Finite Group Databasebeta

← #440927

Subgroups — #440927

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×#2023C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#15128C211yesmaximal, characteristic, complemented
5#11385C212yesmaximal, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11208C212yesmaximal, complemented
8#11066C212yesmaximal
9#10925C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2566—24no—
19#2566—24no—
20C2×#396—21no—
21C2×#396—21no—
22C2×#395—21no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#385—21no—
27C2×#385—21no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2023—24no—
32#1990—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1446—21no—
43#1444—21no—
44#1441—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—24no—
59C2×#125—42no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102Q818yescomplemented
91C2×C4×C8—22no—
92C2×C4×C8—42no—
93#396—28no—
94#396—28no—
95#395—44no—
96#390—22no—
97#390—22no—
98#390—22no—
99#390D414yes—
100#389—24no—
101#389—42no—
102#385—28no—
103#385—28no—
104#384—22no—
105#384D414yes—
106#382—22no—
107#382—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
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—42no—
123#340—42no—
124#340—42no—
125#336—22no—
126#336—24no—
127#336—42no—
128C25—21no—
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—82no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C2×D412yes—
141C22×D4C2×Q814yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
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—22no—
171C22×C8—22no—
172C22×C8C2×Q814yes—
173C4⋊Q8—28no—
174C4⋊Q8—48no—
175C4⋊Q8—44no—
176#125—48no—
177#125—48no—
178C4×D4—48no—
179C4×D4—48no—
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—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
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—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×Q814yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42#4114yes—
250#102—48no—
251#102—48no—
252C4×C8—44no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—82no—
262C24—22no—
263C24—24no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24C22×D411yescharacteristic
274C2×Q8—48no—
275C2×Q8—88no—
276C2×Q8—88no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4C4⋊Q814yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—48no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
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—88no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4#12412yes—
358C22×C4#12412yes—
359C22×C4#13512yes—
360C22×C4#13512yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×Q812yes—
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—42no—
375C2×C8—42no—
376C2×C8—42no—
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—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—88no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—41no—
418C42—41no—
419C42—42no—
420C23—24no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—81no—
434C23—82no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23C2×#12011yescharacteristic
444C23C2×#12411yescharacteristic
445C23#53511yescharacteristic
446C23#54612yes—
447C23D4×Q812yes—
448C23C2×#13511yescharacteristic
449C23C2×#13511yescharacteristic
450C23C2×AGL(1,ℤ/8)11yescharacteristic
451C23C2×#13711yescharacteristic
452C23#57612yes—
453C23#57712yes—
454Q8—816no—
455D4—28no—
456D4—44no—
457D4—44no—
458D4—44no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—816no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#44914yes—
522C2×C4#45214yes—
523C2×C4#52511yescharacteristic
524C2×C4C2×C4⋊Q811yescharacteristic
525C2×C4#53312yes—
526C2×C4#53511yescharacteristic
527C2×C4#53611yescharacteristic
528C2×C4#54212yes—
529C2×C4#54212yes—
530C2×C4#54312yes—
531C2×C4#54612yes—
532C2×C4#54612yes—
533C2×C4#54612yes—
534C2×C4D4×Q812yes—
535C2×C4#55412yes—
536C2×C4#55412yes—
537C2×C4#55612yes—
538C2×C4#56212yes—
539C8—44no—
540C8—82no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—84no—
547C22—84no—
548C22—82no—
549C22—82no—
550C22—84no—
551C22—84no—
552C22#240411yescharacteristic
553C22#243411yescharacteristic
554C22#243511yescharacteristic
555C22#249212yes—
556C22#269212yes—
557C22C2×#44911yescharacteristic
558C22C2×#45211yescharacteristic
559C22#300811yescharacteristic
560C22#300911yescharacteristic
561C22#313111yescharacteristic
562C22#313612yes—
563C22#314111yescharacteristic
564C22#320711yescharacteristic
565C22#321011yescharacteristic
566C22#322412yes—
567C22#329712yes—
568C22#330112yes—
569C22#331812yes—
570C22#332012yes—
571C22#338012yes—
572C22#338112yes—
573C22#338412yes—
574C22#338512yes—
575C4—24no—
576C4—24no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—84no—
583C4—81no—
584C4—84no—
585C4—81no—
586C4#246512yes—
587C4#254912yes—
588C4#269212yes—
589C4#270012yes—
590C2—84no—
591C2#1567111yescharacteristic
592C2#2396311yescharacteristic
593C2#2396411yescharacteristic
594C2#2583511yescharacteristic
595C2#2583911yescharacteristic
596C2#2585711yescharacteristic
597C2#2586511yescharacteristic
598C2#2781712yes—
599C2#2782512yes—
600C2#3152312yes—
601C2#3152512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1990C2semidirect
4. #15128C2semidirect
5. #11385C2semidirect
6. #11371C2semidirect
7. #11208C2semidirect
10. #10874C2semidirect
90. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#20232
C2×#19902
#151282
#113852
#113712
#112082
#110662
#109252
#108742
#107882