Finite Group Databasebeta

← #230413

Subgroups — #230413

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23114C211yesmaximal, characteristic, complemented
2#23113C211yesmaximal, characteristic, complemented
3#19125C211yesmaximal, characteristic
4#18066C211yesmaximal, characteristic, complemented
5#17829C211yesmaximal, characteristic, complemented
6#9847C211yesmaximal, characteristic, complemented
7#9624C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9578C211yesmaximal, characteristic, complemented
10#8996C211yesmaximal, characteristic, complemented
11#8996C211yesmaximal, characteristic, complemented
12#8932C211yesmaximal, characteristic, complemented
13#8932C211yesmaximal, characteristic, complemented
14#8278C211yesmaximal, characteristic, complemented
15#8191C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3035C2211yescharacteristic
18#3035C2211yescharacteristic
19C8×#41C2211yescharacteristic
20#2935C2211yescharacteristic
21#2935C2211yescharacteristic
22#2925C2211yescharacteristic
23#2925C2211yescharacteristic
24#1975—21no—
25#1974—21no—
26#1962C2211yescharacteristic
27#1962C2211yescharacteristic
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1929—21no—
31#1929—21no—
32#1876—21no—
33#1876—21no—
34#1767C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36#1765—21no—
37#1764—21no—
38C4×#104C2211yescharacteristic
39#1744C2211yescharacteristic
40#1682C411yescharacteristic, complemented
41#1682C411yescharacteristic, complemented
42#1681C411yescharacteristic, complemented
43#1681C411yescharacteristic, complemented
44#1674C411yescharacteristic, complemented
45#1673C411yescharacteristic, complemented
46#1668C411yescharacteristic, complemented
47#1667C411yescharacteristic, complemented
48#1589—21no—
49#1581—21no—
50#1581—21no—
51C8×D8—21no—
52#1572—21no—
53#1572—21no—
54#1570C2211yescharacteristic
55#1570C2211yescharacteristic
56#1568—21no—
57#1567C2211yescharacteristic
58#1564—21no—
59C2×#335C2211yescharacteristic
60C2×#335C2211yescharacteristic
61#1540C2211yescharacteristic
62#1540C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65#1480C2211yescharacteristic
66#1480C2211yescharacteristic
67#1476C2211yescharacteristic
68C8×M16C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1278C2211yescharacteristic
72#1277C2211yescharacteristic
73#1277C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#135—21no—
77C2×#131—21no—
78C2×#131—21no—
79#532C2311yescharacteristic
80#532C2311yescharacteristic
81C4×#41C2311yescharacteristic
82#500C2×C411yescharacteristic
83#493C2×C411yescharacteristic
84#486C2×C411yescharacteristic
85#486C2×C411yescharacteristic
86#466—21no—
87#466—21no—
88#464—21no—
89#464—21no—
90#463—21no—
91#463—21no—
92#462C2×C411yescharacteristic
93#462C2×C411yescharacteristic
94#461C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#460C2×C411yescharacteristic
97#460C2×C411yescharacteristic
98#459C2×C411yescharacteristic
99#459C2×C411yescharacteristic
100C8×Q8C2×C411yescharacteristic
101C8×Q8C2×C411yescharacteristic
102C4×D8—21no—
103C4×D8—21no—
104#436—21no—
105#436—21no—
106#435—41no—
107#435—41no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4—21no—
111C8×D4—21no—
112C8×D4C2×C411yescharacteristic
113C8×D4C2×C411yescharacteristic
114#433—41no—
115#433—41no—
116#432—21no—
117#432—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431C2311yescharacteristic
122#431C2311yescharacteristic
123C2×#108—21no—
124C2×#108—21no—
125#423C2311yescharacteristic
126#423C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×#104—21no—
132C2×#104—21no—
133C2×#104—21no—
134C2×#104—21no—
135C2×#104D411yescharacteristic
136C2×#104D411yescharacteristic
137#416—21no—
138#416—21no—
139#408—21no—
140#408—21no—
141#408—21no—
142#408—21no—
143#405—21no—
144#405—21no—
145C4×M16C2311yescharacteristic
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C43C2311yescharacteristic
152#341—21no—
153#341—21no—
154#339D411yescharacteristic
155#339D411yescharacteristic
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335C2×C411yescharacteristic
163#335C2×C411yescharacteristic
164#335C2×C411yescharacteristic
165#335C2×C411yescharacteristic
166#335D411yescharacteristic
167#335D411yescharacteristic
168#335D411yescharacteristic
169#335D411yescharacteristic
170#334—21no—
171#334—21no—
172#334—21no—
173#334—21no—
174#332C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#331C2×C411yescharacteristic
179#331C2×C411yescharacteristic
180#331C2×C411yescharacteristic
181#331C2×C411yescharacteristic
182Q8⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—41no—
189D4⋊C8—41no—
190#322—21no—
191C82—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41C2×D411yescharacteristic
195#135—81no—
196#135—81no—
197#135—81no—
198#135—81no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×SD16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×D8—21no—
206C2×D8—21no—
207C2×D8—21no—
208C2×D8—21no—
209#131—42no—
210#131—42no—
211#131—81no—
212#131—81no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—41no—
216C2×M16—41no—
217C2×M16C2×D411yescharacteristic
218C2×M16C2×D411yescharacteristic
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—41no—
227C22×C8—41no—
228C22×C8#4111yescharacteristic
229C22×C8#4111yescharacteristic
230C4⋊Q8C22×C411yescharacteristic
231C4⋊Q8C22×C411yescharacteristic
232#127C22×C411yescharacteristic
233#127C22×C411yescharacteristic
234#124—21no—
235#124—21no—
236#121—21no—
237#121—21no—
238#121—21no—
239#121—21no—
240C4×Q8C22×C411yescharacteristic
241C4×Q8#4111yescharacteristic
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4C22×C411yescharacteristic
249C4×D4#4111yescharacteristic
250#117—21no—
251#117C2×D411yescharacteristic
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2411yescharacteristic
258#108—81no—
259#108—81no—
260#107—21no—
261#107—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8C22×C411yescharacteristic
274C4⋊C8C22×C411yescharacteristic
275C4⋊C8C22×C411yescharacteristic
276C4⋊C8C22×C411yescharacteristic
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C2×D411yescharacteristic
280C4⋊C8C2×D411yescharacteristic
281C4⋊C8C2×D411yescharacteristic
282C4⋊C8C2×D411yescharacteristic
283#104—22no—
284#104—22no—
285#104—42no—
286#104—42no—
287#104—81no—
288#104—81no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310C22⋊C8—41no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313#97—21no—
314#97—41no—
315#97—41no—
316#97—81no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—41no—
325C4×C8—41no—
326C4×C8—81no—
327C4×C8C22×C411yescharacteristic
328C4×C8C22×C411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331#41—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8C4×D411yescharacteristic
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4C4×D411yescharacteristic
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C42−1+411yescharacteristic
366Q16—81no—
367Q16—81no—
368SD16—81no—
369SD16—81no—
370SD16—81no—
371SD16—81no—
372D8—42no—
373D8—81no—
374D8—81no—
375D8—81no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—82no—
384M16—82no—
385M16#13512yes—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—81no—
416C2×C8—82no—
417C2×C8—82no—
418C2×C8C4×D411yescharacteristic
419C2×C8C4×D411yescharacteristic
420C2×C8C4×D411yescharacteristic
421C2×C8C4×D411yescharacteristic
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4C4×D411yescharacteristic
429C4⋊C4C4×D411yescharacteristic
430C22⋊C4—21no—
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—41no—
444C42—41no—
445C42—81no—
446C42#13512yes—
447C42C23×C411yescharacteristic
448C42C22×D411yescharacteristic
449C42C2×#4111yescharacteristic
450C42H(5,𝔽2)11yescharacteristic
451C23—21no—
452C23—41no—
453C23—41no—
454C23#54711yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461D4—21no—
462D4—21no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—81no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478D4—81no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
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—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4#52011yescharacteristic
505C2×C4D4211yescharacteristic
506C2×C4#54811yescharacteristic
507C2×C4#55311yescharacteristic
508C2×C4C2×#13511yescharacteristic
509C2×C4C2×#13511yescharacteristic
510C2×C4#57611yescharacteristic
511C2×C4#57611yescharacteristic
512C8—21no—
513C8—21no—
514C8—21no—
515C8—21no—
516C8—22no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—81no—
529C8—81no—
530C8—81no—
531C8—81no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22#236111yescharacteristic
544C22#333911yescharacteristic
545C22#334111yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—42no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4C4×#13511yescharacteristic
557C4#294811yescharacteristic
558C4#295511yescharacteristic
559C4#295711yescharacteristic
560C4#328011yescharacteristic
561C4#328711yescharacteristic
562C2—21no—
563C2—21no—
564C2—81no—
565C2—161no—
566C2—161no—
567C2#2301711yescharacteristic
568C2#2302711yescharacteristic
569C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#19125C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23114C2semidirect
2. #23113C2semidirect
4. #18066C2semidirect
5. #17829C2semidirect
6. #9847C2semidirect
7. #9624C2semidirect
8. #9582C2semidirect
9. #9578C2semidirect
10. #8996C2semidirect
11. #8996C2semidirect
12. #8932C2semidirect
13. #8932C2semidirect
14. #8278C2semidirect
15. #8191C2semidirect
40. #1682C4semidirect
41. #1682C4semidirect
42. #1681C4semidirect
43. #1681C4semidirect
44. #1674C4semidirect
45. #1673C4semidirect
46. #1668C4semidirect
47. #1667C4semidirect

Maximal subgroups

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

subgroupindex
#231142
#231132
#191252
#180662
#178292
#98472
#96242
#95822
#95782
#89962
#89322
#82782
#81912