Finite Group Databasebeta

← #211878

Subgroups — #211878

Order 512.

682 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21518C211yesmaximal, characteristic, complemented
2#21517C211yesmaximal, characteristic, complemented
3#19125C211yesmaximal, characteristic
4#17975C211yesmaximal, characteristic, complemented
5#17316C211yesmaximal, characteristic, complemented
6#9719C211yesmaximal, characteristic, complemented
7#9624C211yesmaximal, characteristic, complemented
8#9590C211yesmaximal, characteristic, complemented
9#9590C211yesmaximal, characteristic, complemented
10#9208C211yesmaximal, characteristic, complemented
11#9208C211yesmaximal, characteristic, complemented
12#8667C211yesmaximal, characteristic, complemented
13#8291C211yesmaximal, characteristic, complemented
14#8255C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3035C2211yescharacteristic
18#3035C2211yescharacteristic
19C8×#41C2211yescharacteristic
20C8×#41C2211yescharacteristic
21C2×#444—21no—
22C2×#443—21no—
23#2931C2211yescharacteristic
24#2931C2211yescharacteristic
25C4×#131C2211yescharacteristic
26#1980C2211yescharacteristic
27#1980C2211yescharacteristic
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1929—21no—
31#1929—21no—
32#1876—21no—
33#1876—21no—
34#1806C2211yescharacteristic
35#1806C2211yescharacteristic
36C4×#107C2211yescharacteristic
37#1770C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1765—21no—
40#1764—21no—
41#1682C411yescharacteristic, complemented
42#1682C411yescharacteristic, complemented
43#1681C411yescharacteristic, complemented
44#1681C411yescharacteristic, complemented
45#1674C411yescharacteristic, complemented
46#1673C411yescharacteristic, complemented
47#1668C411yescharacteristic, complemented
48#1667C411yescharacteristic, complemented
49#1581—21no—
50#1581—21no—
51#1581—21no—
52#1581—21no—
53#1567C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1540C2211yescharacteristic
56#1540C2211yescharacteristic
57#1502C2211yescharacteristic
58#1502C2211yescharacteristic
59#1492C2211yescharacteristic
60#1492C2211yescharacteristic
61#1450—21no—
62#1450C2211yescharacteristic
63C8×M16—21no—
64C8×M16—21no—
65C8×M16C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C2×#322—21no—
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×#135—21no—
76C2×#135—21no—
77C2×#131—21no—
78C2×#131—21no—
79C2×#131D411yescharacteristic
80C2×#131D411yescharacteristic
81#532C2311yescharacteristic
82#532C2311yescharacteristic
83C4×#41C2311yescharacteristic
84#500C2×C411yescharacteristic
85#493C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#466—21no—
89#466—21no—
90#464—21no—
91#464—21no—
92#463—21no—
93#463—21no—
94#462C2×C411yescharacteristic
95#462C2×C411yescharacteristic
96#461C2×C411yescharacteristic
97#461C2×C411yescharacteristic
98#460C2×C411yescharacteristic
99#460C2×C411yescharacteristic
100#459C2×C411yescharacteristic
101#459C2×C411yescharacteristic
102C8×Q8—21no—
103C8×Q8C2×C411yescharacteristic
104C8×Q8C2×C411yescharacteristic
105#444—44no—
106#443—44no—
107C4×D8—21no—
108C4×D8—21no—
109#436—21no—
110#436—21no—
111#436—21no—
112#436—21no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—22no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4—21no—
121C8×D4C2×C411yescharacteristic
122C8×D4C2×C411yescharacteristic
123#431—21no—
124#431—21no—
125#431C2311yescharacteristic
126#431C2311yescharacteristic
127C2×#108—21no—
128C2×#108—21no—
129C2×#107—21no—
130#423C2311yescharacteristic
131#423C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×#104—21no—
135C2×#104—21no—
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140#416—21no—
141#416—21no—
142#408—21no—
143#408—21no—
144#405—21no—
145#405—21no—
146#405D411yescharacteristic
147#405D411yescharacteristic
148C4×M16—21no—
149C4×M16C2311yescharacteristic
150C2×#97—21no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8—21no—
154C2×C4×C8—21no—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C4×C4⋊C4C2311yescharacteristic
159C4×C4⋊C4C2311yescharacteristic
160#341—21no—
161#341—21no—
162#340—21no—
163#339—21no—
164#339—21no—
165#336—21no—
166#336—21no—
167#335—21no—
168#335—21no—
169#335C2×C411yescharacteristic
170#335C2×C411yescharacteristic
171#335C2×C411yescharacteristic
172#335C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#331C2×C411yescharacteristic
178#331C2×C411yescharacteristic
179#331C2×C411yescharacteristic
180#331C2×C411yescharacteristic
181#327—22no—
182#327—22no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
193#322D411yescharacteristic
194C82—21no—
195C82—21no—
196C82—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41C2×D411yescharacteristic
200#135—42no—
201#135—42no—
202#135—42no—
203#135—42no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×SD16—21no—
207C2×SD16—21no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214#131—22no—
215#131—22no—
216#131—22no—
217#131—42no—
218#131—42no—
219#131—42no—
220#131—42no—
221#131D814yescomplemented
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—41no—
227C2×M16—41no—
228C2×M16C2×D411yescharacteristic
229C2×M16C2×D411yescharacteristic
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—22no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—41no—
239C22×C8—41no—
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8#4111yescharacteristic
243C22×C8#4111yescharacteristic
244C4⋊Q8C22×C411yescharacteristic
245C4⋊Q8C22×C411yescharacteristic
246#127C22×C411yescharacteristic
247#127C22×C411yescharacteristic
248#124—21no—
249#124—21no—
250#121—21no—
251#121—21no—
252#121—21no—
253#121—21no—
254C4×Q8C22×C411yescharacteristic
255C4×Q8#4111yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4C22×C411yescharacteristic
263C4×D4#4111yescharacteristic
264#117—21no—
265#117C2×D411yescharacteristic
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C2411yescharacteristic
272#108—42no—
273#108—42no—
274#107—21no—
275#107—21no—
276#107—42no—
277C4⋊C8—22no—
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⋊C8C22×C411yescharacteristic
288C4⋊C8C22×C411yescharacteristic
289C4⋊C8C22×C411yescharacteristic
290C4⋊C8C22×C411yescharacteristic
291C4⋊C8C22×C411yescharacteristic
292C4⋊C8C22×C411yescharacteristic
293#104—42no—
294#104—42no—
295#104—42no—
296#104—42no—
297#104—42no—
298#104—42no—
299#103—21no—
300#103—21no—
301#103—21no—
302#103—21no—
303#102—21no—
304#102—21no—
305#102—21no—
306#102—21no—
307#102—21no—
308#102—21no—
309#102—21no—
310#102—21no—
311C22⋊C8—22no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8—41no—
318C22⋊C8—41no—
319C22⋊C8#4112yes—
320#97—21no—
321#97—21no—
322#97—21no—
323#97—42no—
324C4×C8—22no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—42no—
334C4×C8C22×C411yescharacteristic
335C4×C8C22×C411yescharacteristic
336C4×C8C2×D411yescharacteristic
337C4×C8C2×D411yescharacteristic
338C4×C8C2×D411yescharacteristic
339C4×C8C2×D411yescharacteristic
340C4×C8C2×D411yescharacteristic
341C4×C8C2×D411yescharacteristic
342C4×C8#4111yescharacteristic
343C4×C8#4111yescharacteristic
344#95—21no—
345#95—21no—
346#41—21no—
347#41—21no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353#41—41no—
354#41—41no—
355#41—41no—
356#41C2×D812yes—
357#41#13512yes—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8C4×D411yescharacteristic
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4C4×D411yescharacteristic
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381Q16—42no—
382Q16—42no—
383SD16—42no—
384SD16—42no—
385SD16—42no—
386SD16—42no—
387D8—42no—
388D8—42no—
389D8—81no—
390D8—81no—
391M16—24no—
392M16—21no—
393M16—21no—
394M16—24no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—82no—
400M16—82no—
401M16C2×D812yes—
402M16#13512yes—
403C2×C8—24no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—24no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—21no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—81no—
427C2×C8—81no—
428C2×C8—81no—
429C2×C8—82no—
430C2×C8—82no—
431C2×C8—82no—
432C2×C8C4×D411yescharacteristic
433C2×C8C4×D411yescharacteristic
434C2×C8C4×D411yescharacteristic
435C2×C8C4×D411yescharacteristic
436C2×C8#12111yescharacteristic
437C2×C8#12111yescharacteristic
438C2×C8#12111yescharacteristic
439C2×C8#12111yescharacteristic
440C2×C8C2×D812yes—
441C2×C8#13512yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4C4×D411yescharacteristic
453C4⋊C4C4×D411yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42C23×C411yescharacteristic
469C42C22×D411yescharacteristic
470C42C2×#4111yescharacteristic
471C42C2×#4111yescharacteristic
472C23—21no—
473C23—41no—
474C23—41no—
475C23#53211yescharacteristic
476Q8—21no—
477Q8—41no—
478Q8—41no—
479Q8—41no—
480Q8—41no—
481Q8C4×D812yes—
482D4—22no—
483D4—22no—
484D4—21no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—81no—
499D4C4×D812yes—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C4×#4111yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4#52511yescharacteristic
528C2×C4#52911yescharacteristic
529C2×C4C22×D811yescharacteristic
530C2×C4C2×#13511yescharacteristic
531C2×C4C2×#13511yescharacteristic
532C2×C4C2×#13511yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—22no—
538C8—21no—
539C8—22no—
540C8—21no—
541C8—42no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—81no—
551C8—81no—
552C8#46612yes—
553C8#46612yes—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C4×#12111yescharacteristic
566C22#316811yescharacteristic
567C22#316911yescharacteristic
568C4—21no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—82no—
575C4—81no—
576C4—81no—
577C4—161no—
578C4C2×C4×D811yescharacteristic
579C4C4×#13511yescharacteristic
580C4#295611yescharacteristic
581C4#295711yescharacteristic
582C4C2×#46611yescharacteristic
583C4#305011yescharacteristic
584C2—21no—
585C2—21no—
586C2—42no—
587C2—161no—
588C2—161no—
589C2#2151911yescharacteristic
590C2C4×#46611yescharacteristic
591C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#19125C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21518C2semidirect
2. #21517C2semidirect
4. #17975C2semidirect
5. #17316C2semidirect
6. #9719C2semidirect
7. #9624C2semidirect
8. #9590C2semidirect
9. #9590C2semidirect
10. #9208C2semidirect
11. #9208C2semidirect
12. #8667C2semidirect
13. #8291C2semidirect
14. #8255C2semidirect
15. #8251C2semidirect
41. #1682C4semidirect
42. #1682C4semidirect
43. #1681C4semidirect
44. #1681C4semidirect
45. #1674C4semidirect
46. #1673C4semidirect
47. #1668C4semidirect
48. #1667C4semidirect
221. #131D8semidirect

Maximal subgroups

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

subgroupindex
#215182
#215172
#191252
#179752
#173162
#97192
#96242
#95902
#92082
#86672
#82912
#82552
#82512