Finite Group Databasebeta

← #199049

Subgroups — #199049

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#34070C211yesmaximal, characteristic, complemented
3C2×#1963C211yesmaximal, characteristic, complemented
4#17704C211yesmaximal, characteristic, complemented
5#17574C211yesmaximal, characteristic, complemented
6#9300C214yesmaximal, complemented
7#9299C214yesmaximal, complemented
8#7957C211yesmaximal, characteristic, complemented
9C2×#1274C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×C4×#41—21no—
13#3245—22no—
14#3226C412yescomplemented
15#3226C2212yes—
16#3205C412yescomplemented
17#3205C2212yes—
18#3186—22no—
19#3169C414yescomplemented
20#3100—22no—
21#3100—22no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×#104—21no—
26C22×#104—21no—
27#2328C2212yescomplemented
28#2327C2212yescomplemented
29#2326—22no—
30#2325—22no—
31#2325—22no—
32#2297—21no—
33#2290—22no—
34#2290—22no—
35#2285C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1963—22no—
38#1963C2214yes—
39#1547—22no—
40#1546C2212yescomplemented
41#1545C2212yescomplemented
42#1540—22no—
43#1519—22no—
44#1517C2214yes—
45#1502—22no—
46#1286C2214yescomplemented
47#1274C2214yescomplemented
48C22×#41—21no—
49C22×#41—21no—
50C22×M16—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532C2×C412yes—
62#532C2×C412yes—
63#532C2×C412yescomplemented
64#532D412yes—
65#532D412yes—
66#532C2312yes—
67C2×C4⋊Q8C2×C411yescharacteristic, complemented
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2×C411yescharacteristic, complemented
70C2×#127C2311yescharacteristic
71C2×#124—21no—
72C2×#124—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121D412yes—
77C4×#41—44no—
78C4×#41—44no—
79C2×C4×Q8—22no—
80C2×C4×D4—24no—
81C2×C4×D4—21no—
82C2×C4×D4—22no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4D412yes—
86C2×#117—22no—
87C2×#117—21no—
88C22×C42—22no—
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92#500C2×C414yescomplemented
93#493C2×C414yescomplemented
94#486C2×C414yescomplemented
95#486C2×C414yescomplemented
96#480—24no—
97#480—24no—
98#474—24no—
99#474—24no—
100#466—28no—
101#459—24no—
102#459—24no—
103#447—24no—
104#447—24no—
105C4×D8—28no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431C2×C412yescomplemented
112#431C2312yes—
113#431C2312yes—
114#423—21no—
115#423—21no—
116#423—21no—
117#423C2312yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×#104—24no—
123C2×#104—24no—
124C2×#104—44no—
125C2×#104—44no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2×C412yescomplemented
130#391—22no—
131#390—22no—
132#388—24no—
133#388—24no—
134#387—24no—
135#387—24no—
136#386—24no—
137#386—24no—
138#386—24no—
139#385—24no—
140#385—24no—
141#384—21no—
142#382C2312yes—
143#380—21no—
144#380D412yes—
145C4×C4⋊C4—21no—
146C4×C4⋊C4D412yes—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149#376—21no—
150#376—21no—
151#376D412yes—
152#376C2312yes—
153C2×#95—21no—
154C2×#95—21no—
155#332—24no—
156#331—24no—
157#327—28no—
158C2×#41—22no—
159C2×#41—22no—
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×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—42no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×D4—22no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—41no—
183C22×D4C2×D412yes—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—21no—
188C23×C4—41no—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×D8—28no—
193C2×M16—22no—
194C2×M16—41no—
195C2×M16—42no—
196C2×M16—42no—
197C22×C8—22no—
198C22×C8—41no—
199C22×C8—41no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8C22×C412yes—
206C4⋊Q8C22×C412yes—
207C4⋊Q8C2×D412yes—
208#127—21no—
209#127—21no—
210#127—21no—
211#127—21no—
212#127—21no—
213#127C22×C412yes—
214#127C22×C412yes—
215#127C2×D412yes—
216#124—22no—
217#124—22no—
218#124—22no—
219#124—22no—
220#124—22no—
221#124—22no—
222#124—41no—
223#124—41no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—22no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—41no—
235#121—41no—
236#121—41no—
237#121—41no—
238#121C22⋊C414yes—
239#121C22⋊C414yes—
240C4×Q8—44no—
241C4×Q8—44no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—44no—
253C4×D4—416no—
254C4×D4—41no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—44no—
259C4×D4C22⋊C414yes—
260C4×D4C22⋊C414yes—
261#117—48no—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C2×D412yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—48no—
288C2×C42D814yescomplemented
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42#4111yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42C2411yescharacteristic
303#107—28no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311C4⋊C8—41no—
312#104—48no—
313#104—48no—
314#102—28no—
315#102—28no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—41no—
320C4×C8—22no—
321C4×C8—22no—
322C4×C8—41no—
323C4×C8C22×C414yes—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95C2×D414yes—
332C24—22no—
333C24—41no—
334C24#12011yescharacteristic
335#41—44no—
336#41—44no—
337#41—44no—
338#41—44no—
339#41—44no—
340#41—44no—
341#41—44no—
342#41—44no—
343#41—42no—
344#41—42no—
345#41—42no—
346#41—42no—
347#41—44no—
348#41—44no—
349#41—82no—
350#41—82no—
351#41—82no—
352#41—82no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4C2×C22⋊C414yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—81no—
414C22×C4—81no—
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#12011yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12311yescharacteristic
431C22×C4#12311yescharacteristic
432C22×C4C2×D812yes—
433C22×C4#13512yes—
434C22×C4AGL(1,ℤ/8)12yes—
435C22×C4AGL(1,ℤ/8)12yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439D8—88no—
440M16—42no—
441M16—42no—
442M16—82no—
443C2×C8—24no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—82no—
447C2×C8—82no—
448C2×C8—82no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4C2×C22⋊C414yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C42—22no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—48no—
479C42—44no—
480C42—42no—
481C42C2×C22⋊C411yescharacteristic
482C42C2×C22⋊C411yescharacteristic
483C42C4×D411yescharacteristic
484C42C4×D411yescharacteristic
485C42C4×D411yescharacteristic
486C42C4×D411yescharacteristic
487C42#12111yescharacteristic
488C42#12111yescharacteristic
489C42#12111yescharacteristic
490C42#12111yescharacteristic
491C42#12311yescharacteristic
492C42#12311yescharacteristic
493C42C2×D814yes—
494C42C23×C411yescharacteristic
495C42C22×D411yescharacteristic
496C42C2×#4111yescharacteristic
497C42C2×#4111yescharacteristic
498C23—24no—
499C23—41no—
500C23—41no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23#38611yescharacteristic
510C23#38611yescharacteristic
511C23#44712yes—
512C23#44912yes—
513C23C2×#12011yescharacteristic
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—82no—
521D4—44no—
522D4—44no—
523D4—48no—
524D4—48no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—82no—
532D4—82no—
533D4—82no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—84no—
538D4—82no—
539D4—82no—
540D4—82no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—84no—
569C2×C4—816no—
570C2×C4—84no—
571C2×C4—82no—
572C2×C4—84no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4#38611yescharacteristic
576C2×C4#38611yescharacteristic
577C2×C4#38611yescharacteristic
578C2×C4#38611yescharacteristic
579C2×C4#38611yescharacteristic
580C2×C4#38611yescharacteristic
581C2×C4C4×D812yes—
582C2×C4C4×D812yes—
583C2×C4#44212yes—
584C2×C4#44212yes—
585C2×C4#44712yes—
586C2×C4#44912yes—
587C2×C4#45912yes—
588C2×C4#45912yes—
589C2×C4#46312yes—
590C2×C4#46312yes—
591C2×C4C22×C22⋊C411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×#12111yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×#12311yescharacteristic
597C2×C4C22×D811yescharacteristic
598C2×C4C2×#13511yescharacteristic
599C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
600C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
601C8—44no—
602C8—82no—
603C8—82no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—164no—
621C22#187912yes—
622C22#187912yes—
623C22C2×#38611yescharacteristic
624C22C2×#44711yescharacteristic
625C22C2×#44911yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—44no—
635C4—44no—
636C4—84no—
637C4—82no—
638C4—82no—
639C4#187912yes—
640C4#187912yes—
641C4C2×C4×D811yescharacteristic
642C4C2×#44211yescharacteristic
643C4C2×#45911yescharacteristic
644C4C2×#46311yescharacteristic
645C2—21no—
646C2—21no—
647C2—22no—
648C2—22no—
649C2—84no—
650C2—162no—
651C2C2×#187911yescharacteristic
652C2#2099811yescharacteristic
653C2#2101211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#60940C2
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. #34070C2semidirect
3. C2×#1963C2semidirect
4. #17704C2semidirect
5. #17574C2semidirect
6. #9300C2semidirect
7. #9299C2semidirect
8. #7957C2semidirect
9. C2×#1274C2semidirect
14. #3226C4semidirect
16. #3205C4semidirect
19. #3169C4semidirect
27. #2328C22semidirect
28. #2327C22semidirect
40. #1546C22semidirect
41. #1545C22semidirect
46. #1286C22semidirect
47. #1274C22semidirect
63. #532C2×C4semidirect
67. C2×C4⋊Q8C2×C4semidirect
69. C2×#127C2×C4semidirect
92. #500C2×C4semidirect
93. #493C2×C4semidirect
94. #486C2×C4semidirect
95. #486C2×C4semidirect
111. #431C2×C4semidirect
129. C2×C4×C8C2×C4semidirect
288. C2×C42D8semidirect

Maximal subgroups

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

subgroupindex
#609402
#340702
C2×#19632
#177042
#175742
#93002
#92992
#79572
C2×#12742