Finite Group Databasebeta

← #211072

Subgroups — #211072

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×#1586C211yesmaximal, characteristic, complemented
5#17337C211yesmaximal, characteristic, complemented
6#9670C211yesmaximal, characteristic, complemented
7#9609C211yesmaximal, characteristic, complemented
8#9550C211yesmaximal, characteristic, complemented
9#9550C211yesmaximal, characteristic, complemented
10#9097C211yesmaximal, characteristic, complemented
11#9097C211yesmaximal, characteristic, complemented
12#8604C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, 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#1586—21no—
41#1586—21no—
42#1586—21no—
43#1586—21no—
44#1585—21no—
45#1582—21no—
46#1581—21no—
47#1565C2211yescharacteristic
48#1565C2211yescharacteristic
49#1564C2211yescharacteristic
50C2×#334C2211yescharacteristic
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#334—21no—
164#334—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—44no—
395M16—81no—
396M16—81no—
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—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—82no—
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—22no—
518C8—21no—
519C8—21no—
520C8—22no—
521C8—21no—
522C8—21no—
523C8—22no—
524C8—22no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—42no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8—81no—
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
570C4#306811yescharacteristic
571C4#306811yescharacteristic
572C2—21no—
573C2—21no—
574C2—81no—
575C2—82no—
576C2—161no—
577C2#2149011yescharacteristic
578C2#2155511yescharacteristic
579C2#2155811yescharacteristic
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×#1586C2semidirect
5. #17337C2semidirect
6. #9670C2semidirect
7. #9609C2semidirect
8. #9550C2semidirect
9. #9550C2semidirect
10. #9097C2semidirect
11. #9097C2semidirect
12. #8604C2semidirect
13. #8296C2semidirect
14. #8166C2semidirect
15. C8×#100C2semidirect
300. #100M16semidirect

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×#15862
#173372
#96702
#96092
#95502
#90972
#86042
#82962
#81662
C8×#1002