Finite Group Databasebeta

← #392938

Subgroups — #392938

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2275C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1864C212yesmaximal
4C2×#1804C211yesmaximal, characteristic
5C2×#1802C211yesmaximal, characteristic
6#14277C211yesmaximal, characteristic, complemented
7#10908C212yesmaximal, complemented
8#10884C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10833C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23#2275—24no—
24#2275—24no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4—21no—
31C2×#376C2211yescharacteristic
32C22×#95—21no—
33#1948—24no—
34#1864—28no—
35#1804C418yescomplemented
36#1802—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46#1442—21no—
47#1442—21no—
48C4⋊#95C2212yes—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4—21no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×C4×D4D414yes—
71C2×C4×D4Q814yes—
72C2×C4×D4Q814yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8C2×C414yescomplemented
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102C2×C418yes—
105#389—24no—
106#389—24no—
107#384—22no—
108#384—22no—
109#382—22no—
110#382C2×C414yescomplemented
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4D414yes—
117C4×C22⋊C4—28no—
118C4×C22⋊C4—28no—
119C4×C22⋊C4—28no—
120#376—22no—
121#376—22no—
122C2×#95—28no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340C2×C418yescomplemented
140#336—24no—
141#336—24no—
142C25—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4C22×C412yes—
151C22×D4C2×D412yes—
152C22×D4C2×D412yes—
153C22×D4C2×Q812yes—
154C22×D4C2×Q812yes—
155C22×D4#4112yes—
156C22×D4#4112yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—82no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C22×C414yes—
198C4×D4—28no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—28no—
207C4×D4—28no—
208C4×D4—28no—
209C4×D4C4⋊C4116yescomplemented
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×Q812yes—
244C2×C4⋊C4C2×Q812yes—
245C2×C4⋊C4#4112yes—
246C2×C4⋊C4#4112yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—82no—
268C2×C42—82no—
269C2×C42—82no—
270C2×C42C22×C412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×Q812yes—
275C2×C42C2×Q812yes—
276C2×C42#4112yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280#106—48no—
281C4⋊C8—28no—
282C4⋊C8—44no—
283#102—216no—
284#102—48no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—44no—
295#95—82no—
296#95—82no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24C22×D411yescharacteristic
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4C2×C4⋊C414yes—
331C2×D4C4×D414yes—
332C2×D4C4×Q814yes—
333C2×D4#12114yes—
334C2×D4C22⋊Q814yes—
335C2×D4#12514yes—
336C2×D4C4⋊Q814yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—24no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—88no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—88no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4#12012yes—
404C22×C4#12012yes—
405C22×C4#12112yes—
406C22×C4#12112yes—
407C22×C4C22⋊Q812yes—
408C22×C4C22⋊Q812yes—
409C22×C4C22⋊Q812yes—
410C22×C4C22⋊Q812yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×Q811yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C4⋊C4C2×C4⋊C414yes—
460C4⋊C4#12114yes—
461C4⋊C4C22⋊Q814yes—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—84no—
474C42—84no—
475C42—84no—
476C42C2×C4⋊C414yes—
477C42C4×D414yes—
478C42C22⋊Q814yes—
479C42#12314yes—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23C2×C4×D411yescharacteristic
504C23C2×#12011yescharacteristic
505C23C2×#12111yescharacteristic
506C23C2×C22⋊Q811yescharacteristic
507C23C2×C22⋊Q811yescharacteristic
508C23#52511yescharacteristic
509C23#52511yescharacteristic
510C23#57612yes—
511C23#57712yes—
512C23#57712yes—
513C23#57712yes—
514D4—24no—
515D4—24no—
516D4—24no—
517D4—24no—
518D4—24no—
519D4—24no—
520D4—24no—
521D4#38918yes—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—816no—
569C2×C4—816no—
570C2×C4—816no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#38514yes—
583C2×C4#38514yes—
584C2×C4C22×C4⋊C411yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×C4×Q811yescharacteristic
588C2×C4C4×#4111yescharacteristic
589C2×C4C4×#4111yescharacteristic
590C2×C4C2×#12111yescharacteristic
591C2×C4C2×#12111yescharacteristic
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52511yescharacteristic
598C2×C4#52511yescharacteristic
599C2×C4#52511yescharacteristic
600C2×C4C2×#12511yescharacteristic
601C2×C4#52911yescharacteristic
602C2×C4#52911yescharacteristic
603C2×C4#52911yescharacteristic
604C2×C4#52911yescharacteristic
605C2×C4C2×C4⋊Q811yescharacteristic
606C2×C4#53211yescharacteristic
607C2×C4#53211yescharacteristic
608C2×C4#53311yescharacteristic
609C2×C4#53311yescharacteristic
610C2×C4#53311yescharacteristic
611C2×C4#53311yescharacteristic
612C8—44no—
613C8—82no—
614C22—28no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—48no—
623C22—44no—
624C22—44no—
625C22C2×#38511yescharacteristic
626C22C4×#12111yescharacteristic
627C22C4×C22⋊Q811yescharacteristic
628C22#284711yescharacteristic
629C22#284811yescharacteristic
630C22#285111yescharacteristic
631C22#285711yescharacteristic
632C22#294612yes—
633C22#295212yes—
634C22#300811yescharacteristic
635C22#300911yescharacteristic
636C22#300911yescharacteristic
637C22#301011yescharacteristic
638C22#304212yes—
639C22#304512yes—
640C22#308112yes—
641C22#308112yes—
642C22#308412yes—
643C22#308412yes—
644C22#311812yes—
645C22#311912yes—
646C22#312312yes—
647C22#312512yes—
648C4—24no—
649C4—24no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—84no—
658C4—162no—
659C4C2×#38911yescharacteristic
660C4#229611yescharacteristic
661C4C4×#12111yescharacteristic
662C4C4×C22⋊Q811yescharacteristic
663C4#285111yescharacteristic
664C4#285411yescharacteristic
665C4#285511yescharacteristic
666C4#285711yescharacteristic
667C2—48no—
668C2#1469711yescharacteristic
669C2#2087611yescharacteristic
670C2#2088211yescharacteristic
671C2#2151012yes—
672C2#2161912yes—
673C2#3291311yescharacteristic
674C2#3291411yescharacteristic
675C2#3296211yescharacteristic
676C2#3296511yescharacteristic
677C2#3307612yes—
678C2#3320512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1948C2semidirect
6. #14277C2semidirect
7. #10908C2semidirect
8. #10884C2semidirect
9. #10850C2semidirect
10. #10833C2semidirect
35. #1804C4semidirect
98. C2×C4⋊C8C2×C4semidirect
110. #382C2×C4semidirect
139. #340C2×C4semidirect
209. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22752
C2×#19482
C2×#18642
C2×#18042
C2×#18022
#142772
#109082
#108842
#108502
#108332
#107882