Finite Group Databasebeta

← #202835

Subgroups — #202835

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21410C211yesmaximal, characteristic
2#21395C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic, complemented
4C4⋊C4×M16C211yesmaximal, characteristic, complemented
5C2×#1560C211yesmaximal, characteristic
6#9576C211yesmaximal, characteristic, complemented
7#9575C211yesmaximal, characteristic, complemented
8#9572C211yesmaximal, characteristic, complemented
9#9571C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9564C211yesmaximal, characteristic, complemented
13#9563C211yesmaximal, characteristic, complemented
14#8042C211yesmaximal, characteristic, complemented
15#7876C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#428—21no—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×C4×M16—21no—
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1987C2211yescharacteristic
28#1987C2211yescharacteristic
29#1982C2211yescharacteristic
30#1982C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1949—21no—
34#1949—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1895—21no—
40#1895—21no—
41#1874—21no—
42#1874—21no—
43#1771—21no—
44#1771—21no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1762—21no—
50#1762—21no—
51#1762—21no—
52#1762—21no—
53#1736—21no—
54#1728—21no—
55#1560C418yescomplemented
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×#331C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×#41—21no—
69C22×M16—21no—
70C22×M16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73#532—22no—
74#532—22no—
75#532—22no—
76#532C2×C414yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×#127C2311yescharacteristic
79C2×#124—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×#117—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#432—22no—
90#432—22no—
91#428—44no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96#423—22no—
97#423—22no—
98#423C2×C414yes—
99#423C2×C414yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C22⋊C8—21no—
117C4×M16—22no—
118C4×M16—22no—
119C4×M16—44no—
120C2×#97—21no—
121C2×#97—21no—
122C2×#97—21no—
123C2×#97D411yescharacteristic
124C2×#97D411yescharacteristic
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#382—21no—
132#380—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C2×#95—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#333C2×C414yescomplemented
160#333C2×C414yescomplemented
161#331C2×C414yescomplemented
162#331C2×C414yescomplemented
163C2×#41—24no—
164C2×#41—42no—
165C2×#41—82no—
166C2×#41—82no—
167C22×Q8—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—41no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—41no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4111yescharacteristic
178C2×M16—22no—
179C2×M16—24no—
180C2×M16—24no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—41no—
185C2×M16—41no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—42no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C22×C412yes—
204#127—21no—
205#127—21no—
206#127—21no—
207#127C22×C412yes—
208#124—24no—
209#124—42no—
210#121—24no—
211#121—24no—
212#121—42no—
213#121—42no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—42no—
217C4×D4—42no—
218#117—42no—
219#117—42no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—41no—
248C2×C42C22×C412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42C2411yescharacteristic
256#107—42no—
257#107—42no—
258#106—42no—
259#106—42no—
260C4⋊C8—24no—
261C4⋊C8—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8C22×C412yes—
271C4⋊C8C22×C412yes—
272C4⋊C8C22×C412yes—
273C4⋊C8C22×C412yes—
274#102—42no—
275#102—42no—
276#102—42no—
277#102—42no—
278#102—42no—
279#102—42no—
280#102—42no—
281#102—42no—
282C22⋊C8—42no—
283#97—22no—
284#97—22no—
285#97—42no—
286#97—42no—
287#97—42no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—42no—
293C4×C8—42no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—41no—
299#95—41no—
300C24—41no—
301C24#12111yescharacteristic
302#41—88no—
303#41—88no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—81no—
307C2×Q8—81no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—44no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—81no—
324C2×D4—81no—
325C2×D4—82no—
326C2×D4—82no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4#11712yes—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12312yes—
360C22×C4#12412yes—
361C22×C4#12612yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365M16—28no—
366M16—44no—
367M16—48no—
368M16—48no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8AGL(1,ℤ/8)12yes—
398C2×C8#13712yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4AGL(1,ℤ/8)12yes—
422C4⋊C4#13712yes—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—82no—
426C42—42no—
427C42—42no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—82no—
433C42#11712yes—
434C42C4×D412yes—
435C42C4×D412yes—
436C42#12312yes—
437C42#12412yes—
438C42#12612yes—
439C42C23×C411yescharacteristic
440C42C2×#4111yescharacteristic
441C42C2×#4111yescharacteristic
442C42C2×#4111yescharacteristic
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—42no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23#38812yes—
457C23C2×#12111yescharacteristic
458Q8—82no—
459Q8—82no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—82no—
465D4—82no—
466D4—88no—
467D4—81no—
468D4—81no—
469D4—81no—
470D4—81no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—88no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4#38812yes—
514C2×C4#38812yes—
515C2×C4#38812yes—
516C2×C4C2×#11711yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C2×C4C2×#12411yescharacteristic
521C2×C4C2×#12611yescharacteristic
522C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
523C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
524C2×C4C2×#13711yescharacteristic
525C2×C4C2×#13711yescharacteristic
526C2×C4#57511yescharacteristic
527C2×C4#57511yescharacteristic
528C2×C4#57511yescharacteristic
529C2×C4#57511yescharacteristic
530C8—22no—
531C8—22no—
532C8—41no—
533C8—41no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—42no—
538C8—81no—
539C8—81no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—161no—
551C22—161no—
552C22—161no—
553C22—161no—
554C22C2×#38811yescharacteristic
555C22#303611yescharacteristic
556C22#303711yescharacteristic
557C22#305511yescharacteristic
558C22#305611yescharacteristic
559C22#306811yescharacteristic
560C22#306811yescharacteristic
561C4—22no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—82no—
569C4—81no—
570C4—81no—
571C4—161no—
572C4—161no—
573C4C4×AGL(1,ℤ/8)11yescharacteristic
574C4C4×AGL(1,ℤ/8)11yescharacteristic
575C4C4×#13711yescharacteristic
576C4C4×#13711yescharacteristic
577C4#309211yescharacteristic
578C4#309311yescharacteristic
579C4#314111yescharacteristic
580C4#314211yescharacteristic
581C2—42no—
582C2—162no—
583C2#1930511yescharacteristic
584C2#1930611yescharacteristic
585C2#1930911yescharacteristic
586C2#1931011yescharacteristic
587C2#2105911yescharacteristic
588C2#2107811yescharacteristic
589C2#2109311yescharacteristic
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
3. #21170C2semidirect
4. C4⋊C4×M16C2semidirect
6. #9576C2semidirect
7. #9575C2semidirect
8. #9572C2semidirect
9. #9571C2semidirect
10. #9568C2semidirect
11. #9567C2semidirect
12. #9564C2semidirect
13. #9563C2semidirect
14. #8042C2semidirect
15. #7876C2semidirect
55. #1560C4semidirect
159. #333C2×C4semidirect
160. #333C2×C4semidirect
161. #331C2×C4semidirect
162. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#214102
#213952
#211702
C4⋊C4×M162
C2×#15602
#95762
#95752
#95722
#95712
#95682
#95672
#95642
#95632
#80422
#78762