Finite Group Databasebeta

← #211071

Subgroups — #211071

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#471C211yesmaximal, characteristic
2#17871C211yesmaximal, characteristic
3C2×#1588C211yesmaximal, characteristic, complemented
4C8×#137C211yesmaximal, characteristic, complemented
5#17320C211yesmaximal, characteristic, complemented
6#9672C211yesmaximal, characteristic, complemented
7#9610C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9551C211yesmaximal, characteristic, complemented
10#9096C211yesmaximal, characteristic, complemented
11#9095C211yesmaximal, characteristic
12#8605C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, characteristic, complemented
14#8166C211yesmaximal, characteristic, complemented
15#8165C211yesmaximal, characteristic, complemented
16C2×#471C411yescharacteristic
17C8×#41C2211yescharacteristic
18C8×#41C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C4×#137C2211yescharacteristic
21C4×AGL(1,ℤ/8)C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1977C411yescharacteristic
26#1951C411yescharacteristic
27#1931C411yescharacteristic
28#1930C411yescharacteristic
29#1908C411yescharacteristic
30#1907C411yescharacteristic
31#1815C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#101C2211yescharacteristic
34C4×#100C2211yescharacteristic
35#1588—21no—
36#1588—21no—
37#1588—21no—
38#1588—21no—
39#1588—21no—
40#1586—21no—
41#1583—21no—
42#1581—21no—
43#1579—21no—
44SD16⋊C8—21no—
45C8×Q16—21no—
46C8×SD16—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#1452—21no—
60#1452C2211yescharacteristic
61#1451—21no—
62#1451—21no—
63#1451C2211yescharacteristic
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C2×#322—21no—
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
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C22×M16—21no—
81C4×#41C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84#471—24no—
85C8×Q8—21no—
86C8×Q8—21no—
87C8×Q8—21no—
88C8×Q8—21no—
89C8×Q8—21no—
90C8×Q8D411yescharacteristic
91C8×Q8D411yescharacteristic
92#442—21no—
93#441—21no—
94#440—21no—
95#440—21no—
96C4×Q16—21no—
97C4×SD16—21no—
98C4×SD16—21no—
99C4×SD16—21no—
100C4×SD16—21no—
101C4×SD16—21no—
102C4×SD16—21no—
103C4×D8—21no—
104#436—21no—
105#436—21no—
106#436—21no—
107#436—21no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4—21no—
111C8×D4—22no—
112C8×D4D411yescharacteristic
113C8×D4D411yescharacteristic
114#431—21no—
115#431—21no—
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120C2×#108C2×C411yescharacteristic
121C2×#108C2×C411yescharacteristic
122#428C2×C411yescharacteristic
123#428C2×C411yescharacteristic
124C2×#106C2×C411yescharacteristic
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128#419C2×C411yescharacteristic
129#418C2×C411yescharacteristic
130#417C2×C411yescharacteristic
131#417C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#101C2×C411yescharacteristic
135C2×#101C2×C411yescharacteristic
136C2×#100C2×C411yescharacteristic
137C2×#100C2×C411yescharacteristic
138C2×C22⋊C8—21no—
139#405—21no—
140#405—21no—
141#405C2×C411yescharacteristic
142#405C2×C411yescharacteristic
143C4×M16—21no—
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#343C2×C411yescharacteristic
155#343C2×C411yescharacteristic
156#343C2×C411yescharacteristic
157#343C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#341C2×C411yescharacteristic
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335—21no—
165#334—21no—
166#334—21no—
167#334—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180#324—22no—
181#323—22no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322D411yescharacteristic
192#322D411yescharacteristic
193#322D411yescharacteristic
194#322D411yescharacteristic
195C82—21no—
196C2×#41C22×C411yescharacteristic
197C22×Q8C22×C411yescharacteristic
198C22×D4C22×C411yescharacteristic
199C23×C4—21no—
200#137—22no—
201#137M1614yescomplemented
202AGL(1,ℤ/8)—22no—
203AGL(1,ℤ/8)—22no—
204C2×Q16—21no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×SD16—22no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×SD16—41no—
211C2×D8—21no—
212#131—22no—
213#131—22no—
214#131—41no—
215#131—41no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—41no—
219C2×M16—41no—
220C2×M16—82no—
221C2×M16C22×C411yescharacteristic
222C2×M16C22×C411yescharacteristic
223C2×M16C22×C411yescharacteristic
224C2×M16C22×C411yescharacteristic
225C2×M16C22×C411yescharacteristic
226C2×M16C22×C411yescharacteristic
227C22×C8—22no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8#4111yescharacteristic
238C22×C8#4111yescharacteristic
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8C2×D411yescharacteristic
244C4×Q8#4111yescharacteristic
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4C2×D411yescharacteristic
252C4×D4#4111yescharacteristic
253#117—21no—
254#117C22×C411yescharacteristic
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C2411yescharacteristic
262#108—22no—
263#108—41no—
264#107—21no—
265#107—21no—
266#106—21no—
267#106—21no—
268#106—21no—
269#106—21no—
270C4⋊C8—22no—
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—41no—
281C4⋊C8—41no—
282C4⋊C8C2×D411yescharacteristic
283C4⋊C8C2×D411yescharacteristic
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#101—22no—
297#101—41no—
298#100—22no—
299#100—41no—
300C22⋊C8—22no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8#4112yes—
309#97—21no—
310#97—21no—
311#97—21no—
312#97—21no—
313#97—21no—
314C4×C8—22no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—41no—
328C4×C8—41no—
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8C2×D411yescharacteristic
337C4×C8#4111yescharacteristic
338C4×C8#4111yescharacteristic
339C24—21no—
340#41—21no—
341#41—21no—
342#41C2×M1612yes—
343#41#13112yes—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8C4×D411yescharacteristic
350C2×Q8C2×M1612yes—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4C4×D411yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4C23×C411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372Q16—24no—
373SD16—24no—
374SD16—42no—
375SD16—42no—
376SD16—42no—
377D8—42no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—42no—
390M16—42no—
391M16—82no—
392M16—82no—
393M16C2×M1612yes—
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—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—44no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—81no—
420C2×C8—81no—
421C2×C8—81no—
422C2×C8—81no—
423C2×C8C4×D411yescharacteristic
424C2×C8C4×D411yescharacteristic
425C2×C8C4×D411yescharacteristic
426C2×C8C4×D411yescharacteristic
427C2×C8#12111yescharacteristic
428C2×C8#12111yescharacteristic
429C2×C8#12111yescharacteristic
430C2×C8#12111yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4C4×D411yescharacteristic
440C4⋊C4C4×D411yescharacteristic
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—41no—
450C42—41no—
451C42C22×D411yescharacteristic
452C42C22×D411yescharacteristic
453C42C2×#4111yescharacteristic
454C42C2×#4111yescharacteristic
455C23—21no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—81no—
460C23—81no—
461C23C4×#4111yescharacteristic
462Q8—21no—
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8#43512yes—
468D4—22no—
469D4—22no—
470D4—21no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4#43512yes—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4#52511yescharacteristic
501C2×C4#52911yescharacteristic
502C2×C4#53211yescharacteristic
503C2×C4C22×M1611yescharacteristic
504C2×C4C2×#13111yescharacteristic
505C2×C4#56811yescharacteristic
506C2×C4#56811yescharacteristic
507C8—22no—
508C8—22no—
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—21no—
516C8—21no—
517C8—42no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8—81no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—161no—
539C22—161no—
540C22C4×#12111yescharacteristic
541C22#296811yescharacteristic
542C22#297011yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—81no—
551C4—81no—
552C4C2×#43511yescharacteristic
553C4#293211yescharacteristic
554C4#293311yescharacteristic
555C4D4×M1611yescharacteristic
556C4#306811yescharacteristic
557C4#306911yescharacteristic
558C2—21no—
559C2—21no—
560C2—42no—
561C2—161no—
562C2#2148911yescharacteristic
563C2#2155711yescharacteristic
564C2#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. C2×#1588C2semidirect
4. C8×#137C2semidirect
5. #17320C2semidirect
6. #9672C2semidirect
7. #9610C2semidirect
8. #9552C2semidirect
9. #9551C2semidirect
10. #9096C2semidirect
12. #8605C2semidirect
13. #8296C2semidirect
14. #8166C2semidirect
15. #8165C2semidirect
201. #137M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4712
#178712
C2×#15882
C8×#1372
#173202
#96722
#96102
#95522
#95512
#90962
#90952
#86052
#82962
#81662
#81652