Finite Group Databasebeta

← #440909

Subgroups — #440909

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11385C212yesmaximal, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11095C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2441—24no—
19#2441—24no—
20C2×#399—21no—
21C2×#396—21no—
22C2×#393—21no—
23C2×#389—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#387—21no—
27C2×#384C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#2058—24no—
32#1990—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—24no—
58C2×#125—42no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8Q818yescomplemented
90C2×C4×C8—22no—
91C2×C4×C8—42no—
92#399—28no—
93#396—28no—
94#393—44no—
95#389—28no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389D414yes—
100#387—28no—
101#384—22no—
102#384D414yes—
103#380—22no—
104#380—22no—
105#380—24no—
106#380—42no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—42no—
128C25C2×D411yescharacteristic
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—82no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q814yes—
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
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×C8C2×Q814yes—
167C22×C8C2×Q814yes—
168C4⋊Q8—28no—
169C4⋊Q8—48no—
170C4⋊Q8—44no—
171#125—48no—
172#125—44no—
173#125—44no—
174#117—48no—
175#117—44no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
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—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4#4114yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—44no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#106—48no—
245C22⋊C8—48no—
246C4×C8—44no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—44no—
253#95—44no—
254#95—42no—
255#95—42no—
256#95—82no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24C4⋊Q812yes—
268C24C22×D411yescharacteristic
269C2×Q8—48no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×Q8—84no—
273C2×Q8—84no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
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#12012yes—
343C22×C4#12012yes—
344C22×C4C4⋊Q812yes—
345C22×C4#13512yes—
346C22×C4#13512yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4#13712yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×Q812yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4#12414yes—
398C22⋊C4—48no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—84no—
402C22⋊C4—88no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—48no—
407C42—41no—
408C42—41no—
409C42—42no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23C2×#12011yescharacteristic
434C23C2×C4⋊Q811yescharacteristic
435C23#53611yescharacteristic
436C23#54612yes—
437C23#54612yes—
438C23C2×#13511yescharacteristic
439C23C2×#13511yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13711yescharacteristic
442C23#57712yes—
443C23#57712yes—
444Q8—816no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#44914yes—
507C2×C4#45214yes—
508C2×C4#52511yescharacteristic
509C2×C4C2×#12411yescharacteristic
510C2×C4#53312yes—
511C2×C4#53511yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#54212yes—
514C2×C4#54212yes—
515C2×C4#54312yes—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4D4×Q812yes—
519C2×C4D4×Q812yes—
520C2×C4#55412yes—
521C2×C4#55412yes—
522C2×C4#55612yes—
523C2×C4#56212yes—
524C8—44no—
525C8—44no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22#240411yescharacteristic
538C22#243411yescharacteristic
539C22#243511yescharacteristic
540C22#254912yes—
541C22#270012yes—
542C22C2×#44911yescharacteristic
543C22C2×#45211yescharacteristic
544C22#300911yescharacteristic
545C22#301011yescharacteristic
546C22#316011yescharacteristic
547C22#316312yes—
548C22#316511yescharacteristic
549C22#322811yescharacteristic
550C22#323111yescharacteristic
551C22#324412yes—
552C22#329612yes—
553C22#330112yes—
554C22#330312yes—
555C22#330412yes—
556C22#331012yes—
557C22#332012yes—
558C22#333112yes—
559C22#333212yes—
560C4—24no—
561C4—24no—
562C4—44no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4—162no—
571C4#246512yes—
572C4#249212yes—
573C4#269212yes—
574C4#269212yes—
575C2—82no—
576C2#1567111yescharacteristic
577C2#2396411yescharacteristic
578C2#2396611yescharacteristic
579C2#2581211yescharacteristic
580C2#2582011yescharacteristic
581C2#2588611yescharacteristic
582C2#2588711yescharacteristic
583C2#2877412yes—
584C2#2878312yes—
585C2#3174112yes—
586C2#3174312yes—
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×#2058C2semidirect
3. C2×#1990C2semidirect
5. #11385C2semidirect
6. #11357C2semidirect
7. #11220C2semidirect
10. #10880C2semidirect
89. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#20582
C2×#19902
#151252
#113852
#113572
#112202
#110952
#109212
#108802
#107312