Finite Group Databasebeta

← #202811

Subgroups — #202811

Order 512.

824 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21988C211yesmaximal, characteristic, complemented
2#21893C211yesmaximal, characteristic
3#21410C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic, complemented
5C2×#1551C211yesmaximal, 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#8045C211yesmaximal, characteristic, complemented
15#7876C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#405—21no—
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1987C2211yescharacteristic
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1895—21no—
37#1895—21no—
38#1895—21no—
39#1895—21no—
40#1854—21no—
41#1854—21no—
42#1847—21no—
43#1836—21no—
44#1802—21no—
45#1802—21no—
46#1771—21no—
47#1771—21no—
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53#1762—21no—
54#1729—21no—
55#1551C418yescomplemented
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×M16D411yescharacteristic
71C22×M16D411yescharacteristic
72C23×C8—21no—
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—
85C2×#117—21no—
86C22×C42C2311yescharacteristic
87#432—22no—
88#432C2×C414yes—
89C2×#107—21no—
90C2×#107—21no—
91C2×#106—21no—
92C2×#106—21no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2×C414yes—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×C22⋊C8—21no—
114#405—44no—
115C4×M16—22no—
116C4×M16—22no—
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97D411yescharacteristic
121C2×#97D411yescharacteristic
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#384D411yescharacteristic
127#384D411yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131#382—21no—
132#380—21no—
133#380—21no—
134#380—21no—
135#380D411yescharacteristic
136#380D411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142#340—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#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#333C2×C414yes—
159#333C2×C414yes—
160#331C2×C414yes—
161#331C2×C414yes—
162C2×#41—24no—
163C2×#41—42no—
164C2×#41—82no—
165C2×#41—82no—
166C22×Q8—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—41no—
171C23×C4—41no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—41no—
180C2×M16—41no—
181C2×M16—42no—
182C2×M16—81no—
183C2×M16—81no—
184C22×C8—24no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8C22×C412yes—
201#127—21no—
202#127—21no—
203#127—21no—
204#127C22×C412yes—
205#124—24no—
206#124—42no—
207#121—24no—
208#121—24no—
209#121—42no—
210#121—42no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—42no—
214C4×D4—42no—
215#117—42no—
216#117—42no—
217#117—81no—
218#117—81no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—41no—
243C2×C42C22×C412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C2×C42C2411yescharacteristic
251#107—81no—
252#107—81no—
253#106—81no—
254#106—81no—
255C4⋊C8—24no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—42no—
265C4⋊C8C22×C412yes—
266C4⋊C8C22×C412yes—
267C4⋊C8C22×C412yes—
268C4⋊C8C22×C412yes—
269#102—42no—
270#102—42no—
271#102—42no—
272#102—42no—
273#102—81no—
274#102—81no—
275#102—81no—
276#102—81no—
277C22⋊C8—24no—
278#97—24no—
279#97—22no—
280#97—22no—
281#97—42no—
282#97—42no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—42no—
288C4×C8—42no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—41no—
294#95—41no—
295C24—41no—
296C24#12111yescharacteristic
297#41—88no—
298#41—88no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—81no—
302C2×Q8—81no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—44no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—81no—
319C2×D4—81no—
320C2×D4—82no—
321C2×D4—82no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4#11712yes—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12312yes—
349C22×C4#12412yes—
350C22×C4#12612yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4111yescharacteristic
354M16—28no—
355M16—44no—
356M16—84no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—48no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8AGL(1,ℤ/8)12yes—
387C2×C8#13712yes—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—81no—
411C4⋊C4—81no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—82no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—82no—
424C42#11712yes—
425C42C4×D412yes—
426C42C4×D412yes—
427C42#12312yes—
428C42#12412yes—
429C42#12612yes—
430C42C23×C411yescharacteristic
431C42C2×#4111yescharacteristic
432C42C2×#4111yescharacteristic
433C42C2×#4111yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—42no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23#38812yes—
448C23C2×#12111yescharacteristic
449Q8—82no—
450Q8—82no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—82no—
456D4—82no—
457D4—88no—
458D4—81no—
459D4—81no—
460D4—81no—
461D4—81no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—88no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4#38812yes—
503C2×C4#38812yes—
504C2×C4#38812yes—
505C2×C4C2×#11711yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4C2×#12411yescharacteristic
510C2×C4C2×#12611yescharacteristic
511C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
512C2×C4C2×#13711yescharacteristic
513C2×C4#57511yescharacteristic
514C2×C4#57511yescharacteristic
515C2×C4#57611yescharacteristic
516C2×C4#57711yescharacteristic
517C2×C4#57711yescharacteristic
518C2×C4#57811yescharacteristic
519C8—22no—
520C8—22no—
521C8—41no—
522C8—41no—
523C8—42no—
524C8—42no—
525C8—42no—
526C8—42no—
527C8—81no—
528C8—81no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—161no—
540C22—161no—
541C22—161no—
542C22—161no—
543C22C2×#38811yescharacteristic
544C22#303611yescharacteristic
545C22#303711yescharacteristic
546C22#306211yescharacteristic
547C22#306311yescharacteristic
548C22#306811yescharacteristic
549C22#306811yescharacteristic
550C4—22no—
551C4—22no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—82no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—161no—
561C4—161no—
562C4C4×AGL(1,ℤ/8)11yescharacteristic
563C4C4×#13711yescharacteristic
564C4#295111yescharacteristic
565C4#295211yescharacteristic
566C4#309211yescharacteristic
567C4#309311yescharacteristic
568C4#313611yescharacteristic
569C4#313711yescharacteristic
570C2—42no—
571C2—162no—
572C2#1936911yescharacteristic
573C2#1937011yescharacteristic
574C2#1937311yescharacteristic
575C2#1937411yescharacteristic
576C2#2105911yescharacteristic
577C2#2107811yescharacteristic
578C2#2108711yescharacteristic
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
1. #21988C2semidirect
4. #20441C2semidirect
6. #9576C2semidirect
7. #9575C2semidirect
8. #9572C2semidirect
9. #9571C2semidirect
10. #9568C2semidirect
11. #9567C2semidirect
12. #9564C2semidirect
13. #9563C2semidirect
14. #8045C2semidirect
15. #7876C2semidirect
55. #1551C4semidirect

Maximal subgroups

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

subgroupindex
#219882
#218932
#214102
#204412
C2×#15512
#95762
#95752
#95722
#95712
#95682
#95672
#95642
#95632
#80452
#78762