Finite Group Databasebeta

← #189128

Subgroups — #189128

Order 512.

1120 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2935C211yesmaximal, characteristic
2#21893C211yesmaximal, characteristic
3#21869C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6#8656C211yesmaximal, characteristic, complemented
7#8654C211yesmaximal, characteristic, complemented
8#8652C211yesmaximal, characteristic, complemented
9#8650C211yesmaximal, characteristic, complemented
10#8648C211yesmaximal, characteristic, complemented
11#8646C211yesmaximal, characteristic, complemented
12#8644C211yesmaximal, characteristic, complemented
13#8642C211yesmaximal, characteristic, complemented
14#8045C211yesmaximal, characteristic, complemented
15#8044C211yesmaximal, characteristic, complemented
16C22×#131—21no—
17C2×#532C2211yescharacteristic
18#2935—24no—
19#2935C418yescomplemented
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×#435—21no—
24C2×C8×D4C2211yescharacteristic
25C2×C8×D4C2211yescharacteristic
26C2×#432C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C2×#408—21no—
30C2×C4×M16C2211yescharacteristic
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#1993—21no—
34#1987—21no—
35#1982—21no—
36#1961—21no—
37#1895—21no—
38#1895—21no—
39#1875—21no—
40#1875—21no—
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1809—21no—
46#1809—21no—
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54#1802C2211yescharacteristic
55#1776—21no—
56#1776—21no—
57#1569—24no—
58C2×#335C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#334C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70C22×#41—21no—
71C2×#131—28no—
72C2×#131—44no—
73C2×#131—44no—
74C22×M16—21no—
75C22×M16—21no—
76C22×M16—21no—
77C22×M16D411yescharacteristic
78C22×M16D411yescharacteristic
79C23×C8—21no—
80C23×C8—21no—
81C23×C8—21no—
82#532—22no—
83#532—22no—
84#532C2×C414yes—
85#532Q814yes—
86C2×C4⋊Q8D411yescharacteristic
87C2×#127D411yescharacteristic
88C2×#124—21no—
89C2×#121—21no—
90C2×#121D411yescharacteristic
91C2×#121D411yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4C2311yescharacteristic
94C2×C4×D4C2311yescharacteristic
95C2×#117—21no—
96C2×#117—21no—
97C22×C42C2311yescharacteristic
98#436—22no—
99#436—22no—
100#436C2×C414yes—
101#436C2×C414yes—
102#435—28no—
103#435—28no—
104C8×D4—22no—
105C8×D4—22no—
106C8×D4C2×C414yes—
107C8×D4C2×C414yes—
108#432—22no—
109#432—22no—
110C2×#107—21no—
111C2×#107—21no—
112C2×#106—21no—
113C2×#106—21no—
114#423—22no—
115#423—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2×C414yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#102—21no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×#102—21no—
131C2×#102—21no—
132C2×#102—21no—
133C2×#102—21no—
134#408—28no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C4×M16C2×C414yes—
141C4×M16D414yes—
142C2×#97—21no—
143C2×#97D411yescharacteristic
144C2×#97D411yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#382—21no—
152#382—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#340—21no—
172#340—21no—
173#335—22no—
174#335—22no—
175#334—22no—
176#334—22no—
177C2×#41—24no—
178C2×#41—24no—
179C2×#41—42no—
180C2×#41—42no—
181C2×#41—42no—
182C2×#41—42no—
183C22×Q8—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C2×D411yescharacteristic
189C22×D4C2×D411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4#4111yescharacteristic
195#131—432no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—41no—
201C2×M16—41no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—42no—
207C2×M16—81no—
208C2×M16—81no—
209C2×M16C22⋊C414yes—
210C2×M16C22⋊C414yes—
211C22×C8—24no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—42no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C4⋊Q8—22no—
231C4⋊Q8C4⋊C414yes—
232#127—22no—
233#127C4⋊C414yes—
234#124—24no—
235#124—24no—
236#121—24no—
237#121—22no—
238#121—22no—
239#121—24no—
240#121C4⋊C414yes—
241#121C4⋊C414yes—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4C22×C412yes—
249C4×D4C22×C412yes—
250C4×D4C2×Q812yes—
251C4×D4C2×Q812yes—
252#117—81no—
253#117—81no—
254#117—82no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42C22×C412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×Q812yes—
283C2×C42#4111yescharacteristic
284C2×C42#4112yes—
285C2×C42C2411yescharacteristic
286#107—42no—
287#107—42no—
288#106—42no—
289#106—42no—
290C4⋊C8—24no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—42no—
302C4⋊C8C22×C412yes—
303C4⋊C8C22×C412yes—
304#102—81no—
305#102—81no—
306#102—81no—
307#102—81no—
308#102—81no—
309#102—81no—
310#102—81no—
311#102—81no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—42no—
317#97—42no—
318#97C22⋊C414yes—
319#97C22⋊C414yes—
320C4×C8C22×C412yes—
321C4×C8C22×C412yes—
322C4×C8C2×D412yes—
323C4×C8C2×D412yes—
324#95—41no—
325#95—41no—
326#95—41no—
327#95—41no—
328C24—21no—
329C24—21no—
330C24#12111yescharacteristic
331#41—48no—
332#41—48no—
333#41—48no—
334#41—48no—
335C2×Q8—24no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4C2×C4⋊C412yes—
361C2×D4C2×C4⋊C412yes—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4#11712yes—
393C22×C4#12112yes—
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4C22⋊Q812yes—
398C22×C4C22⋊Q812yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4111yescharacteristic
402M16—28no—
403M16—44no—
404M16—48no—
405M16—48no—
406M16—84no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—48no—
416C2×C8—48no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—48no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8C2×C22⋊C412yes—
435C2×C8C2×C22⋊C412yes—
436C2×C8C2×C22⋊C412yes—
437C2×C8C2×C22⋊C412yes—
438C2×C8AGL(1,ℤ/8)12yes—
439C2×C8AGL(1,ℤ/8)12yes—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—24no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4C2×C4⋊C412yes—
464C4⋊C4C2×C4⋊C412yes—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C42—82no—
474C42—82no—
475C42C2×C22⋊C412yes—
476C42C2×C4⋊C412yes—
477C42#11712yes—
478C42#12112yes—
479C42C22⋊Q812yes—
480C42C22⋊Q812yes—
481C42C23×C411yescharacteristic
482C42C22×D411yescharacteristic
483C42C22×Q811yescharacteristic
484C42C2×#4111yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23#38012yes—
506C23C2×#12111yescharacteristic
507Q8—48no—
508D4—22no—
509D4—22no—
510D4—22no—
511D4—22no—
512D4—48no—
513D4—48no—
514D4—48no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—84no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4#38012yes—
565C2×C4#38012yes—
566C2×C4#38012yes—
567C2×C4C22×C22⋊C411yescharacteristic
568C2×C4C22×C4⋊C411yescharacteristic
569C2×C4C2×#11711yescharacteristic
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×C22⋊Q811yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
574C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
575C2×C4#57511yescharacteristic
576C2×C4#57511yescharacteristic
577C2×C4#57611yescharacteristic
578C2×C4#57711yescharacteristic
579C2×C4#57711yescharacteristic
580C2×C4#57811yescharacteristic
581C8—22no—
582C8—22no—
583C8—22no—
584C8—42no—
585C8—42no—
586C8—81no—
587C8—81no—
588C8—81no—
589C8—81no—
590C8#41814yes—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×#38011yescharacteristic
613C22#303611yescharacteristic
614C22#303611yescharacteristic
615C22#306211yescharacteristic
616C22#306311yescharacteristic
617C22#306811yescharacteristic
618C22#306911yescharacteristic
619C4—22no—
620C4—22no—
621C4—21no—
622C4—21no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—161no—
627C4—161no—
628C4—161no—
629C4—161no—
630C4C2×#41811yescharacteristic
631C4#289611yescharacteristic
632C4#291211yescharacteristic
633C4#291311yescharacteristic
634C4#306211yescharacteristic
635C4#306311yescharacteristic
636C4#307711yescharacteristic
637C4#307711yescharacteristic
638C2—42no—
639C2—82no—
640C2—82no—
641C2#1936811yescharacteristic
642C2#1936911yescharacteristic
643C2#1937011yescharacteristic
644C2#1937111yescharacteristic
645C2#2064611yescharacteristic
646C2#2066411yescharacteristic
647C2#2067911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20824C2semidirect
5. C2×#1569C2semidirect
6. #8656C2semidirect
7. #8654C2semidirect
8. #8652C2semidirect
9. #8650C2semidirect
10. #8648C2semidirect
11. #8646C2semidirect
12. #8644C2semidirect
13. #8642C2semidirect
14. #8045C2semidirect
15. #8044C2semidirect
19. #2935C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29352
#218932
#218692
#208242
C2×#15692
#86562
#86542
#86522
#86502
#86482
#86462
#86442
#86422
#80452
#80442