Finite Group Databasebeta

← #440787

Subgroups — #440787

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4#15150C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11095C212yesmaximal
8#10970C212yesmaximal
9#10926C211yesmaximal, characteristic, complemented
10#10921C211yesmaximal, characteristic
11#10789C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2423—24no—
20#2423—24no—
21C2×#399—21no—
22C2×#398—21no—
23C2×#397—21no—
24C2×#397—21no—
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2096—24no—
35#1834—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—42no—
91C2×C4×C8—24no—
92C2×C4×C8—22no—
93#399—28no—
94#398—28no—
95#397—44no—
96#397—44no—
97#391—22no—
98#391D414yes—
99#389—22no—
100#389—22no—
101#388—28no—
102#387—28no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384D414yes—
107#380—22no—
108#380—22no—
109#380—42no—
110#380—42no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#340—42no—
129#336—22no—
130#336—42no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D412yes—
139C22×Q8#4114yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
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—22no—
171C22×C8—22no—
172C22×C8#4114yes—
173C22×C8#4114yes—
174C22⋊Q8—48no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177C4×Q8—48no—
178C4×Q8—44no—
179C4×Q8—44no—
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—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
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—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
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—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4SD1614yes—
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—
245#103—48no—
246#103—48no—
247C22⋊C8—48no—
248C4×C8—28no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—44no—
259#95—42no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24C2×SD1612yes—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8#12414yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
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—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4#12012yes—
357C22×C4#12012yes—
358C22×C4C2×SD1612yes—
359C22×C4C2×SD1612yes—
360C22×C4#13512yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+412yes—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C22⋊C4—28no—
409C22⋊C4—48no—
410C22⋊C4—44no—
411C22⋊C4—84no—
412C22⋊C4—88no—
413C42—22no—
414C42—22no—
415C42—24no—
416C42—48no—
417C42—41no—
418C42—41no—
419C42—42no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23C2×#12011yescharacteristic
444C23#53211yescharacteristic
445C23#53511yescharacteristic
446C23#54612yes—
447C23#54712yes—
448C23C22×SD1611yescharacteristic
449C23C2×#13511yescharacteristic
450C23C2×AGL(1,ℤ/8)11yescharacteristic
451C23C2×#13711yescharacteristic
452C23#57712yes—
453C23#57812yes—
454Q8—28no—
455Q8—44no—
456Q8—44no—
457Q8—44no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
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—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—82no—
518C2×C4—82no—
519C2×C4#44814yes—
520C2×C4#44914yes—
521C2×C4#52511yescharacteristic
522C2×C4C2×#12411yescharacteristic
523C2×C4#52912yes—
524C2×C4#53511yescharacteristic
525C2×C4#53611yescharacteristic
526C2×C4#53912yes—
527C2×C4#53912yes—
528C2×C4#54012yes—
529C2×C4#54612yes—
530C2×C4#54612yes—
531C2×C4#54712yes—
532C2×C4#55012yes—
533C2×C4#55112yes—
534C2×C4#55112yes—
535C2×C4#55212yes—
536C2×C4#56212yes—
537C8—44no—
538C8—44no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—82no—
549C22—82no—
550C22#240611yescharacteristic
551C22#243111yescharacteristic
552C22#243511yescharacteristic
553C22#255312yes—
554C22#270612yes—
555C22C2×#44811yescharacteristic
556C22C2×#44911yescharacteristic
557C22#300911yescharacteristic
558C22#301111yescharacteristic
559C22#316711yescharacteristic
560C22#317812yes—
561C22#318211yescharacteristic
562C22#320811yescharacteristic
563C22#321011yescharacteristic
564C22#322512yes—
565C22#329812yes—
566C22#330412yes—
567C22#332812yes—
568C22#333212yes—
569C22#333512yes—
570C22#333612yes—
571C22#334312yes—
572C22#334412yes—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—44no—
577C4—82no—
578C4—81no—
579C4—82no—
580C4—82no—
581C4—81no—
582C4—162no—
583C4—162no—
584C4#247112yes—
585C4#248812yes—
586C4#269412yes—
587C4#270612yes—
588C2—82no—
589C2#1566911yescharacteristic
590C2#2409411yescharacteristic
591C2#2409711yescharacteristic
592C2#2559811yescharacteristic
593C2#2561311yescharacteristic
594C2#2588411yescharacteristic
595C2#2588711yescharacteristic
596C2#2884712yes—
597C2#2886712yes—
598C2#3193212yes—
599C2#3193512yes—
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×#2096C2semidirect
3. C2×#1834C2semidirect
5. #11357C2semidirect
6. #11220C2semidirect
9. #10926C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24232
C2×#20962
C2×#18342
#151502
#113572
#112202
#110952
#109702
#109262
#109212
#107892