Finite Group Databasebeta

← #211888

Subgroups — #211888

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22166C211yesmaximal, characteristic, complemented
2#22165C211yesmaximal, characteristic, complemented
3#19125C211yesmaximal, characteristic
4#17981C211yesmaximal, characteristic, complemented
5#17331C211yesmaximal, characteristic, complemented
6#9721C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9590C211yesmaximal, characteristic, complemented
9#9590C211yesmaximal, characteristic, complemented
10#9210C211yesmaximal, characteristic, complemented
11#9210C211yesmaximal, characteristic, complemented
12#8666C211yesmaximal, characteristic, complemented
13#8289C211yesmaximal, characteristic, complemented
14#8255C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3035C2211yescharacteristic
18#3035C2211yescharacteristic
19#2965C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2931C2211yescharacteristic
22#2931C2211yescharacteristic
23#2922C2211yescharacteristic
24#1980C2211yescharacteristic
25#1980C2211yescharacteristic
26#1975—21no—
27#1974—21no—
28#1950C2211yescharacteristic
29#1950C2211yescharacteristic
30#1938—21no—
31#1937—21no—
32#1876—21no—
33#1876—21no—
34#1866—21no—
35#1866—21no—
36#1806C2211yescharacteristic
37#1806C2211yescharacteristic
38C4×#106C2211yescharacteristic
39#1770C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
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#1454—21no—
62#1453—21no—
63#1452—21no—
64#1451C2211yescharacteristic
65#1450C2211yescharacteristic
66C8×M16—21no—
67C8×M16C2211yescharacteristic
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—
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
100#446—21no—
101C8×Q8C2×C411yescharacteristic
102C8×Q8C2×C411yescharacteristic
103C4×D8—21no—
104C4×D8—21no—
105#436—21no—
106#436—21no—
107#436—21no—
108#436—21no—
109#436—21no—
110#435—22no—
111C8×D4—21no—
112C8×D4—21no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—21no—
117C8×D4C2×C411yescharacteristic
118C8×D4C2×C411yescharacteristic
119#433—21no—
120#433—21no—
121#431—21no—
122#431—21no—
123#431—21no—
124#431—21no—
125#431C2311yescharacteristic
126#431C2311yescharacteristic
127C2×#108—21no—
128C2×#108—21no—
129C2×#106—21no—
130#424D411yescharacteristic
131#424D411yescharacteristic
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140C2×#104—21no—
141C2×#104—21no—
142#416—21no—
143#416—21no—
144#408—21no—
145#408—21no—
146#408D411yescharacteristic
147#408D411yescharacteristic
148C4×M16—21no—
149C4×M16C2311yescharacteristic
150C2×#97—21no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C4×C4⋊C4C2311yescharacteristic
157C4×C4⋊C4C2311yescharacteristic
158#341—21no—
159#341—21no—
160#340—21no—
161#339—21no—
162#339—21no—
163#336—21no—
164#336—21no—
165#335C2×C411yescharacteristic
166#335C2×C411yescharacteristic
167#335C2×C411yescharacteristic
168#335C2×C411yescharacteristic
169#334—21no—
170#334—21no—
171#332C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#331C2×C411yescharacteristic
176#331C2×C411yescharacteristic
177#331C2×C411yescharacteristic
178#331C2×C411yescharacteristic
179#327—22no—
180#327—22no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
193C82—21no—
194C82—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41C2×D411yescharacteristic
198#135—81no—
199#135—81no—
200#135—81no—
201#135—81no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×SD16—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212#131—81no—
213#131—81no—
214#131—81no—
215#131—81no—
216C2×M16—22no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—41no—
220C2×M16—41no—
221C2×M16C2×D411yescharacteristic
222C2×M16C2×D411yescharacteristic
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—41no—
229C22×C8—41no—
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8#4111yescharacteristic
233C22×C8#4111yescharacteristic
234C4⋊Q8C22×C411yescharacteristic
235C4⋊Q8C22×C411yescharacteristic
236#127C22×C411yescharacteristic
237#127C22×C411yescharacteristic
238#124—21no—
239#124—21no—
240#121—21no—
241#121—21no—
242#121—21no—
243#121—21no—
244C4×Q8C22×C411yescharacteristic
245C4×Q8#4111yescharacteristic
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4C22×C411yescharacteristic
253C4×D4#4111yescharacteristic
254#117—21no—
255#117C2×D411yescharacteristic
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C2411yescharacteristic
262#108—81no—
263#108—81no—
264#107—21no—
265#107—21no—
266#106—42no—
267C4⋊C8—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283C4⋊C8C22×C411yescharacteristic
284C4⋊C8C22×C411yescharacteristic
285C4⋊C8C22×C411yescharacteristic
286C4⋊C8C22×C411yescharacteristic
287#104—81no—
288#104—81no—
289#104—81no—
290#104—81no—
291#104—81no—
292#104—81no—
293#103—21no—
294#103—21no—
295#103—21no—
296#103—21no—
297#102—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301#102—21no—
302#102—21no—
303#102—21no—
304#102—21no—
305C22⋊C8—22no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8—21no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8#4112yes—
318#97—22no—
319#97—21no—
320#97—81no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—81no—
328C4×C8C22×C411yescharacteristic
329C4×C8C22×C411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8#4111yescharacteristic
337C4×C8#4111yescharacteristic
338#95—21no—
339#95—21no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—21no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8C4×D411yescharacteristic
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4C4×D411yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375Q16—81no—
376Q16—81no—
377SD16—81no—
378SD16—81no—
379SD16—81no—
380SD16—81no—
381D8—81no—
382D8—81no—
383D8—81no—
384D8—81no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—44no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—82no—
395M16—82no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—81no—
421C2×C8—81no—
422C2×C8—82no—
423C2×C8—82no—
424C2×C8C4×D411yescharacteristic
425C2×C8C4×D411yescharacteristic
426C2×C8C4×D411yescharacteristic
427C2×C8C4×D411yescharacteristic
428C2×C8#12111yescharacteristic
429C2×C8#12111yescharacteristic
430C2×C8#12111yescharacteristic
431C2×C8#12111yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C4×D411yescharacteristic
443C4⋊C4C4×D411yescharacteristic
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42C23×C411yescharacteristic
459C42C22×D411yescharacteristic
460C42C2×#4111yescharacteristic
461C42C2×#4111yescharacteristic
462C23—21no—
463C23—41no—
464C23—41no—
465C23#53211yescharacteristic
466Q8—21no—
467Q8—21no—
468Q8—41no—
469Q8—41no—
470Q8—41no—
471Q8—41no—
472D4—21no—
473D4—21no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—81no—
485D4—81no—
486D4—81no—
487D4—81no—
488D4—81no—
489D4—81no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C4×#4111yescharacteristic
516C2×C4C2×#12111yescharacteristic
517C2×C4#52511yescharacteristic
518C2×C4#52911yescharacteristic
519C2×C4#57611yescharacteristic
520C2×C4#57611yescharacteristic
521C2×C4#57711yescharacteristic
522C2×C4#57711yescharacteristic
523C8—22no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—42no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—81no—
543C8—81no—
544C22—21no—
545C22—21no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C4×#12111yescharacteristic
556C22#317911yescharacteristic
557C22#317911yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—82no—
565C4—81no—
566C4—81no—
567C4—161no—
568C4#294811yescharacteristic
569C4#295211yescharacteristic
570C4#295611yescharacteristic
571C4#295711yescharacteristic
572C4#306211yescharacteristic
573C4#306211yescharacteristic
574C2—21no—
575C2—21no—
576C2—81no—
577C2—161no—
578C2—161no—
579C2#2151911yescharacteristic
580C2#2152911yescharacteristic
581C2#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. #22166C2semidirect
2. #22165C2semidirect
4. #17981C2semidirect
5. #17331C2semidirect
6. #9721C2semidirect
7. #9625C2semidirect
8. #9590C2semidirect
9. #9590C2semidirect
10. #9210C2semidirect
11. #9210C2semidirect
12. #8666C2semidirect
13. #8289C2semidirect
14. #8255C2semidirect
15. #8251C2semidirect
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
#221662
#221652
#191252
#179812
#173312
#97212
#96252
#95902
#92102
#86662
#82892
#82552
#82512