Finite Group Databasebeta

← #197603

Subgroups — #197603

Order 512.

1389 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60415C211yesmaximal, characteristic
2#60276C211yesmaximal, characteristic, complemented
3C2×#1806C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5#17485C211yesmaximal, characteristic, complemented
6#9208C218yesmaximal, complemented
7#8079C211yesmaximal, characteristic, complemented
8#7901C211yesmaximal, characteristic, complemented
9C2×#575—21no—
10C2×#532C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3040—22no—
13#3039C412yescomplemented
14#3038C412yescomplemented
15#3037—21no—
16#3037C412yescomplemented
17#3036—21no—
18#3036C412yescomplemented
19#3035—22no—
20#3035C2214yes—
21#2988—22no—
22#2988—22no—
23#2982—21no—
24#2972—22no—
25#2972—21no—
26C8×#41—22no—
27C8×#41C2214yescomplemented
28C2×#431—22no—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C2×#423C2211yescharacteristic
33C22×#104—21no—
34#2896—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1806—22no—
37#1806C2214yescomplemented
38C2×#339—21no—
39#1502—22no—
40#1502C2214yescomplemented
41C2×#327—22no—
42#1492C2214yescomplemented
43#1486—22no—
44#1480—22no—
45#1301C2214yescomplemented
46#1279C2214yescomplemented
47C22×#41—21no—
48C22×#41D411yescharacteristic
49#575—44no—
50#575—44no—
51C2×#137—22no—
52C2×#137—22no—
53C2×AGL(1,ℤ/8)—22no—
54C2×AGL(1,ℤ/8)—22no—
55C2×#135—24no—
56C22×M16—21no—
57C22×M16—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532C2×C412yes—
65#532C2312yes—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×#127C2×C411yescharacteristic
68C2×#124—21no—
69C2×#121—21no—
70C2×#121—21no—
71C4×#41—21no—
72C4×#41—21no—
73C4×#41—21no—
74C4×#41—21no—
75C4×#41C2×C412yes—
76C4×#41D412yes—
77C4×#41D412yes—
78C4×#41C2312yes—
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8D411yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×#117—21no—
88C2×#117D411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#464—24no—
94#464—24no—
95#463—24no—
96#463—24no—
97#462—22no—
98#462C2×C414yescomplemented
99#461—22no—
100#461C2×C414yescomplemented
101#460—22no—
102#460C2×C414yescomplemented
103#459—22no—
104#459C2×C414yescomplemented
105C8×Q8—24no—
106C8×Q8—24no—
107C8×D4—28no—
108C8×D4—24no—
109C8×D4—24no—
110C8×D4—28no—
111#431—24no—
112#431—24no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—24no—
118#431D412yes—
119#431D412yes—
120#431D412yes—
121#431C2312yes—
122#423—21no—
123#423—21no—
124#423—21no—
125#423—21no—
126#423—21no—
127#423C2×C412yes—
128#423C2×C412yes—
129#423C2312yes—
130C2×C4⋊C8—24no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8C2312yes—
137C2×#104—24no—
138C2×#104—44no—
139#419—22no—
140#418—22no—
141#417—22no—
142#417—22no—
143#416—24no—
144C2×C22⋊C8—24no—
145C2×C22⋊C8—21no—
146C2×C22⋊C8D412yes—
147C2×C4×C8—21no—
148C2×C4×C8—24no—
149C2×C4×C8C2312yes—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2312yes—
154C2×#95—21no—
155#339—24no—
156#327—28no—
157D4⋊C8—28no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—41no—
165C2×#41—41no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C2×#41—42no—
170C2×#41C22⋊C412yes—
171C2×#41D814yes—
172C2×#41SD1614yes—
173C2×#41C2×D412yes—
174C22×Q8—21no—
175C22×Q8C22⋊C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—41no—
181C22×D4C22⋊C411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190#137—48no—
191#137—48no—
192AGL(1,ℤ/8)—48no—
193AGL(1,ℤ/8)—48no—
194#135—48no—
195#135—48no—
196C2×Q16—24no—
197C2×SD16—24no—
198C2×SD16—24no—
199C2×D8—24no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—41no—
206C2×M16—42no—
207C2×M16—42no—
208C22×C8—22no—
209C22×C8—28no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—41no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8C22×C412yes—
217#127—21no—
218#127—21no—
219#127—21no—
220#127C22×C412yes—
221#124—22no—
222#124—22no—
223#124—41no—
224#124—41no—
225#121—22no—
226#121—22no—
227#121—22no—
228#121—22no—
229#121—41no—
230#121—41no—
231#121—41no—
232#121—41no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8C22⋊C412yes—
238C4×Q8C22⋊C412yes—
239C4×Q8C22×C412yes—
240C4×Q8C2×D412yes—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4C22⋊C412yes—
262C4×D4C22⋊C412yes—
263C4×D4C22×C412yes—
264C4×D4C2×D412yes—
265#117—21no—
266#117—21no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117C22⋊C412yes—
272#117C2×D412yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—41no—
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42#4111yescharacteristic
307C2×C42#4111yescharacteristic
308C2×C42#4111yescharacteristic
309C2×C42C2411yescharacteristic
310C4⋊C8—28no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8C22×C414yes—
323#104—48no—
324#103—24no—
325#103—24no—
326#102—24no—
327#102—24no—
328C22⋊C8—28no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8C2×D414yes—
333C4×C8—28no—
334C4×C8—22no—
335C4×C8—22no—
336C4×C8—22no—
337C4×C8C2×D414yes—
338#95—22no—
339#95—41no—
340C24—21no—
341C24—41no—
342C24#12111yescharacteristic
343#41—22no—
344#41—22no—
345#41—24no—
346#41—24no—
347#41—24no—
348#41—24no—
349#41—24no—
350#41—42no—
351#41—42no—
352#41—42no—
353#41—42no—
354#41—44no—
355#41—44no—
356#41—82no—
357#41—82no—
358#41—82no—
359#41—82no—
360#41#10218yes—
361#41C2×D814yes—
362#41C2×SD1614yes—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8#10214yes—
370C2×Q8C2×C22⋊C412yes—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—82no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4#10214yes—
392C2×D4C2×C22⋊C412yes—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4—81no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C4×D411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4#12011yescharacteristic
424C22×C4#12011yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12311yescharacteristic
431C22×C4#12311yescharacteristic
432C22×C4C2×D812yes—
433C22×C4C2×SD1612yes—
434C22×C4#13512yes—
435C22×C4#13512yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4111yescharacteristic
439Q16—48no—
440SD16—48no—
441SD16—48no—
442D8—48no—
443M16—44no—
444M16—44no—
445M16—42no—
446M16—42no—
447M16—82no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—82no—
455C2×C8—82no—
456C2×C8—88no—
457C2×C8—82no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4C2×C22⋊C412yes—
472C4⋊C4C2×C22⋊C412yes—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—42no—
490C42—42no—
491C42—82no—
492C42C2×C22⋊C411yescharacteristic
493C42C2×C22⋊C411yescharacteristic
494C42C4×D411yescharacteristic
495C42C4×D411yescharacteristic
496C42C4×D411yescharacteristic
497C42C4×D411yescharacteristic
498C42#12011yescharacteristic
499C42#12011yescharacteristic
500C42#12111yescharacteristic
501C42#12111yescharacteristic
502C42#12311yescharacteristic
503C42#12311yescharacteristic
504C42C23×C411yescharacteristic
505C42C22×D411yescharacteristic
506C42C22×D411yescharacteristic
507C42C2×#4111yescharacteristic
508C23—22no—
509C23—24no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23#38611yescharacteristic
519C23#38611yescharacteristic
520C23#46512yes—
521C23#46612yes—
522C23C2×#12111yescharacteristic
523Q8—22no—
524Q8—24no—
525Q8—42no—
526Q8—42no—
527Q8—82no—
528Q8C2×#10214yes—
529D4—28no—
530D4—24no—
531D4—24no—
532D4—22no—
533D4—24no—
534D4—42no—
535D4—42no—
536D4—44no—
537D4—82no—
538D4—82no—
539D4—82no—
540D4—84no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4C2×#10214yes—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—88no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4#38611yescharacteristic
580C2×C4#38611yescharacteristic
581C2×C4#38611yescharacteristic
582C2×C4#38611yescharacteristic
583C2×C4#38611yescharacteristic
584C2×C4#38611yescharacteristic
585C2×C4C2×#10212yes—
586C2×C4C2×#10212yes—
587C2×C4#41612yes—
588C2×C4#41612yes—
589C2×C4#46512yes—
590C2×C4#46512yes—
591C2×C4#46512yes—
592C2×C4#46612yes—
593C2×C4#46612yes—
594C2×C4#46612yes—
595C2×C4C22×C22⋊C411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×#12011yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12311yescharacteristic
601C2×C4C22×D811yescharacteristic
602C2×C4C22×SD1611yescharacteristic
603C2×C4C2×#13511yescharacteristic
604C2×C4C2×#13511yescharacteristic
605C8—44no—
606C8—44no—
607C8—44no—
608C8—82no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—24no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—164no—
625C22#189312yes—
626C22#189312yes—
627C22C2×#38611yescharacteristic
628C22C2×#46511yescharacteristic
629C22C2×#46611yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—84no—
641C4—82no—
642C4#189312yes—
643C4#189312yes—
644C4C22×#10211yescharacteristic
645C4C2×#41611yescharacteristic
646C4#295612yes—
647C4C2×#46511yescharacteristic
648C4C2×#46611yescharacteristic
649C2—21no—
650C2—22no—
651C2—22no—
652C2—21no—
653C2—44no—
654C2—162no—
655C2#2091811yescharacteristic
656C2C2×#189311yescharacteristic
657C2#2100011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#60415C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #60276C2semidirect
3. C2×#1806C2semidirect
4. C2×#1502C2semidirect
5. #17485C2semidirect
6. #9208C2semidirect
7. #8079C2semidirect
8. #7901C2semidirect
13. #3039C4semidirect
14. #3038C4semidirect
16. #3037C4semidirect
18. #3036C4semidirect
27. C8×#41C22semidirect
37. #1806C22semidirect
40. #1502C22semidirect
42. #1492C22semidirect
45. #1301C22semidirect
46. #1279C22semidirect
98. #462C2×C4semidirect
100. #461C2×C4semidirect
102. #460C2×C4semidirect
104. #459C2×C4semidirect

Maximal subgroups

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

subgroupindex
#604152
#602762
C2×#18062
C2×#15022
#174852
#92082
#80792
#79012