Finite Group Databasebeta

← #211066

Subgroups — #211066

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#472C211yesmaximal, characteristic
2#17891C211yesmaximal, characteristic, complemented
3#17869C211yesmaximal, characteristic, complemented
4C2×#1581C211yesmaximal, characteristic, complemented
5#17337C211yesmaximal, characteristic, complemented
6#9670C211yesmaximal, characteristic, complemented
7#9610C211yesmaximal, characteristic, complemented
8#9550C211yesmaximal, characteristic, complemented
9#9550C211yesmaximal, characteristic, complemented
10#9097C211yesmaximal, characteristic, complemented
11#9097C211yesmaximal, characteristic, complemented
12#8605C211yesmaximal, characteristic, complemented
13#8297C211yesmaximal, characteristic, complemented
14#8166C211yesmaximal, characteristic, complemented
15C8×#100C211yesmaximal, characteristic, complemented
16C2×#472C411yescharacteristic
17#2965C2211yescharacteristic
18#2965C2211yescharacteristic
19C4×AGL(1,ℤ/8)C2211yescharacteristic
20C4×AGL(1,ℤ/8)C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1978C411yescharacteristic
26#1952C411yescharacteristic
27#1931C411yescharacteristic
28#1931C411yescharacteristic
29#1907C411yescharacteristic
30#1907C411yescharacteristic
31#1816C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#100C2211yescharacteristic
34C4×#100C2211yescharacteristic
35#1594—21no—
36#1592—21no—
37#1590—21no—
38#1589—21no—
39#1586—21no—
40#1585—21no—
41#1582—21no—
42#1581—21no—
43#1581—21no—
44#1581—21no—
45#1581—21no—
46#1581—21no—
47#1565C2211yescharacteristic
48#1565C2211yescharacteristic
49#1564C2211yescharacteristic
50C2×#335C2211yescharacteristic
51#1478C2211yescharacteristic
52#1478C2211yescharacteristic
53#1477C2211yescharacteristic
54#1477C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1460C2211yescharacteristic
58#1460C2211yescharacteristic
59#1455—21no—
60#1453—21no—
61#1452C2211yescharacteristic
62#1451—21no—
63#1451C2211yescharacteristic
64C8×M16—21no—
65C8×M16C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×AGL(1,ℤ/8)C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C22×M16—21no—
81C23×C8—21no—
82C4×#41C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C2×C4×D4C2311yescharacteristic
85#472—24no—
86#446—21no—
87#446D411yescharacteristic
88#446D411yescharacteristic
89#442—21no—
90#442—21no—
91#440—21no—
92#440—21no—
93C4×SD16—21no—
94C4×SD16—21no—
95C4×D8—21no—
96C4×D8—21no—
97C4×D8—21no—
98C4×D8—21no—
99C4×D8—21no—
100C4×D8—21no—
101#436—21no—
102#436—21no—
103#436—21no—
104#436—21no—
105#436—21no—
106#436D411yescharacteristic
107#436D411yescharacteristic
108#435—21no—
109#435—21no—
110#435—22no—
111C8×D4—21no—
112C8×D4—21no—
113C8×D4—21no—
114C8×D4—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×#107C2×C411yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128#418C2×C411yescharacteristic
129#418C2×C411yescharacteristic
130#417C2×C411yescharacteristic
131#417C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#100C2×C411yescharacteristic
135C2×#100C2×C411yescharacteristic
136C2×#100C2×C411yescharacteristic
137C2×#100C2×C411yescharacteristic
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140#405—21no—
141#405—21no—
142#405C2×C411yescharacteristic
143#405C2×C411yescharacteristic
144C4×M16—21no—
145C4×M16—21no—
146C4×M16—21no—
147C4×M16C2311yescharacteristic
148C4×M16C2311yescharacteristic
149C4×M16C2311yescharacteristic
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—
170D4⋊C8—21no—
171D4⋊C8—21no—
172D4⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180#323—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#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
193C82—21no—
194C82—21no—
195C82—21no—
196C2×#41C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C22×D4C22×C411yescharacteristic
199C23×C4—21no—
200C23×C4—21no—
201AGL(1,ℤ/8)—22no—
202AGL(1,ℤ/8)—22no—
203AGL(1,ℤ/8)—41no—
204AGL(1,ℤ/8)—41no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—22no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—41no—
213#131—22no—
214#131—22no—
215#131—41no—
216#131—41no—
217C2×M16—22no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—41no—
223C2×M16—41no—
224C2×M16—82no—
225C2×M16C22×C411yescharacteristic
226C2×M16C22×C411yescharacteristic
227C2×M16C22×C411yescharacteristic
228C2×M16C22×C411yescharacteristic
229C2×M16C22×C411yescharacteristic
230C2×M16C22×C411yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—41no—
237C22×C8—41no—
238C22×C8—82no—
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8#4111yescharacteristic
242C22×C8#4111yescharacteristic
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×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4C2×D411yescharacteristic
256C4×D4#4111yescharacteristic
257#117—21no—
258#117C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42—21no—
264C2×C42C2411yescharacteristic
265#108—22no—
266#108—41no—
267#107—21no—
268#107—21no—
269#107—21no—
270#107—21no—
271#106—21no—
272#106—21no—
273C4⋊C8—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8C2×D411yescharacteristic
284C4⋊C8C2×D411yescharacteristic
285#103—21no—
286#103—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297#100—22no—
298#100—22no—
299#100—22no—
300#100M1614yescomplemented
301C22⋊C8—22no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310C22⋊C8—41no—
311C22⋊C8#4112yes—
312#97—22no—
313#97—21no—
314#97—21no—
315#97—21no—
316#97—21no—
317#97—21no—
318#97—21no—
319#97—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8C2×D411yescharacteristic
337C4×C8C2×D411yescharacteristic
338C4×C8#4111yescharacteristic
339C4×C8#4111yescharacteristic
340C24—21no—
341C24—21no—
342#41—21no—
343#41—21no—
344#41—21no—
345#41—21no—
346C2×Q8C4×D411yescharacteristic
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4C4×D411yescharacteristic
360C2×D4C2×M1612yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4C23×C411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377SD16—42no—
378SD16—42no—
379D8—42no—
380D8—42no—
381D8—42no—
382D8—42no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—21no—
391M16—21no—
392M16—42no—
393M16—42no—
394M16—42no—
395M16—42no—
396M16—82no—
397M16—82no—
398M16—82no—
399M16C2×M1612yes—
400M16C2×M1612yes—
401M16#13112yes—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—82no—
426C2×C8—81no—
427C2×C8—82no—
428C2×C8—81no—
429C2×C8—81no—
430C2×C8C4×D411yescharacteristic
431C2×C8C4×D411yescharacteristic
432C2×C8C4×D411yescharacteristic
433C2×C8C4×D411yescharacteristic
434C2×C8#12111yescharacteristic
435C2×C8#12111yescharacteristic
436C2×C8#12111yescharacteristic
437C2×C8#12111yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4C4×D411yescharacteristic
445C4⋊C4C4×D411yescharacteristic
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42C22×D411yescharacteristic
457C42C22×D411yescharacteristic
458C42C2×#4111yescharacteristic
459C42C2×#4111yescharacteristic
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23C4×#4111yescharacteristic
472Q8—21no—
473Q8—21no—
474D4—21no—
475D4—22no—
476D4—22no—
477D4—21no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—41no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×#12111yescharacteristic
510C2×C4#52511yescharacteristic
511C2×C4#52911yescharacteristic
512C2×C4#53211yescharacteristic
513C2×C4C22×M1611yescharacteristic
514C2×C4C2×#13111yescharacteristic
515C2×C4#56811yescharacteristic
516C2×C4#56811yescharacteristic
517C8—21no—
518C8—21no—
519C8—22no—
520C8—21no—
521C8—21no—
522C8—22no—
523C8—22no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—41no—
529C8—42no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—81no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8#47212yescomplemented
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—161no—
554C22—161no—
555C22C4×#12111yescharacteristic
556C22#296511yescharacteristic
557C22#296711yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4#293311yescharacteristic
567C4D4×M1611yescharacteristic
568C4D4×M1611yescharacteristic
569C4#293511yescharacteristic
570C4C2×#47211yescharacteristic
571C4#306711yescharacteristic
572C2—21no—
573C2—21no—
574C2—81no—
575C2—82no—
576C2—161no—
577C2#2149011yescharacteristic
578C2C4×#47211yescharacteristic
579C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×#472C4
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
2. #17891C2semidirect
3. #17869C2semidirect
4. C2×#1581C2semidirect
5. #17337C2semidirect
6. #9670C2semidirect
7. #9610C2semidirect
8. #9550C2semidirect
9. #9550C2semidirect
10. #9097C2semidirect
11. #9097C2semidirect
12. #8605C2semidirect
13. #8297C2semidirect
14. #8166C2semidirect
15. C8×#100C2semidirect
300. #100M16semidirect
539. C8#472semidirect

Maximal subgroups

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

subgroupindex
C4×#4722
#178912
#178692
C2×#15812
#173372
#96702
#96102
#95502
#90972
#86052
#82972
#81662
C8×#1002