2×C4C2×#11711yescharacteristic
521C2×C4C2×C4×D411yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×#12411yescharacteristic
525C2×C4C2×#12611yescharacteristic
526C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
527C2×C4C2×#13711yescharacteristic
528C2×C4#57511yescharacteristic
529C2×C4#57511yescharacteristic
530C8—42no—
531C8—42no—
532C8—42no—
533C8—42no—
534C8—42no—
535C8—42no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8—81no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#38811yescharacteristic
554C22#309211yescharacteristic
555C22#309311yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4C4×AGL(1,ℤ/8)11yescharacteristic
573C4C4×#13711yescharacteristic
574C4#295512yes—
575C4#303611yescharacteristic
576C4#303711yescharacteristic
577C2—21no—
578C2—21no—
579C2—41no—
580C2—41no—
581C2—161no—
582C2#2105911yescharacteristic
583C2#2108011yescharacteristic
584C2#2110011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
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
2. #20431C2semidirect
3. #17807C2semidirect
6. #9584C2semidirect
7. #9583C2semidirect
8. #9582C2semidirect
9. #9581C2semidirect
10. #9580C2semidirect
11. #9579C2semidirect
12. #9578C2semidirect
13. #9577C2semidirect
14. #8078C2semidirect
15. #7868C2semidirect
45. #1561C4semidirect
46. #1560C4semidirect
47. #1559C4semidirect
48. #1557C4semidirect

Maximal subgroups

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

subgroupindex
#214182
#204312
#178072
#177372
#174662
#95842
#95832
#95822
#95812
#95802
#95792
#95782
#95772
#80782
#78682