Finite Group Databasebeta

← #211884

Subgroups — #211884

Order 512.

666 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#17981C211yesmaximal, characteristic, complemented
5#17320C211yesmaximal, characteristic, complemented
6#9721C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9592C211yesmaximal, characteristic, complemented
9#9592C211yesmaximal, characteristic, complemented
10#9208C211yesmaximal, characteristic, complemented
11#9208C211yesmaximal, characteristic, complemented
12#8666C211yesmaximal, characteristic, complemented
13#8289C211yesmaximal, characteristic, complemented
14#8257C211yesmaximal, characteristic, complemented
15#8253C211yesmaximal, 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×#106C2211yescharacteristic
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#1586—21no—
50#1586—21no—
51#1586—21no—
52#1586—21no—
53#1566C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1540C2211yescharacteristic
56#1540C2211yescharacteristic
57#1502C2211yescharacteristic
58#1502C2211yescharacteristic
59#1492C2211yescharacteristic
60#1492C2211yescharacteristic
61#1452—21no—
62#1452C2211yescharacteristic
63#1451—21no—
64#1451—21no—
65#1451C2211yescharacteristic
66#1451C2211yescharacteristic
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×#106—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#335C2×C411yescharacteristic
168#335C2×C411yescharacteristic
169#335C2×C411yescharacteristic
170#335C2×C411yescharacteristic
171#334—21no—
172#334—21no—
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#322—21no—
191#322—21no—
192#322—21no—
193#322D411yescharacteristic
194#322D411yescharacteristic
195#322D411yescharacteristic
196#322D411yescharacteristic
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—41no—
217#131—41no—
218#131—42no—
219#131—42no—
220#131—42no—
221#131—42no—
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—22no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
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#106—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#41AGL(1,ℤ/8)12yes—
357#41AGL(1,ℤ/8)12yes—
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—21no—
392M16—21no—
393M16—21no—
394M16—21no—
395M16—42no—
396M16—42no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—82no—
402M16—82no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
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—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—44no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—81no—
431C2×C8—81no—
432C2×C8—82no—
433C2×C8—82no—
434C2×C8C4×D411yescharacteristic
435C2×C8C4×D411yescharacteristic
436C2×C8C4×D411yescharacteristic
437C2×C8C4×D411yescharacteristic
438C2×C8#12111yescharacteristic
439C2×C8#12111yescharacteristic
440C2×C8#12111yescharacteristic
441C2×C8#12111yescharacteristic
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—
481Q8#44212yes—
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—
499D4#44212yes—
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×C4C2×AGL(1,ℤ/8)11yescharacteristic
530C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
531C2×C4#57511yescharacteristic
532C2×C4#57511yescharacteristic
533C8—22no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—42no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—81no—
553C8—81no—
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#318011yescharacteristic
567C22#318011yescharacteristic
568C4—21no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—82no—
575C4—81no—
576C4—81no—
577C4—161no—
578C4C2×#44211yescharacteristic
579C4#294311yescharacteristic
580C4#295611yescharacteristic
581C4#295711yescharacteristic
582C4#305511yescharacteristic
583C4#305511yescharacteristic
584C2—21no—
585C2—21no—
586C2—42no—
587C2—161no—
588C2—161no—
589C2#2151911yescharacteristic
590C2#2152611yescharacteristic
591C2#2155811yescharacteristic
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. #17981C2semidirect
5. #17320C2semidirect
6. #9721C2semidirect
7. #9625C2semidirect
8. #9592C2semidirect
9. #9592C2semidirect
10. #9208C2semidirect
11. #9208C2semidirect
12. #8666C2semidirect
13. #8289C2semidirect
14. #8257C2semidirect
15. #8253C2semidirect
41. #1682C4semidirect
42. #1682C4semidirect
43. #1681C4semidirect
44. #1681C4semidirect
45. #1674C4semidirect
46. #1673C4semidirect
47. #1668C4semidirect
48. #1667C4semidirect

Maximal subgroups

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

subgroupindex
#215182
#215172
#191252
#179812
#173202
#97212
#96252
#95922
#92082
#86662
#82892
#82572
#82532