Finite Group Databasebeta

← #6470263

Subgroups — #6470263

Order 512.

3016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3007C211yesmaximal, characteristic, complemented
2C2×#3006C211yesmaximal, characteristic
3#35019C211yesmaximal, characteristic, complemented
4C2×#2341C211yesmaximal, characteristic, complemented
5#32909C214yesmaximal, complemented
6#29718C214yesmaximal, complemented
7#24547C214yesmaximal, complemented
8#24543C214yesmaximal, complemented
9#24375C214yesmaximal, complemented
10#24371C214yesmaximal, complemented
11#21934C211yesmaximal, characteristic, complemented
12#21765C211yesmaximal, characteristic, complemented
13#21764C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#137C2212yescomplemented
16C22×AGL(1,ℤ/8)C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×Q8C2212yescomplemented
20C2×#464C2218yescomplemented
21#3007—28no—
22#3006—28no—
23C2×#452C2214yes—
24C2×#449C2214yes—
25C2×#451C2212yescomplemented
26C2×#448C2212yes—
27C2×#450C2212yescomplemented
28C2×#447C2212yes—
29C2×#417C2212yes—
30C2×#417C2212yescomplemented
31C2×#407C2211yescharacteristic
32C2×#407C2211yescharacteristic
33#2850C2214yes—
34#2639—21no—
35#2639—21no—
36#2591—22no—
37#2561—22no—
38#2504C2212yescomplemented
39#2503C2212yescomplemented
40#2436C2212yescomplemented
41#2435C2212yescomplemented
42#2341—22no—
43#2341—22no—
44#2341C2214yes—
45#2341C2214yescomplemented
46#2308—22no—
47C2×#390—22no—
48C2×#390C2212yescomplemented
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52C2×C4×C22⋊C4C2211yescharacteristic
53#2073C2218yescomplemented
54#2055C2218yescomplemented
55#2054C2218yescomplemented
56#2012C2218yescomplemented
57#1895C2214yescomplemented
58#1894C2214yescomplemented
59#1880C2218yescomplemented
60#1878C2218yescomplemented
61#1874C2214yes—
62#1872C2214yescomplemented
63#1851C2214yescomplemented
64#1851C2214yescomplemented
65#1824C2212yescomplemented
66#1823C2212yescomplemented
67#1805C2218yescomplemented
68#1797C2214yescomplemented
69#1787C2212yescomplemented
70#1786C2212yescomplemented
71C22×#41—21no—
72C22×#41—21no—
73C22×#41—24no—
74C22×#41—48no—
75C22×#41D418yes—
76C22×#41C2312yes—
77C22×#41C2312yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C2×#137—28no—
87C2×#137—48no—
88C2×AGL(1,ℤ/8)—28no—
89C2×AGL(1,ℤ/8)—48no—
90C22×Q16C2314yes—
91C22×SD16C2314yes—
92C22×SD16C2314yes—
93C22×D8C2314yes—
94C22×M16C2312yes—
95C22×M16C2312yes—
96C2×#124C2314yes—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—24no—
106C2×#121—24no—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C2×C4×Q8—24no—
110C2×C4×Q8—48no—
111C2×C4×Q8C2318yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4D412yes—
123C22×C42—22no—
124C22×C42C2312yes—
125#464—416no—
126#452—216no—
127#451—28no—
128#450—28no—
129#449—216no—
130#448—28no—
131#447—28no—
132C2×C4⋊C8C2318yes—
133#417—28no—
134#417—28no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2318yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2318yes—
141#407—24no—
142#407—24no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147#400—22no—
148#400—22no—
149#397—22no—
150#397—22no—
151#394—22no—
152#394—22no—
153#393—22no—
154#393—22no—
155#391C2314yes—
156#390—28no—
157#390—28no—
158#390—24no—
159#390C2314yes—
160#390C2318yes—
161#388—24no—
162#388—24no—
163#386—24no—
164#386—24no—
165#386—24no—
166#386—24no—
167#382C2314yes—
168#381—24no—
169#380—21no—
170#380—21no—
171#380—24no—
172#380—24no—
173#380C2312yes—
174#380C2312yes—
175#379—24no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4—24no—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4D412yes—
180C4×C22⋊C4C2312yes—
181C4×C22⋊C4C2312yes—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2318yes—
186#340C2318yes—
187#336C2318yes—
188C25—21no—
189C25—21no—
190C25C2×D411yescharacteristic
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—28no—
196C2×#41—28no—
197C2×#41—44no—
198C2×#41—44no—
199C2×#41—432no—
200C2×#41—432no—
201C2×#41—48no—
202C2×#41—48no—
203C2×#41C2×D418yes—
204C2×#41C2×D418yes—
205C22×Q8—28no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—44no—
212C22×Q8C2×D414yes—
213C22×Q8C2412yes—
214C22×Q8C2412yes—
215C22×D4—28no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—44no—
226C22×D4—44no—
227C22×D4—48no—
228C22×D4—48no—
229C22×D4—44no—
230C22×D4C2×D414yes—
231C22×D4C2412yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—28no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—44no—
242C23×C4—48no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D414yes—
249C23×C4#4114yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253#137—432no—
254AGL(1,ℤ/8)—432no—
255C2×Q16—28no—
256C2×Q16—216no—
257C2×SD16—28no—
258C2×SD16—216no—
259C2×SD16—28no—
260C2×SD16—48no—
261C2×D8—28no—
262C2×D8—48no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—44no—
266C2×M16—44no—
267C22×C8C2414yes—
268C22×C8C2414yes—
269#124—48no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—44no—
281#121—44no—
282C4×Q8—216no—
283C4×Q8—48no—
284C4×Q8—416no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—84no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C42—28no—
322C2×C42—44no—
323C2×C42—48no—
324C2×C42C2×D414yes—
325C2×C42C2414yes—
326C4⋊C8—48no—
327#103—28no—
328#103—28no—
329#103—48no—
330#102—28no—
331#102—28no—
332#102—48no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—41no—
352C24—42no—
353C24—41no—
354C24—42no—
355C24—44no—
356C24—42no—
357C24—44no—
358C24—44no—
359C24—42no—
360C24—82no—
361C24—81no—
362C24—81no—
363C24#12114yes—
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366C24C22×D411yescharacteristic
367#41—216no—
368#41—216no—
369#41—48no—
370#41—48no—
371#41—416no—
372#41—416no—
373#41—432no—
374#41—432no—
375C2×Q8—28no—
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—28no—
384C2×Q8—48no—
385C2×Q8—42no—
386C2×Q8—82no—
387C2×Q8—84no—
388C2×Q8C22×D414yes—
389C2×Q8H(5,𝔽2)18yes—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—432no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—42no—
411C2×D4—84no—
412C2×D4—82no—
413C2×D4—816no—
414C2×D4C22×D414yes—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—432no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—82no—
451C22×C4#12114yes—
452C22×C4#12114yes—
453C22×C4#12114yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2511yescharacteristic
466Q16—416no—
467SD16—416no—
468SD16—416no—
469D8—416no—
470M16—48no—
471M16—48no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—28no—
475C2×C8—28no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—28no—
481C4⋊C4—416no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—416no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—88no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—28no—
502C42—44no—
503C42—416no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—28no—
516C23—24no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—48no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—84no—
532C23—84no—
533C23—88no—
534C23—88no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—164no—
539C23—164no—
540C23C2×#12112yes—
541C23C2×#12112yes—
542C23C2×#12112yes—
543C23#57514yes—
544C23C23×D411yescharacteristic
545Q8—216no—
546Q8—24no—
547Q8—28no—
548Q8—44no—
549Q8—44no—
550Q8—84no—
551Q8#53418yes—
552D4—24no—
553D4—24no—
554D4—416no—
555D4—416no—
556D4—48no—
557D4—44no—
558D4—44no—
559D4—44no—
560D4—84no—
561D4—88no—
562D4—88no—
563C2×C4—216no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4C2×#12112yes—
603C2×C4C2×#12112yes—
604C2×C4C2×#12112yes—
605C2×C4C23×D411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)14yes—
608C8—44no—
609C8—44no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—88no—
628C22—816no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—84no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22—168no—
637C22#304014yes—
638C22C22×#12111yescharacteristic
639C22#348412yes—
640C22C2×#57512yes—
641C22#358511yescharacteristic
642C22#358611yescharacteristic
643C4—28no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—82no—
650C4—84no—
651C4—82no—
652C4—82no—
653C4C2×#53412yes—
654C4#349012yes—
655C2—24no—
656C2—42no—
657C2—42no—
658C2—84no—
659C2—162no—
660C2#3498111yescharacteristic
661C2C2×#304011yescharacteristic
662C2#6042511yescharacteristic
663C2#6157712yes—
664C2#6158312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3007C2semidirect
3. #35019C2semidirect
4. C2×#2341C2semidirect
5. #32909C2semidirect
6. #29718C2semidirect
7. #24547C2semidirect
8. #24543C2semidirect
9. #24375C2semidirect
10. #24371C2semidirect
11. #21934C2semidirect
12. #21765C2semidirect
13. #21764C2semidirect
15. C22×#137C22semidirect
19. C22×C4×Q8C22semidirect
20. C2×#464C22semidirect
25. C2×#451C22semidirect
27. C2×#450C22semidirect
30. C2×#417C22semidirect
38. #2504C22semidirect
39. #2503C22semidirect
40. #2436C22semidirect
41. #2435C22semidirect
45. #2341C22semidirect
48. C2×#390C22semidirect
53. #2073C22semidirect
54. #2055C22semidirect
55. #2054C22semidirect
56. #2012C22semidirect
57. #1895C22semidirect
58. #1894C22semidirect
59. #1880C22semidirect
60. #1878C22semidirect
62. #1872C22semidirect
63. #1851C22semidirect
64. #1851C22semidirect
65. #1824C22semidirect
66. #1823C22semidirect
67. #1805C22semidirect
68. #1797C22semidirect
69. #1787C22semidirect
70. #1786C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30072
C2×#30062
#350192
C2×#23412
#329092
#297182
#245472
#245432
#243752
#243712
#219342
#217652
#217642