Finite Group Databasebeta

← #366794

Subgroups — #366794

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#1901C211yesmaximal, characteristic
3C2×#1901C211yesmaximal, characteristic
4#20837C211yesmaximal, characteristic
5C2×#1821C211yesmaximal, characteristic
6C22⋊C4×M16C211yesmaximal, characteristic
7C22×#343C211yesmaximal, characteristic
8#11354C212yesmaximal, complemented
9#11045C212yesmaximal, complemented
10#10963C212yesmaximal, complemented
11#10857C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#423C2212yes—
17C22×#101C2211yescharacteristic
18C22×#101C2212yes—
19C22×#101C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×C4×M16C2212yes—
23#2502—22no—
24#2502—22no—
25#2502—21no—
26#2502—21no—
27#2502—22no—
28C2×#398C414yes—
29C2×#391C2212yes—
30C2×#385C414yes—
31C2×#384C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1901—24no—
34#1901—24no—
35#1851—22no—
36#1850—22no—
37#1821—24no—
38#1753—22no—
39#1752—21no—
40C8×C22⋊C4—21no—
41C2×#343—24no—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2211yescharacteristic
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2211yescharacteristic
48C2×#343C2212yes—
49#1728C2212yes—
50#1728C2212yes—
51C23×Q8C2311yescharacteristic
52C24×C4C2×C412yes—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C22×M16—21no—
57C22×M16—22no—
58C22×M16—24no—
59C22×M16—24no—
60C22×M16—22no—
61C22×M16—42no—
62C22×M16—42no—
63C22×M16D414yes—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2312yes—
67C22×M16C2311yescharacteristic
68C22×M16C2312yes—
69C22×M16C2311yescharacteristic
70C23×C8—21no—
71C23×C8—21no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C412yes—
90C22×C4⋊C4C2×C418yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2×C412yes—
94C22×C22⋊C4C2×C412yes—
95C22×C42C2312yes—
96#423—24no—
97#423—42no—
98C2×C4⋊C8—24no—
99C2×#101—24no—
100C2×#101—22no—
101C2×#101—22no—
102C2×#101—24no—
103C2×#101—24no—
104C2×#101—42no—
105C2×#101—42no—
106#407—22no—
107#407—22no—
108#407—22no—
109#407D414yes—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C4×M16—24no—
115C4×M16—42no—
116C2×#97—22no—
117C2×C4×C8—22no—
118#398—24no—
119#398—28no—
120#398—24no—
121#391—22no—
122#391—22no—
123#391—24no—
124#385—24no—
125#385—28no—
126#385—24no—
127#384—22no—
128#384—22no—
129#384—24no—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C4×C22⋊C4D411yescharacteristic
133C4×C22⋊C4D411yescharacteristic
134C4×C22⋊C4D412yes—
135C2×#95C2×C418yes—
136C2×#95C2×C412yes—
137#343—24no—
138#343—24no—
139#343—24no—
140#343—22no—
141#343—22no—
142#343—24no—
143#343—24no—
144#336—24no—
145#336—22no—
146#336—22no—
147C25C22×C411yescharacteristic
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—84no—
167C23×C4C22×C412yes—
168C23×C4C22×C414yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C414yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—24no—
184C2×M16—21no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—28no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—28no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—44no—
208C2×M16—44no—
209C2×M16—44no—
210C2×M16—48no—
211C2×M16—48no—
212C2×M16—44no—
213C2×M16—42no—
214C2×M16C2×D412yes—
215C2×M16C2×D414yes—
216C2×M16C2×D412yes—
217C2×M16#4114yes—
218C2×M16#4114yes—
219C22×C8—21no—
220C22×C8—24no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—44no—
227C22×C8—44no—
228C22×C8—42no—
229C22⋊Q8—42no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—88no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4C22⋊C412yes—
274C2×C22⋊C4C22⋊C414yes—
275C2×C22⋊C4C22⋊C412yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D414yes—
283C4⋊C8—48no—
284#101—48no—
285#101—44no—
286#101—44no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291#97—44no—
292C4×C8—44no—
293#95—28no—
294#95—28no—
295#95C22⋊C412yes—
296#95C22⋊C414yes—
297#95C22⋊C412yes—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—42no—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24C4×D412yes—
305C24C4×D412yes—
306C24C23×C411yescharacteristic
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C414yes—
372C22×C4C4×D412yes—
373C22×C4C4×D412yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12412yes—
379C22×C4#12412yes—
380C22×C4#12712yes—
381C22×C4#12712yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392M16—22no—
393M16—24no—
394M16—22no—
395M16—24no—
396M16—22no—
397M16—24no—
398M16—24no—
399M16—24no—
400M16—48no—
401M16—48no—
402M16—42no—
403M16—42no—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—42no—
412M16—42no—
413M16#12314yes—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—48no—
420C2×C8—48no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C22⋊C4—24no—
445C22⋊C4—28no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4#10118yescomplemented
449C42—22no—
450C42—22no—
451C42—24no—
452C42—22no—
453C42—24no—
454C42—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—48no—
470C23—48no—
471C23#39012yes—
472C23#39012yes—
473C23C2×#10114yes—
474C23C22×C22⋊C411yescharacteristic
475C23C2×C4×D411yescharacteristic
476C23C2×C4×D411yescharacteristic
477C23C4×#4112yes—
478C23#52012yes—
479Q8—81no—
480Q8—81no—
481Q8—82no—
482Q8—82no—
483Q8—81no—
484Q8—81no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—88no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4#39012yes—
525C2×C4#39012yes—
526C2×C4C2×#10118yes—
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C2×C4#52511yescharacteristic
531C2×C4C2×#12411yescharacteristic
532C2×C4#52912yes—
533C2×C4C2×#12711yescharacteristic
534C2×C4#53411yescharacteristic
535C2×C4#53611yescharacteristic
536C2×C4#53912yes—
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#54612yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#54712yes—
543C2×C4#55112yes—
544C2×C4#55212yes—
545C8—24no—
546C8—42no—
547C8—44no—
548C8—44no—
549C8—81no—
550C8—81no—
551C22—22no—
552C22—24no—
553C22—24no—
554C22—28no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—44no—
564C22—48no—
565C22—44no—
566C22—44no—
567C22#196614yes—
568C22C2×#39011yescharacteristic
569C22C4×#12311yescharacteristic
570C22#232211yescharacteristic
571C22#233912yes—
572C22#235912yes—
573C22C22×#10112yes—
574C22#288812yes—
575C4—22no—
576C4—22no—
577C4—28no—
578C4—44no—
579C4—82no—
580C4—82no—
581C4—84no—
582C4#241211yescharacteristic
583C4#241311yescharacteristic
584C4#242512yes—
585C4#248812yes—
586C4#250111yescharacteristic
587C4#253911yescharacteristic
588C2—21no—
589C2—22no—
590C2—24no—
591C2—21no—
592C2—42no—
593C2#1399611yescharacteristic
594C2C2×#196611yescharacteristic
595C2#2121911yescharacteristic
596C2#2248114yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11354C2semidirect
9. #11045C2semidirect
10. #10963C2semidirect
11. #10857C2semidirect
448. C22⋊C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#19012
#208372
C2×#18212
C22⋊C4×M162
C22×#3432
#113542
#110452
#109632
#108572