Finite Group Databasebeta

← #404929

Subgroups — #404929

Order 512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1853C212yesmaximal
5#15150C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11199C212yesmaximal
9#11042C211yesmaximal, characteristic
10#10908C212yesmaximal, complemented
11#10807C211yesmaximal, characteristic, complemented
12C23×C4⋊C4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C2×#397—21no—
18C2×#397—21no—
19#2345—24no—
20#2345—24no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2084—24no—
34#1948—24no—
35#1853—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88#397—44no—
89#397—44no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389D414yes—
94#387—28no—
95#387—28no—
96#384—22no—
97#384—22no—
98#380—28no—
99#380—22no—
100#380—22no—
101#380—24no—
102#380—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C4×C22⋊C4—28no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#336—24no—
129#336—22no—
130C25C2×D411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8#4114yes—
171C22×C8#4114yes—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4#4112yes—
220C2×C4⋊C4#4112yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4SD1614yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—82no—
242C2×C42—82no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#106—48no—
246#106—48no—
247C4⋊C8—28no—
248C22⋊C8—216no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—42no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—44no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24#12112yes—
270C24C2×SD1612yes—
271C24AGL(1,ℤ/8)12yes—
272C24C22×D411yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—216no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4#12112yes—
352C22×C4C2×SD1612yes—
353C22×C4C2×SD1612yes—
354C22×C4#13712yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4112yes—
363C22×C42−1+412yes—
364C22×C42−1+412yes—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—216no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4#12114yes—
405C4⋊C4#12114yes—
406C22⋊C4—28no—
407C22⋊C4—44no—
408C22⋊C4—48no—
409C22⋊C4—88no—
410C42—22no—
411C42—22no—
412C42—24no—
413C42—41no—
414C42—41no—
415C42—42no—
416C42—84no—
417C42—84no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23#46014yes—
441C23C2×#12011yescharacteristic
442C23C2×#12111yescharacteristic
443C23#52511yescharacteristic
444C23#54712yes—
445C23#54712yes—
446C23C22×SD1611yescharacteristic
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×#13711yescharacteristic
449C23#57511yescharacteristic
450C23#57712yes—
451C23#57712yes—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—816no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#44814yes—
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4#52511yescharacteristic
521C2×C4#52511yescharacteristic
522C2×C4#52512yes—
523C2×C4#52511yescharacteristic
524C2×C4#52912yes—
525C2×C4#53211yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#53611yescharacteristic
528C2×C4#53612yes—
529C2×C4#54112yes—
530C2×C4#54712yes—
531C2×C4#54812yes—
532C2×C4#55212yes—
533C2×C4#55212yes—
534C2×C4#55212yes—
535C2×C4#55612yes—
536C8—44no—
537C8—44no—
538C22—28no—
539C22—24no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—44no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22#240611yescharacteristic
550C22#241312yes—
551C22#243211yescharacteristic
552C22#247812yes—
553C22#284711yescharacteristic
554C22C2×#44811yescharacteristic
555C22#300611yescharacteristic
556C22#300911yescharacteristic
557C22#300911yescharacteristic
558C22C2×#46011yescharacteristic
559C22#303711yescharacteristic
560C22#304212yes—
561C22#310311yescharacteristic
562C22#311211yescharacteristic
563C22#312312yes—
564C22#333512yes—
565C22#333512yes—
566C22#334412yes—
567C22#334512yes—
568C22#334512yes—
569C22#335012yes—
570C22#335112yes—
571C22#335612yes—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—44no—
577C4—44no—
578C4—82no—
579C4—81no—
580C4—82no—
581C4—162no—
582C4—162no—
583C4#239211yescharacteristic
584C4#241512yes—
585C4#247512yes—
586C4#259911yescharacteristic
587C4#261111yescharacteristic
588C4#262111yescharacteristic
589C2—44no—
590C2#1490711yescharacteristic
591C2#2409711yescharacteristic
592C2#2410111yescharacteristic
593C2#2457812yes—
594C2#2459012yes—
595C2#2570211yescharacteristic
596C2#2571411yescharacteristic
597C2#2745812yes—
598C2#2747112yes—
599C2#3290311yescharacteristic
600C2#3291411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
3. C2×#1948C2semidirect
6. #11228C2semidirect
7. #11220C2semidirect
10. #10908C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C2×#20842
C2×#19482
C2×#18532
#151502
#112282
#112202
#111992
#110422
#109082
#108072