Finite Group Databasebeta

← #211115

Subgroups — #211115

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23718C211yesmaximal, characteristic
2#17891C211yesmaximal, characteristic
3#17870C211yesmaximal, characteristic, complemented
4C2×#1588C211yesmaximal, characteristic, complemented
5#17330C211yesmaximal, characteristic, complemented
6#9667C211yesmaximal, characteristic, complemented
7#9608C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9551C211yesmaximal, characteristic, complemented
10#9096C211yesmaximal, characteristic, complemented
11#9095C211yesmaximal, characteristic
12#8606C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, characteristic, complemented
14#8166C211yesmaximal, characteristic, complemented
15C8×#101C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C4×#137C2211yescharacteristic
20C4×AGL(1,ℤ/8)C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×#436C2211yescharacteristic
23#2922C2211yescharacteristic
24#2090C411yescharacteristic
25#2079C411yescharacteristic
26#2064C411yescharacteristic
27#2063C411yescharacteristic
28#2052C411yescharacteristic
29#2051C411yescharacteristic
30#2046C411yescharacteristic
31#2019C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#101C2211yescharacteristic
34C4×#100C2211yescharacteristic
35#1594—21no—
36#1592—21no—
37#1590—21no—
38#1589—21no—
39#1588—21no—
40#1588—21no—
41#1588—21no—
42#1588—21no—
43#1588—21no—
44#1587—21no—
45#1584—21no—
46#1583—21no—
47#1565C2211yescharacteristic
48#1565C2211yescharacteristic
49#1564C2211yescharacteristic
50C2×#335C2211yescharacteristic
51#1479C2211yescharacteristic
52#1478C2211yescharacteristic
53#1477C2211yescharacteristic
54#1477C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1461C2211yescharacteristic
58#1460C2211yescharacteristic
59#1454—21no—
60#1453—21no—
61#1451—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450—21no—
65#1450C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#137C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×SD16C2×C411yescharacteristic
78C22×M16—21no—
79C4×#41C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82#446D411yescharacteristic
83#446D411yescharacteristic
84C8×Q8—21no—
85C8×Q8—21no—
86C8×Q8—21no—
87C8×Q8—21no—
88C8×Q8—21no—
89#442—21no—
90#441—21no—
91#440—21no—
92#440—21no—
93C4×Q16—21no—
94C4×SD16—21no—
95C4×SD16—21no—
96C4×SD16—21no—
97C4×SD16—21no—
98C4×SD16—21no—
99C4×SD16—21no—
100C4×D8—21no—
101#436—21no—
102#436—21no—
103#436—21no—
104#436—21no—
105#436D411yescharacteristic
106#436D411yescharacteristic
107#435—21no—
108#435—21no—
109C8×D4—21no—
110C8×D4—22no—
111#433—21no—
112#433—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108C2×C411yescharacteristic
122C2×#108C2×C411yescharacteristic
123#428C2×C411yescharacteristic
124#428C2×C411yescharacteristic
125C2×#106C2×C411yescharacteristic
126#424C2×C411yescharacteristic
127#424C2×C411yescharacteristic
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131#419C2×C411yescharacteristic
132#418C2×C411yescharacteristic
133#417C2×C411yescharacteristic
134#417C2×C411yescharacteristic
135C2×#103C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#101C2×C411yescharacteristic
138C2×#101C2×C411yescharacteristic
139C2×#100C2×C411yescharacteristic
140C2×#100C2×C411yescharacteristic
141#408C2×C411yescharacteristic
142#408C2×C411yescharacteristic
143C2×C22⋊C8—21no—
144C4×M16—21no—
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#343C2×C411yescharacteristic
154#343C2×C411yescharacteristic
155#343C2×C411yescharacteristic
156#343C2×C411yescharacteristic
157#341C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#335—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167Q8⋊C8—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179#324—22no—
180#323—22no—
181#322—21no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192C82—21no—
193C82—21no—
194C82—21no—
195C2×#41C22×C411yescharacteristic
196C22×Q8C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199#137—22no—
200#137—41no—
201AGL(1,ℤ/8)—22no—
202AGL(1,ℤ/8)—41no—
203C2×Q16—21no—
204C2×SD16—21no—
205C2×SD16—22no—
206C2×SD16—21no—
207C2×SD16—21no—
208C2×SD16—21no—
209C2×SD16—41no—
210C2×D8—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—41no—
214C2×M16—41no—
215C2×M16—82no—
216C2×M16C22×C411yescharacteristic
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8C22×C411yescharacteristic
227C22×C8C22×C411yescharacteristic
228C22×C8#4111yescharacteristic
229C22×C8#4111yescharacteristic
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C2×D411yescharacteristic
235C4×Q8#4111yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C2×D411yescharacteristic
243C4×D4#4111yescharacteristic
244#117—21no—
245#117C22×C411yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42C2411yescharacteristic
253#108—22no—
254#108—41no—
255#107—21no—
256#107—21no—
257#106—21no—
258#106—21no—
259#106—21no—
260#106—21no—
261C4⋊C8—22no—
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⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#102—21no—
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291#101—22no—
292#101—22no—
293#100—41no—
294#100—41no—
295C22⋊C8—22no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8#4112yes—
308#97—21no—
309#97—21no—
310#97—21no—
311#97—21no—
312#97—21no—
313C4×C8—22no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—41no—
323C4×C8—41no—
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8#4111yescharacteristic
333C4×C8#4111yescharacteristic
334C24—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338#41—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8C4×D411yescharacteristic
345C2×Q8#13112yes—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4C4×D411yescharacteristic
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4C23×C411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367Q16—42no—
368SD16—42no—
369SD16—42no—
370SD16—42no—
371SD16—42no—
372D8—42no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—81no—
385M16—81no—
386M16—82no—
387M16—82no—
388M16#13112yes—
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—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—22no—
403C2×C8—21no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—44no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—81no—
413C2×C8—81no—
414C2×C8C4×D411yescharacteristic
415C2×C8C4×D411yescharacteristic
416C2×C8C4×D411yescharacteristic
417C2×C8C4×D411yescharacteristic
418C2×C8#12111yescharacteristic
419C2×C8#12111yescharacteristic
420C2×C8#12111yescharacteristic
421C2×C8#12111yescharacteristic
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4C4×D411yescharacteristic
431C4⋊C4C4×D411yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42C22×D411yescharacteristic
443C42C22×D411yescharacteristic
444C42C2×#4111yescharacteristic
445C42C2×#4111yescharacteristic
446C23—21no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—81no—
451C23—81no—
452C23C4×#4111yescharacteristic
453Q8—22no—
454Q8—21no—
455Q8—22no—
456Q8—21no—
457Q8—41no—
458Q8—41no—
459D4—21no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4#52511yescharacteristic
492C2×C4#52911yescharacteristic
493C2×C4#53211yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4#56811yescharacteristic
497C2×C4#56811yescharacteristic
498C8—22no—
499C8—21no—
500C8—21no—
501C8—22no—
502C8—21no—
503C8—21no—
504C8—21no—
505C8—21no—
506C8—21no—
507C8—21no—
508C8—42no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—81no—
518C8—81no—
519C8—81no—
520C8—81no—
521C22—21no—
522C22—21no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—161no—
530C22—161no—
531C22C4×#12111yescharacteristic
532C22#296611yescharacteristic
533C22#296711yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4#293311yescharacteristic
544C4#293311yescharacteristic
545C4#293311yescharacteristic
546C4#293511yescharacteristic
547C4#306811yescharacteristic
548C4#306911yescharacteristic
549C2—21no—
550C2—21no—
551C2—81no—
552C2—161no—
553C2#2149611yescharacteristic
554C2#2155711yescharacteristic
555C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×AGL(1,ℤ/8)C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
3. #17870C2semidirect
4. C2×#1588C2semidirect
5. #17330C2semidirect
6. #9667C2semidirect
7. #9608C2semidirect
8. #9552C2semidirect
9. #9551C2semidirect
10. #9096C2semidirect
12. #8606C2semidirect
13. #8296C2semidirect
14. #8166C2semidirect
15. C8×#101C2semidirect

Maximal subgroups

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

subgroupindex
#237182
#178912
#178702
C2×#15882
#173302
#96672
#96082
#95522
#95512
#90962
#90952
#86062
#82962
#81662
C8×#1012