Finite Group Databasebeta

← #360566

Subgroups — #360566

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3C2×#1850C211yesmaximal, characteristic
4C2×#1803C212yesmaximal, complemented
5C2×#1787C211yesmaximal, characteristic
6#13880C211yesmaximal, characteristic
7#11077C212yesmaximal, complemented
8#11050C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103—28no—
18C22×#103—24no—
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26Q8×C22⋊C4—24no—
27Q8×C22⋊C4C418yescomplemented
28C2×#391—22no—
29C2×#391C2212yes—
30C2×#385—21no—
31C2×#385—21no—
32C2×#384C2212yes—
33C2×#380—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1850—24no—
39#1803—28no—
40#1787—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46#1444—22no—
47C4⋊#95C2212yes—
48C23×Q8—22no—
49C23×Q8—21no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—24no—
53C23×Q8—24no—
54C23×Q8—42no—
55C23×Q8—42no—
56C23×Q8C2×C414yes—
57C23×Q8D414yes—
58C23×Q8Q814yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2311yescharacteristic
61C24×C4—21no—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8—22no—
65C23×C8—24no—
66C23×C8—22no—
67C23×C8—22no—
68C23×C8—42no—
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2312yes—
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—42no—
76C2×C4×Q8—42no—
77C2×C4×Q8C2×C418yescomplemented
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—22no—
95C22×C42C2312yes—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—42no—
99C2×#103—232no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—24no—
111C2×#103—42no—
112C2×#103—42no—
113C2×#103D418yes—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117#391—24no—
118#391—216no—
119#391C2×C418yescomplemented
120#385—44no—
121#385—44no—
122#384—24no—
123#384—42no—
124#380—22no—
125#380—28no—
126#380—22no—
127#380—22no—
128#380C2×C414yes—
129C4×C22⋊C4—28no—
130C4×C22⋊C4C2×C414yes—
131C4×C22⋊C4D414yes—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95D412yes—
139#340—24no—
140#340—24no—
141#340—42no—
142#340—42no—
143#336—24no—
144C25C2×D411yescharacteristic
145C22×Q8—24no—
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—28no—
149C22×Q8—24no—
150C22×Q8—28no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—216no—
160C22×Q8—28no—
161C22×Q8—216no—
162C22×Q8—28no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8—48no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8—48no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8—48no—
179C22×Q8C4⋊C4116yescomplemented
180C22×Q8C22×C412yes—
181C22×Q8C22×C412yes—
182C22×Q8C2×D412yes—
183C22×Q8C2×D412yes—
184C22×Q8C2×D414yes—
185C22×Q8C2×Q812yes—
186C22×Q8C2×Q812yes—
187C22×Q8#4114yes—
188C23×C4—21no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—82no—
208C23×C4C22×C412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×Q812yes—
215C23×C4#4112yes—
216C23×C4#4112yes—
217C23×C4C2411yescharacteristic
218C22×C8—28no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—21no—
226C22×C8—22no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—48no—
236C22×C8—48no—
237C22×C8—44no—
238C22×C8—44no—
239C22×C8C2×D414yes—
240C4×Q8—216no—
241C4×Q8—48no—
242C4×Q8—48no—
243C4×Q8—48no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4C22×C414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4C22⋊C414yes—
287C2×C22⋊C4C22⋊C414yes—
288C2×C22⋊C4SD1614yes—
289C2×C22⋊C4Q1614yes—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C22×C414yes—
298C2×C42C2×D414yes—
299C4⋊C8—48no—
300#103—216no—
301#103—216no—
302#103—44no—
303#103—44no—
304#103—44no—
305#103—44no—
306C22⋊C8—44no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95C22⋊C418yes—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C2×C22⋊C412yes—
319C24#12112yes—
320C24C22⋊Q812yes—
321C24C2×SD1612yes—
322C24C2×Q1612yes—
323C24AGL(1,ℤ/8)12yes—
324C24#13712yes—
325C24C22×D411yescharacteristic
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—216no—
343C2×Q8—232no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—24no—
352C2×Q8—416no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—416no—
356C2×Q8—416no—
357C2×Q8—432no—
358C2×Q8—416no—
359C2×Q8—416no—
360C2×Q8—432no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8C2×C4⋊C414yes—
368C2×Q8C2×C4⋊C414yes—
369C2×Q8C4×D418yes—
370C2×Q8#12014yes—
371C2×Q8C22⋊Q818yes—
372C2×Q8#12314yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—416no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4C2×C22⋊C414yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C414yes—
446C22×C4C2×C4⋊C414yes—
447C22×C4#11714yes—
448C22×C4#12114yes—
449C22×C4#12112yes—
450C22×C4C22⋊Q814yes—
451C22×C4C22⋊Q812yes—
452C22×C4C2×SD1614yes—
453C22×C4C2×Q1614yes—
454C22×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×Q811yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—416no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C22⋊C4—28no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C22⋊C4#103116yescomplemented
504C42—24no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—416no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23#38014yes—
526C23C2×#10314yes—
527C23#41714yes—
528C23#46014yes—
529C23#46214yes—
530C23#47514yes—
531C23#47714yes—
532C23C22×C22⋊C411yescharacteristic
533C23C2×#12111yescharacteristic
534C23C2×C22⋊Q811yescharacteristic
535C23D4212yes—
536C23#54612yes—
537C23C22×SD1611yescharacteristic
538C23C22×Q1611yescharacteristic
539C23C2×AGL(1,ℤ/8)11yescharacteristic
540C23C2×#13711yescharacteristic
541C23#57712yes—
542C23#57812yes—
543Q8—24no—
544Q8—28no—
545Q8—24no—
546Q8—24no—
547Q8—28no—
548Q8—24no—
549Q8—24no—
550Q8—432no—
551Q8—44no—
552Q8—42no—
553Q8—42no—
554Q8—44no—
555Q8#38518yescomplemented
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—432no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—816no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4#38014yes—
616C2×C4C2×#10318yescomplemented
617C2×C4C22×C4⋊C411yescharacteristic
618C2×C4C2×#11711yescharacteristic
619C2×C4C2×C4×D412yes—
620C2×C4#51812yes—
621C2×C4C2×#12011yescharacteristic
622C2×C4C2×#12111yescharacteristic
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4C2×C22⋊Q812yes—
625C2×C4C2×#12311yescharacteristic
626C2×C4#53411yescharacteristic
627C2×C4#53411yescharacteristic
628C2×C4#54312yes—
629C2×C4D4212yes—
630C2×C4#54612yes—
631C2×C4#54612yes—
632C2×C4#54612yes—
633C2×C4#55112yes—
634C2×C4#55112yes—
635C8—44no—
636C8—81no—
637C8—81no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22#180318yescomplemented
649C22C2×#38011yescharacteristic
650C22D4×C22⋊C412yes—
651C22#241011yescharacteristic
652C22#241211yescharacteristic
653C22#243712yes—
654C22C22×#10311yescharacteristic
655C22C2×#41711yescharacteristic
656C22#289812yes—
657C22C2×#46011yescharacteristic
658C22C2×#46211yescharacteristic
659C22#304311yescharacteristic
660C22#304411yescharacteristic
661C22C2×#47511yescharacteristic
662C22C2×#47711yescharacteristic
663C22#308212yes—
664C22#326512yes—
665C22#327012yes—
666C22D4×SD1612yes—
667C22D4×Q1612yes—
668C22#329812yes—
669C22#329912yes—
670C22#330812yes—
671C22#331612yes—
672C4—24no—
673C4—28no—
674C4—28no—
675C4—44no—
676C4—44no—
677C4—84no—
678C4—84no—
679C4—161no—
680C4—161no—
681C4C2×#38511yescharacteristic
682C4#231511yescharacteristic
683C4#234212yes—
684C4#241011yescharacteristic
685C4#241211yescharacteristic
686C4#243712yes—
687C2—28no—
688C2#1387711yescharacteristic
689C2C2×#180311yescharacteristic
690C2#2065111yescharacteristic
691C2D4×#10312yes—
692C2#2242112yes—
693C2#2435611yescharacteristic
694C2#2436011yescharacteristic
695C2#2452811yescharacteristic
696C2#2453211yescharacteristic
697C2#2595512yes—
698C2#2596312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
4. C2×#1803C2semidirect
7. #11077C2semidirect
8. #11050C2semidirect
9. #10789C2semidirect
27. Q8×C22⋊C4C4semidirect
77. C2×C4×Q8C2×C4semidirect
119. #391C2×C4semidirect
179. C22×Q8C4⋊C4semidirect
503. C22⋊C4#103semidirect
555. Q8#385semidirect
616. C2×C4C2×#103semidirect
648. C22#1803semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×Q8×C22⋊C42
C2×#18502
C2×#18032
C2×#17872
#138802
#110772
#110502
#107892
#107302
#107212